Discontinuity, Nonlinearity, and Complexity

Volume 4 Issue 3 September 2015

ISSN 2164‐6376 (print) ISSN 2164‐6414 (online)

An Interdisciplinary Journal of

Discontinuity, Nonlinearity, and Complexity

Discontinuity, Nonlinearity, and Complexity Editors Valentin Afraimovich San Luis Potosi University, IICO-UASLP, Av.Karakorum 1470 Lomas 4a Seccion, San Luis Potosi, SLP 78210, Mexico Fax: +52 444 825 0198 Email: [email protected]

Lev Ostrovsky Zel Technologies/NOAA ESRL, Boulder CO 80305, USA Fax: +1 303 497 5862 Email: [email protected]

Xavier Leoncini Centre de Physique Théorique, Aix-Marseille Université, CPT Campus de Luminy, Case 907 13288 Marseille Cedex 9, France Fax: +33 4 91 26 95 53 Email: [email protected]

Dimitry Volchenkov The Center of Excellence Cognitive Interaction Technology Universität Bielefeld, Mathematische Physik Universitätsstraße 25 D-33615 Bielefeld, Germany Fax: +49 521 106 6455 Email: [email protected]

Associate Editors Marat Akhmet Department of Mathematics Middle East Technical University 06531 Ankara, Turkey Fax: +90 312 210 2972 Email: [email protected]

Ranis N. Ibragimov Applied Statistics Lab GE Global Research 1 Research Circle, K1-4A64 Niskayuna, NY 12309 Email: [email protected]

Jose Antonio Tenreiro Machado ISEP-Institute of Engineering of Porto Dept. of Electrotechnical Engineering Rua Dr. Antonio Bernardino de Almeida 4200-072 Porto, Portugal Fax: +351 22 8321159 Email: [email protected]

Dumitru Baleanu Department of Mathematics and Computer Sciences Cankaya University, Balgat 06530 Ankara, Turkey Email: [email protected]

Nikolai A. Kudryashov Department of Applied Mathematics Moscow Engineering and Physics Institute (State University), 31 Kashirskoe Shosse 115409 Moscow, Russia Fax: +7 495 324 11 81 Email: [email protected]

Denis Makarov Pacific Oceanological Institute of the Russian Academy of Sciences,43 Baltiiskaya St., 690041 Vladivostok, RUSSIA Tel/fax: 007-4232-312602. Email: [email protected]

Marian Gidea Department of Mathematics Northeastern Illinois University Chicago, IL 60625, USA Fax: +1 773 442 5770 Email: [email protected]

Gennady A. Leonov Department of Mathematics and Mechanics St-Petersburg State University 198504, Russia Email: [email protected]

Josep J. Masdemont Department of Matematica Aplicada I Universitat Politecnica de Catalunya (UPC) Diagonal 647 ETSEIB 08028 Barcelona, Spain Email: [email protected]

Juan Luis García Guirao Department of Applied Mathematics Technical University of Cartagena Hospital de Marina 30203-Cartagena, SPAIN Fax:+34 968 325694 E-mail: [email protected]

E.N. Macau Lab. Associado de Computção e Mateática Aplicada, Instituto Nacional de Pesquisas Espaciais, Av. dos Astronautas, 1758 12227-010 Sáo José dos Campos –SP Brazil Email: [email protected]

Michael A. Zaks Technische Universität Berlin DFG-Forschungszentrum Matheon Mathematics for key technologies Sekretariat MA 3-1, Straße des 17. Juni 136, 10623 Berlin, Germany Email: [email protected]

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Ravi P. Agarwal Department of Mathematics Texas A&M University – Kingsville, Kingsville, TX 78363-8202 USA Email: [email protected]

Editorial Board Vadim S. Anishchenko Department of Physics Saratov State University Astrakhanskaya 83, 410026, Saratov, Russia Fax: (845-2)-51-4549 Email: [email protected]

Continued on back materials

An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 4, Issue 3, September 2015

Editors Valentin Afraimovich Xavier Leoncini Lev Ostrovsky Dimitry Volchenkov

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Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 219–223

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Modeling Fluid Dynamics in the Ocean and Atmosphere S.V. Prants† Laboratory of Nonlinear Dynamical Systems, Pacific Oceanological Institute of the Russian Academy of Sciences, 43 Baltiiskaya st., 690041 Vladivostok, Russia Submission Info Communicated by Valentin Afraimovich Received 27 February 2015 Accepted 2 March 2015 Available online 1 October 2015 Keywords Geophysical fluid dynamics Coherent structures Vortices Dynamical systems

Abstract This Special Issue collects together works on analytic solutions and numerical simulation of fluid dynamics in the ocean and atmosphere. The contributed papers address a variety of problems in geophysical fluid dynamics including formation of coherent structures in random hydrodynamic flows, hyperbolicity in the ocean, mesoscale surface and deep vortices in the ocean, formation of vocalized atmospheric vortices and motion of tropical cyclones, convective instability and nonlinear structures in systems with a multi-component convection, instability development in shear stratified flows and others.

©2015 L&H Scientific Publishing, LLC. All rights reserved.

The ocean and atmosphere are highly turbulent media with a variety of dynamical phenomena of different space and time scales ranging from millimeters to a few thousands of kilometers and from milliseconds to thousands of years. In spite of a variety of small- and large- scale random perturbations, there appear well ordered and long-lived coherent structures like large-scale eddies and jet currents which are easily visible in satellite images. As to the ocean, one can monitor day by day in different websites (http://www.aviso.oceanobs.com, http://oceancolor.gsfc.nasa.gov, http://www.nodc.noaa.gov and others) sea color, surface temperature, chlorophyll concentration, salinity, wind velocity and many other things allowing researchers to analyze the state of the ocean and try to forecast its future condition. Naturally occurring large-scale flows in the ocean and atmosphere are studied by geophysical fluid dynamics that is about dynamics of stratified (layered) and turbulent fluid on the rotating Earth. This discipline combines methods and approaches of mathematics and theoretical physics with advanced numerical modeling. Many valuable break-downs in geophysics have been often theory-based rather than experiment-based because in situ measurements are extremely difficult, expensive and even impossible. Recently, new ideas and methods were introduced to geophysical fluid dynamics due to dynamical systems theory approach. The basic ones have been borrowed from the phenomenon of chaotic advection in fluids, an analogue of dynamical Hamiltonian chaos in mechanics. They have been successfully explored in physical oceanography and physics of the atmosphere for the last two-three decades to study advective mixing and transport of water and air masses (for recent reviews see [1–3]). It is well known from dynamical systems theory that solutions of deterministic advection equations can be chaotic in the sense of exponential sensitivity to small † Corresponding

author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.09.001

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S.V. Prants / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 219–223

variations in initial conditions and/or control parameters. It means that even a simple and regular Eulerian velocity field may cause practically unpredictable fluid-particle’s trajectories, the phenomenon known as chaotic advection [4]. Since the phase space of advection equations is the physical space for fluid particles, many abstract mathematical objects from dynamical systems theory (stagnation points, KAM tori, stable and unstable manifolds, periodic and chaotic orbits, etc.) have their material analogues in fluid flows. It is well known that besides “trivial” elliptic stagnation points, the motion around which is stable, there are hyperbolic stagnation points which organize fluid motion in their neighborhood in a specific way. In a steady flow the hyperbolic points are typically connected by the separatrices which are their stable and unstable invariant manifolds. In an unsteady flow they are replaced by the corresponding hyperbolic trajectories with two associated invariant manifolds which in general intersect each other transversally resulting in a complex manifold structure known as homo- or heteroclinic tangles, seeds of chaos. The existence of large-scale quasi-deterministic coherent structures in turbulent flows has long been recognized. Before the coherent structures were found, it was a common opinion that turbulent flows are determined only by irregular vortical fluid motion. Although up to now there is no consensus on a strict definition of coherent structures, they can be considered as connected turbulent fluid masses with phase-correlated (i.e., coherent) vorticity over the spatial extent of the shear layer. Thus, turbulence consists of coherent and phase-random (incoherent) motions with the latter ones to be superimposed on the former ones. Lagrangian motion may be strongly influenced by those coherent structures that support distinct regimes in a given turbulent flow. The discovery that turbulent flows are not fully random but embody orderly organizing structures was a kind of revolution in fluid mechanics. As to complicated but not totally random flows, including large-scale geophysical ones, it was Haller [5] who proposed a concept of Lagrangian coherent structures with the boundaries delineated by distinguished material lines or surfaces and advected with the flow. To extract these structures, he proposed to compute finitetime Lyapunov exponents. The Lagrangian coherent structures are the most influential attracting and repelling hyperbolic material surfaces (curves) in 3D (2D) velocity fields. They are Lagrangian because they are invariant manifolds consisting of the same fluid particles. They are coherent because they are comparatively long lived and more robust than the other adjacent structures. The Lagrangian coherent structures are connected with stable and unstable invariant manifolds of hyperbolic stagnation points. The dynamical systems approach in physical oceanography is useful not only to identify organizing structures in oceanic flows but also to plan research vessel cruises. Before choosing the track of a planed cruise, it is instructive to find locations of hyperbolic and elliptic stagnation points in a studied region and compute forwardand backward-in-time Lagrangian maps of some tracer’s indicators in the daily provided satellite-derived velocity field such as finite-time Lyapunov exponents, absolute, zonal and meridional displacements, vorticity and others [2, 6]. Those maps allow to visualize practically important structures in the region. For example, it has been shown in [7] how with the help of drift Lagrangian maps to delineate Lagrangian fronts with favorable fishery conditions. The same approach has been shown to be effective to identify mesoscale eddies with a risk of contamination by Fukushima-derived radionuclides [8, 9]. This Special Issue addresses a variety of problems in geophysical fluid dynamics including formation of coherent structures in random hydrodynamic flows, hyperbolicity in the ocean, mesoscale surface and deep vortices in the ocean, formation of vocalized atmospheric vortices and motion of tropical cyclones, convective instability and nonlinear structures in systems with a multi-component convection, instability development in shear stratified flows and a hydrodynamical problem of free surface flow past a flat plate. The issue is opened with the paper “Clustering of a positive random field — what is this?” by Klyatskin where he consider a surprising phenomenon of clusterization in stochastic dynamic systems. Clustering in a random field means the emergence of compact areas with large values of this field as compared with a background where these values are fairly low. This spatial pattern is permanently changing. Statistical averaging, of course, completely destroys that clustering because it happens in different spatial areas in individual real-


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izations. Analyzing this phenomenon in terms of traditional statistical characteristics, such as moment and correlation functions of arbitrary order, is meaningless because clustering either exists or not. The main point of the Klyatskin’s analysis is that the clusterization appears with probability equal to one, i.e., almost in all realizations of the dynamic systems under consideration. This type of statistical coherence may be considered as a kind of self-organization in complex dynamic systems. The retrieval of statistically stable characteristics is similar to the concept of coherence like the self-organization of a multicomponent system that evolves from random interactions of its elements. Similar problems arise in geophysical fluid dynamics, magnetohydrodynamics, physics of plasma, astrophysics and radiophysics. In the next paper “Equilibrium distributions for hydrodynamic flows” Klyatskin considers coherent structures of vortex formation (the vortex genesis) in stochastic quasi-geostrophic flows related to the Earth’s rotation and random bottom topography. Hyperbolicity of a dynamical system is a property that is characterized by the presence of expanding and contracting directions in its phase space. This is a situation with phase trajectories converging in one direction and diverging in the other one. Manifestations of that property in the ocean are addressed by Prants et al in their paper “Hyperbolicity in the ocean”. The authors have shown how to identify hyperbolic points, hyperbolic trajectories and their stable and unstable manifolds in a satellite-derived velocity field in the ocean. To validate simulation results they used available tracks of oceanic drifters following near surface currents in some areas in the Northwestern Pacific Ocean. The tracks illustrate how drifters “feel” the presence of hyperbolic points, hyperbolic trajectories and stable and unstable manifolds and change abruptly their trajectories when approaching a hyperbolicity region. In particular, they discuss how dynamical systems methods can be helpful in planning drifter’s launches in the ocean. In order to study, for example, transport and mixing of water masses in a given region, the launch sites should be located nearby potentially long-lived hyperbolic stagnation points because those drifters would pass a complex way approaching at first such a point along its stable manifold and moving then away along the corresponding unstable manifold. One may anticipate that those drifters would pass regions with very different properties, from stagnation ones nearby hyperbolic stagnation points to Lagrangian coherent structures and Lagrangian fronts with a large energetics. The task of motion of a tropical cyclone within the framework of a proposed hydromechanical model is solved by B. Schmerlin and M. Schmerlin in the paper “Application of the hydromechanical model for a description of tropical cyclones motion”. The model contains parameters describing tropical cyclones and its interaction with wind field. The diagnostic, quasi-prognostic and prognostic calculations of movement of tropical cyclones have been carried out. Diagnostic calculations show that the model correctly describes peculiarities of the motion. Quasi-prognostic calculations show that model parameters may be rather correctly defined during a preliminary “preprognostic” period. The oceans and seas are populated with various types of mesoscale (from tenths to a few hundreds of km) eddies that transfer heat, salt, nutrients, carbon, pollutants and other tracers across the ocean. Those eddies may persist for the periods ranging from a few weeks to a few years and have a strong influence on the local climate, hydrography and fishery. Most of them are extended from the surface to some depth and even to the bottom. However, there exist very interesting vortex objects in the ocean called meddies. Warm and salty Mediterranean water flows out via the Strait of Gibraltar beneath cooler incoming water and moves along the continental slope. At a depth of around 1000 meters, that water reaches neutral buoyancy and separates from the slope. Some masses of this water pinch off and drift southwestward in the Atlantic Ocean as clockwise-rotating lenses of salty and warm water reaching neither the surface nor the bottom. Meddies are long-lived coherent structures because they rotate rapidly and translate slowly through the calm waters of the Canary Basin. Some of them may live up to 5 years. The problem is how to recognize and identify meddies and other deep eddies, living at depths below 100 m, with surface measurements or from satellites which can observe processes only at the sea surface. Ciani et al in the paper “Influence of deep vortices on the ocean surface” study the influence of deep vortices on the ocean surface in terms of sea-surface elevation. The authors use several mathematical and numerical models, from the most idealized configurations with point vortices to realistic ones with finite-volume vortices

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and determine analytically the surface influence of vortices at rest (a steady signature) and in motion (a dynamical signature). Then, using a nonlinear numerical hydrodynamic model for oceanic vortices, they determine the growth with time of a dynamical signature for drifting vortices without steady signature. The authors come to conclusion on the possibility to detect several types of oceanic vortices with surface measurements. Different self-organized and ordered convective structures in the form of periodic cloud rolls (convective cells) are often observed in the Earth atmosphere in satellite images. Another example of a self-organized convective structure is tropical cyclones (hurricanes), large scale atmospheric vortices formed over the ocean surface in the tropics. Tornadoes are also ordered convective structures of a smaller horizontal scale. These phenomena are traditionally considered to be the realization of a moist convective instability of the atmosphere. B. Schmerlin et al in the paper “The formation of localized atmospheric vortices of different spatial scales and ordered cloud structures” address those issues. The classical Rayleigh theory of convective instability of a viscous and heat conductive rotating atmospheric layer is generalized in their paper to the case of phase transitions of water vapor both for the precipitation and nonprecipitation convection. Instability region on the plane of model parameters turned out to generally consist of two subregions. In one subregion localized axisymmetric disturbances with a tropical cyclone (hurricane) structure have the highest growth rate. In case of precipitation convection, the ascending motions along the axis of symmetry correspond to such disturbances. In case of nonprecipitation convection, a spontaneous growth of localized vortices both with ascending and descending motions along that axis become possible. Under other parameters values in case of precipitation convection spatially periodic cloud structures (convective rolls or closed cloud cells) have the highest growth rate and in case of nonprecipitation convection there appear mesoscale systems of convective rolls or mesoscale cloud clusters with annular cloud structures. Phenomena of a convective instability near bifurcation points are important not only in the Earth’s atmosphere but in the atmosphere of other planets and in the ocean as well. 3D double-diffusive convection process in a layer of salty and rotating water column, governing by equations for momentum and diffusion of temperature and salt, is analyzed in the Kozitsky’s paper “An approach to the modeling of nonlinear structures in systems with a multi-component convection”. He considers the 3D multi-component convection in a horizontally infinite layer in an incompressible fluid slowly rotating around a vertical axis. A family of complex Ginzburg-Landau amplitude equations is derived by multiple-scaled method in the neighborhood of Hopf bifurcation points. They are numerically simulated in case of three-mode convection at large Rayleigh numbers. It is shown that the convection typically takes a form of hexagonal structures for localized initial conditions. The rotation of the system prevents a spread of the convective structures on the whole area. The approach developed is applied to modeling the Saturn’s polar hexagon. The Saturn’s hexagon, discovered by the Voyager mission in 1981 and 1982 and revisited since 2006 by the Cassini mission, is a persisting hexagonal cloud pattern around the north pole of Saturn. The sides of the hexagon are about 13800 km long. The hexagon does not shift in longitude like other clouds in the visible atmosphere. The double-diffusive convection plays in the Jupiter’s atmospheres the important role. The atmosphere there is a mixture of hydrogen with helium, and in the upper atmosphere there exists a vertical negative gradient of temperature due to hot lower layers. Thus we have a diffusive type of double-diffusive convection in a rotation system described by similar amplitude equations as in the ocean but with another coefficients. In theory of hydrodynamic stability of high-Reynolds-number plane-parallel shear flows an important role is played by so-called critical layer where the flow velocity is equal to the phase velocity of a disturbance. Inside that layer fluid particles move together with the wave and, therefore, an intensive exchange of energy and momentum occurs between the individual (fluid particles) and collective (wave) degrees of freedom, the phenomenon known as a wave-flow resonance. As a result, even at small wave amplitude, the motion of fluid particles within a critical layer is rearranged significantly and affects the development of the wave. In particular, it is the wave-flow resonance that is usually responsible for the wave (and, hence, flow) instability in linear approximation. Churilov in the paper “Instability development in shear flow with an inflection-free velocity profile and thin pycnocline” analyzes weakly stratified flows of the class with a wide 3D spectrum of the most


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unstable waves with very close growth rates and phase velocities so that their individual critical layers merge into a common one. The analysis of evolution equations for those waves has shown that throughout a weakly nonlinear stage of development their amplitudes grow explosively. During the first (three-wave) phase, the most rapidly growing are low-frequency waves whereas at the next phase, when numerous and diverse higher-order wave interactions come into play, the growth of high-frequency waves is accelerated and they overtake lowfrequency waves. The results obtained are illustrated by numerical calculations for some ensembles of waves. Two-dimensional free surface flows past a plate have attracted an attention for the last three decades. A transient free surface irrotational flow past a two-dimensional semi-infinite flat plate in the fluid of a finite depth has been considered in the linear inviscid and incompressible approximations by Ogilat and Stepanyants in their paper “Transient free surface ow past a two-dimensional flat stern”. The plate was suddenly submerged at relatively small depth below the free surface into the fluid uniformly moving with a constant velocity. The linearized problem was solved for relatively small Froude numbers using the Laplace and Fourier transforms, as well as the Wiener-Hopf technique. It has been shown that eventually at large time, the transient solution approaches asymptotically the steady-state solution. Normally, cavitation can lead to severe damage in vessels and hydraulic machinery. Therefore, the prevention of cavitation is an important concern. However, cavitation may be useful in some important applications, e.g., in reduction of friction drag for underwater vehicles. In fact, cavitating flow patterns and vapor structures are often unstable, and they often violently collapse as reaching a region with increased pressure. In the paper ‘Cavitating flow between two shear moving parallel plates and its control” by Yan Liu, Sheng Ren and Jiazhong Zhang a method for producing cavitation with the help of two parallel plates moving in opposite direction is proposed. The evolution of the cavitation and phase transition is studied there in detail by a developed scheme based on the lattice Boltzmann method. First, authors introduce the main features of the method and potential models for a single component multiphase flow. Then, the numerical simulation of evolution of phase transition, induced by shear motions of two parallel plates, is carried out, and complicated patterns of cavitating flows are analyzed in such a micro- and multiphase dynamical system. It is shown that the shear motion of two parallel plates could induce cavitation, and the cavitating flow patterns could be controlled efficiently by means of such parameters as initial density and velocity of motion. The guest editor would like to thank all of the authors who contributed to this Special Issue. Also, I am grateful to Valentine Afraimovich, Albert Luo and to the DNC editorial staff who made this issue possible. References [1] Koshel, K.V. and Prants, S.V. (2006) Chaotic advection in the ocean, Physics-Uspekhi, 49(11), 1151–1178. [2] Prants, S.V. (2014), Chaotic Lagrangian transport and mixing in the ocean, The European Physical Journal Special Topics, 223(13), 2723–2743. [3] Wiggins, S. (2005), The dynamical systems approach to Lagrangian transport in oceanic flows, Annual Review of Fluid Mechanics, 37(1), 295–328. [4] Aref, H. (1984), Stirring by chaotic advection, Journal of Fluid Mechanics, 143(1), 1–21. [5] Haller, G. (2002), Lagrangian coherent structures from approximate velocity data, Physics of Fluids, 14(6), 1851–1861. [6] Prants, S.V. (2013), Dynamical systems theory methods to study mixing and transport in the ocean, Physica Scripta, 87(3), 038115. [7] Prants, S.V., Budyansky, M.V., and Uleysky, M.Yu. (2014), Identifying Lagrangian fronts with favourable fishery conditions, Deep Sea Research Part I: Oceanographic Research Papers, 90, 27–35. [8] Budyansky, M.V., Goryachev, V.A., Kaplunenko, D.D., Lobanov, V.B., Prants, S.V., Sergeev, A.F., Shlyk, N.V., and Uleysky, M.Yu. (2015), Role of mesoscale eddies in transport of Fukushima-derived cesium isotopes in the ocean, Deep Sea Research Part I: Oceanographic Research Papers, 96, 15–27. [9] Prants, S.V., Budyansky, M.V., and Uleysky, M.Yu. (2014), Lagrangian study of surface transport in the Kuroshio Extension area based on simulation of propagation of Fukushima-derived radionuclides, Nonlinear Processes in Geophysics, 21(1), 279–289.



Clustering of a Positive Random Field –What is This? V.I. Klyatskin† A. M. Obukhov Institute of Atmospheric Physics RAS, Moscow, Pyzhevsky per. 3, 119017, Russia Submission Info Communicated by S.V. Prants Received 22 November 2014 Accepted 2 March 2015 Available online 1 October 2015

Abstract It is shown that, in parametrically excited stochastic dynamic systems described by partial differential equations, spatial structures (clusters) can appear with probability one, i.e., almost in every system realization, due to rare events happened with probability approaching to zero. The problems of such type arise in hydrodynamics, magnetohydrodynamics, physics of plasma, astrophysics, and radiophysics.

Keywords Intermittency Typical realization curve Dynamical localization Clustering


. . . Chaos is the place which serves to contain all things; for if this had not subsisted neither earth nor water nor the rest of the elements, nor the Universe as a whole, could have been constructed.. . . Sextus Empiricus, Against the Physics, against the Ethicists, R. G. Bury, p. 217, Harvard University Press, 1997. 1 Introduction. The main problem of the statistical analysis of stochastic dynamic systems First of all, I formulate the main problem of the statistical analysis of stochastic dynamic systems as I interpret it: revealing the fundamental features of such systems, which appear with probability one, i.e., almost in all realizations of the dynamic systems under consideration, on the basis of the corresponding statistical analysis [1]. These physical processes and phenomena occurring with probability one will be referred to as coherent processes and phenomena. This type of statistical coherence may be viewed as some organization of a complex dynamic system, and retrieval of its statistically stable characteristics is similar to the concept of coherence as self-organization of multicomponent systems that evolve from the random interactions of their elements [2]. I note a curious fact that nontrivial situations can be realized even in Gaussian random fields. Such a situation is realized, for example, in the two-dimensional problems of geophysical hydrodynamics in rotating fluids with random bottom topography (see e.g. [3–6], monographs [7–9] and the paper [10]). † Corresponding


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V.I. Klyatskin / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 225–242

Two features are characteristic of parametric excitation in dynamic systems described by partial differential equations: 1. On the one hand, at the initial stages of dynamic system evolution, such a parametric excitation is accompanied by an increase of all statistical characteristics of the problem solution (such as moment and correlation functions of any order) with time; 2. On the other hand, separate field realizations can show the stochastic nonstationary phenomenon of clustering in phase and physical spaces. Clustering of a field is identified as the emergence of compact areas with large values of this field against the residual background of areas where these values are fairly low. Their spatial pattern is permanently changing. Naturally, statistical averaging completely destroys all data on clustering, because, in individual realizations, it happens in different spatial areas. It is quite natural that the notion of clustering by itself is related to the spatial behavior of a dynamic system in individual realizations! Consideration of clustering in terms of traditional statistical characteristics such as moment and correlation functions of arbitrary order is meaningless! Clustering either exists or not exists. Parametrically excited dynamical systems are encountered in all branches of physics. And this excitation is commonly associated with instability of such systems relative to fluctuations in the initial conditions. A similar situation takes place for deterministic initial conditions under fluctuation of some parameters of the system. In this case, dynamic systems can be described by either ordinary or partial differential equations. First, we consider the statistical description of lognormal Markovian processes. 2 Lognormal Markovian random process, intermittency, typical realization curve and dynamical localization The simplest system of such type defines the lognormal Markovian random process y(t) and is described by the first-order ordinary differential equation d y(t; α ) = −α y(t; α ) + z(t)y(t; α ), y(0) = 1, dt

(1)

where z(t) is the Gaussian white noise process with the parameters z(t) = 0, Bz (t − t ) = 2Dδ (t − t ). The solution to this equation has the following stochastic structure ˆ t d τ z(τ )}. (2) y(t; α ) = exp{−α t + 0

Figure 1 displays realizations of lognormal random processes y(t; α ) (2) for positive and negative parameters α for the parameter ratio |α |/D = 1 (the dashed curves show the functions exp{−Dt} and exp{Dt}, correspondingly). The figure shows the presence of rare but strong fluctuations relative to the dashed curves towards both large values and zero. Such a property of random processes is called intermittency. The curve with respect to which the fluctuations are observed is referred to as the typical realization curve. The one-time probability density of the lognormal process (2) P(t; y) = δ (y(t) − y is described by the Fokker-Plank equation

∂ ∂ ∂ ∂ P(y,t; α ) = (α y + D y y)P(y,t; α ), ∂t ∂y ∂y ∂y

P(y, 0; α ) = δ (y − 1),

(3)

¡ whose solution depends, naturally, on parameter α P(y,t; α ) =

ln2 (yeα t ) 1 √ exp{− }. 4Dt 2y π Dt

(4)


4 y(t)

30

α >0

y(t)

227

α 0 and α < 0 for the parameter ratio |α |/D = 1.

From Eq. (1), one can easily obtain the equality α = − lim ∂ ln y(t; α ) /∂ t. Consequently, parameter α in t→∞

Eq. (2) is the Lyapunov characteristic index for the lognormal random process y(t; α ). We note that changing the sign of parameter α in (2) is statistically equivalent to switch to the process 1/y(t; α ) [7–9]. We discuss now the concept of typical realization curve of random process z(t). This concept concerns the fundamental features of the behavior of a separate process realization as a whole for temporal intervals of arbitrary duration. 2.1

Typical realization curve

Statistical characteristics of a random process z(t) at a fixed instant t are described by the one-time probability density P(z,t) and the probability distribution function ˆ F(t, Z) = P(z(t) < Z) =

Z

−∞

dzP(z,t).

The typical realization curve of random process z(t) is defined as the deterministic curve z∗ (t), which is the median of the integral distribution function and is determined as the solution to the algebraic equation ˆ

∗

F (t, z (t)) =

z∗ (t) −∞

1 dz P(z,t) = . 2

(5)

z∗ (t) z(t)

Δt3

{

t1

Δt2

{

{

Δt1

t2

t

Fig. 2 To the definition of the typical realization curve of a random process.

228


This means, on the one hand, that for any t the probabilities P{z(t) > z∗ (t)} = P{z(t) < z∗ (t)} = 1/2. On the other hand, this curve has a specific property that, for any time interval (t1 ,t2 ), the random process z(t) ’winds’ around the curve z∗ (t) such that the mean times are 1 Tz(t)>z∗ (t) = Tz(t) 0, and it exponentially increases in the opposite case of α < 0. At α = 0 intermittency takes place about the straight line f ∗ (t) = 1. 2.2

Dynamical localization

I specially point to the fact that, for the one-dimensional problems, positive values of the Lyapunov characteristic index α , just correspond to the physical phenomenon of dynamic localization (clustering). Among such dynamic systems are the one-dimensional stationary Schrödinger equation for eigenfunctions (Anderson’s dynamic localization) and the one-dimensional problem on waves in layered random media (dynamic localization of waves). 3 Lognormal random fields, intermittency, statistical topography and clustering 3.1

Lognormal random fields

Consider now the positive lognormal random field f (r,t), whose one-point probability density P(r,t; f ) = δ ( f (r,t)− f ) satisfies the equation

∂ ∂ ∂2 ∂ ∂ P(r,t; f ) = {D0 2 + α f +Df f f }P(r,t; f ), ∂t ∂r ∂f ∂f ∂f

P(r, 0; f ) = δ ( f − f0 (r)) ,

(6)

where D0 is the coefficient of diffusion in r-space and coefficients α and D f characterize diffusion in f -space. Here, parameter α may be positive, negative, and equal to zero (the critical case). For the one-point characterf (r,t) = 1/ f (r,t). The solution to this istics, changing the sign of α means switching from field f (r,t) to field equation has the form P(r,t; f ) =

ln2 [ f eα t / f0 (r)] ∂2 1 exp{D0 t 2 } exp{− }. ∂r 4D f t 2 f πD f t

(7)

´ ´ For a positive conservative random field f (r,t) with dr f (r,t) = dr f0 (r), we have α = D f , and Eq. (6) assumes the form ∂2 ∂ ∂2 P(r,t; f ) = (D0 2 + α 2 f 2 )P(r,t; f ). (8) ∂t ∂r ∂f


229

It is clear that property of intermittency is characteristic of arbitrary random field f (r,t). First of all, temporal evolution f (r,t) at any fixed spatial point r is a random process, for which holds all aforesaid material. In the case of the spatially homogeneous problem corresponding to the initial field distribution f0 (r) = f0 , all one-point statistical characteristics of field f (r,t) are independent of point r, and positive value of characteristic index α = − lim ∂ ln f (r,t) /∂ t means that realizations of this field decrease with time at arbitrary spatial t→∞ point notwithstanding the large occasional spikes characteristic of lognormal process. Characteristic time of field decaying is t ∼ 1/α . But if this field decreases almost everywhere, it must concentrate somewhere, i.e., clustering must appear. For negative values of parameter α , the field increases at any fixed point of space. Thus, in the context of the spatially homogeneous problem, one-point statistical characteristics of random field f (r,t) are statistically equivalent to statistical characteristics of lognormal process f (t; α ) (4). Phenomenon of spatial structure formation in individual realizations of random fields can be revealed and described only by analyzing the one-time and one-point probability densities of these fields on the basis of ideas of statistical topography. 4 Elements of the statistical topography of random fields Randomness of medium parameters in dynamic systems gives rise to a stochastic behavior of physical fields. Individual samples of scalar two-dimensional fields f (R,t), where R = (x, y), resemble a rough mountainous terrain with randomly scattered peaks, troughs, ridges, and saddles. Similarly to common topography of mountain ranges, the statistical topography studies the systems of contours (level lines in the two-dimensional case and surfaces of constant values in the three-dimensional case) specified by the equality f (r,t) = f . Figure 3 shows examples of realizations of two random fields characterized by different statistical structures. a

b

4

2

2 1

0

0 −2 140

140 100

100 60

−1

3

140 100

−2

60

120

120

100

100

80

80

60

60 100

100

120

140

60

80

2 1

60

60

140

80

3

140

140

60

4

2 1

100

120

140

Fig. 3 Realizations of the fields governed by (a) Gaussian and (b) lognormal distributions and the corresponding topographic level lines.

230


Phenomenon of clustering random fields can be detected and described only on the basis of the ideas of statistical topography. For analyzing a system of contours (for simplicity, we will deal with the two-dimensional case and assume r = R), we introduce the singular indicator function ϕ (R,t; f ) = δ ( f (R,t) − f ) concentrated on these contours. The convenience of this function consists, in particular, in the fact that it allows simple expressions for quantities such as the total area of regions where f (R,t) > f (i.e., within level lines f (R,t) = f ) ˆ ∞ ˆ ˆ df dR ϕ (R,t; f ), S(t; f ) = θ ( f (R,t) − f )dR = f

and the total ’mass’ of the field within these regions ˆ ˆ M(t; f ) = f (R,t)θ ( f (R,t) − f )dR =

∞

f df

f

ˆ

dR ϕ (R,t; f ),

where θ ( f (R,t) − f ) is the Heaviside theta function. The mean value of indicator function of random field f (R,t) determines the one-time (in time) and one-point (in space) probability density P(R,t; f ) = δ ( f (R,t)− f ) . Consequently, this probability density immediately determines ensemble-averaged values of the above expressions S(t; f ) and M(t; f ): ˆ ∞ ˆ ˆ ˆ ∞ df f df dR P(R,t; f ), M(t; f ) = dR P(R,t; f ). S(t; f ) = f

f

In the case of the spatially homogeneous field f (R,t), the corresponding probability density P(R,t; f ) is independent of R. In this case, statistical averages of the above expressions (without integration over R) will characterize the corresponding specific (per unit area) values of these quantities. In this case, random field f (R,t) is statistically equivalent to the random process whose statistical characteristics coincide with the spatial one-point characteristics of field f (R,t). Consider now the conditions of occurrence of stochastic structure formation [1, 7–9, 11–13]. 4.1

On the criterion of stochastic structure formation

It is clear that, for a positive field f (R,t), the condition of clustering with a probability of one, i.e., almost in all realizations, is formulated in the general case as simultaneous tendency of fulfillment of the following asymptotic equalities for t → ∞ ˆ S(t; f ) → 0, M(t; f ) → d R f (R,t) . On the contrary, simultaneous tendency of fulfillment of the asymptotic equalities for t → ∞ ˆ S(t; f ) → ∞, M(t; f ) → d R f (R,t) corresponds to the absence of structure formation. In the case of a spatially homogeneous field f (R,t), the corresponding probability density P(R,t; f ) is independent of R. In this case, statistical averages of the above expressions (without integration over R) will characterize the corresponding specific (per unit area) values of these quantities. So, the specific mean area shom (t; f ) over which the random field f (R,t) exceeds a given level f , coincides with the probability of the event f (R,t) > f at any spatial point, i.e., shom (t; f ) = θ ( f (R,t) − f ) = P{ f (R,t) > f }


231

and therefore the mean specific area offers a geometric interpretation of the probability of the event f (R,t) > f , which is apparently independent of the point R. Consequently, in the case of a homogeneous field, conditions of clustering are reduced to the tendency of asymptotic equalities for t → ∞ shom (t; f ) = P{ f (r,t) > f } → 0,

mhom (t; f ) → f (t) .

Absence of clustering corresponds to the tendency of asymptotic equalities for t → ∞ shom (t; f ) = P{ f (r,t) > f } → 1,

mhom (t; f ) → f (t) .

Thus, in spatially homogeneous problems, clustering is a universal (not only physical) phenomenon inherent in nature (with some provisos, of course) and realized with probability one, i.e., occurred in almost all realizations of a positive random field generated by a rare event whose probability tends to zero. It is noteworthy that these conditions have transparent mathematical meaning and can be described at a relatively elementary mathematical level based on the ideas of statistical topography. Namely availability of these rare events is the trigger that starts the process of structure formation. In the conditions of developed clustering, the field is simply absent in the most part of space! As for setup time of such spatial structure formation, it depends on limiting behavior of the right-hand expressions in all above asymptotic equalities. It is clear that the above conditions of presence and absence of clustering field f (R,t) bear no relation to parametric growth in time of the field statistical characteristics such as moment and correlation functions of arbitrary order. The above criterion of ’ideal’ clustering (analogously to ideal hydrodynamic) describes dynamics of cluster formation in the dynamic systems described in general by the first-order partial differential equations. This ideal structure originates in the form of very thin belts (in the two-dimensional case) or very thin tubes (in the three-dimensional case). As for actual physical systems, various additional factors come to play with time; they are related to generation of random field spatial derivatives like spatial diffusion or diffraction, which deform the pattern of clustering, but not dispose it. In particular, a possible situation can occur when the probability density rapidly approaches its steady-state regime P(R; f ) for t → ∞. In this case, functionals like ˆ ∞ ˆ ˆ ˆ ∞ df f df dR P(R; f ) and M( f ) = dR P(R; f ) S( f ) = f

f

cease to describe further deformation of the clustering pattern, and we must study temporal evolution of functionals related to the spatial derivatives of field f (R,t). 4.2

Statistical topography of the lognormal random fields

Knowing the one-point probability density of random field f (r,t) (7), we can obtain some general information about the spatial structure of f (r,t). In particular, functionals of f (r,t) such as the total mean volume (in the three-dimensional case) or area (in the two-dimensional case) of the domain in which f (r,t) > f and the total mean ’mass’ of the field confined within that domain are given by the formulas ˆ ˆ ∞ ˆ ˆ ∞ d f P(r,t; f ), M(t, f ) = dr d f f P(r,t; f ). V (t, f ) = dr f

f

The values of these functionals are independent of the diffusion in the r space (of the coefficient D0 ), and in the case of probability distribution (7), we obtain the asymptotic form of the mean volume as t → ∞ decreases in time for α > 0

ˆ D 1 α /D −α 2 t/4D e dr f0 (r). V (t, f ) ≈ α π f α /Dt

232


For α < 0, the mean volume occupies all the space as t → ∞. The asymptotic form of the total mean ’mass’ as t → ∞ is (α < 2D)

ˆ f (2D−α )/D −(2D−α )2t/4 1 D ( ) e ]. M(t, f ) ≈ e(D−α )t dr f0 (r)[1 − (2D − α ) π t f0 (r) Therefore, for α > 0, all the mean ’mass’ is collected in clusters in the limit t → ∞. For homogeneous initial conditions we have the asymptotic expressions as t → ∞

⎧ ⎪ 1 D f0 α /D −α 2t/4D ⎪ ⎪ (α > 0), ⎪ ⎨ α πt ( f ) e

vhom (t, f ) = P{ f (r,t) > f } ≈ ⎪ ⎪ 1 D f |α |/D −α 2t/4D ⎪ ⎪ ( ) e (α < 0), ⎩1− |α | π t f0

1 D f (2D−α )/D −(2D−α )2t/4D ( ) e ]. mhom (t, f ) ≈ f0 e(D−α )t [1 − (2D − α ) π t f0

(9)

(10)

Hence, for α > 0, the specific total volume tends to zero, and the specific total ’mass’ contained in that volume tends to the mean ’mass’ of the entire space, which corresponds to the criterion for structure formation with probability one for ’ideal clustering’ of the field f (r,t) under consideration. In these conditions, random field f (r,t) is practically absent in the most part of space. In addition, at each fixed point of space, the characteristic decay time of the fieldis α t ∼ 1, and the characteristic time of cluster structure formation of the field is α t ∼ max 4ξ , 4ξ /(2ξ − 1)2 , where ξ = D/α . For α < 0, clustering is absent, and only the general increase of random field f (r,t) occurs everywhere in the space. In this case, therefore, chaos remains chaos! Only clustering of the zeros of the field f (r,t) occurs. We have the following theorem [1]: In a statistically homogeneous problem, a conservative positive parametrically excited random log-normal field always experiences clustering with probability one, i.e., for almost all realizations of this field. Here, as noted above, α = D for a conservative f (r,t) (see Eq. (8)). Therefore, the characteristic time of cluster structure formation α t ∼ 4, which is four times greater than the characteristic decay time of the field at almost every point of space. 5 Some examples of parametric excitation of stochastic dynamic systems The simplest examples are the Anderson localization for wave eigenfunctions of the stationary one-dimensional Schrödinger equation with a random potential and the dynamic localization of wave field intensity in a wave problem of propagation in randomly layered medium (Helmholtz stochastic equation). Clearly, both examples are characterized by exponential growth of the moments of wave field intensity with the distance in medium from the source [7–9]. Moreover, in a number of cases, clustering of both passive scalar tracer (density field) and vector tracer (magnetic field energy) can occur in problems on turbulent transfer in the scope of kinematic approximation! The basic stochastic equations for the density field ρ (r,t) and the nondivergent magnetic field H(r,t) at the kinematic stage are the scalar continuity equation

∂ ∂ + u(r,t))ρ (r,t) = μρ Δρ (r,t), ∂t ∂r and the vector induction equation (

(

ρ (r, 0) = ρ0 (r)

∂ ∂ ∂ + u(r,t))H(r,t) = (H(r,t) · )u(r,t) + μH ΔH(r,t), ∂t ∂r ∂r

(11)

(12)


233

where μρ is the dynamical diffusion coefficient for the density, μH is the dynamical diffusion coefficient for the magnetic field, which is related to the medium conductivity, and u(r,t) is the field of turbulent velocities. Dynamic system density (11) and´ magnetic field (12) are conservative, and both the total scalar mass ´ M = drρ (r,t) and the magnetic flux dr H(r,t) remain constant during the evolution. For homogeneous initial conditions ρ (r, 0) = ρ0 , H(r, 0) = H0 , that we consider here, the following equalities are a corollary of the conservatism of dynamic systems (11) and (12) ρ (r,t) = ρ0 , H(r,t) = H0 . A specific feature of Eqs. (11) and (12) is the parametric excitation with time in each realization of both the density field ρ (r,t) (for a compressible fluid flow) and the magnetic field energy E(r,t) = H2 (r,t) (for a turbulent fluid flow), which is called the stochastic dynamo [7–9]. These equations are fairly complicated and depend on a large body of parameters. For a homogeneous initial condition, one can explicitly distinguish two temporal intervals in which problem solutions differ fundamentally. At the first (initial) stage, fields are generated in every particular realization. Effects related to dynamic diffusion are clearly inessential at this stage, and one can omit the corresponding terms in Eqs. (11) and (12). Thus, at the first stage we arrive at the equations ( (

∂ ∂ + u(r,t))ρ (r,t) = 0, ∂t ∂r

ρ (r, 0) = ρ0 ,

∂ ∂ ∂ + u(r,t))H(r,t) = (H(r,t) · )u(r,t), ∂t ∂r ∂r

H(r, 0) = H0 .

(13) (14)

However, it is namely the interval during which spatial structure formations can originate in separate realizations! Note that the partial differential equations (Eulerian description) (13), (14) are equivalent to the system of characteristic equations for particles (Lagrangian description) which are the simplest purely kinematic equations d r(t) = u(r(t),t), dt

r(0) = r0 .

Numerical simulations show that the behavior of a system of particles essentially depends on whether the random field of velocities is nondivergent or divergent (see e.g. monographs [7–9]). Fig. 4 illustrate structure formation in density field by the photos of cluster structure of cloudy sky.a Discussion of this photos see in Sect. 6.2. The point is that the nature of stochastic behaviour of water masses is completely inessential here. It can be both developed convection and atmospheric turbulence. According to the time of this photo session the second mechanism was most probable. I give also a pattern taken from Internet that shows the cluster structure of the Universe and is seemingly a direct consequence of clustering the cosmic matter in random velocity field (Fig. 5) As an illustration of ’ideal’ and ’deformed’ clustering we mention also the lava lakes Fig. 6. I illustrate structure formation in magnetic field by the extract from an internet-page: What does puzzle astrophysicists so strongly?b Contrary to hypotheses formed for fifty years, at the boundary of planetary system observers encountered a boiling foam of locally magnetized areas each of hundreds of millions kilometers in extent, which form a nonstationary cellular structure in which magnetic field lines are permanently breaking and recombining to form new areas—magnetic ”bubbles” (fig. 7). a These

photos were obtained by V.A. Dovzhenko on June 15 and August 2, 2013 at the coast of the Azov Sea (Russia) at 21:00. Atkinson, Voyagers Find Giant Jacuzzi-like Bubbles at Edge of Solar System, http://www.universetoday.com/86446/voyagers-find-giant-jacuzzi-like-bubbles-at-edge-of-solar-system/ b Nancy

234


a

b

Fig. 4 Nearly ’ideal’ clustering (a) and ’deformed’ clustering (b).

Fig. 5 Cluster structure of the Universe.

Another example of problems pertaining to parametric excitation of dynamic systems is the problem on propagation of a monochromatic plane wave in random multidimensional media in terms of the complex scalar Leontovich parabolic equation [7–9] i ik ∂ u(x, R) = ΔR u(x, R) + ε (x, R)u(x, R), ∂x 2k 2

u(x, R) = u0 ,

(15)

where function ε (x, R) is the fluctuating portion (deviation from unity) of dielectric permittivity, x–axis is directed along the initial direction of wave propagation, and vector R denotes the coordinates in the transverse plane. Note that this equation is the Schrödinger equation with a random potential ε (x, R), where coordinate x plays the role of time t. Introducing the amplitude-phase representation of the wavefield in Eq. (15) by the formula u(x, R) = A(x, R)eiS(x,R) , we can write the equation for the wavefield intensity I(x, R) = |u(x, R)|2 in the form 1 ∂ I(x, R) + ∇R {∇R S(x, R)I(x, R)} = 0, ∂x k

I(0, R) = I0 (R).

(16)


a

235

b

Fig. 6 The lava lakes in the depths of Nyiragongo Crater (in the heart of the Great Lakes region of Africa) (a), and Kilauea Crater (b) (Hawaiian National Volcano Park).

Fig. 7 Artist’s interpretation depicting the new view of the heliosphere (left). Computer simulation of the magnetic reconnection in the heliosheats, which look like bubbles, or sausages (right).

From this equation follows that the power of a wave in plane x = const is conserved in the general case of arbitrary incident wave beam: ˆ ˆ E0 = I(x, R)dR = I0 (R)dR. Equation (16) coincides in form with Eq. (13). Consequently, we can treat it as the equation of transfer of conservative tracer in the potential velocity field. Consequently realizations of intensity must show cluster behavior, which manifests itself in the appearance of caustic structures. For example, Fig. 8 shows photos of the transverse sections of the laser beam propagating in turbulent medium for different magnitudes of dielectric permittivity fluctuations (laboratory investigations). Figure 9 shows similar photos. These photos were simulated numerically. Both figures clearly show the appearance of caustic structures in the wave field. Note that the nonlinear parabolic equation describing self-action of a harmonic wave field in multidimensional random media, i ik ∂ u(x, R) = ΔR u(x, R) + ε (x, R; I(x, R))u(x, R), u(0, R) = u0 (R) (17) ∂x 2k 2 coincides in form with the nonlinear Schrödinger equation. Consequently, clustering of wave field energy must occur in this case too, because Eq. (16) is formally independent of the shape of function ε (x, R; I(x, R)).

236


a

b

Fig. 8 Transverse section of laser beam propagating in turbulent medium in the regions of (a) strong focusing and (b) strong (saturated) fluctuations. Experiment in laboratory conditions.

a

b

Fig. 9 Transverse section of laser beam propagating in turbulent medium in the regions of (a) strong focusing and (b) strong (saturated) fluctuations. Numerical simulations.

In particular, random initial condition u0 (R) yields the caustic structure formation even if ε (x, R) = 0. In this case, Eqs. (15) and (17) are simplified and assume the form i ∂ u(x, R) = ΔR u(x, R), ∂x 2k

u(0, R) = u0 (R)

(18)

that allows an analytical solution, k ix u(x, R) = exp{ ΔR }u0 (R) = 2k 2π ix

ˆ

dR exp{

ik (R − R )2 }u0 (R ). 2x

(19)

For the plane incident wave, the initial condition to Eqs. (18) has the form |u0 (R)| = 1, i.e., u0 (R) = eiS0 (R) , where S0 (R) is the field of the random initial phase. Here, the spatial fluctuations in the initial distribution of the phase are transformed into the caustic structure of the wave field intensity (random phase screen), which is well known and is regularly observed both in water pools and shallow waters (see fig. 10). 5.1

Waves on sea surface

Consider now the behavior of water boundary at the sea surface. In this case, equations of hydrodynamics must be supplemented with the kinematic boundary condition on the free sea surface z = ξ (R,t), where R denotes the coordinates in the plane perpendicular to the vertical axis z (Fig. 11), which has the form

∂ ξ (R,t) ∂ ξ (R,t) + ui (R, ξ (R,t),t) = wz (R, ξ (R,t);t), ∂t ∂ Ri

(20)


237

Fig. 10 Caustics in a pool and in a sea.

ξ (R,t) 0 z H(R) R

−H

Fig. 11 Disturbance of water surface.

where u(R, z,t) and w(R, z;t) are horizontal and vertical components of the velocity field. We can consider condition (20) as a closed stochastic quasilinear equation in the kinematic approximation, i.e., at given statistical characteristics of the velocity field {u(R, z,t), w(R, z;t)}. At the same time, Eq. (20) describes generation of waves on the sea surface which are driven by the vertical component of the hydrodynamic velocity field. Differentiating Eq. (20) with respect to R, we obtain an equation in the gradient of surface displacement pk (R,t) = ∂ ξ (R,t)/∂ Rk , which is a characteristic of surface slopes,

∂ pk (R,t) ∂ ui (R, z;t) ∂ ui (R, ξ (R,t);t) +[ pk (R,t)]pi (R,t) |z=ξ (R,t) + ∂t ∂ Rk ∂z ∂ pi (R,t) ∂ wz (R, z;t) ∂ wz (R, ξ (R,t);t) pk (R,t). (21) = |z=ξ (R,t) + + ui (R, ξ (R,t),t) ∂ Rk ∂ Rk ∂z The problem considered here has the second boundary condition associated with inhomogeneity of bottom topography (see Fig. 11). In the scope of the kinematic approximation, this boundary condition appears in the functional form; namely, the variational derivatives of the solution ξ (R,t) and p(R,t) assume the form δ ξ (R,t) ∼ θ z − H(R) θ t − t , δ u(R , z ,t )

δ p(R,t) ∼ θ z − H(R) θ t − t , δ u(R , z ,t )

(22)

where θ (z) is the Heaviside theta function. So statistical solution of this problem must answer the question whether information about the existence of anomalous large structures is contained in the quasilinear equation under consideration.

238


Note that some huge waves – called also rogue waves – are occasionally observed on the sea and ocean surface (see, e.g., [14, 15]). Figures 12 and 13 show three photos of an unusual narrow and lengthy immobile structure of about 4-5-m height observed on June 11, 2006 near Kamchatka Pacific coast at a distance of 1-1.5 km from the shoreline [15].

a

Fig. 12 Rogue wave. (a) Side view.

b

c

Fig. 13 Rogue wave. Front view of (b) the head of the wave and (c) the middle of the wave .

The author of these photos, M.M. Sokolovsky described this phenomenon in the following words: ’It was certainly a strange wave, because it appeared and then disappeared a few times. No waves were observed around this wave, peace and quite.’


239

6 Stochastic transport phenomena in a random velocity field 6.1

On the statistical characteristics of the velocity field

The random velocity field u(r,t) is assumed to be a divergent Gaussian field with correlation and spectral tensors: ˆ Bi j (r − r ,t − t ) = ui (r,t)u j (r ,t ) = dk fi j (k,t − t )eik(r−r ) , ˆ (23) 1 p −ikr s (r,t)e , f (k,t) = f (k,t) + f (k,t), dr B fi j (k,t) = i j i j ij ij (2π )d where d is the dimension of the space and the spectral component of the tensor of the velocity field have the structure ki k j ki k j p s s fi j (k,t) = f (k,t) δi j − 2 , fi j (k,t) = f p (k,t) 2 . k k In addition, we assume the random field u(r,t) to be delta-correlated in time and let us introduce the following parameters related to the velocity field [7–9]: ˆ ∞ ˆ ∞ 1 s p d τ ω (r,t + τ )ω (r,t), D = d τ div u(r,t + τ )div u(r,t) , (24) D = d−1 0 0 where ω (r,t) = ∇ × u(r,t) is the vortex of the velocity field. 6.2

Clustering of the density field in a random velocity field

Stochastic structure formation in a spatially homogeneous statistical problem of the diffusion of a density field ρ (r,t) in a random velocity field is described by Eq. (13). In this case, the one-point probability density of the field ρ (r,t), independent of the spatial coordinates r, is described by the equation [7–9]

∂ ∂2 P(t; ρ ) = Dρ 2 ρ 2 P(t; ρ ), P(0; ρ ) = δ (ρ − ρ0 ), ∂t ∂ρ where the diffusion coefficient in the ρ -space Dρ = Dp is given by relation (24) and, therefore, the one-point probability density of the density field is lognormal with the following probability density and the corresponding integral distribution function: P(τ ; ρ ) =

1 √

2ρ πτ

exp{−

ln2 (ρ eτ /ρ0 ) }, 4τ

F(τ ; ρ ) = Pr(

ln (ρ eτ /ρ0 ) √ ), 2 τ

where τ = Dρ t is a parameter. As pointed out above, the problem for the one-point characteristics of the density field ρ (r,t), is statistically equivalent to the analysis of a random process for which all the moment functions, both for n > 0, and n < 0, exponentially increase in time: ρ (r,t) = ρ0 ,

ρ n (r,t) = ρ0n en(n−1)τ ,

while the typical realization curve of the density field coinciding with the Lyapunov exponential at any fixed point in space exponentially decreases in time

ρ ∗ (t) = eln(ρ (r,t) = ρ0 e−τ , which attests to the cluster nature of fluctuations of the density field with probability 1 (i.e., in almost all of its realizations) in arbitrary divergent flows. Besides, the characteristic decay time of the density field is τ ∼ 1 at each fixed point of space.

240


Note that, even for an incompressible fluid in hydrodynamic flows, the density field will clusterize for a ’floating’ impurity, in view of the finite inertia of impurity fields and for multiphase flows of a fluid, i.e., always when the potential component of the spectrum arises in the velocity field of the impurity which is distinct from the velocity field of the fluid itself. Thus, clustering of the density field in the Eulerian description must occur if we deal with the diffusion of the density of inertialess buoyant tracer. This situation corresponds, for example, to the cloudy sky in Fig. 4. 6.3

On clustering of magnetic field energy

Consider the probabilistic description of the magnetic field on the basis of the dynamic equation (14) [7–9]. The probability density of the magnetic field energy E(r,t) = H2 (r,t) for a spatially homogeneous problem satisfies the equation ∂ ∂ ∂ ∂ P(t; E) = {α E +D E E}P(t; E), ∂t ∂E ∂E ∂E with initial condition P(0; E) = δ (E − E0 ), where coefficients α and D characterize diffusion in E-space

α =2

d −1 p (D − Ds ) , d +2

D = 4(d − 1)

(d + 1) Dp + Ds , d(d + 2)

and the one-point statistical characteristics of energy E(r,t)/E0 is statistically equivalent to the characteristics of the random process E(t; α )/E0 = y(t; α ) (2) with the probability density of magnetic field energy P(E,t; α ) =

ln2 (Eeα t /E0 ) 1 √ exp{− }. 4Dt 2E π Dt

(25)

Distribution (25) has a long flat tail for Dt 1, which is indicative of an increased role of great peaks of process E(t; α ) in the formation of the one-point statistics. And all moments of the energy are functions exponentially increasing with time E n (t) = E0n exp{−2n

d −1 p (d + 1) Dp + Ds (D − Ds )t + 4n2 (d − 1) t} d +2 d(d + 2)

and, in particular case of n = 1, the specific average energy is given by E(t) = E0 exp{2

d −1 s (D + Dp )t}. d

The typical realization curve of random process E(t) is the exponential function E ∗ (t) = E0 e−α t = E0 exp{−2

d −1 p (D − Ds )t} d +2

that can both increase and decrease with time. Indeed, for α < 0 (Dp < Ds ), the typical realization curve exponentially increases with time, which is evidence of general increase of magnetic energy at every spatial point. Otherwise, for α > 0 (Dp > Ds ), the typical realization curve exponentially decreases at every spatial point, which is indicative of cluster structure of magnetic energy field. We noted that all statistical characteristics of magnetic field energy (moment and correlation functions of different order) involves the spectral components of the velocity field Dp and Ds additively. This is of course a consequence of linearity of Eqs. (12) and (14), but this fact implies that all principal (functional) laws of such a statistical description do not distinguish between the effects of the solenoidal and potential components of the random velocity field. In other words, all obtained laws for these statistical quantities have the same structure for both an incompressible flow (Dp = 0) and a purely potential flow (Ds = 0).


241

For an incompressible flow, no clustering occurs, while for a potential flow, in contrast, clustering does occur. It is therefore absolutely clear that the above statistical characteristics contain no information about stochastic structure formation in individual realizations of the magnetic field energy, i.e., about clustering. In addition, the original induction equation (12) is applicable within the applicability range of the kinematic approximation. In the presence of clustering, when the magnetic field is absent in most of the space, it is natural that its aftereffect on the velocity field is inessential. In the absence of clustering, when the magnetic field is generated in the whole of space, we can expect that the kinematic approximation is applicable only on a time interval so short that, in our opinion, there is no point whatsoever in discussing the effect of the dynamical diffusion coefficient on the formation of statistics of the magnetic field energy. 7 Conclusions We have considered the process of stochastic structure formation in dynamical systems with parametric excitation of positive random fields f (r,t) described by partial differential equations. Such a structure formation in space and time either does or does not exist! And if it is realized in space, then this occurs in specific realizations almost always, i.e., with probability one (exponential fast) and consist in the following for a spatially homogeneous statistical problem:

• As time passed, the field decreases at almost all points of space (with some fluctuations, of course).

• In the space itself, small-volume domains occur where this field is concentrated (clustered), and stochastic structure formation is caused by diffusion of random field f (r,t) in its phase space { f }.

In the case under consideration, clustering of a field f (r,t) of any nature is a general property of dynamical fields, and we can say that structure formation for any such random field is a law of Nature. In this paper, we found the conditions under which such structure formation is realized. Notably, these conditions have a clear physical-mathematical meaning and can be described at a relatively elementary mathematical level based on the ideas of statistical topography. To conclude with, I note that a point commonly accepted in many works suggests that, for an event to take place, it is required that this event was the most probable one. For example, in recent work, Prof. Ivanitskii G.R. calculated certain probabilities and came out with a hypothesis on origin of life from the perspective of physics [16]: ’Life can be defined as resulting from a game involving interactions of matter one part of which acquires the ability to remember the success (or failure) probabilities from the previous rounds of the game, thereby increasing its chances for further survival in the next rounds. This part of matter is currently called living matter.’ I cannot agree with the idea that origin of life is a game process. I believe that origin of life is an event happened with probability one [11]. Acknowledgement This work was supported by the RSF 14-27-00134.

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References [1] Klyatskin, V.I. (2013), Clustering of a random positive field as a law of Nature, Theoretical and Mathematical Physics, 176(3), 1252–1266. [2] Nicolis, G. and Prigogin, I. (1989), Exploring Complexity, an Introduction, Freeman W.H. and Company, New York. [3] Klyatskin, V.I. (1969), On Statistical theory of two-dimensional turbulence, Journal of Applied Mathematics and Mechanics, 33(5), 864–866. [4] Klyatskin, V.I. (1995), Equilibrium states for quasigeostrophic flows with random topography, Izvestiya, Atmospheric and Oceanic Physics, 31(6), 717–722. [5] Klyatskin, V.I. and Gurarie, D. (1996), Random topography in geophysical models, in: Stochastic Models in Geosystems, eds. Molchanov, S.A. and Woyczynski, W.A. IMA Volumes in Math. and its Appl. 85, 149–170. N.Y. SpringerVerlag. [6] Klyatskin, V.I. and Gurarie D. (1996), Equilibrium states for quasigeostrophic flows with random topography, Physica D, 98, 466–480. [7] Klyatskin, V.I. (2011), Lectures on Dynamics of Stochastic Systems, Elsevier, Boston, MA. [8] Klyatskin, V.I. (2014), Stochastic Equations: Theory and Applications in Acoustics, Hydrodynamics, Magnetohydrodynamics, and Radiophysics, Volume 1 (Basic Concepts, Exact Results, and Asymptotic Approximations), Springer: Complexity, Springer Berlin. [9] Klyatskin, V.I. (2014), Stochastic Equations: Theory and Applications in Acoustics, Hydrodynamics, Magnetohydrodynamics, and Radiophysics, Volume 2 (Coherent Phenomena in Stochastic Dynamic Systems), Springer: Complexity, Springer Berlin. [10] Klyatskin, V.I. (2015), Equilibrium distributions for hydrodynamic flows, Discontinuity, Nonlinearity, and Complexity, 4(3), 243–255. [11] Klyatskin, V.I. (2012), Spatial structures can form in stochastic dynamic systems due to near-zero-probability events: (comment on ’21st century: what is life from the perspective of physics?), Physics-Uspekhi, 55(11), 1152–1154. [12] Klyatskin, V.I. (2013), On the criterion of stochastic structure formation in random media, Proceedings of the 4th International Interdisciplinary Chaos Symposium, 69–73, Springer-Verlag, Berlin. [13] Klyatskin V.I. (2013), On the Statistical Theory of Spatial Structure Formation in Random Media, Russian Journal of Mathematical Physics, 20(3), 295–314. [14] Kharif C., Pelinovskyy E. and Slyunaen A. (2009), Rogue Waves in the Ocean, Springer, Berlin. [15] Klyatskin V.I. (2014), Anomalous waves as an object of statistical topography. Problem statement, Theoretical and Mathematical Physics, 180(1), 850–861. [16] Ivanittskii, G.R. (2010), 21st century: what is life from the perspective of physics, Physics-Uspekhi, 53(4), 327–356.



Equilibrium Distributions for Hydrodynamic Flows V.I. Klyatskin † A. M. Obukhov Institute of Atmospheric Physics RAS, Moscow, Pyzhevsky per. 3, 119017, Russia Submission Info Communicated by S.V. Prants Received 20 November 2014 Accepted 2 March 2015 Available online 1 October 2015

Abstract This paper deals with the problem of stochastic structure formation in random hydrodynamic flows. In particular, starting from an analysis of the steady-state probability density, it considers coherent structures of vortex formation (vortex genesis) in stochastic quasi-geostrophic flows, which are related to rotation and random topography of the bottom.

Keywords Stochastic structure formation Coherent structures Vortex genesis Quasi-geostrophic flows


1 Introduction Consider the turbulent motion model that assumes the presence of external forces f(r,t) acting on the fluid. Such a model is evidently only imaginary, because no actual analogues exist for these forces. However, assuming that forces f(r,t) ensure an appreciable energy income only to large-scale velocity components, we can expect that, within the concepts of the theory of local isotropic turbulence, the imaginary nature of field f(r,t) will only slightly affect statistical properties of small-scale turbulent components. Consequently, this model is quite appropriate for describing small-scale properties of turbulence. Motion of an incompressible fluid under external forces is governed by the Navier–Stokes equation (

1 ∂ ∂ ∂ + u(r,t) )u(r,t) = − p(r,t) + ν Δu(r,t) + f(r,t), ∂t ∂r ρ0 ∂ r ∂ ∂ u(r,t) = 0, f(r,t) = 0. ∂r ∂r

(1)

Here, ρ0 is the density of the fluid, ν is the kinematic viscosity, and pressure field p(x,t) is expressed in terms of the velocity field at the same instant by the relationship ˆ ∂ 2 (ui (r ,t)u j (r ,t)) dr , (2) p(r,t) = −ρ0 Δ−1 r, r ∂ ri ∂ rj † Corresponding


244


where Δ−1 (r, r ) is the integral operator inverse to the Laplace operator (repeated indices assume summation). Neglecting the effect of viscosity and external random forces in Eq. (1), we arrive at the equation 1 ∂ u(r,t) + (u(r,t) · ∇ )u(r,t) = − ∇ p(r,t) ∂t ρ

(3)

that describes dynamics of perfect fluid and is called the Euler equation. If we substitute Eq. (2) in Eq. (1) to exclude the pressure field, then we obtain in the three-dimensional case that the Fourier transform of the velocity field with respect to spatial coordinates ˆ ˆ 1 −ikr , ui (r,t) = dk ui (k,t)eikr , ui (k,t) = drui (r,t)e (2π )3 ( u∗i (k,t) = ui (−k,t)) satisfies the nonlinear integro-differential equation ˆ ˆ i ∂ αβ ui (k,t) + dk1 dk2 Λi (k1 , k2 , k) uα (k1 ,t) uβ (k2 ,t) − ν k2 ui (k,t) = fi (k,t) , ∂t 2

(4)

where αβ

Λi (k1 , k2 , k) =

1 (2π )3

kα Δiβ (k) + kβ Δiα (k) δ (k1 + k2 − k),

Δi j (k) = δi j −

ki k j k2

(i, α , β = 1, 2, 3),

and f(k,t) is the spatial Fourier harmonic of external forces, ˆ f(k,t) = drf(r,t)e−ikr , f(r,t) =

1 (2π )3

ˆ

dkf(k,t)eikr .

A specific feature of the three-dimensional hydrodynamic motions consists in the existence of the integral of energy under the condition that external forces and effects related to the molecular viscosity are absent. We can obtain by standard way that functional ˆ ϕ [t; z(k )] = ϕ [t; z] = exp{i dk u(k ,t)z(k )} satisfies the linear variational differential Hopf equation in functional space

∂ ϕ [t; z] = − ∂t

ˆ

dkzi (k){ν k2

δ −i fi (k,t)}ϕ [t; z] δ zi (k) ˆ ˆ ˆ δ 2 ϕ [t; z] 1 αβ dkzi (k) dk1 dk2 Λi (k1 , k2 , k) . (5) − 2 δ zα (k1 ) δ zβ (k2 )

A consequence of Eq. (5) is the equality

δ ϕ [t; z] = iz(k)ϕ [t; z]. δ f(k,t − 0)

(6)

Average Eq. (5) over an ensemble of realizations of random force f(k,t). If we assume now that f(x,t) is the Gaussian random field homogeneous and isotropic in space and stationary in time with the correlation tensor Bi j (x1 − x2 ,t1 − t2 ) = fi (x1 ,t1 ) f j (x2 ,t2 ) ,


245

then the field f(k,t) will also be the Gaussian stationary random field with the correlation tensor

1 f j (k ,t) = Fi j (k, τ )δ (k + k), fi (k,t + τ ) 2 where Fi j (k, τ ) is the spatial spectrum of the external force given by the formula ˆ 3 Fi j (k, τ ) = 2(2π ) dxBi j (x, τ )e−ikx . In view of the fact that forces are spatially isotropic, we have Fi j (k, τ ) = F(k, τ )Δi j (k). As long as field f(k,t) is delta-correlated in time, we have F(k, τ ) = F(k)δ (τ ), so we can obtain the equation for one-time characteristic functional of turbulent flow Φ[t; z] = ϕ [t; z] 1 ∂ Φ[t; z] = − ∂t 4

ˆ dkF(k)Δi j (k)zi (k)z j (−k)Φ[t; z] ˆ −

1 dkzi (k){ 2

ˆ

ˆ dk1

dk2 Λi,αβ (k1 , k2 , k)

δ2 δ + ν k2 }Φ[t; z]. (7) δ zα (k1 )δ zβ (k2 ) δ zi (k)

Eq. (7) is the diffusion equation in the infinite-dimensional space, because of which it is the variational differential equation. The diffusion coefficient can be different for different wave components; it is given by the spectral tensor of external forces F(k)Δi j (k). In the conditions of absent molecular viscosity and random external forces, the problem on evolution of the velocity field specified at the initial moment becomes meaningful. In the context of this problem, the characteristic functional of velocity satisfies the equation ˆ ˆ ˆ 1 ∂ δ 2 Φ[t; z] αβ Φ[t; z] = − . (8) dkzi (k) dk1 dk2 Λi (k1 , k2 , k) ∂t 2 δ zα (k1 ) δ zβ (k2 ) Note that this equation was considered by E. Hopf in his classic paper [1] and is called now the Hopf equation (see also [2, 3]). At the same time, in the absence of molecular viscosity and random external forces, the input integro-differential equation assumes the form ˆ ˆ i ∂ αβ ui (k,t) + dk1 dk2 Λi (k1 , k2 , k) uα (k1 ,t) uβ (k2 ,t) = 0 ∂t 2 and describes the motion of the ideal fluid. It can have a number of integrals of motion, which may result in the existence of the solution to Eq. (8) steady-state for t → ∞ and independent of initial values. Such a solution is called the equilibrium distribution. For the two-dimensional and three-dimensional velocity fields, these distributions appear to be significantly different. Consider Eq. (8). In view of multiple nonlinear interactions between different harmonics of random velocity field, we can expect that the steady-state distribution of the velocity field exists for t → ∞ and satisfies the steady-state Hopf equation ˆ ˆ ˆ δ 2 Φ[z] αβ = 0. dkzi (k) dk1 dk2 Λi (k1 , k2 , k) δ zα (k1 ) δ zβ (k2 ) The unique solution to this equation in the class of the Gaussian functionals is the functional [4] ˆ γ dkΔi j (k)zi (k)z j (−k)} Φ[z(k)] = exp{− 2 corresponding to the uniform energy distribution over wave numbers (the white noise).

(9)

246


Note that solution (9) can satisfy the initial equation (7) if random forces are specially fit to compensate the molecular viscosity. Indeed, substituting functional (9) in Eq. (7), we see that the term with the second variational derivative vanishes (which is a consequence of the fact that the integral of motion — energy — exists in the case of the ideal fluid) and other terms — they correspond to the linearized initial-value problem — are mutually canceled only if (10) Fi j (k) = 4νγ k2 Δi j (k). This relationship corresponds to the so-called fluctuation–dissipation theorem for hydrodynamic flows. In the case of the two-dimensional perfect fluid, one more, the second integral of motion quadratic in velocities exists in addition to the energy integral; it is the square of the vorticity of the velocity field. In this case, there appears the equilibrium distribution different from the white noise (10) and characterized by a number of features, the main of which consists in the existence of coherent structures which are described by spectral density proportional to the delta-function. 2 Two-dimensional hydrodynamics In the simplest case, the incompressible fluid flow in the two-dimensional plane R = (x, y) is described by the stream function ψ (R,t) satisfying equation, that has the following form [5]

∂ Δψ (R,t) = J {Δψ (R,t); ψ (R,t)} , ∂t where J {ψ (R,t); ϕ (R,t)} =

ψ (R, 0) = ψ0 (R),

(11)

∂ ψ (R,t) ∂ ϕ (R,t) ∂ ϕ (R,t) ∂ ψ (R,t) − ∂x ∂y ∂x ∂y

is the Jacobian of two functions. Nonlinear interactions must bring the hydrodynamic system (11) to statistical equilibrium. In view of the fact that establishing this equilibrium requires a great number of interactions between the disturbances of different scales, we can suppose that, in the simplest case of statistically homogeneous and isotropic initial random field ψ0 (R), this distribution will be the Gaussian distribution, so that our task consists in the determination of this distribution parameters. During the evolution, random stream function ψ (R,t) remains a homogeneous and isotropic function. Because the stream function is defined to an additive constant, we can describe its statistical characteristics by the one-time structure function 2

= 2 Bψ (0,t) − Bψ (R − R,t) , Dψ (R − R ,t) = ψ (R,t) − ψ (R ,t) where

Bψ (R − R ,t) = ψ (R,t)ψ (R ,t)

is the spatial correlation function of field ψ (R,t). We will seek the steady-state (equilibrium) distribution on the class of the Gaussian distributions of statistically homogeneous and isotropic field ψ (R,t) described by the structure function Dψ (R) = lim Dψ (R,t). With t→∞ this goal in view, we consider the three-point equality

∂ Δψ (R1 ,t)Δψ (R2 ,t)Δψ (R3 ,t) = 0 ∂t for t → ∞ from which follows that

∂ Δψ (R1 ,t)Δψ (R2 ,t)Δψ (R3 ,t) = {1} + {2} + {3} = 0, ∂t

(12)


247

where by {1} we designate the variable {1} = J {Δψ (R1 ,t); ψ (R1 ,t)Δψ (R2 ,t)Δψ (R3 ,t)} ,

(13)

while the variables {2} and {3} correspond to cyclic permutation on the vectors {R1 , R2 , and R3 }. Expression (13) can be rewritten in the form

∂ Δψ (R1 ,t) ∂ ψ (R1 ,t) ∂ Δψ (R1 ,t) ∂ ψ (R1 ,t) Δψ (R2 ,t)Δψ (R3 ,t) − Δψ (R2 ,t)Δψ (R3 ,t) . {1} = ∂ x1 ∂ y1 ∂ y1 ∂ x1 Next we split the quantic correlation in Eq. (13) in the product of pair correlations using Gaussianity of the field ψ (R,t)

∂ Δψ (R1 ,t) ∂ ψ (R1 ,t) ∂ Δψ (R1 ,t) ∂ ψ (R1 ,t) Δψ (R2 ,t) Δψ (R3 ,t) + Δψ (R3 ,t) Δψ (R2 ,t) {1} = ∂ x1 ∂ y1 ∂ x1 ∂ y1

∂ Δψ (R1 ,t) ∂ ψ (R1 ,t) ∂ Δψ (R1 ,t) ∂ ψ (R1 ,t) Δψ (R2 ,t) Δψ (R3 ,t) − Δψ (R3 ,t) Δψ (R2 ,t) . − ∂ y1 ∂ x1 ∂ y1 ∂ x1 (14) Here we dropped the product terms containing J {Δψ (R,t); ψ (R,t)} = 0 as it vanishes after the ensemble averaging. In addition, we can replace derivatives ∂ ∂R1 with ( ∂ ∂R2 , ∂ ∂R3 ) and, consequently, express quantity {1} in terms of the correlation function of current {1} = (

∂2 ∂2 − )ΔR2 ΔR3 (ΔR2 − ΔR3 ) Bψ (R1 − R2 )Bψ (R1 − R3 ). ∂ x2 ∂ y3 ∂ x3 ∂ y2

Introducing vectors q1 = R1 − R2 , q2 = R2 − R3 , q3 = R3 − R1

(q1 + q2 + q1 = 0)

and scalars qi = |qi | we replace all R-partial derivatives through the q-partial derivatives x1 − x2 ∂ ∂ =− , ∂ x2 q1 ∂ q1 x3 − x1 ∂ ∂ =− , ∂ x3 q1 ∂ q1

y1 − y2 ∂ ∂ =− , ∂ y2 q1 ∂ q1 y3 − y1 ∂ ∂ =− . ∂ y3 q1 ∂ q1

The off-shot is the following equation {1} = −[q3 × q1 ]X (q3 ; q1 ), where we denote by [q3 × q1 ] the wedge-product of two vectors, and X (q3 ; q1 ) is given by the expression X (q3 ; q1 ) =

1 ∂2 2 Δq3 Dψ (q3 )Δq1 Dψ (q1 ) − Δq3 Dψ (q3 )Δ2q1 Dψ (q1 ) , q3 q1 ∂ q3 ∂ q1 2

(15)

where Dψ (q) is the structure function of current and operator Δq = ∂∂q2 + 1q ∂∂q is the radial part of the Laplace operator. As a result, the fundamental equality (12) reduces to the expression [q3 × q1 ]X (q3 ; q1 ) + [q2 × q3 ]X (q2 ; q3 ) + [q1 × q2 ]X (q1 ; q2 ) = 0.

248


In view of equalities [q3 × q1 ] = [q2 × q3 ] = [q1 × q2 ], it grades into the final functional equation X (q1 ; q2 ) + X (q2 ; q3 ) + X (q3 ; q1 ) = 0

(16)

in arbitrary scalar coordinates q1 = |R1 − R2 |, q2 = |R2 − R3 |, q3 = |R3 − R1 |. Then, multiplying Eq. (16) by q1 q2 and applying the differential operator ∂ 4 /∂ q21 ∂ q22 to the product, we can eliminate variable q3 and convert it to the equation for function Dψ , ∂6 2 Δq1 Dψ (q1 )Δq2 Dψ (q2 ) − Δq1 Dψ (q1 )Δ2q2 Dψ (q2 ) = 0. 3 3 ∂ q1 ∂ q2 Assuming now that function Δq Dψ (q) → 0 for q → ∞, we obtain the equation Δ2q1 Dψ (q1 )Δq2 Dψ (q2 ) − Δq1 Dψ (q1 )Δ2q2 Dψ (q2 ) = 0,

(17)

which can be solved by the method of separation of variables. As a result, we arrive at the equation of the form (Δq + λ ) Δq Dψ (q) = 0,

(18)

where λ is the separation constant with the dimension of the inverse square of length and Δq is the radial part of the Laplace operator. There are two possible solutions to Eq. (18), depending on whether constant λ is positive (λ = k02 > 0) or negative (λ = −k02 < 0). If λ = k02 > 0, Eq. (18) can be reduced to the equation Δq Dψ (q) = CJ0 (k0 q), where J0 (z) is the Bessel function of the first kind. In this case, structure function Dψ (q) is determined as the solution to the Laplace equation, and we obtain the spectral density of structure function in the form [6] E(k) = E δ (k − k0 ). The delta-like behavior of spectral density is evidence of the fact that fields ψ (R,t) are highly correlated, which suggests that coherent structures can exist in the developed turbulent flow of the two-dimensional fluid (in the sense of the existence of the corresponding eigenfunctions slowly decaying with distance). This pattern corresponds to random structures characterized by certain fixed spatial scale. In the problem under consideration, such structures are vortices, which means that structure formation is realized here in the form of vortex genesis. In the case λ = −k02 < 0, Eq. (18) can be reduced to the similar equation Δq Dψ (q) = CK0 (k0 q). However, the right-hand side of this equation is proportional to the McDonalds function K0 (z) with the dimensional parameters k0 and C. The corresponding spectral density of structure function is now given by the well known formula [7] (see also [8–10] E0 , E(k) = 2 k + k02 which corresponds to the Gibbs distribution with two integrals of motion (energy and squared curl of the velocity field). Note that, in the three-dimensional problem with only one integral of motion (energy), the Gibbs equilibrium distribution corresponds to the uniform energy distribution with respect to wave numbers (the white noise). Note that the steady-state solution to the initial stochastic dynamic equation (11) satisfies the equation Δψ (R) = F (ψ (R)) ,


249

where F (ψ (R)) is arbitrary function determined from boundary conditions at infinity. In the simplest case of the Fofonoff flow corresponding to the linear function F (ψ (R)) = −λ ψ (R), this equation assumes the form [11] Δψ (R) = −λ ψ (R).

(19)

Considering formally Eq. (19) as the stochastic equation, we can easily obtain that the structure function of field ψ (R) satisfies the equation coinciding with Eq. (18). This means that the Gaussian equilibrium state is statistically equivalent to the stochastic Fofonoff flow of the fluid. Of course, the realizations of dynamic systems (11) and (19) are different. Thus, despite strong nonlinearity of the input equation (11), the equilibrium regime (for t → ∞) appears statistically equivalent to the linear equation in which the nonlinear interactions are absent. 3 Random geophysical hydrodynamic flows Consider now the description of hydrodynamic flows on the rotating Earth in the so-called quasi-geostrophic approximation [12]. In the simplest case of the one-layer model, the incompressible fluid flow in the twodimensional plane R = (x, y) is described by the stream function that satisfies the equation

∂ ∂ Δψ (R,t) + β0 ψ (R,t) = J {Δψ (R,t) + h(R); ψ (R,t)} , ∂t ∂x

(20)

ψ (R, 0) = ψ0 (R), where parameter β0 is the derivative of the local Coriolis parameter f0 with respect to latitude, J{ψ , ϕ } is the Jacobian of two functions ψ (R,t) and ϕ (R,t) J {ψ (R,t); ϕ (R,t)} =

∂ ψ (R,t) ∂ ϕ (R,t) ∂ ϕ (R,t) ∂ ψ (R,t) − , ∂x ∂y ∂x ∂y

h(R) relative to its average thickness H0 h(R)/H0 is the deviation of bottom topography and function h(R) = f0 (Fig. 1a).

a

b H0

ρ0

H0 h(x)

H1

ρ1

H2

ρ2

x

h(x)

x

Fig. 1 Diagrammatic views of (a) one-layer and (b) two-layer models of hydrodynamic flows.

The velocity field is expressed in terms of the stream function by the relationship v (R,t) = (−

∂ ψ (R,t) ∂ ψ (R,t) , ). ∂y ∂x

Note that, under the neglect of Earth’s rotation and effects of underlying surface topography, Eq. (23) reduces to the standard equation of two-dimensional hydrodynamics (see, e.g., [5])

∂ Δψ (R,t) + J {Δψ (R,t)); ψ (R,t)} , ∂t

ψ (R, 0) = ψ0 (R).

(21)

250


Equation (23) describes the barotropic motion of a fluid. In the more general case of baroclinic motions, investigation is usually carried out within the framework of the two-layer model of hydrodynamic flows described by the system of equations [12]

∂ ∂ [Δψ1 − α1 F (ψ1 − ψ2 )] + β0 ψ1 = J {Δψ1 − α1 F (ψ1 − ψ2 ) ; ψ1 } , ∂t ∂x ∂ ∂ [Δψ2 − α2 F (ψ2 − ψ1 )] + β0 ψ2 = J {Δψ2 − α2 F (ψ2 − ψ1 ) + f0 α2 h; ψ2 } , ∂t ∂x where additional parameters F = f02 ρ /g(Δρ ),

(22)

Δρ /ρ = (ρ2 − ρ1 )/ρ0 > 0

are introduced and α1 = 1/H1 and α2 = 1/H2 are the inverse thicknesses of layers (Fig. 1b). In the simplest case of the one-layer model, the incompressible fluid flow in the two-dimensional plane R = (x, y) is described by the stream function that satisfies the equation

∂ ∂ Δψ (R,t) + β0 ψ (R,t) = J {Δψ (R,t) + h(R); ψ (R,t)} , ∂t ∂x

(23)

where parameter β0 is the derivative of the local Coriolis parameter f0 with respect to latitude, and function h(R) relative to layer average thickness H0 . This h(R)/H0 is the deviation of bottom topography h(R) = f0 equation describes the barotropic motion of a fluid. In the more general case of baroclinic motions, investigation is usually carried out within the framework of the two-layer model of hydrodynamic flows described by the system of equations

∂ ∂ [Δψ1 − α1 F(ψ1 − ψ2 )] + β0 ψ1 = J {Δψ1 − α1 F(ψ1 − ψ2 ); ψ1 } , ∂t ∂x ∂ ∂ [Δψ2 − α2 F(ψ2 − ψ1 )] + β0 ψ2 = J {Δψ2 − α2 F(ψ2 − ψ1 ) + f0 α2 h; ψ2 } , ∂t ∂x

(24)

where additional parameters F = f02 ρ /g(Δρ ) and Δρ /ρ = (ρ2 − ρ1 )/ρ0 > 0 are introduced and α1 = 1/H1 and α2 = 1/H2 are the inverse thicknesses of layers. Here, in the framework of the one-layer model described by Eq. (23), consideration of the steady-state equation for the correlation function of current also results in Eq. (16) with function X (q1 ; q2 ) given by the expression X (q1 ; q2 ) =

1 ∂ 2 2 Δq1 Dψ (q1 )Δq2 Dψ (q2 ) − Δq1 Dψ (q1 )Δ2q2 Dψ (q2 ) q1 q2 ∂ q1 ∂ q2 − 2 Δq1 Bhψ (q1 )Δq2 Dψ (q2 ) − Δq1 Dψ (q1 )Δq2 Bhψ (q2 ) , (25)

where Bhψ (|R − R |) = lim h(R)ψ (R ,t) is the steady-state cross-correlation function of bottom surface and t→∞

function of current, rather than by Eq. (15). Derivation of Eq. (25) assumes that joint statistics of fields ψ (R,t) and h(R) is Gaussian. Multiplying now Eq. (16) by q1 q2 and applying differential operator ∂ 4 /∂ q21 ∂ q22 , we obtain the following equation for functions Bhψ (q) and Dψ (q) ∂ 6 2 2 D (q )Δ D (q ) − Δ D (q )Δ D (q ) Δ ψ 1 q ψ 2 q ψ 1 ψ 2 q q 2 1 1 2 3 ∂ q3 ∂ q2 − 2 Δq1 Bhψ (q1 )Δq2 Dψ (q2 ) − Δq1 Dψ (q1 )Δq2 Bhψ (q2 ) = 0. (26)


251

Separating variables in Eq. (26), we arrive at the nonclosed equation in the structure function of the function of current (27) (Δq + λ ) Δq Dψ (q) − 2Δq Bhψ (q) = 0. We will seek the solution to Eq. (27) under the assumption that functions Dψ (q) and Bhψ (q) belong to the class of bounded functions. In this case Eq. (27) can be rewritten in the form (Δq + λ ) Bψψ (q) + Bhψ (q) = 0,

(28)

where Bψψ (q) is the correlation function of the function of current and we assume that ψ = 0. To derive an equation for function Bhψ (q), we consider evolution of the three-point correlator Δψ (R1 ,t)h(R2 )h(R3 ). In the case of statistically stationary fields,

∂ Δψ (R1 ,t)h(R2 )h(R3 ) = 0 ∂t for t → ∞. Proceeding similarly to the above derivation of Eqs. (16) and (26), we obtain the partial differential equation of the form (29) (Δq + λ1 ) Bhψ (q) + Bhh (q) = 0 with separation constant λ1 different in the general case from λ . The system of equations (28) and (29) is now a closed one. However, two key questions arise. 1. Is the separation constant λ1 (and the corresponding length scale) really independent of the parameter λ in Eq. (27)? 2. Does the equilibrium Gaussian ensemble exist in reality (the fundamental assumption)? To answer these questions, we must consider the joint characteristic functional of the two random fields ψ (R,t) and h(R)

ˆ (30) Φt [v(R ), κ (R )] = exp{i dR ψ (R ,t)v(R ) + h(R )κ (R ) } = exp{i [(ψ |v) + (h|κ )]} . Here, ( f |g) denotes the scalar product of functions f (R) and g(R) in space {R}. The characteristic functional Φt [v(R ), κ (R )] determines all statistical characteristics of the field ψ (R,t) at an instant t. Differentiating the functional Φt [v(R ), κ (R )] with respect to time, we obtain the linear equation in variational derivatives in an infinite-dimensional functional space named the Hopf equation

∂ δ δ δ Φt [v(R ), κ (R )] = −i(v|Δ−1 J(Δ + ; ))Φt [v(R ), κ (R )]. ∂t δv δκ δv

(31)

It is obvious that for an equilibrium state (when t → ∞) lim

∂

t→∞ ∂ t

Φt [v(R ), κ (R )] = 0,

and, consequently, the equilibrium functional is characterized by the steady-state equation (v|Δ−1 J(Δ

δ δ δ + ; ))Φ∞ [v(R ), κ (R )] = 0. δv δκ δv

(32)

We show now that the Gaussian characteristic functional with the parameters Bψψ (q), Bhψ (q) and Bhh (q) defined from Eqs. (28) and (14) does satisfy Eq. (32). In fact, the Gaussian characteristic functional Φ∞ [v(R ), κ (R )] has the structure Φ∞ [v(R ), κ (R )] = exp{−

1 v|Bψψ |v + 2 v|Bhψ |κ + (κ |Bhh | κ ) }, 2

252


where paranthesis (· · · |B| · · · ) denotes the convolutions of pairs of functions {v; v}; {v; κ }; {κ ; κ }, for example, ¨ dRdR v(R)Bhψ (R − R )κ (R ). v|Bhψ |κ = Substituting Φ∞ [v(R ), κ (R )] on the left-hand side of Eq. (32), we obtain Δ−1 v|J v|ΔBψψ + κ |ΔBhψ + (κ |ΔBhh ) + v|ΔBhψ ; v|ΔBψψ + κ |ΔBhψ = 0.

(33)

Substituting now the values of ΔBψψ (q) and ΔBhψ (q) in the system of equations (28) and (29) in Eq. (33), we see that the Jacobian J(...) = 0 under the condition that λ = λ1 [13]. This condition is the sufficient condition for the steady-state Gaussian distribution be possible to exist. Hence two separation constants in Eqs. (28) and (29) must coincide, and the problem is characterized by the unique length scale determined by parameter λ [13]. In this case, the solution Eqs. (28) and (29) is expressed in terms of the correlation function of topography inhomogeneities Bhψ (q) by the equality Bhψ (q) = (Δ + λ )−1 Bhh (q), Bψψ (q) = B0ψψ (q) + (Δ + λ )−2 Bhh (q),

(34)

where the notation B0ψψ (q) denotes the correlation function of the field ψ when the bottom is flat, i.e., when h = 0. Notice that the steady-state solution of the quasi-geostrophic Eq. (34) is statistically equivalent to a steadystate Fofonoff fluid flow (35) Δψ (R) + h(R) = −λ ψ (R). The use of the method outlined above indicated that this situation takes place in more general cases as well. In this case a steady-state equilibrium Gaussian state may exist as well when t → ∞, which is statistically equivalent to the steady-state solution of Eqs. (24) for the case of the simplest Fofonoff flow Δψ1 (R,t) − α1 F (ψ1 (R,t) − ψ2 (R,t)) = −λ1 ψ1 (R,t), Δψ2 (R,t) − α2 F (ψ2 (R,t) − ψ1 (R,t)) + f0 α2 h(R) = −λ1 ψ2 (R,t).

(36)

Now, the statistically equilibrium state will contain two different parameters λ1 and λ2 characterizing the corresponding spatial scales. System (36) can be rewritten in the form 2 ψ1 (R,t) = α1 α2 F f0 h(R), L L2 ψ2 (R,t) = −α2 f0 (Δ + λ1 − α1 F)h(R), where the operator

(37)

L2 = Δ2 + [λ1 + λ2 − F (α1 + α2 ) Δ] + λ1 λ2 − F (α2 λ1 + α1 λ2 ) .

The corollary of Eqs. (37) is the fact that coherent structures may now have two different spatial scales, which means that the two-layer liquid is generally characterized by two fixed sizes [14, 15]. It seems that similar structures were observed in the experiments with rotating fluid (see, e.g., papers [16–18] and monograph [19]). I cite Figs. 2 and 3 from monograph [19], Fig. 4 from [17], and Fig. 5 from [20], as examples which, in my opinion, correspond to the described situation. Figure 6 shows an example of structure formation in the field of surface streams in the Baltic Sea [21, 22].


253

b

Fig. 2 Convective vortex grid on a rotating platform: (a) plain view, (b) inclined view.

Fig. 3 An example of irregular convective vortex pattern for greater speed of rotation.

4 Conclusions A characteristic feature of all above solutions consists in the fact that they predict the possibility for coherent states to exist in the developed turbulent flow. Nothing can be said about the stability of these states. However, we note that the above Gaussian equilibrium ensemble forms the natural noise in a number of geophysical systems described in the quasi-geostrophic approximation and is similar to the thermal noise in the statistical physics. For this reason, this noise may play very important and sometimes determinative role in the statistical theory of quasi-geostrophic flows of fluid. A characteristic feature of all above solutions consists in the fact that they predict the possibility for coherent states to exist in the developed turbulent flow. Nothing can be said about the stability of these states. However, we note that the above Gaussian equilibrium ensemble forms the natural noise in a number of geophysical systems described in the quasi-geostrophic approximation and is similar to the thermal noise in the statistical physics. For this reason, this noise may play very important and sometimes determinative role in the statistical theory of quasi-geostrophic flows of fluid.

254


Fig. 4 Velocity field of baroclinic turbulence. The particle streaks represents instantaneous velocities. The photograph was taken about 25 rotation periods after onset of baroclinic instability.

Fig. 5 Shear instability-driven vortex genesis on a sphere.

Acknowledgement This work was supported by the RSF 14-27-00134.


255

Fig. 6 Manifestation of the ’black’ small-scale eddies in the Baltic Sea using satellite synthetic aperture radar (SAR) images.

References [1] Hopf, E. (1952), Statistical hydromechanics and functional calculus, J. Rat. Mech. Anal., 1, 87–123. [2] Hopf, E. (1957), On the application of functional calculus to the statistical theory of turbulence, in: Proceedings of Symposia in Applied Mathematics , 7, 41–50, Am. Math. Society. [3] Hopf, E. (1962), Remarks on the functional-analytic approach to turbulence, in: Proceedings of Symposia in Applied Mathematics, 13, 157–163, Am. Math. Society. [4] Hopf, E. and Titt, E.W. (1953), On certain special solution of the equation of statistical hydrodynamics, J. Rat. Mech. Anal., 2, 587. [5] Landau, L.D. and Lifshitz, E.M. (1987), Fluid Mechanics, Second edition, Pergamon Press, London. [6] Klyatskin, V.I. (1969), On Statistical theory of two-dimensional turbulence, Journal of Applied Mathematics and Mechanics, 33(5), 864–866. [7] Kraichnan, R.H. (1967), Inertial waves in two-dimensional turbulence, Physics of Fluids, 10, 1417–1423. [8] Kraichnan, R.H. (1975), Statistical dynamics of two-dimensional flows, Journal of Fluid Mechanics, 67, 155–175. [9] Kraichnan, R.H. and Montgomery, D. (1980), Two-dimensional turbulence, Reports on Progress in Physics, 43, 547– 619. [10] Miller, J., Weichman P. and Cross, M. (1992), Statistical mechanics, Euler’s equation and Jupiter’s red spot, Physical Review, A45(4), 2328–2359. [11] Fofonoff, N. (1954), Steady flow in a frictionless homogeneous ocean, Journal Of Marine Research, 13, 254–262. [12] Pedlosky, J. (1982), Geophysical Fluid Dynamics, Springer-Verlag, New York. [13] Klyatskin, V.I. (1995), Equilibrium states for quasigeostrophic flows with random topography, Izvestiya, Atmospheric and Oceanic Physics, 31(6), 717–722. [14] Klyatskin, V.I. and Gurarie, D. (1996), Random topography in geophysical models, in: Stochastic Models in Geosystems, eds. Molchanov, S.A. and Woyczynski, W.A. IMA Volumes in Math. and its Appl. 85, 149–170. N.Y. SpringerVerlag. [15] Klyatskin, V.I. and Gurarie D. (1996), Equilibrium states for quasigeostrophic flows with random topography, Physica D, 98, 466–480. [16] Hopfinger, E.J. and Browand, F.K. (1982), Vortex solitary waves in rotating, turbulent flow, Nature, 295(5848), 393– 394. [17] Hopfinger, E.J. (1989), Turbulence and vortices in rotating fluids, in: Theoretical and Applied Mechanics, 117–138. Germain, P., Piau, M. and Caillerie, D. (Editors), IUTAM, Elsevier Science Publishers B.V. (North-Holland). [18] Boubnov, B.M. and Golitsyn, G.S. (1986), Experimental study of convective structures in rotating fluids, Journal of Fluid Mechanics, 167(6), 503-531. [19] Boubnov, B.M. and Golitsyn, G.S. (1995), Convection in Rotating Fluids, Ser. Fluid Mechanics and its Applications, Vol. 29, Dordrecht, Boston, London: Kluver Academic Publishers. [20] Pavlov, V., Buisine, D. and Goncharov, V. (2001), Formation of vortex clusters on a sphere, Nonlinear Processes In Geophysics, 8, 9–19. [21] Karimova, S.S., Lavrova, O.Yu. and Solov’ev, D.M. (2012), Observation of Eddy Structures in the Baltic Sea with the use of Radiolocation and Radiometric Sattelite Data, Izvestiya, Atmos. and Oceanic Phys., 49(9), 1006–1013. [22] Karimova, S. (2012), Spiral eddis in the Baltic, Black and Caspian seas as seen by satellite radar data, Advances in Space Research, 50, 1107–1124.



Hyperbolicity in the Ocean S.V. Prants1†, M.V. Budyansky1, M.Yu. Uleysky1, J. Zhang2 1 Laboratory

of Nonlinear Dynamical Systems, Pacific Oceanological Institute of the Russian Academy of Sciences, 43 Baltiiskaya st., 690041 Vladivostok, Russia 2 School of Energy and Power Engineering, Xi-an Jiaotong University, 710049, P.R. China Submission Info Communicated by Xavier Leoncini Received 26 January 2015 Accepted 2 March 2015 Available online 1 October 2015 Keywords Hyperbolic points in the ocean Lyapunov exponents Altimetric velocity field Drifter tracks

Abstract Some manifestations of hyperbolicity in the ocean, the important concept in dynamical systems theory, are discussed. It is shown how to identify hyperbolic points, hyperbolic trajectories and their stable and unstable manifolds solving advection equations for passive scalars in a satellite-derived AVISO velocity field and computing finite-time Lyapunov exponents by the singular-value decomposition method. To validate our simulation we use available tracks of oceanic drifters following near surface currents in some areas in the Northwestern Pacific Ocean. The tracks illustrate how drifters “feel” the presence of hyperbolic points, hyperbolic trajectories and stable and unstable manifolds and change abruptly their trajectories when approaching a hyperbolicity region. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Hyperbolic points, hyperbolic trajectories and their stable and unstable manifolds Hyperbolicity is an important concept in dynamical systems theory (see, e.g., [1–4]). It is characterized by the presence of expanding and contracting directions in the phase space of a dynamical system. This is a situation with phase trajectories converging in one direction and diverging in the other one. In unperturbed Hamiltionian systems the separatrices (if they exist) connect either two hyperbolic stagnation point (HSPs) or belongs to a single HSP. In the case of a heteroclinic connection, the stable branch of the separatrice of one stagnation point (SP) coinsides with the unstable branch of the separatrice of the other point and vice versa. In the case of a homoclinic connection, stable and unstable branches of the separatrice of a single HSP coinside. In Fig. 1 we plot the schematic phase portrait of a flow nearby a HSP with phase trajectories converging in one direction and diverging in the other one. Under a periodic perturbation, HSPs transform into periodic hyperbolic trajectories (HTs) with time-dependent separatrices called stable, Ws (γ ), and unstable, Wu (γ ), invariant manifolds. If a trajectory has nonzero Lyapunov exponents it is said to be hyperbolic [5]. In the extended phase space with time as the third coordinate, Ws (γ ) and Wu (γ ) are surfaces filled in trajectories approaching asymptotically to γ (t) at t → ∞ (Ws ) and t → −∞ (Wu ). Those surfaces intersect each other on the section plane t = 0 in an infinite number of homoclinic points producing a heteroclinic or a homoclinic tangle which is a seed of dynamical chaos (see, e.g., [1, 2, 4]). HSPs, † Corresponding author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.09.004

258

S.V. Prants, M.V. Budyansky, M.Yu. Uleysky / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 257–270

HTs and stable and unstable invariant manifolds are hyperbolic objects with the dimensions of n = 0, n = 1 and n ≤ m − 1, respectively, where m is the dimension of the phase space. A blob in the phase space experiences stretching and folding that typically give rise to chaotic motion even in purely deterministic dynamical systems, the remarkable phenomenon known as deterministic or dynamical chaos [1, 2, 4]. The ultimate reason of that chaos is the instability of trajectories that is expressed in terms of the hyperbolicity conditions. In this paper we deal with the motion of passive scalars in vector velocity fields from the Lagrangian point of view. By a passive scalar or a tracer one means the particle that acquires rapidly the velocity of a flow and does not affect on the flow. By an instantaneous SP one means a point in space where the velocity is zero at a fixed instant of time. As is well known, local stability properties of the SPs can be characterized by eigenvalues of the Jacobian matrix of a velocity field evaluated at that instant of time. For 2D flows, if the two eigenvalues are real and of opposite sign, then the SP is a HSP. If they are pure imagine and complex conjugated, then one gets an ESP. The two zero eigenvalues of the Jacobian matrix mean the existence of a parabolic SP. In 3D fluid flows invariant manifolds are material lines or 2D surfaces composing of the same fluid particles in the course of time. In 2D fluid flows stable and unstable manifolds of a HT γ (t) are material lines consisting of the set of points through which at instant of time t pass trajectories asymptotical to γ (t) at t → ∞ (Ws ) and t → −∞ (Wu ). They are complicated curves infinite in time and space that act as boundaries to fluid transport and partition the flow in topologically different regions. Theory of chaotic motion of particles in fluids, known as chaotic advection [6], is well developed in periodic 2D flows where periodic HTs can be seen as fixed points using Poincaré maps. The stable and unstable manifolds of HTs can be computed using different techniques and are known to form complex homoclinic or heteroclinic tangles which are “seeds” of chaotic advection. Chaotic advection is due to the action of both the deformation part of the velocity field (the strain field), which permanently stretches and expands any patch of tracer, and the vorticity part, which tends to fold any patch of tracer. In elliptic regions, for example inside eddies, where the relative vorticity dominates over the strain, any tracer pattern is weakly deformed. Hyperbolical objects are deformation regions where the strain dominates. In these regions any tracer pattern is horizontally stretched into elongated and thin filaments. Typical geophysical flows in the ocean and atmosphere and other natural flows are aperiodic. There is no analog of the KAM theorem in aperiodic vector fields. Nevertheless, the notion of hyperbolicity and stable and unstable manifolds are known for those fields in the mathematical community because hyperbolicity is not connected with the nature of the considered time dependence (see, e.g., [5, 7, 8] and the other references mentioned in [3]). If hyperbolicity is determined by any means, then the nature of the time dependence plays no role. Once a hyperbolic trajectory is located, the stable and unstable manifold theorem for hyperbolic trajectories immediately applies [3]. On the other hand, the notions of hyperbolicity, SPs, HTs and their stable and unstable manifolds and the very phenomenon of dynamical chaos are strictly defined in the infinite-time limit. In geophysical flows given in terms of experimentally or numerically generated velocity fields, the hyperbolic objects are of transient nature. HSPs in steady flows and periodic HTs in idealistic periodic flows are persistent features. In real-life flows trajectories can gain or lose hyperbolicity over time, i.e., they may be hyperbolic for one time interval and not to be hyperbolic for the other one. The very definition of stable and unstable manifolds requires an infinitetime limit which is irrelevant for geophysical flows. It is a challenge even to define hyperbolicity, HTs and their stable and unstable manifolds in aperiodic flows. The generalization of the dynamical system theory to aperiodic velocity fields, defined on a space-time grid, has been developed recently [3, 9–11]. The aim of this paper is to show how abstract notion of hyperbolicity is manifested in surface oceanic flows. In Sec. 2 we describe our method for computing finite-time Lyapunov exponents in a n-dimensional vector field [12] via singular values of the evolution matrix for linearized equations of motion. In Sec. 3 we introduce advection equations which govern the motion of tracers in a velocity field on the Earth’s sphere provided by the AVISO group (http://www.aviso.oceanobs.com) That altimetric subsurface velocity field is derived in


259

the geostrophic approximation using satellite-altimeter measurements of sea level anomalies. Identification of “instantaneous” HSPs and ESPs in daily provided altimetric velocity fields and their bifurcations are briefly described in Sec. 4. In order to validate our simulation we use tracks of oceanic drifters following near surface currents, measuring different properties of sea water and sending the data to passing satellites. Drifter observations now cover most areas of the world’s oceans. In Sec. 5 we illustrate manifestations of hyperbolicity in some areas in the Northwestern Pacific Ocean with the help of a few drifter’s tracks which “feel” the presence of HSPs and their stable and unstable manifolds and change abruptly their trajectories when approaching the HSPs. In order to identify unstable manifolds of HTs we use in Sec. 6 a local approach based on seeking for individual HTs and a global approach based on computing a backward-in-time Lyapunov exponents by the method described in Sec. 2 and in Appendix A. In Sec. 7 we discuss briefly some applications of the dynamical systems approach in physical oceanography.

Fig. 1 The schematic phase portrait of a flow nearby a hyperbolic stagnation point with phase trajectories converging in one direction and diverging in the other one.

2 The Lyapunov exponent as a measure of hyperbolicity We describe in this section a general method to compute Lyapunov exponents, introduced in Ref. [12], which is valid for n-dimensional vector fields. We start with a n-dimensional set of nonlinear ordinary differential equations in the vector form x˙ = f(x,t),

x = (x1 , . . . , xn ),

f(x,t) = ( f1 (x1 , . . . , xn ,t), . . . , fn (x1 , . . . , xn ,t)).

(1)

The Lyapunov exponent at an arbitrary point x0 is given by Λ(x0 ) = lim

lim

t→∞ kδ x(0)k→0

ln(kδ x(t)k/kδ x(0)k) , t

(2)

where δ x(t) = x1 (t) − x0 (t), x0 (t) and x1 (t) are solutions of the set (1), x0 (0) = x0 . The limit exists, is the same for almost all the choices of δ x(0) and has a clear geometrical sense: trajectories of two nearby particles diverge (converge) in time exponentially (in average) with the coefficients given by the Lyapunov exponents.

260


Due to smallness of δ x one can linearize the set (1) in a vicinity of some trajectory x0 (t) and obtain a set of time-dependent linear equations [13]     δ x˙1 δ x1  .  = J(t)  .  , (3) δ x˙n δ xn where J(t) is the Jacobian matrix of the set (1) along the trajectory x0 (t) 

 ∂ f1 (x0 (t),t) ∂ f1 (x0 (t),t) ...   ∂ x1 ∂ xn   J(t) =  ............. .  ∂ f (x (t),t)  ∂ f (x (t),t) n 0 n 0 ... ∂ x1 ∂ xn Solution of the linear set (3) can be found with the help of the evolution matrix G(t,t0 )     δ x1 (t) δ x1 (t0 )  . . .  = G(t,t0 )  . . .  . δ xn (t) δ xn (t0 )

(4)

(5)

The evolution matrix obeys the differential equation which can be obtained after substituting (5) into (3) G˙ = JG,

(6)

with the initial condition G(t0 ,t0 ) = I, where I is the unit matrix. Any evolution matrix has the important multiplicative property G(t,t0 ) = G(t,t1 )G(t1 ,t0 ). (7) One can write decompose the evolution matrix in the following form: G(t,t0 ) = U (t,t0 )Σ(t,t0 )V T (t,t0 ),

(8)

which is known as a singular-value decomposition. Here U , V are orthogonal and Σ = diag(σ1 , . . . , σn ) is diagonal matrices. The quantities σ1 , . . . , σn are called singular values of the matrix G. x(t)k The maximal value lim kkδδx(0)k for the set (3) equals to σ1 (G(t)), the maximal singular value of the kδ x(0)k→0 σ2 (G(t)) lim = 0, t→∞ σ1 (G(t))

matrix G(t). If where σ2 (G(t)) is the next (smaller) singular value of the matrix G(t) in magnitude, then (2) can be redefined as follows: Λmax = lim

t→∞

The quantity Λ=

ln σ1 (G(t)) . t − t0

ln σ1 (G(t)) t − t0

(9)

(10)

is called the finite-time Lyapunov exponent (FTLE). It is the ratio of the logarithm of a maximal possible stretching of a vector to a time interval t − t0 . The instantaneous Lyapunov exponent Λ0 is a Lyapunov exponent of the set of linear equations     δ x˙1 δ x1  .  = J(0)  .  . (11) δ x˙n δ xn It is the rate of exponential diverging of trajectories at a given point at a given instant of time.


261

In Appendix A we analyze in detail the singular-value decomposition of a 2 × 2-matrix which in fact is used to compute the Lyapunov exponents in this paper. Geometric meaning of the singular-value decomposition in 2D case is shown in Fig. 7. In Appendix B we find a solution of the 2D set of linear equations with time-independent coefficients. The FTLE maps is now a commonly used tool to visualize hyperbolic objects in the ocean and atmosphere [14, 15]. They are computed here in the 2D AVISO velocity field by the method [12] as follows. A studied area is seeded with a large number of synthetic particles for each of which advection equations (13) are integrated backward in time for a given period of time. The FTLE value Λ is calculated in accordance with (10) and coded by color.

3 Advection equations for tracers in 2D oceanic flows If advected particles rapidly adjust their own velocity to that of a background flow and do not affect the flow properties, then they are called passive particles or tracers and satisfy simple equations of motion dr = v(r,t), dt

(12)

where r = (x, y, z) and v = (u, v, w) are the position and velocity vectors at a point (x, y, z). This equation means that the Lagrangian velocity of a passive particle (the left side of Eq. 12) equals to the Eulerian velocity of the flow at the location of that particle (the right side of Eq. 12). In fluid mechanics by passive particles one means small parcels of water with their properties or small foreign bodies in a flow. Horizontal velocities in the ocean are much greater that the vertical ones, typically by four orders of magnitude (10−1 vs 10−5 m/s). Motion of a fluid particle in a 2D flow is the trajectory of a dynamical system with given initial conditions governed by the velocity field dx = u(x, y,t), dt

dy = v(x, y,t), dt

(13)

where (x, y) is the location of a tracer, u and v are zonal (east–west direction) and meridional (north–south direction) components of its velocity. There are two common approaches in hydrodynamics, the Eulerian and Lagrangian ones. In the Eulerian approach one measures or computes the velocity of fluid parcels and other their characteristics in a number of locations. The more convenient approach to study transport and mixing in the ocean is the Lagrangian one when one integrates trajectories for a large number of synthetic particles advected by an Eulerian velocity field using Eq. 12. The velocity field in the ocean can be computed either by solving the corresponding master equations in simple kinematic and dynamic models or as the output of a numerical circulation model or derived from a measurement. The launch of Earth-observing satellites with altimeters on board in the 1990s opened a new era for studying ocean surface circulation. A satellite radar measures precisely the distance from the radar antenna to the ocean surface by computing the round-trip travel time of a microwave signal. Dynamic topography refers to the topography of the sea surface related to the dynamics of its own flow. In hydrostatic equilibrium, the surface of the ocean would have no topography, but due the ocean currents, its maximum dynamic topography is on the order of two meters and are influenced by ocean circulation, temperature and salinity. A clockwise rotation (anticyclone) is found around elevations on the ocean surface in the northern hemisphere and depressions in the southern hemisphere. Conversely, a counterclockwise rotation (cyclone) is found around depressions in the northern hemisphere and elevations in the southern hemisphere. Combined with precise satellite location data, altimetry measurements yield sea-surface heights which, in turn, allow to infer ocean currents under conditions of the geostrophic balance. Away from the surface and bottom layers, horizontal pressure gradients in the ocean almost exactly balance the Coriolis force. The resulting

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flow is known as geostrophic. The major currents, such as the Gulf Stream, the Kuroshio and the Antarctic Circumpolar Current, are examples of geostrophic currents. Daily geostrophic velocities for the world’s oceans, provided by the AVISO database (http://www.aviso.oceanobs.com), approximate geostrophic ocean currents. The velocity data covers the period from 1992 to the present time with daily data on a 1/3◦ Mercator grid. All the results in this paper are obtained by solving advection equations (13) for a large number of synthetic tracers. Tracer’s coordinates x and y are related with their latitude φ and longitude λ in degrees as follows: π y x 180 λ= , φ= arcsin tanh + y0 , 60 π 180 60 (14) π 180 y0 = artanh sin φ0 , φ0 = −82. π 180 We use the transformation (14) because the AVISO grid is homogeneous in those coordinates. The velocities u and v in Eq. 13 are expressed through the latitudinal Uφ and longitudinal Uλ components of the linear velocity U in cm/s as follows: 0.466 10800 86400 u= U ≈ U , π RE cos φ 100000 λ cos φ λ (15) 0.466 10800 86400 v= Uφ ≈ Uφ , π RE cos φ 100000 cos φ where RE is the Earth radius in km. Bicubical spatial interpolation and third order Lagrangian polynomials in time are used to provide fine-gridded data and accurate numerical results. Lagrangian trajectories are computed by integrating the advection equations (13) with a fourth-order Runge-Kutta scheme.

4 Identification of “instantaneous” stagnation points in the altimetric velocity field We compute each day positions of “instantaneous” SPs which are defined as geographical locations where the altimetric velocity equals to zero. The SPs are Eulerian features of frozen time velocity fields, and they typically move in a time-dependent velocity field. However, they are not fluid particle trajectories. In fact, the velocity of motion of stagnation points may even approach infinity (see the tutorial paper [3]). The example of the interpolated altimetric velocity field in the North Western Pacific on 14 November 2010 is shown in Fig. 2 with overlaid positions of hyperbolic (crosses) and elliptic (circles) SPs. The plot makes an impression on complexity of subsurface velocity field in dynamically active region in the ocean. The curved dark band with large eastward velocities is the Kuroshio Extension current. It is an extension of the strong jet-like Kuroshio current carrying warm and salty water from the south to the Japan coast. The ring-like structures in Fig. 2 are large mesoscale eddies with the diameter of the order of 100 – 300 km and lifetimes of the order of a few months and more. Most of them have been pinched off from large meanders of the Kuroshio Extension jet. The elliptic points, situated mainly in the centers of eddies, are those stagnation points around which the motion is stable and circular. The saddle-type HSPs are situated between the eddies of the same or opposite polarity (dipoles) and nearby single eddies. They are important because they are features attracting water in one direction and expelling water in the other one. In spite of nonstationarity of the velocity field some of the SPs may exist for weeks and more. The stagnation points are typically moving Eulerian features in a frozen-time velocity field which may undergo bifurcations in the course of time. Bifurcation theory, among other things, is interested in behavior of fixed points of vector fields as a parameter is varied. In our case the role of the parameter plays time. In other words, one monitors positions of HSPs and ESPs day by day and look for their movement around in the surface ocean flow. Nothing interesting, besides a rearrangement of the flow, occurs if they do not change their stability type. When they do that there are, in principle, a few possibilities [3]. In the saddle-node bifurcation two stagnation points, a HSP and a ESP, collide and annihilate each other as one increases time. The opposite process could occur as well: in the course of time two stagnation points are born suddenly, one HSP and one


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ESP. After the collision stagnation points may move apart without changing in number (transcritical bifurcation) or split into three ones (pitchfork bifurcation).

5 Manifestations of hyperbolicity in the oceanic drifter’s tracks In the last 25 years, the deployment of satellite-tracked surface drifting buoys (drifters) has increased drastically. The modern drifter consists of a surface buoy and a drogue which is centered at a depth of 15 meters beneath the sea surface. The drifters follow near surface currents, measure temperature and other properties of sea water and send the data to passing satellites. Drifter observations now cover most areas of the world’s oceans at sufficient density to map mean currents at one degree resolution (see http://www.aoml.noaa.gov). They are Lagrangian instruments providing direct real-time observation of lateral advection. Among the other things, drifters can be used to identify positions of HSPs in the ocean where stable and unstable manifolds intersect each other. A series FTLE maps in Fig. 3, computed by formula (10), illustrates oceanographic situation in the first half of 2012 in the Northwestern Pacific to the east of the Kuril Islands (Russia). Those maps clearly show mesoscale eddies in the region with elliptic points at their centers and so-called Lagrangiam coherent structures [14] demarcated by black “ridges”. Those “ridges” on backward-in-time FTLE maps are known to approximate unstable manifolds of HTs present in the region for the integration time. The FTLE “ridges” are locations of particles on a given day with maximal (locally) values of the FTLE. Fluid particles on both sides of “a ridge” diverge maximally in the future or in the past. It means that the water parcels on one side of the “ridge” are involved in the vortex motion around an elliptic point, whereas the ones on the other side move away from the eddy. Thus, “ridges” on FTLE maps approximate the eddy boundaries. We impose on the FTLE maps in Fig. 3 tracks of two drifters for a few days before the date indicated on each panel. They clarify the role of hyperbolic points in organizing oceanic flows. It is seen on Figs. 3a and b on 27 and 28 April how two drifters approach the HSP, located at (46◦ N, 152.3◦ E), along its stable manifold from the south and north. After approaching that point, they begin to move away from it along its unstable manifold to the west and east (Figs. 3c, d, e and f on 29 and 30 April and 1 and 2 May). After all, this plot validate our calculation of locations of SPs in the altimetric velocity field. Another example of manifestation of hyperbolic regions in the ocean is provided by tracks of the drifters deployed on 11 June 2011 in the Kuroshio Extension area [16]. Four drifters were deployed on 11 June 2011 at the longitude ≈ 144◦ E. The drifters 2, 1 (4) and 3 were released in the jet core and just to the south and north of the core, respectively. Pieces of their trajectories from 14 to 22 June are plotted in Fig. 4. In the first days all they moved more or less coherently to the southeast (Fig. 4a). After appearing on 13 June a HSP (≈ 34◦ N, ≈ 148◦ E) at the periphery of the Kuroshio cyclonic eddy they approached that point along its stable manifold. Drifter no. 1 (squares in Fig. 4), deployed

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Fig. 3 Lyapunov maps in the Northwestern Pacific region to the east of the Kuril Islands (Russia). The values of FTLE Λ, computed for a month backward in time from the day indicated in each panel, are coded by nuances of the grey color in days−1 . “Instantaneous” elliptic and hyperbolic points, to be present in the area on fixed days, are indicated by circles and crosses, respectively. Tracks of the drifters are shown by full circles for a few days before the date indicated.

on the jet’s southern flank was the first reaching that HSP (Fig. 4a). Moving along the unstable direction of the corresponding HT, it was captured by the eddy performing cyclonic rotations around the eddy center (Figs. 4e and f). The trajectory of the drifter no. 2 (circles in Fig. 4) was cardinally different. Being released in the jet core, the drifter no. 2 approached the HSP later than the drifter no. 1 (Fig. 4b). After that, it began to move in the main jet in the opposite direction as compared to the drifter no. 1 (Figs. 4c, d, e and f). That fast diverging of the drifters nos. 1 and 2 after reaching the HSP can be interpreted as a manifestation of hyperbolicity. As to the drifters no. 3 and 4, they also felt the influence of the HSP (Figs. 4e and f) and continued eastward along the main current.

S.V. Prants, M.V. Budyansky, M.Yu. Uleysky / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 257–270 2011−06−14

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Fig. 4 Trajectories of the drifters deployed on 11 June 2011 in the Kuroshio Extension area. The plot illustrates manifestation of a HSP in drifter’s tracks. Squares, circles, triangles and stars mark tracks of the drifters nos.1, 2, 3 and 4, respectively.

6 Identification of unstable manifolds in altimetric velocity fields The stable and unstable manifolds of HTs in a studied region can be detected by local and global methods. In the local approach, one finds firstly positions of all the HSPs. Then it is necessary to identify HTs which are situated, as a rule, nearby the HSPs. It can be done by the different ways. We prefer to use a HSP as the first guess, then place at it on a fixed day a few material segments oriented under different angles to each other and compute the FTLEs for the particles constituting those segments. Coordinates of the particle with the maximal FTLE values give us approximate position of the HT nearest to that HSP on that day. Then we place a large number of synthetic particles in the form of a patch with the center at the HT position and evolve it forward in time.

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Fig. 6 Lyapunov map in the region to the east off the Hokkaido Island and the Kuril Islands on 1 October 2004 computed backward in time for two weeks. Locations of saury catches are shown by full circles with the amount of catch proportional to the circle’s radius [17, 18].

It is shown in Fig. 5 how the method works in the northwestern part of the Pacific Ocean east of the Hokkaido Island (Japan) and the Southern Kuril Islands (Russia), where the waters of the cold subarctic Oyashio Current encounter the waters of the warm Kuroshio Current. This region is known to be one of the richest fishery in the world. This figure shows how the corresponding unstable manifolds evolve from five tracer patches placed on 15 September 2004 near the five chosen HTs. Already by 25 September, the patches elongate and display the strongest unstable manifolds in the region. In the global approach, one seeds the whole region under study with a large number of tracers and compute different Lagrangian quantities. It has been proved [14] that the curves of local maxima (“ridges”) of the FTLE field attributed to initial tracer’s positions approximate stable manifolds when computing advection equations forward in time and unstable ones when computing them backward in time. In order to check that all the pathes in Fig. 5 in the course of time delineate the corresponding unstable manifolds, we compute in Fig. 6 the FTLE field backward in time for two weeks. This plot clearly demonstrates a large eddy with the center at (42.5◦ N, 147.5◦ E) which has been shown in [17, 18] to be a warm-core anticyclonic Kuroshio ring. The black “ridged” around that eddy approximate the unstable manifolds of the selected five HTs and coincide with the corresponding curves in Fig. 5. It has been studied by Lagrangian methods in [17, 18] that those manifolds are


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so-called Lagrangian fronts attracting the passive marine organisms which is a food for saury.

7 On some applications of the dynamical systems approach in physical oceanography The dynamical systems approach in physical oceanography is useful not only to identify organizing structures in oceanic flows but also to plan research vessel cruises. Before choosing the track of a planed cruise, it is instructive to find locations of HSPs and ESPs in a studied region and compute forward- and backward-in-time Lagrangian maps of some tracer’s indicators in the daily provided AVISO velocity field such as FTLE, absolute, zonal and meridional displacements, vorticity, etc. Those maps allow to visualize practically important structures in the region. For example, it has been shown in [17,18] how with the help of drift Lagrangian maps to delineate Lagrangian fronts with favorable fishery conditions. The method proposed seems to be useful in forecasting potential fishing grounds in different regions of the World Ocean. On the other hand, the same methods may help to recognize areas where marine organisms prefer to congregate and create protectable marine reservations there. Those methods have been shown in [19–23] to be effective to identify mesoscale eddies with a risk of contamination by Fukushima-derived radionuclides. The dynamical systems methods seem to be helpful in planning drifter’s launches. After identifying HSPs and ESPs, as well as eddies and jets in a studied region, we know where the drifters should be launched to give a valuable information about the oceanic flow. If the intention is to study properties of a given eddy or a given jet current, then the drifters should be launched well in the interior of such features. If the aim is to study mixing, then the launch sites should be located in the hyperbolic regions. With this aim it is better to launch drifters nearby potentially long-lived HSPs because those ones would pass a complex way approaching at first a HSP along itsstable manifold and moving then away along the corresponding unstable manifold. One may anticipate that those drifters would pass regions with very different properties, from stagnation ones nearby HSPs to Lagrangian coherent structures and Lagrangian fronts with a large energetics.

Acknowledgments This work was supported by the Russian Foundation for Basic Research (project no. 13–01–12404ofim) and by the Program “Dalniy Vostok” of the Far-Eastern Branch of the Russian Academy of Sciences (project nos. 15-I-1-003 o, 15-I-1-047 o, and 15-I-4-041). Appendix A. Singular-value decomposition of a 2 × 2-matrix The singular-value decomposition is a representation of any m × n-matrix in the form M = U ΣV,

(16)

where U and V are m × m and n × n unitary matrices, respectively, Σ is a diagonal m × n-matrix. The diagonal elements of Σ are singular values of the matrix M. The eigenvectors u and v, such that Mv = σ u and M ∗ u = σ v (σ is a singular value of M), are, respectively, left and right singular vectors of the matrix M. If M is real-valued then its singular values are real as well. U and V are orthogonal matrices. The matrix Σ and its singular-value decomposition are defined to an accuracy of the permutation of singular values. Therefore, one may require to order the singular values of Σ as a nonincreasing sequence, and such a decomposition is single. If the matrix M is squared then its singular-value decomposition has a simple geometric meaning. Action of any matrix to a vector can be represented as the following three successive transformations: the first rotation/reflection by the matrix V , a stretching/contraction along the coordinate axis by the matrix Σ and the second rotation/reflection by the matrix U . Thus, the matrix M transforms a sphere with the unit radius in an ellipsoid

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with the semiaxis to be equal to singular values directed along the left singular vectors. The right singular vectors are correspondingly pre-images of the ellipsoid’s semiaxis. Let us consider now a 2D flow with 2 × 2 evolution matrix with the singular-value decomposition ab cos φ2 − sin φ2 σ1 0 cos φ1 − sin φ1 G = U DV ⇒ = . (17) cd 0 σ2 sin φ2 cos φ2 sin φ1 cos φ1 Transformations of a sphere with the unit radius by those matrices and singular vectors are shown in Fig. 7. Reflection matrices are not used in this decomposition, therefore singular values can be negative. However, it is clear from general considerations that the evolution matrix of a continuous flow cannot contain reflections. It can be proven directly for the matrix (26). y ϕ1

y j

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Multiplying the matrices, one gets the set with four equations and four variables ab σ1 cos φ1 cos φ2 − σ2 sin φ1 sin φ2 −σ1 sin φ1 cos φ2 − σ2 cos φ1 sin φ2 = . cd σ1 cos φ1 sin φ2 + σ2 sin φ1 cos φ2 −σ1 sin φ1 sin φ2 + σ2 cos φ1 cos φ2

(18)

Let us introduce the following notations:

α = a + d, β = a − d, γ = c + b, δ = c − b, ξ = σ1 + σ2 , η = σ1 − σ2 , Φ = φ1 + φ2 , Ψ = φ2 − φ1 .

(19)

Adding and deducting Eqs. 18 and using the notations (19), we get

α = ξ cos Φ, Solution of the set (20) is p ξ = α 2 + δ 2,

η=

β = η cos Ψ, p

β 2 + γ 2,

γ = η sin Ψ,

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Φ = arctan2 (δ , α ),

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where arctan2 (y, x) is an angle between the vector (x, y) and the axis x which can be defined as ( arctan (y/x), x ≥ 0, arctan2 (y, x) = arctan (y/x) + π , x < 0.

(20)

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The final solution is p p (a + d)2 + (c − b)2 + (a − d)2 + (b + c)2 , σ1 = 2 arctan2 (c − b, a + d) − arctan2 (c + b, a − d) , φ1 = 2

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p

p (a + d)2 + (c − b)2 − (a − d)2 + (b + c)2 , σ2 = 2 arctan2 (c − b, a + d) + arctan2 (c + b, a − d) . φ2 = 2 (23) It is evident from the solution that the singular values are ordered in a nonincreasing way, i.e., σ1 ≥ σ2 . The product σ1 σ2 defines the ratio of the final area to the initial and equals to Det M. It follows from the definition of a singular-value decomposition that kMxk σ1 > > σ2 , (24) kxk where k · k is the Euclidean norm. In other words, the length of any vector x is changed under the action of the matrix M in σ2 times as minimum and in σ1 times as maximum. Appendix B. Evolution matrix for the 2D set linear equations with time-independent coefficients Here we find a solution of the 2D set of linear equations with time-independent coefficients x˙ x ab =J , J= . y˙ y cd The evolution matrix G of the set (25) is  D > 0, A1 eλ1 t + A2 eλ2 t ,      K  λt D = 0, e I+ t , G(t) = 2    K    eλ t I cos (ω t) + √ sin (ω t) , D < 0, −D where

10 I= , 01 D = 4bc + (a − d)2 ,

√ √ DI + K DI − K a − d 2b √ √ K= , A1 = , A2 = , 2c d − a 2 D 2 D √ √ √ −D a+d a+d + D a+d − D λ= , ω= , λ1 = , λ2 = . 2 2 2 2

(25)

(26)

(27)

The eigenvalues of the evolution matrix are eλ1 t and eλ2t . The flow (25), in general, does not preserve an area. The ratio of the areas at t and at the initial instant of time is given by the determinant of the evolution matrix Det G(t): S(t) = Det G(t) = e(a+d)t = eTr Jt . (28) S(0) Therefore, the flow (25) conserves the area if and only if the trace of the velocity matrix is zero. It is equivalent to the condition with zero sum of eigenvalues of the velocity matrix. The exact expressions for the maximal singular value σ1 of the evolution matrix (26) and for the finite-time Lyapunov exponent are rather complicated. We write down here only the asymptotics at t → ∞: s  (b + c)2 + (a − d)2 λ1t   e , D > 0,   4bc + (a − d)2 σ1 (t) = (29)  |b − c|teλ t , D = 0,     F(t)eλ t , D < 0,

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where F(t) is a limited periodic function with the period ω /2. Taking into account definition (9), we define the Lyapunov exponent of the set (25).

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Application of the Hydromechanical Model for a Description of Tropical Cyclones Motion Boris Shmerlin†, Mikhail Shmerlin Federal State Budgetary Institution “Research and Production Association Typhoon”, 4 Pobedy street, 249038 Obninsk Kaluga region, Russia Submission Info Communicated by S.V. Prants Received 27 January 2015 Accepted 2 March 2015 Available online 1 October 2015 Keywords Tropical cyclone Tropical cyclone track forecasting Track forecast errors Hydromechanical model

Abstract Within the framework of the hydromechanical model (HMM), proposed by one of the authors, a tropical cyclone (TC) motion is defined by a largescale wind field and a TC intensity. The model contains parameters describing TC and its interaction with wind field. The diagnostic, quasi-prognostic and prognostic calculations of TC movement are carried out. Diagnostic and quasi-prognostic calculations mean that an objective analysis of a large scale wind field and an objective analysis of a TC intensity is used during a TC whole lifetime. In case of diagnostic calculations, model parameters (constants for each TC) are defined from the best coincidence between the real and calculated track of a TC during a TC whole lifetime; for quasiprognostic calculations they are defined during the preliminary “preprognostic” period. Diagnostic calculations show that the HMM rather correctly describes peculiarities of a TC motion. Quasi-prognostic calculations show that model parameters may be rather correctly defined during a preliminary “preprognostic” period. The results of the diagnostic, quasi-prognostic and prognostic calculations are presented. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction On the basis of the analysis of barotropic and baroclinic models ignoring the friction of a tropical cyclone (TC) against the underlying surface (US) the opinion has been formed that TCs deviate from the surrounding large scale flow called the steering flow (SF) only insignificantly [1]. As it often contradicts the observation data [2], the efforts of scientists were directed towards the search of causes of a considerable deviation of a TC from a SF within the framework of the mentioned above models. However the friction of a TC against a US is the factor providing deceleration of a TC as a whole and leading to the considerable deviation of a TC from a SF. As it has been stated in our papers [3-10], the characteristic time of deceleration of a TC due to the friction against a US is about 12 hours, the time of entrainment of a TC by a SF due to the friction of a TC against a SF is about 60 hours. At first sight a TC must be nearly motionless and as opposed to the traditional approach we should look for reasons due to which a TC in some or other cases moves with the velocity that is close to the velocity of a † Corresponding

author. Email address: [email protected]


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Boris Shmerlin, Mikhail Shmerlin / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 271–279

SF. In such a situation it seems reasonable to use general formulae for the forces acting on a circular cylinder running in an arbitrary way in the arbitrary two-dimensional vortex flow for the description of a TC motion. This approach was realized in the hydromechanical model (HMM) [3-10]. We used the results obtained in the works of Yakimov [11] in 1970 and Petrov [12] in 1978. The similar approach was used in the works of Kuo [13] in 1969 and Jones [14] in 1977, but then researches with its application were stopped. We are going to deal with the difference between the HMM and models [13,14] in more details in the next section. 2 Structure of the hydromechanical model Equation describing the motion of a TC within the framework of the HMM has the form: dV/dt = dV0 /dt + [(V − V0) × Ω 0 ] − α · (V − V0 ) − α1 · V0 + Fβ , Ω 0 = iΩ0 , Ω0 = 2Ω⊥ (1 − A) − Aω , Ω⊥ = Ω sin ϕ , ω = Vϕ (L)/L, ˆ Ω cos ϕ L α = (b + c)/2, α1 = (b − a)/2, Fβ = f Vϕ (r)r2 dr, L2 R 0

(1)

where V − a TC velocity; V0 (x, y,t) − a field of a steering flow velocity (vectors V and V0 (x, y,t) directed horizontally); i−a unit vector, directed upward vertically; 2Ω⊥ − the Coriolis parameter; ω − an angular velocity of a TC rotation; Ω− the angular velocity of the Earth rotation; ϕ − a latitude of a TC center; L − a TC radius; Vϕ (r) − a radial distribution of a tangential wind velocity of a TC; A − a coefficient characterizing the distinction of circulation within a TC from a purely horizontal two-dimensional one (is known [13-15], so that in case of a purely two-dimensional flow around a cylinder the Coriolis parameter vanishes from the equation of motion, that is A = 1, however in case of a vortex ring with ascending motion on the axis the Coriolis parameter substantially determines the displacement of a vortex ring, and in this case A < 1); b − a coefficient of a TC friction against an underlying surface; a − a coefficient of a steering flow friction against an underlying surface; c − a coefficient of a TC friction against a steering flow; Fβ − a northward Rossby force, Fβ -its modulus; f − a dimensionless parameter of the model; R-the Earth radius. We don’t give the corresponding equation in a spherical coordinate system though it was used in the work. In Kuo [13] and Jones [14] models, mentioned in the introduction, exact formulae for the forces acting on the cylinder in the homogeneous stationary flow of the ideal liquid with a constant horizontal velocity shift were used and equations of a TC motion were formulated for this case. A special attention was also paid there to the importance of taking into consideration the friction of a TC against a US in case of describing a TC motion. In (1) we used the results [11, 12], which are valid to within the accuracy O(L/L )3 , where L -a scale of inhomogenuity of a SF. Associated with this the additional rather important summand, SF acceleration (full), has appeared in the equation (1). As a result it has become possible to apply this equation to the description of a TC motion in a SF of any space configuration and time dependence. Rossby force accounting for the influence of β -effect upon a TC motion is also additionally included in (1) [16]. The parameterizations used in the model have been changed completely (look further). At last the important difference between HMM and models [13, 14] is the presence of the Coriolis parameter in the formula for Ω0 , as the direction of the deviation of a TC from a SF is defined by the sign of Ω0 [3-10]. In Kuo [13] and Jones [14] models Ω0 = −ω < 0, and a TC can deviate only to the right of a SF. This conflicts with TC motion observation data [2]. Within the framework of the HMM the parameter Ω0 can change its sign owing to the change of the angular velocity of a TC rotation in the process of a TC development and changing of the Coriolis parameter because of a latitude TC displacement. It leads to the fact that a TC can deviate both to the right and to the left of a SF. The regularities of a TC deviation from a SF in the framework of the HMM turn out to be quite different from those in other models ignoring the friction of a TC against a US. Thus, for the stationary TC motion with Ω0 ≡ const in a homogeneous stationary SF with a velocity V0 from (1) we have: Ω0 × V0 ] + [Ω Ω0 × Fβ ] − α Fβ }(α 2 + Ω20 )−1 . V = V0 − {α1 α V0 − α1 [Ω

(2)


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The characteristic relaxation time of a TC velocity to the stationary velocity (2) is τ ≈ α −1 . If, for example, ˙ 0 (t)| α 2 , then the expression (2) with Ω0 = Ω0 (t) determines Ω0 = Ω0 (t) and Ω0 (t) changes slowly enough, |Ω velocity of a TC, which may be called quasistationary and may be used for the description of a TC motion in this case. Neglecting β -effect for a modulus of a stationary velocity of a TC and an angle γ between the vectors V and V0 we obtain: α 2 (1 − D)2 + Ω20 1 α DΩ0 α1 ] 2 , tgγ = 2 , D= , (3) V = V0 [ 2 2 2 α α + Ω0 α (1 − D) + Ω0 where γ > 0 corresponds to the deviation of a TC to the left of a SF. The stated dependences are presented in Fig.1. For example one can see, that when Ω0 (t) passes through zero the velocity of a TC reduces significantly if D ≈ 1 and the direction of a TC motion changes significantly too even in a homogeneous stationary SF [5].

Fig. 1 Dependences of V /V0 and tgγ on Ω0 . 1

Due to the β -effect the additional velocity component of a TC has appeared Vβ = Fβ (α 2 + Ω20 )− 2 , which forms an angle ϕ with the direction to the north, tgϕ = Ω0 /α , ϕ > 0 corresponds to the north-eastern direction. In particular if b a, b c, then α1 ≈ α , D ≈ 1, and in case Ω0 = 0 we obtain V = Fβ /α for any velocity Ω0 × Fβ ]/Ω20 . of a SF V0 . If |Ω0 | α , then V ≈ V0 − [Ω Judging on the literature further researches with the application of Kuo [13] and Jones [14] models were stopped. Therefore at present the HMM stands apart from the models of various types which are used for TC motion prediction. We’ll give the schemes of parameterizations used in the HMM. A modified Renkin vortex was chosen by us as a vortex modeling a TC . The radial distribution of a tangential wind velocity in it is defined by the formula Vϕ (x)/Vm = v(x) = 2x/(1 + x2 ), where x = r/Rm , Vm -maximum wind velocity in a TC, Rm -the radius on which a wind velocity reaches its maximum. As it is seen from (1) the most important characteristic of a TC is a dimension or a radius of a TC L. In the known radial distributions of TC parameters a characteristic TC outer radius is absent. Nevertheless such an outer radius can be introduced. It can be the radius of an area of ascending motions (cloud cover) of a TC. The Ooiama formula [17] allows to connect the vertical velocity w(x) and the radial distribution of a tangential velocity and from condition that w(x) = 0 to find a non-dimensional radius of an area of ascending motions x0 [18]: w(x) =

d Vm cd d 2 2 {x v (x)/ [xv(x) + Ro−1 x2 ]}, x dx dx

Ro =

Vm ; Rm Ω sin ϕ

x0 = [1 + (4 + 3Ro)1/2 ]1/2 .

(4)

Here cd ≈ 1.1 × 10−3 -is the drag coefficient, Ro-is the Rossby number. We suppose L = x0 Rm . After

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that it is easy to obtain the formula for values entering (1). We at once find ω = 2Vm /[Rm (1 + x20 )], Fβ = f Ω cos ϕ Vm Rm [x20 − ln(1 + x20 )]/(Rx20 ). Friction coefficients are the proportionality coefficients before a velocity (or relative velocity) in the formula for the friction force divided by a TC mass. Introducing the height of homogenous atmosphere H ≈ 8 × 103 m for the friction coefficient of a SF against a US we get a = cd V0 /H. Considering the friction force of a TC against a SF to be proportional to a relative momentum flux through the cross section of a TC we have c = 2k|V − V0 |/(πRm x0 ), k ≈ 1-is the second dimensionless parameter of the model. We’ll calculate a friction coefficient of a TC against a US considering that the quiescent air from a boundary layer with a vertical velocity w(x) enter into´the area occupied by a TC and then is entrained into a horizontal x motion with the TC velocity. Then b = x2sH 0 0 w(x)xdx = 2scHd Vm 1+Ro−1x0(1+x2 )2 ≡ sb0 , s ≈ 1-is the third dimen0 0 sionless parameter of the model. The value τv = b−1 0 is a ratio of a TC mass to a mass flux into the area occupied by a TC from a boundary layer that is characteristic time of a TC “ventilation”. The turn-around time of a rotating coordinate system is τr = (Ω sin ϕ )−1 . The ratio τr /τv characterizes the distinction of circulation within a TC from a purely twodimensional one. We consider that A = exp{−p[b0 /(Ω sin ϕ )]}, where p-is the last non-dimensional parameter of the model. In case of a purely horizontal two-dimensional circulation within a TC b0 = 0 and A = 1. Thus in the general case the model contains four parameters f , k, s, p, which are considered to be constants for each concrete TC. The method of a TC motion forecast on the basis of the HMM includes three blocks: the block of a TC intensity and radial structure which provides an objective analysis and prognosis of values Vm and Rm ; the SF block providing an objective analysis and prognosis of a SF velocity field; properly the block of the forecast using the HMM. The information of telegrams - storm warnings is used in the first block. They are broadcast four times a day by the National Hurricanes Centre (NHC) for the Atlantic and the North-East Pacific and by the Joint Typhoon Warning Centre (JTWC) for the North-West Pacific. The telegrams contain an objective analysis and a forecast of TC coordinates and a maximum wind velocity Vm for a duration of prognostic periods up to five days inclusive and of a radius R1 , on which a wind velocity reaches V1 = 34 knots for a duration of prognostic periods up to three days inclusive. Herefrom we obtain Rm = R1 [Vm /V1 + (Vm /V1 )2 − 1]−1 . An objective analysis and a forecast (for a duration of prognostic periods up to six days inclusive) of wind fields on the standard levels in the GRIB code on the grid 2.5 × 2.5 deg is used in a SF block. The referenced information is given by the Global Operational Model of the Hydrometeorological Center of The Russian Federation twice a day. The wind field which is weighted average through a height of a troposphere in the layer from the surface to 200 mb is obtained for each forecast time. It contains the circulation of a TC. A window including a TC circulation is cut in the neighborhood of a TC and the interpolation of a wind field from an outer region is carried out inside the window. The received wind field is considered to be a SF velocity field V0 (x, y,t) for the corresponding time moment. It must be emphasized that a choice of an optimal height of averaging of large scale wind fields was not done by us and this rather time consuming task must be solved in the course of further researches. As an example you can see a wind field with a TC circulation in Fig. 2 on the left and a SF velocity field on the right, the TC is situated in the point with coordinates 125 deg of eastern longitude and 25 deg of northern latitude. The linear interpolation of corresponding values in time is used for time intervals between nearest prognosis of Vm and R1 and a SF velocity fields. A concrete TC motion within the framework of HMM in case of prepared in advance SF fields is calculated in split second on a personal computer. It gave us an opportunity to create mathematical support capable of calculating all TCs motion of a few seasons at once and comparing success of various parameterizations, algorithms of choosing model parameters etc from the condition of minimum mean over a few seasons calculation errors. In particular a lot of sensible physical hypotheses as to the defining of a TC dimension L can be suggested. Above


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Fig. 2 A large scale wind field before and after the exclusion of a TC circulation. On the left there is a window including a TC circulation.

we have described the approach to the defining of a TC dimension and a corresponding set of parameterizations which demonstrated the best results in the calculations for a few TCs seasons. 3 Diagnostic calculations of a TC motion Diagnostic calculations mean that an objective analysis of SF velocity fields and values Vm and R1 is used during the TC whole lifetime [6]. In case of diagnostic calculations model parameters (constants for each TC) are defined from condition of the best coincidence between the real and calculated track of a TC during the whole lifetime. We have stored all the necessary for calculations archives of wind fields and telegrams about TCs for the seasons of 2001, 2003. 2010 and 2011. There were about 130 cyclones with the lifetime from 4 to 15 days within these seasons. The diagnostic calculations were carried out for them. For all these TCs the mean along the trajectory deviation of the calculated position from the real one doesn’t exceed 150 km and for most of the TCs, it is less than 100 km which can be compared with the accuracy of defining of the TC actual position. Diagnostic calculations reproduce not only the general character of a TC trajectory but characteristic peculiarities of most of the trajectories: characteristic form of different trajectories near the recurvature points, loops, hanging about and so on. Thus the HMM rather correctly describes a TC motion. Examples of the diagnostic calculations for TCs with atypical trajectories are shown in Fig. 3. The first four digits in the name of a TC mean the year, the next two digits mean the serial regional number of the TC of this year, symbol “W” corresponds to the North-West Pacific and symbol “A” corresponds to the North-West Atlantic. Hereinafter labels on the trajectories are set every 12 hours, the trajectories with circles are the calculated ones. The horizontal axis shows the longitude (negative values correspond to the west longitude) and the vertical axis shows the latitude in degrees. 4 Quasi-prognostic calculations of a TC motion Quasi-prognostic calculations mean that an objective analysis of SF velocity fields and values Vm and R1 is used during the TC whole lifetime as well. In addition to this however model parameters (constants for each TC) are defined from condition of the best coincidence between the real and the calculated track of a TC during the preliminary “preprognostic” period. The duration of a preprognostic period in case of a real forecast is defined by the information about the previous TC motion that is available at the moment of the forecast [7, 8]. Quasiprognostic calculations differ from prognostic calculations by the fact that in case of a real prognosis during the

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Fig. 3 Examples of diagnostic calculations: TC 2003 21W is on the left, lifetime is 10 days, mean along the trajectory error is 87 km; TC 2011 08W is in the centre, lifetime is 10 days, mean along the trajectory error is 118 km; TC 2011 18W is on the right, lifetime is 9.5 days, mean along the trajectory error is 103 km.

prognostic period instead of the objective analysis of SF velocity fields and values Vm and R1 their prognosis is used. So the problem of choosing of an optimal algorithm for defining model parameters during a “preprognostic” period is put forward on this stage. Earlier while carrying out theoretical researches within the framework of the HMM the instability of trajectories of vortices modeling TCs was observed: in certain cases small changes of vortex parameters or parameters of model synoptic situations lead to the change of a vortex trajectory type, for example a parabolic trajectory is occurred instead of a direct one [3, 4]. The same phenomenon is demonstrated by diagnostic [6] and quasiprognostic calculations [7, 8]. As a rule the errors during the “preprognostic” period are small and close to each other in the whole domain Σ of values of model parameters. Two situations are possible in this case. In most cases quasi-prognostic trajectories calculated under various values of model parameters from domain Σ are of the same type and rather close to each other. We can speak about this as about the trajectories stability. In this case as a rule a satisfactory criterion of model parameters choice is the condition of a minimum calculation error during a “preprognostic” period. The examples of such quasi-prognoses are shown in Fig. 4. Along with that there are situations when quasi-prognostic trajectories calculated under various values of model parameters from the domain Σ differ from each other significantly and can be of different type. We can speak about this as about the trajectories instability with respect to small changes of model parameters. Nevertheless even in this case the HMM under corresponding parameters values in a diagnostic calculations describes a TC motion rather correctly [5, 6]. So a conclusion can be made from this: the instability in a stated above sense is an inner characteristic feature of a TC motion. It can be an objective reason of a poor quality of forecasts in definite situations. It is important to underline that the problem of trajectory stability or instability can be solved in each concrete case of a forecast in advance by means of enumeration of trajectories corresponding to various values of model parameters from the domain Σ. We would like to emphasize that we have dealt with the trajectories instability at the moment of the forecast and for a period of the forecast. Thus if a TC has passed the point of instability then the domain Σ corresponding to small errors of a “preprognostic” period decreases rapidly and the trajectory itself can be quite stable hereafter. We have carried out the comparison of effectiveness of several physically sensible algorithms for defining model parameters during a “preprognostic” period which do not contain the analysis of trajectories stability. As a result it turned out that minimum mean over a few seasons errors of quasi-prognosis for 72 hours take place when the HMM parameters f , k, s, p are the same for all TCs and are not chosen at all during a “preprognostic” period. Thus, these parameters values are defined from the condition of minimum mean over several seasons error of quasi-prognostic calculations for 72 hours.


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Fig. 4 Examples of quasi-prognoses, “preprognostic” period is 48 hours: TC 2003 10A is on the left, quasi-prognostic period is 9 days (calculation from the initial point of the trajectory for 11 days); TC 2003 13A is in the centre, quasiprognostic period is 10.5 days (calculation from the initial point of the trajectory for 12.5 days); TC 2003 26W is on the right, quasi-prognostic period is 9.5 days (calculation from the point of the trajectory corresponding to the beginning of the quasi-prognosis for 9.5 days).

Mean forecast errors of the quasi-prognostic calculations for the North-West Pacific are: 217, 272, 258. 257. 267 km for 3, 4 . . . 7 days correspondingly in TC season of 2010 which are significantly less than mean official forecast errors of JTWC (315, 450, 540 km for 3, 4 and 5 days). In general the quasi-prognostic calculations allow to define the HMM parameters values f , k, s, p, which provide acceptable quasi-prognostic errors. These values are used in prognostic calculations hereafter. At the end of this section it is necessary to underline that the information about a “preprognostic” period is not used at all for determining the HMM parameters in case of such an approach. The information about initial coordinates and velocity of a TC for prognostic calculations is only used. We hope that in future we will be able to suggest an algorithm for determining model parameters during a “preprognostic” period which will take into consideration maximum of all this information. Such an algorithm development will allow to reduce mean quasi-prognostic errors and consequently the errors of forecasts of a TC motion. 5 Prognostic calculations of a TC motion In TCs season of 2010 prognostic calculations within the HMM where carried out [9]. The examples of the recurvature point forecasts are shown in Fig. 5. In TCs season 2011 the duration of prognostic period was increased to 5 days (120 hours) [10]. The forecasts examples are shown in Fig. 6. A maximum duration of prognostic period within the framework of the HMM is defined by the used duration of prognostic period of the radial TC structure (value R1 ) and it mustn’t exceed 3 days (72 hours). Nevertheless in season 2010 the forecasts were calculated for 3.5 days and in season 2011 for 5 days inclusive despite of the absence of a radial structure forecast under the condition that there was at least one forecast of value R1 during the first three days. In particular it was considered that beyond the radial structure forecast, R1 is equal to the last prognostic value if prognostic value Vm > V1 . If however the condition Vm < V1 started to be fulfilled from some moment then the radius of the maximum wind Rm preserved the last calculated value. We would like to remind that the mean official JTWC forecast errors for the North-West Pacific for the TC season of 2010 were 315, 450, 540 km for 3, 4 and 5 days respectively. As it is known the mean seasonal official forecast errors are 30-50 km less than the mean errors of any other forecast methods. While calculating the mean

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Fig. 5 Examples for 3.5 days (84 hours) forecasts: TC 2010 15W is on the left, TC 2010 06A is in the centre, TC 2010 11A is on the right.

Fig. 6 Examples for 5 days (120 hours) forecasts: TC 2011 11W is on the left, forecast errors are 134, 263, 382, 355, 195 km for 1-5 days; TC 2011 18W is in the centre, forecast errors are 278, 207, 164, 146, 335 km for 1-5 days; TC 2011 15W is on the right, forecast errors are 59, 249, 116, 126, 120 km for 1-5 days.

seasonal errors JTWC takes into consideration only the errors of those forecasts for which the whole predicted part of a real TC trajectory is situated within the tropical zone that is to the south of the 30th degree of northern latitude. The forecast errors increase significantly beyond the tropical zone. The mean forecast error within the HMM for the North-West Pacific for the TC season of 2010 for 3 days (72 hours) is 350 km with account of errors beyond the tropical zone. The mean HMM forecast errors for the North-West Pacific for the TC season of 2011 are 397 (116), 457 (89), 574 (66) km for 3, 4, 5 days respectively with account of errors within the extratropical zone. The number of forecasts of corresponding forecast-time interval is indicated in brackets. Evidently it can be affirmed that the HMM forecast errors are within the limits of the errors of the official American forecasts by JTWC and less than the forecast errors of the most developed dynamical prediction models at least for the 4-days and 5-days forecasts.


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6 Conclusions We have suggested the hydromechanical model of a TC motion. The results of the diagnostic calculations demonstrate that this model describes the real TC motion rather correctly. Quasi-prognostic calculations allow us to define the values of the HMM parameters, which ensure acceptable quasi-prognostic errors (217, 272, 258, 257, 267 km for 3, 4, . . . 7 days respectively). The forecast errors within the HMM is nearly 130–250 km, depending on a duration of a prognostic period, more than the quasi-prognoses errors. It is connected only with the fact that the forecasts of a large scale wind field and values Vm and R1 differ from the corresponding objective analysis. As the indicated forecasts come close to the objective analysis the HMM forecast errors will come close to the quasi-prognoses errors. It seems the quasi-prognoses errors in its turn can be reduced in case of creating an algorithm of defining the model parameters during the “preprognostic” period which maximally takes into consideration the “preprognostic” information. One more source of reducing of the forecast errors of a TC motion can be the choice of an optimal height of averaging of large scale wind fields. References [1] Chan, J.C.L. (2005), The physics of tropical cyclone motion, Annu. Rev. Fluid Mech., 37, 99–128. [2] Dong, K. (1986), The relationship between tropical cyclone motion and environmental geostrophic flows, Mon. Wea. Rev., 114(1), 115–122. [3] Shmerlin, B.Ya. (1981), A study of patterns in the movement of large scale vortices relative to the pure zonal flow, Meteorologiya i Gidrologiya, No. 7, 27-35 (in Russian). (Engl. Transl. Soviet Meteorology and Hydrology, No. 7–12.) [4] Shmerlin, B.Ya. (1987), Some investigations of TC’s trajectories stability within the framework of the hydromechanical model, in Tropical meteorology. Proceedings of the third international symposium, eds. U.S. Sedunov, et al. Gidrometeoizdat, Leningrad, 292–307 (in Russian). [5] Shmerlin, B.Ya. (1989), On confirmation of adequacy of the hydromechanical model of a tropical cyclone motion, in Tropical meteorology. Proceedings of the fourth international symposium, eds. V.M. Zakharov, et al. Gidrometizdat, Leningrad, 179–186 (in Russian). [6] Shmerlin, B.Ya., et al. (2004), Diagnostic calculations of a TCs motion in the 2001 year season within the framework of the hydromechanical model, in The International Conference MSS-04 “Mode Conversion, Coherent Structures and Turbulence. 23-25 November 2004. Conference Proceedings, eds. N.S. Erokhin, et al. POXOC, Moscow, 284–289 (in Russian). [7] Shmerlin, B.Ya., et al. (2008), Quasi-prognostic calculations of a tropical cyclones motion within the frameworks of the hydromechanical model, in International conference “Fluxes and structures in fluids”. St. Petersburg, Russia, July 02-05, 2007. Selected papers, eds. Yuli D. Chashechkin and Vasily G. Baydulov. IPM RAS, Moscow, 269–274 (in Russian). [8] Shmerlin, B.Ya., et al. (2009), Quasi-prognostic calculations of a tropical cyclones motion, Ukrainian Hydrometeorological Journal, N 4, 67–74 (in Russian). [9] Shmerlin, B.Ya. and Shmerlin, M.B. (2011), Application of the hydromechanical model for a description of tropical cyclones motion, Vestnik of Lobachevsky State University of Nizhni Novgorod, 2011, No. 4, Part 2, 564–566 (in Russian). [10] Shmerlin, B.Ya. and Shmerlin, M.B. (2012), Hydromechanical model of a tropical cyclones motion, Actual problems in remote sensing of the Earth from space, 9(2), 243–248 (in Russian). [11] Yakimov, Y.L. (1970), Motion of a cylinder in arbitrary planar ideal incompressible fluid flow, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, 5(2), 202-204 (in Russian). (Engl. Transl. Fluid Dynamics, 5(2), 350–353.) [12] Petrov, A.G. (1978), Reactions acting on a small body in two-dimensional vortex flow, Doklady Akademii Nauk SSSR, 238(1), 33-35 (in Russian). (Engl. Transl. Soviet Physics–Doklady, 23(1), 18–19.) [13] Kuo, H.L. (1969), Motion of vortices and circulating cylinder in shear flow with friction, J. Atmos. Sci., 26, 390–398. [14] Jones, R.W. (1977), Vortex motion in a tropical cyclone model, J. Atmos. Sci., 34, 1518–1527. [15] Batchelor, G.K. (1973), An Introduction to Fluid Dynamics, Cambridge University Press. [16] Chan, J.C. and Williams R. (1987), Analytical and numerical studies of the Beta-Effect in tropical cyclone motion. Part 1: Zero mean flow, J. Atmos. Sci., 44(9), 1257–1265. [17] Ooyama, K. (1969), Numerical simulation of the life-cycle of tropical cyclones, J. Atmos. Sci., 26, 1–43. [18] Kalashnik, M.V. (1994), On the maximum wind velocity in the tropical cyclone, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana, 30(1), 26-30 (in Russian). (Engl. Transl. Izv. Atmos. Ocean. Phys., 30, 23–27.)



Influence of Deep Vortices on the Ocean Surface Daniele Ciani†, Xavier Carton1 , Igor Bashmachnikov2, Bertrand Chapron3 , Xavier Perrot4 1 Laboratoire

de Physique des Océans, UMR6523, Université de Bretagne Occidentale, 29200, Brest, France and Environmental Sciences Centre (MARE), Faculdade de Ciências, Universidade de Lisboa, Lisbon, Portugal 3 Laboratoire d’Oc´ eanographie Spatiale, IFREMER, Centre de Brest, Plouzané, France 4 Laboratoire de M´ etéorologie Dynamique, Ecole Normale Supérieure, Paris, France 2 Marine

Submission Info Communicated by S.V. Prants Received 16 January 2015 Accepted 2 March 2015 Available online 1 October 2015 Keywords Vortex dynamics Quasigeostrophic theory Potential vorticity inversion Sea-surface elevation

Abstract We study the influence of deep vortices on the ocean surface in terms of sea-surface elevation, a quantity related to a fluid stream function. We use several mathematical and numerical models, from the most idealized configurations (point vortices) to realistic ones (finite volume vortices). We determine analytically the surface influence of vortices at rest (steady signature) and in motion (dynamical signature). Then, using a nonlinear, numerical hydrodynamic model for oceanic vortices, we determine the growth with time of a dynamical signature for drifting vortices without steady signature. We conclude on the possibility to detect several types of oceanic vortices with surface measurements, using the results from our theory and experiments. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The oceanic motions at mesoscale are characterized by horizontal scales ranging from 20 to 200 km and time scales ranging from a few days to a few weeks. The expression of this dynamics is mostly given by gyres, vortices (eddies), meanders of unstable currents and other turbulent features. Mesoscale motions are long-lived, contain more than 80% of the ocean eddy kinetic energy and are globally more energetic than the general circulation. Hence, their impact on transport of heat and salt in the global ocean is relevant. For example, gyres and vortices are recirculating features which trap water masses from their origination areas and carry them over large distances across the ocean [1, 2]. Observations of the ocean show that mesoscale features do not exhaustively describe the ocean dynamics at scales equal to or lower than 200 km. This is also suggested by modeling experiments, in which the modeled eddy kinetic energy increases as the model horizontal spatial resolution increases [3]. Indeed, a finer spatial resolution allows one to take into account submesoscale features and to see that they significantly contribute to the ocean energy budget and circulation. Submesoscale structures have horizontal scales between 0.5 and 20 † Corresponding

author. Daniele Ciani: Laboratoire de Physique des Océans, UMR6523, Université de Bretagne Occidentale, 29200, Brest, France, Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.09.006

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Daniele Ciani et al. / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 281–311

km and time scales from a few hours to a few days. Physically, they are given by small vortices, filaments or turbulent patches, and they can result from hydrodynamic instabilities and from the straining and shearing of pre-existing mesoscale structures. The ocean is highly populated by mesoscale and submesoscale vortices at depths ranging from 0 to 1000 m. For instance, deep vortices are the ones living at depths between 100 and 1000 m. Occurrences of these structures have been documented by observations and numerical models and show that deep vortices can be observed in the Atlantic, Indian Ocean as well as the Mediterranean Sea. They can result from mixing processes, or from the exchanges between semi-closed highly evaporative basins and the open ocean, which can take place at different levels [4]. The first mechanism explains the existence of subsurface EDDIES in the North-Western Atlantic Ocean ( [5–7]), while the second one generates Mediterranean Water Eddies (MEDDIES) off of Gibraltar Strait [8], REDDIES from the Red Sea outflow and PEDDIES from the Persian Gulf outflow (REDDIES=Red Sea Water EDDIES, PEDDIES=Persian Gulf Water EDDIES). These last two types of eddies evolve in the NW Indian Ocean, [9], [10]. Ocean currents can also generate vortices while flowing down a sloping topography, or by sinking processes due to their density anomaly with respect to the background environment. These last two mechanisms constitute the generation processes for Slope Water Eddies (SWODDIES) in the Bay of Biscay and for Levantine Intermediate Water Eddies (LEDDIES) in the western Mediterranean Sea [11], respectively. In general, vortices living in proximity of the sea surface and characterized by mesoscale horizontal scales, have a recognizable and largely investigated signature on remotely sensed oceanic fields, [12,13]. Such fields are given by satellite-derived Sea-Surface Temperature (SST) and Sea-Surface Height (SSH), the latter expressing the sea-surface elevation induced by dynamical structures. Vortices’ signature on the ocean surface is the key for their detection and tracking, which is essential to estimate the eddy influence on oceanic energy, heat and salinity budgets. The smaller the vortex horizontal scale and the larger its depth, the less efficient its detection becomes. In fact, the surface-induced signal increasingly weakens and it is difficult to isolate it from the background surface dynamics. In recent years, several works have attempted to infer subsurface dynamics from observations of the sea-surface. Lapeyre and Klein [14] and Ponte and Klein [15], using Suface Quasi-Geostrophic (SQG) theory, showed that this task can be successful if one restricts this method to study the ocean in the first 500 m of the water column and for features of horizontal lengths in the mesoscale range. These results, considering typical scales of deep vortices shown in Table 1, point out a limit of the SQG theory, which would not be suitable for identifying features as deep vortices, whose radii can go down to the submesoscale range and depths can be way beyond 500 m. Table 1 Most common deep vortices in the world ocean (Anticyclones: clockwise rotation in the Northern Hemisphere) Basin

Depth (m)

Thickness (m)

Radius (km)

Rotation (ms−1 )

Mediterranean Sea

200 − 1000

300-1000

25-50

0.2

MEDDIES

NE Atlantic Ocean

600 − 1200

300-1000

20-50

0.3-0.4

Subsurface EDDIES (NW Atlantic)

NW Atlantic Ocean

700 − 1500

100-1000

15-100

0.2-0.5

LEDDIES

PEDDIES

NW Indian Ocean

250 − 400

100

20-30

0.3

REDDIES

NW Indian Ocean

400 − 800

400-500

20-50

0.15

SWODDIES

NE Atlantic Ocean

70 − 300

200

30-50

0.3

Alternative approaches at inferring subsurface dynamics from sea-surface observations are provided by the combined use of satellite-derived and in-situ observations. Separately, the two kinds of measurements cannot give an exhaustive description of the subsurface dynamics. Satellites provide a synoptic though very noisy signal, since background surface dynamics can have intensities comparable to or larger than the signals coming from lower levels. On the other hand, in-situ measurements can give a precise information on the ocean prop-


283

erties at different depths, but they remain very limited in space and time. In the last fifteen years, many have attempted at combining satellite-derived fields and in-situ data to catch vortices living at depth. In 1991 Stammer et al. [16], stated that Meddies can be detected using the combination of in-situ and altimetric observations (referred to measurements of the deformation of the sea surface). Meddies, as other subsurface anticyclones, are in fact associated to positive, monopolar anomalies in altimetric fields [5]. Recently, Ienna et al. [17] confirmed this behavior. Other hints for Meddy detection can come from combining in-situ, altimetric and satellite-derived surface temperature fields. In fact subsurface anticyclones, beyond their effect on altimetric fields, may lower SST due to their three-dimensional density structure [5]. Taking advantage of these characteristics, Bashmachnikov et al. [18] showed that Meddies can be associated to SST cold anomalies in clockwise rotation. Even in this last case, the use of co-located in-situ observations was fundamental to strengthen informations coming from remotely-sensed fields. Since one is interested in studying deep vortices to evaluate their impact on the global ocean, an automatic and synoptic detection would constitute the ideal scenario for such an investigation. The perspective of a future satellite altimetric mission (Surface Water and Ocean Topography, SWOT [19]) which will improve present-day satellite altimetry performances, encouraged us in further investigating the impact of deep vortices on the ocean surface (in terms of sea-level anomaly). The SWOT mission will provide global altimetric observations of the ocean at submesoscale horizontal resolutions and with precision measurements smaller than 2 cm. Today, such horizontal resolutions are not achievable and the 2 cm measurement precision characterizes only a fraction of the global altimetric observations [20]. The study on the impact of deep vortices on the ocean surface will be carried out in an idealized context. Even though occurrence of both cyclonic and anticyclonic deep vortices is possible, we choose to focus on anticyclones, which represent a large amount of subsurface rotating motions [5–11]. Supposing the existence of isolated deep vortices living in a background rest dynamics, we will try to derive their signature at the sea surface using both analytical models in the frame of the Quasi-Geostrophic theory (QG hereafter) [21] and numerical models. The numerical models will be the 3D Combined Lagrangian Advection Method with buoyancy surface boundary conditions (CLAM-3DQG hereafter, [22]), which simulates the ocean dynamics in a QG context, and the Regional Oceanic Modeling System (ROMS, [23]) which solves numerically the whole set of prognostic equations for the three components of oceanic motions, as well as temperature and salinity. In Section 2, the Material and Methods used for the investigation will be illustrated. In particular, we will recall some elements of the Quasi-Geostrophic framework as well as the main principles of the CLAM-3DQG and ROMS numerical models. In Sections 3 and 4, we will show the results issued from the analytical and numerical studies, respectively. Finally, Section 5 will provide a discussion of the results and of possible future investigations. 2 Material and methods 2.1

Quasi-Geostrophic framework

In the QG theory, which is a suitable framework for mesoscale coherent vortices [7], a vortex can be represented by its potential vorticity, a quantity taking into account its rotation and the conjugated effects of Coriolis acceleration (imposed by Earth’s rotation) and of buoyancy. Assuming a uniformly stratified ocean, the potential vorticity ”Q” is expressed by Equation (1): ∇2H ψ + 2

2

f02 ∂ 2 ψ = Q(x, y, z), N02 ∂ z2

(1)

where ∇2H = ∂∂x2 + ∂∂y2 , f0 is the Coriolis parameter, N0 is the Brunt-Väisälä frequency and ψ = ψ (x, y, z) is the stream function, [21]. Generally, the first term on the left-hand side of Equation (1) is generally referred to as relative vorticity of a QG fluid, while the second one is called stretching. In a Cartesian frame, the stream

284


function is related to the fluid horizontal motion and satisfies u = k × ∇ψ , where k = (0, 0, 1) [21]. In a QG context, the determination of ψ (x, y, z) is then crucial for evaluation of geostrophic currents. Furthermore, this quantity allows one to estimate the sea-surface elevation and sea-surface temperature, the first being proportional to ψ (x, y, 0) and the latter to ∂z ψ |z=0 . Looking at Equation (1), it is evident that the determination of the stream function will depend on the inversion of the potential vorticity Q. If one manipulates Equation (1), rescaling the vertical coordinate as Z = (N0 / f0 )z, the following expression for Q is obtained: ∇2 ψ = Q, 2

2

(2) 2

where the Laplacian operator is now given by ∂∂x2 + ∂∂y2 + ∂∂Z 2 . In this case, analytical inversion of Equation (2) is possible and the stream function ψ can be evaluated through use of Green’s Functions [24]. Normally, this technique is suitable for vortices mathematically represented as point vortices, in which the vortex potential vorticity looses its extension and is concentrated at a geometrical point of a three-dimensional ocean (see Section 3.1.1 for further details). If the three-dimensional finite structure of a vortex has to be represented, alternative approaches are considered, which rely on the decomposition of real functions (describing Q(x, y, z)) on complete and orthogonal basis of functions, e.g. Fourier or Bessel functions (see Section 3.1.2 for further details). Since our aim is to evaluate the sea-surface expression of deep vortices, such structures will be thought as potential vorticity anomalies living in a background dynamics at rest. Such anomalies will be placed at a finite distance from the sea-surface and will be characterized by a variable rotation, thickness and radius in order to account for the whole class of oceanic deep vortices listed in Table 1. The associated stream function ψ will be obtained using the two aforementioned methods and will be evaluated at the sea surface for inferring the associated sea-surface elevation. 2.2

QG numerical model

The CLAM-3DQG model [22] simulates the following system of equations: ⎧ DQ int ⎪ ⎪ ⎪ Dt = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ds bs ⎪ ⎪ =0 ⎪ ⎪ ⎪ Dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 ψint ∂ 2 ψint ∂ f02 ∂ ψint ⎪ ⎪ ( ) = Qint + + ⎪ 2 ⎪ ∂ y2 ∂ z N2 ∂ z ⎪ ⎨ ∂x ⎪ ∂ ψint ⎪ ⎪ =0 ⎪ ⎪ ∂ z z=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 ψs ∂ 2 ψs ∂ f02 ∂ ψs ⎪ ⎪ ⎪ )=0 + + ( 2 ⎪ ⎪ ∂ x2 ∂ y2 ∂z N ∂z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ψs ⎪ ⎪ ⎩ f0 = b|z=0 ∂z

(3)

z=0

D = ∂t ()+u · ∇() and the subscript s indicates sea-surface related quantities. The system of Equations (3) where Dt is actually known as coupled QG-SQG (Quasi Geostrophic and Surface Quasi Geostrophic) dynamics [14] and it allows one to take into account the contribution of surface buoyancy (notice that b = f0 ∂z ψ ) and internal sources of potential vorticities Qint for the determination of the three-dimensional motion. The total stream function resulting from Equation (3) will be ψ = ψint + ψs . Numerically, the stream function derivation is achieved through Fourier transform in the horizontal. Along the vertical coordinate, a 4th-order compact-difference solution is


285

used. Its expression in Fourier space is: [

∂2 − (k2 + l 2 )]ψˆ (k, l, z) = q(k, ˆ l, z) ∂ z2

(4)

for each horizontal wavevector (k, l). The assumption of a constant Brunt-Väisälä frequency N0 , and Coriolis parameter f0 , leads to the scaling of vertical coordinate z by the ratio N0 / f0 . The physical initialization consists of a three-dimensional isolated subsurface potential vorticity anomaly q(r, z) representing a deep isolated anticyclone: π (z − z0 ) π 2 )[( ) + 1]}, (5) q(r, z) = −q0 J0 (αR r){1 + cos( H αR H where J0 is the zero order Bessel function, z0 indicates the position of the isolated vortex along the vertical, q0 is the intensity of the potential vorticity, H is the vortex thickness and αR is the parameter through which the horizontal size of the vortex can be controlled. Note that, in order to get a bounded structure, q(r, z) is set to zero for all r verifying αR r > λ01 , with λ01 the first root of the zero order Bessel function. The resulting structure has the shape shown in Fig. 1. The motivation for choosing this profile is to study how the surface signature of the vortex will develop with time. In fact, if the quantity z0 + H/2 is smaller than the total depth of the considered oceanic basin, Equation 5 will assure an initial zero sea-surface elevation. The runs are performed in a flat-bottomed ocean in a f -plane configuration. The three-dimensional volume 2 in which the simulations are run, measures H L (as shown in Fig. 1) and is adimensionalized and taken to be equal to (2π )3 . H depends on the chosen aspect ratio, which in our case is equal to 0.7 (considering the N0 / f0 stretch along the vertical). The domain is doubly periodic. Simulations with the CLAM-3DQG model required the introduction of a deep current in order to observe the sea-surface response in a dynamic context (see Section 3.2 for further details) and a cartoon of this configuration is given in Fig. 2. The current consists of a single cosine-like vertical variation and has maximum intensity at the depth of the vortex core z0 (i.e. where the minimum PV is located). Its analytical expression is given by:

π U (z) = U0 cos[ (z − z0 )]Heav(z − z0 + 1)Heav(z0 − z + 1) 2 where U0 has been arbitrarily chosen as 20% of the vortex rotational speed, which is initially

(6) √

u2 + v2 .

Initial state for CLAM−3DQG model − Potential Vorticity (du) 6

0.5

Sea Surface

0

5

−0.5

z

4 H

3

−1 −1.5

2 1 0 −3

−2

L −2

−1

0 x

1

2

3

−2.5

Fig. 1 Potential Vorticity (dimensionless units) of a subsurface vortex for numerical simulation with CLAM-3DQG (Vertical Section).

286


Fig. 2 A translating subsurface vortex for numerical simulations with the CLAM-3DQG model. Translation is operated by means of the analytical function U(z).

Integration in time has been performed for several configurations. For each run, the parameters characterizing the subsurface vortex (q0 , αR , H and z0 ) have been changed in order to test the sea surface sensibility to their variation. In particular, surface stream function fields, directly proportional to sea-surface elevation, have been extracted and observed. 2.3

Primitive equation model

The ROMS model [23] is a primitive equation ocean model simulating the Reynolds-averaged Navier-Stokes equations with the assumptions of hydrostatic and Boussinesq approximation [21]. The model has been used in idealized configuration. Simulations have been run in a flat-bottomed east-west periodic oceanic basin in the β -plane approximation in absence of atmospheric forcing. This choice is justified by the interest in isolating the dynamics of the system given by the vortex plus the sea-surface. The oceanic basin under consideration can have two sizes: 600km × 600km × 4km is suitable to study deep anticyclones in the submesoscale range and with rotational speeds around 0.2 m/s, while an extension in the horizontal to 600 km ×1000 km is necessary for mesoscale and faster rotating anticyclones. The latter, in fact, experience a more efficient southwestern drift [25], and need a meridionally larger basin in order to study their evolution far from the southern wall of the domain. The domain resolution is 2 km in the horizontal and the vertical discretization is given by 80 non-uniformly spaced levels. A finer resolution has been chosen for levels closer to the sea-surface. The model is initialized by means of a background density stratification described by an analytical function, exponentially decaying with depth, whose parameters can account for the geography of the Rossby radius of deformation [26]. In our case, the Rossby radius is around 40 km, corresponding to vortices living at latitudes of about 20◦ N. The analytical expression for mean density profiles is given by Equation (7),

ρ (z) = ρ0 exp[−αρ z/H]

(7)

with αρ =0.24, H = 4 km (the vertical extension of the basin) and ρ0 =1027.5 kg m−3 . Deep anticyclones are introduced in such a rest stratification making use of cyclogeostrophic balance. This approach allows one to take into account the combined effects of Earth’s rotation, pressure gradient field and the spinning of the vortex (a non-negligible feature if anticyclones with Ro 1 are dealt with [7]). The density anomalies associated to the deep anticyclones have then been derived from the general form of the cyclogeostrophic balance, which, in polar coordinates, takes the form of Equation (8), v2θ 1 d p(r, z) + f0 vθ = , r ρ0 dr

(8)


287

Table 2 Table of parameters and derived quantities for initialization of ROMS experiments

vθ Initial Speed

CHOSEN PARAMETERS R H Radius Thickness

z0 Depth

DERIVED QUANTITIES p (r, z) ρ (r, z) Pressure Anomaly Density Anomaly

with r the distance from the center of the vortex, vθ the azimuthal velocity, f0 the Coriolis parameter and d p/dr the pressure gradient field. This relation, once chosen the velocity field, contains the information on the density structure associated to the vortex. The velocity field has been chosen in the form prescribed by Equation (9) and it represents an analytical approximation of a deep anticyclone: vθ =

−v0 r − r22 − (z−z20 )2 e R e H , R

(9)

where v0 is the intensity of the field (in rads−1 ), R its Radius, z0 its position along the vertical and H the thickness. Integrating Equation (8) between r and ∞, and, considering that only coherent vortices are treated in this work, the pressure field can be assumed to be zero far from the vortex center (namely at r = ∞) and one can write that: ˆ ∞ d p (r , z) dr = p (∞) − p (r, z) = −p (r, z), (10) dr r finally getting the Equation (11) for the pressure field: p (r, z) = β e−

2(z−z0 )2 H2

2r2

e− R2 − γ e−

(z−z0 )2 H2

r2

e− R2 ,

(11)

with:

β= γ=

ρ0V02 4

ρ0 f0V0 R 2

Derivation of Equation (11) with respect to the z variable yields the density anomaly associated to the vortex and the latter is superposed to rest stratification to get the total three-dimensional field. In other words, the hydrostatic equilibrium is being imposed and it prescribes density anomalies directly from pressure anomaly p (r, z): d p (r, z) = −ρ (r, z) (12) gdz where g is the gravity acceleration. Table 2 summarizes the possible choices of parameters and all the derived quantities. A panel showing typical initial configurations for a ROMS experience is presented. In particular, the initial state relative to a submesoscale anticyclone rotating at 0.2 m/s (maximum tangential speed) is shown. Initial states for other configurations will all resemble the one in Fig. 3, with different characteristics dependent on the parameters’ choice. We recall that the varying vortex parameters are the key to account for different types of oceanic deep anticyclones (see Table 1). As in the previous case, integration in time has been performed for several configurations. For each run, the parameters characterizing the deep anticyclone, shown in Table 2 have been varied. Differently from the CLAM-3DQG model, ROMS provides a direct physical information on the elevation of the sea-surface. Results will be shown in Section 4.

288

Daniele Ciani et al. / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 281–311 3 Density Field (Vertical Section) (kg/m )

−100

v (m/s) v (m/s)

1027 1026.5

−0.1−0.2 5

0.2 0.1

z (m)

1025.5

−0.1

0.15

−300

1026

−0.1

0.1

−200

1025

−400

1024.5 1024

−500

1023.5 1023

−600 100

150

200

250 x(km)

300

350

400

Fig. 3 Schematic of the Initial State for a ROMS experiment. Submesoscale anticyclone rotating at a maximum tangential speed of 0.2 m/s. Here, the sea surface is at z = 0.

3 Theoretical analysis Determination of the surface effects induced by deep vortices is analytically evaluated in a QG framework. We will distinguish the case of steady and translating anticyclones. Though the second ones represent a more frequent geophysical feature, steady anticyclones are suitable for analytical inversion of Equation 2. Despite this, qualitative analytical evaluations are possible in the dynamic case as well. 3.1

Surface influence of steady vortices

In recent years, Meddies have been studied with the point vortex model [24]. This model is useful and provides analytical solutions if the ocean is considered as infinite in all directions. Otherwise, boundary conditions have to be implemented and the method of image vortices can be used to satisfy them. In particular, the oceanatmosphere interface is a physical boundary, as is the ocean bottom (the lateral boundaries of the ocean are often remote enough to be neglected). For upper ocean vortices, the influence of the bottom can be overlooked in the deep ocean, except if special boundary conditions are imposed there (a case considered below). Therefore we consider here mostly the effect of the ocean-atmosphere interface. This interface can be neglected and then be considered as infinite if continuity of the stream function, of its vertical derivative and of the Brunt Väisälä frequency are assured. Since Brunt-Väisälä frequencies in the upper ocean and in the lower atmosphere are comparable ( [21]), the main consequence of this constraint is the existence of an atmospheric circulation induced by the subsurface vortex. This problem can be overcome noticing that the intensity of winds in the lower atmosphere ( O(ms−1 )) is much more intense than typical ocean surface circulations induced by interior vortices ( O(cms−1 )),( [8, 18]). Hence, the eventuality of such a circulation would be negligible compared to atmospheric motion. Taking inspiration from [24] two cases of subsurface potential vorticity anomalies are studied in order to derive the associated stream functions. In order to improve the aforementioned approach, the models rely on point vortex-like configurations with zero total potential vorticity in an infinite ocean, and will model subsurface isolated anticyclones, as shown in [27]. In a further analytical model, an interior vortex will be represented by a three-dimensional potential vorticity living in a vertically bounded ocean. This will allow to treat the problem in a more realistic way. For this extended vortex, the surface expression will also result from the choice of boundary conditions at the bottom of the ocean.


3.1.1

289

Point Vortices

A description of interior QG anticyclones [27], states that if Equation (13) is valid and Equation (1) is taken into account, deep anticyclones can be classified as rotational (R) or stretching (S) vortices. These vortices satisfy a condition of null integral of potential vorticity over the domain volume vT , ˚ Q(x, y, z)dvT = 0 (13) vT

If rotational effects dominate on stretching, the resulting structure will be called R-vortex and the potential vorticity anomaly will be in the form of an internal core surrounded by a ring of opposite sign. The S-vortex will be the reference model for the opposite case, and the potential vorticity anomaly will exhibit a tripolar vertical structure given by an inner core and opposite sign anomalies at the bottom and on top of it, as shown by Fig. 4. We now idealize these R and S configurations into equivalent point vortex distributions. Given a potential vorticity distribution, (14) Q(x, y, z) = Γδ (x)δ (y)δ (z) indicating a point vortex located in the origin of a reference frame, the corresponding stream function (in polar coordinates) is: ˚ ψPV (r) = G(r − r ) ∗ Γδ (r )d 3 r , (15) V

which, for

r

= 0 and G(r) = −1/(4π r), yields:

ψPV (r) = − where Γ =

´´´

v Qdv,

Γ , 4π r

(16)

and v would be given by the volume of a realistic vortex.

Fig. 4 a) R-Vortex and b) S-Vortex, adapted after [27].

R-Vortices A schematic point R-vortex is given in Fig. 5. A potential vorticity ring surrounds the central point vortex. We specify that: • the Cartesian to spherical coordinate transformation yields x = rcosφ cos λ , y = rcosφ sinλ and Z = rsinφ , where λ and φ are the azimuthal and zenithal angles;

290


• the radius r, considering the scaling of the vertical coordinate will be given by r = gives the distance from the center of the reference frame;

x2 + y2 + Z 2, and it

• the radius rθ indicates the distance from the center of the reference to the potential vorticity ring (whose radius is ar ) and it is given by rθ = (x − ar cos θ )2 + (y − ar sin θ )2 + Z 2 and using the spheric coordinates one yields rθ = r2 + a2r − 2ar r cos φ cos θ , with θ = θ − λ ;

Fig. 5 Schematics of the potential vorticity distribution given by a point vortex and a potential vorticity ring.

Using Equation 16 and performing a linear integration, the stream function associated to the potential vorticity ring becomes: ˆ

ψRING (r, φ , λ ) =

−λ +2π −λ

Γ Γ dθ = 2 2 8π rθ 8π σ

ˆ

−λ +2π

−λ

(1 − ζ cos θ )−1/2 d θ

(17)

with σ = r2 + a2r and ζ = 2rar cos φ /(r2 + a2r ). Since solutions for r > ar are searched, the expression of ζ justifies a Taylor expansion for the integrand. Furthermore, without loss of generality, the dependence on the azimuthal variable can be eliminated, as the integral is anyway calculated over a 2π interval. These considerations lead to the following expression for the RING stream function: ˆ 2π Γ 3 3 ζ ψRING (r, φ ) = 2 (1 + ζ 2 + cos θ + cos 2θ )d θ . (18) 8π σ 0 16 2 16 Considering that the integrand contains periodic functions and that the integral extrema are between 0 and 2π , all the terms multiplied by the cosine function do not give any contribution. Using the explicit expression for the ζ parameter, integrating over the φ angle and adding the contribution of the point vortex stream function one can get the stream function for the total configuration shown in Fig. 5, which is given by the following expression:

ψT OT = ψPV + ψRING = −

Γ Γ 3a2r r2 + [1 + ]. 4π r 4(r2 + a2r )1/2 8(r2 + a2r )2

(19)


291

Thus one can calculate the sea-surface elevation above such a vortex as:

η=

f0 ψT OT (x, y, z = H), g

(20)

where H indicates the distance between the vortex and the sea surface and f0 and g are the Coriolis parameter and gravity acceleration, respectively. S-Vortices In order to model the S-vortex-like structure, a vertical tripole of potential vorticity is used. We first calculate ψ for a vertical dipole (Fig. 6), which consists of two point vortices of charge Γ separated by a distance δ . If anticyclonic vortices are studied, Γ < 0, hence the positive and negative poles will be indicated by −Γ and Γ, respectively.

Fig. 6 Schematics of the potential vorticity distribution given by a vertical dipole of point vortices.

We will calculate the stream function at the sea surface, at a point P(x, y, z). We define • α , the angle between the vertical coordinate and the line connecting a point vortex to the point P; • r+/− , the distance between the two point vortices and the point P; • r, which for the chosen configuration (i.e. r >> δ ) will satisfy r r+/− The total stream function will be the superposition of the solutions for the two point vortices:

ψDipole = ψΓ+ + ψΓ− =

Γ 1 1 Γ r+ − r− Γ δ r cos α ( − )= ( )= ( ) 4π r− r+ 4π r+ r− 4π r3

(21)

Γδ Z ( 2 ) 2 4π (x + y + Z 2 )3/2

(22)

which, in a cartesian frame yields:

ψDipole = ψΓ+ + ψΓ− =

292


Fig. 7 Schematics of the potential vorticity distribution given by a vertical tripole of point vortices.

This framework makes the calculus for the vertical tripole completely analogous. We can consider that the tripole is the addition of two dipoles with a common central vortex, thus:

ψTripole = with:

Γ/2 1 Γ/2 1 Γ + ( )− ( ) 4π rd 4π r 4π ru

⎧ ⎨ r − ru = δ cos α , rd − r = r − ru , ⎩ rd − ru = 2δ cos α .

(23)

(24)

The total stream function, can be written as: Γ rru − 2rd ru + rrd ( ) 4π rd rru

(25)

Γ δ 2 r2 cos2 α ( 5 ), 4π r − δ 2 r(r cos α )2

(26)

Γ δ 2Z 2 ( 2 ). 4π (x + y2 + Z 2 )5/2 − δ 2 (x2 + y2 + Z 2 )1/2 Z 2

(27)

ψTripole = and, after some manipulations, one can get:

ψTripole = which, in cartesian coordinates becomes:

ψTripole =

In analogy with the R-vortex case, the QG expression for the induced sea-surface elevation will be:

η=

f0 ψTripole (x, y, z = H), g

(28)

where g and f0 are the same as for Equation (20). Summarizing, the models proposed in this section lead to analytical predictions on the stream function generated by deep vortices (indicated by their potential vorticity distributions). Knowledge of the stream function


293

allows one to determine the vortices sea-surface expression. In fact, in the quasi-geostrophic framework, the sea-surface elevation is related to the surface stream function of the considered oceanic basin. As shown in Section 4.1, the sea-surface elevation induced by a steady isolated deep anticyclone (living in an oceanic basin at rest) is always positive and monopolar. Equations (19) and (27) (for R and S-Vortices respectively) exhibit˝ respectively a 1/r and 1/r3 decay of the stream function from the vortex center and a linear dependence on Γ = v Qdv, related to the potential vorticity of the vortex. To avoid the divergence of ψ at the origin and of the vortex integral kinetic energy, expressed by Equation (29), results of this section are to be used in the r > 0 region. ˚ (∇ψ (x, y, z))2 dxdydz (29) E= V

3.1.2

Finite volume vortices

Another analytical model to estimate surface signatures of deep vortices is based on a finite extent of the potential vorticity anomaly characterizing the deep vortex. Furthermore, the vortex surface signature is studied in a vertically bounded ocean of depth H, allowing implementation of boundary conditions at the ocean bottom. The following, separable expression for the potential vorticity is used: 2

− r2

Q(r, z) = q0 e

[1 − (

Rv

z − z0 2 −( z−z0 )2 ) ]e H = q0 f (r)g(z) H

(30)

where Rv , H, z0 and q0 are the vortex radius, thickness, depth and potential vorticity, respectively. Here we have chosen a S-vortex, but note that R-vortices, as well as any other vortex described by a separable potential vorticity distribution could be studied. To invert potential vorticity into stream function we use projection of the g(z) on sines and cosines in the vertical, and of f (r) on Bessel functions in the horizontal. Under the assumption of constant stratification (N = N0 ), the dimensionless form of Equation (1) is: Q(R, Z) = where: R=

∂ψ ∂ 2ψ 1 ∂ (R )+ R ∂R ∂R ∂ Z2

(31)

r f0 π ∈ [0, ∞), N0 H

(32)

zπ ∈ [0, π ]. H

(33)

Z=

Under the assumption of separability for the stream function,

ψ (R, Z) =

∞

∑ [φn (R)cos(nZ) + χn (R) sin(nZ)],

(34)

n=0

and the potential vorticity can be expressed as: Q(R, Z) =

∞

∑ [Φn (R)cos(nZ) + Xn(R) sin(nZ)]

(35)

1 ∂ ∂2 − n2 ](φn (R), χn (R)) + 2 ∂R R ∂R

(36)

n=0

where: (Φn , Xn ) = [ The radial functions above are:

J0 (λ0m R),

m = 1, . . . , ∞

(37)

294


with λ0m the m-th root of the zero order Bessel function (J0 ), so that:: ∞

Q(R, Z) =

∞

∑ ∑ J0(λ0m R)[Anm cos(nZ) + Bnm sin(nZ)],

(38)

n=0 m=1

ψ (R, Z) =

∞

∞

J0 (λ0m R) [Anm cos(nZ) + Bnm sin(nZ)] . 2 2 n=0 m=1 −(λ0m + n )

∑∑

(39)

The Anm and Bnm coefficients, using orthogonality properties of sines, cosines and Bessel functions, take the form: ˆ 0 ˆ 1 [cos(nZ), sin(nZ)]g(Z)dZ R f (R)J0 (λ0m R)dR (40) (Anm , Bnm ) = Θ −π

0

with: Θ= g(Z) = q0 [1 − (

4

(41)

π [J1 (λ0m )]2

H 2 Z − Z0 2 −[ H ( Z−Z0 )]2 ) ]e π H , ) ( π H

and −

f (R) = e

N0 HR π f0 Rv

(42)

2

.

(43)

Implementation of the boundary conditions is now possible and it results from the arbitrary choice of having no currents and/or no temperature anomalies at the ocean bottom. This can be achieved introducing additional solutions to Equation (39).

Fig. 8 Image vortex.

• The absence of currents at the ocean bottom can be assured using the method of image vortex. Using this approach, an additional “mirror” vortex (whose potential vorticity is opposite with respect to the one in Equation (30)) is supposed to exist beyond the ocean bottom. A schematics of this particular configuration is provided in Fig. 8. Since the inversion of potential vorticity is completely analogous to the one we


295

already performed in this section and the vertical structure of Q(r, z) is symmetrical for the transformation z0 → −z0 (see Equation (30)), the resulting stream function at the ocean bottom will be equal to zero. In fact, at the interface between the real basin (the one for z > 0) and the ”mirror” basin (the one for z < 0) we will be summing two identical functions of opposite sign. The problem of this additional solution is that, even though the absence of current is assured, this is not the case for the temperature anomaly. In fact, being Equation (39) given by a sum of sines and cosines, its derivative with respect to the Z coordinate will not be zero at the ocean bottom. In order to assure this, a further additional solution is required; • the vertical derivative of the stream function at the ocean bottom will assume the zero value if an additional stream function ψ (R, Z) is added to the one described by Equation (39). This solution, as shown by Equation (44), originates from a zero potential vorticity distribution, f 2 ∂ 2ψ ∂ ψ 1 ∂ (R ) + 02 =0 R ∂R ∂R N0 ∂ Z 2

(44)

Assuming separability of the stream function with respect to the radial and vertical coordinates, we can write: ψ (R, Z) = F(R)G(Z) (45) Substitution of Equation (45) into (44) lets us conclude that the total stream function can be expressed as the product of a Bessel function and of an exponential in the radial and vertical coordinate respectively, as expressed by Equation (46): ∞

ψ (R, Z) =

∑ CpJ0 (λ0pR)e−λ

0p Z

.

(46)

p=1

The determination of the C p coefficients will yield from the following expression: ∞ ∞ ∞ ∂ (ψ + ψ ) J0 (λ0m R)Bnm =0 = ∑ −C p λ0p J0 (λ0p R) − ∑ ∑ 2 2 ∂Z Z=0 p=1 n=0 m=1 (λ0m + n )

(47)

To conclude, the system of Equations (46), (47), and (39), (40) allows one to accomplish inversion of the potential vorticity Q(R, Z) (see Equation (38)) with the requirements of specific boundary conditions at the ocean bottom. Values of the stream function can thus be estimated at a distance Z = H from the ocean bottom. These quantities will represent the sea-level anomaly induced by the deep isolated anticyclone. 3.2

Surface influence of translating vortices

Predictions on the development of the surface signature of a deep vortex are possible in a dynamical framework, i.e., in presence of a vortex drifting in an oceanic basin. One of the main mechanisms causing vortex-drift is known as self-advection and it results from the variation of the Coriolis parameter with latitude [24], [25]. Advection currents are also responsible of vortex displacement. In general, a drifting vortex is a more realistic schematization of oceanic vortices compared to the steady case. The theoretical estimates of the dynamical signature will rely on QG theory and on the conservation of potential vorticity, which is a realistic assumption during the early stages of the vortex evolution. In the case of a layered ocean (as in Fig. 9 and 10), if the layer above the eddy (considered as a localized volume of potential vorticity anomaly) is homogeneous, its potential vorticity (PV hereafter) can be given by its shallow-water expression [21]: Q= The Shallow-water PV anomaly is:

ζ + f0 . h

(48)

f0 ζ + f0 − ∗, (49) h H where h is the height of a local column and H ∗ the height of the unperturbed one. Then two cases can occur:

δQ =

296


1. There is initially a signature above the drifting eddy (a steady signature in terms of sea-surface elevation for instance). The upper layer fluid can have zero potential vorticity anomaly (everywhere horizontally) and in that case, the column above the eddy will have a smaller height (h H1 − ΔH, since ηs 0. The fluid column which climbs up the eddy will have negative stretching and therefore negative vorticity, leading to: f0 f 0 + ζ0 , (57) Q2 = ∗ = ∗ H H − ΔH with ζ0 < 0. Q1 =

Therefore, from zero steady signature, a dipolar dynamical signature develops. A bump in sea-surface elevation will appear in the leading half of the eddy and a trough will appear in the trailing one. Note nevertheless that this depends strongly on the initial potential vorticity distribution. Fig. 10 shows schematically the explained dynamics and the resulting (final) signature, which, hereafter will be referred to as dipolar signature.


297

Fig. 9 Dynamics of a monopolar signature. Case of a pre-existing steady signature.

Fig. 10 Dynamics of a dipolar signature. ηs represents the final sea-surface elevation.

In this section, the surface signature of deep vortices has been investigated taking advantage of analytical models relying on the QG framework. The characterization has been presented in both a steady and dynamic case, referred to a steady and drifting deep vortex respectively. Considering a vortex three-dimensional potential vorticity distribution, the stream function (a quantity related to sea-surface elevation and sea surface temperature in the ocean) can be obtained in idealized configurations. Steady deep anticyclones, and vortices in general, can be modeled by means of point vortices, a schematization allowing one to consider their realistic potential vorticity structure and to translate it in equivalent idealized distributions. Such point vortex distributions allow the use of Green’s functions for the determination of the associated stream function. Hence, an analytical form of the surface effects related to a deep anticyclone has been obtained for realistic R-vortices and S-vortices [27] and it is shown by Equations (19) and (27). In both cases the sea-surface elevation depends linearly on the vortex potential vorticity q and the thickness H, quadratically on the vortex radius R (these parameters are included in the quantity Γ = qHR2 ) and it decreases with the vortex depth.

298


The determination of the surface expression of a deep anticyclone can even be accomplished if finite volume vortices, instead of point vortices, are considered. This implies a decomposition of the three-dimensional potential vorticity structures on cosines and sines in the vertical and on Bessel functions along the radial coordinate. A QG framework, provided conservation of potential vorticity in a layered ocean, can also allow analytical qualitative predictions on the shape of the surface signature of a translating deep anticyclone. Two cases have been treated: the one in which a pre-existing steady signature was present and the one in which this feature was zero. In the first case, as the deep anticyclone translates in an oceanic basin, it carries the initial signature along its path (at the sea surface). Its signature is monopolar and constitutes a positive anomaly in the SSH field. In the second case, a dipolar signature develops, and it translates with the deep anticyclone as well. Since we want to investigate the surface expression of vortices with initial zero surface signature, the dynamics of the dipolar signature will be studied numerically in Section 4. 4 Numerical results 4.1

Surface signature of deep anticyclones: steady case

In a numerical model, deep anticyclones in an oceanic basin at rest have a sea-surface signature which can be calculated. Here, the numerical model ROMS will be used. We choose the analytical velocity field (Equation 9) and, using cyclogeostrophic balance (Equation 8), we get the analytical density distribution. Furthermore, we require the velocity field to be identically zero at the ocean bottom, which is a realistic assumption for intrathermocline eddies (as the one shown in Fig. 3). The estimation of the sea-surface elevation induced by the deep anticyclone depends on its density structure and is expressed by Equation 58. ˆ 0 ρd η (r) = dz, (58) H ρs where ρd is the density anomaly associated to the deep anticyclone, ρs is the background density structure and H indicates the ocean depth (Fig. 11). The steady signature of a deep anticyclone in SSH fields, in absence of bottom velocities, has always the form of a monopolar positive anomaly superimposed on the flat sea surface (justifying, in this context, the equivalence between sea-surface elevation and sea-level anomaly, which will be used hereafter); it results from the three-dimensional density structure of the anticyclone. In an idealized f -plane configuration and in the absence of external forcing, a coherent isolated vortex will remain invariant, so will its surface signature. A parametric investigation of the sea-level anomaly induced by a subsurface vortex is achieved. The properties of this feature are summarized by Fig. 12, indicating: • inversely quadratic dependence with respect to the vortex depth z0 ; • linear dependence with respect to the vortex radius, as shown by the upper-right panel of Fig. 12); • linear dependence with respect to the vortex thickness. • linear dependence with respect to the vortex initial angular velocity, which, according to Equation (1) can be related to the vortex potential vorticity ; Furthermore, if anticyclones with velocity fields around 0.2 m/s are studied (this is a typical velocity for deep anticyclones, as suggested by Table 1) the minimum requirements for the sea-level anomaly to be O (cm) are: 1. a mesoscale range for the deep anticyclone, i.e., when the vortex diameter is slightly larger than the Rossby radius of deformation (40 km at a latitude of 20◦ N) ; 2. provided condition 1, the ratio between the vortex depth and thickness has to be around 0.9;


299

Fig. 11 Steady monopolar signature of a subsurface anticyclone in the mesoscale range. The vortex parameters are illustrated in Table 2 SLA prescribed by f−plane ROMS modeling

SLA prescribed by f−plane ROMS modeling

3

2

Thickness=400m Radius=40km −1 Rotation=0.79 rads

2.5

1.8

Depth=400m Thickness=400m −1 Rotation=0.79 rads

1.6 SLA (cm)

SLA (cm)

2 1.5 1

1.4 1.2 1 0.8

0.5 0 200

0.6 400

600 Depth (m)

800

0.4 15

1000


20

25 30 Radius (km)

35

40


2

6.5 Depth=400m Radius=40km −1 Rotation=0.79 rads

6

Depth=400m Thickness=300m Radius=40km

5.5

SLA (cm)

SLA (mm)

1.5

5 4.5

1

4 0.5 300

350

400 Thickness (m)

450

500

3.5 0.5

0.6

0.7 0.8 Rotation (rads−1)

0.9

1

Fig. 12 Parametric study of the maximum sea-level anomaly (SLA) induced by a deep anticyclone. Dependences are studied as a function of the vortex depth (upper-left), thickness (upper-right), radius (lower-left) and initial angular velocity (lower-right).

A threshold in the centimeter range is appropriate for present-day and future satellite altimetry (in the future, the measurement precision of satellite-derived altimetric maps will go down to 2 cm on a global scale [19], presently, this is achieved only for a fraction of the world ocean [20]); this could allow the detection of an oceanic vortex from space, assuming nevertheless the idealized requirement of an ocean at rest. 4.2

Comparison with Point vortex model

A comparison between the point vortex theory and a realistic vortex is presented here. The potential vorticity structure (Ertel PV anomaly [21]) of the analytically modeled vortex (in the static case), is similar to the one

300


shown in Fig. 13-a. The comparison is then operated by means of Equation (19), that models a deep vortex as a point-vortex surrounded by a potential vorticity ring. A preliminary analysis suggests a scaling of the quantities in use, i.e., the Ertel PV anomaly (PVEa) and the stream function ψ . The PVEa has dimensions of (m−1 s−1 ), while the one appearing in Equation (19) (via the Γ quantity) relies on quasi-geostrophic theory, in which the PV (PVQG in this context) has dimensions of s−1 . The scale factor between the two quantities is N2 /g, leading to the following relation PV QG PV Ea

g . N2

(59)

We also recall that in the QG framework, the sea-surface elevation is related to the surface stream function ψs via Equation (20) . In particular, the maximum sea-level anomalies induced by the point-vortex structure (Fig. 5) have been observed as a function of vortex depth and initial potential vorticity. In fact, parameters such as thickness and radius cannot be used in a point-vortex context, where the key quantity is the integral of the PV anomaly over the volume occupied by the vortex motion. We study the case of a vortex with the following parameters: radius=40 km, thickness=300 m (and 400 m), depths between 200 and 1000 m and angular velocities between 0.5 and 1.06rads−1 (about 0.2 to 0.4 ms−1 ), shown in Fig. 13. The analysis confirms that the sea-level anomaly maintains a monopolar structure and that its maximum intensities, provided some restrictions, respect the values observed for realistic vortices. Fig. 13-c-d indicate that the restrictions for the theory to be used are the following: • Velocity restriction: the point-vortex-like structure has to be considered valid in all cases in which the corresponding real vortex is well described by the quasi-geostrophic (QG) theory, i.e., the Rossby number Ro has to be lower than unity. This will assure applicability of Equation (19), that relies on QG framework and that has been used for validation. Fig. 13-c clearly shows how, for a fixed radius and thickness, an increase in initial angular velocity (i.e., relative vorticity) corresponds to an increase in Rossby number, taking the vortex far from geostrophy [7] and making the theoretical description divergent from the realistic case; • Depth restriction: the point-vortex-like structure is an appropriate approximation of a vortex when the corresponding realistic structure is fully immersed in the ocean. (i.e. PV anomaly isosurfaces do not intersect the sea surface). If this condition is not verified, the point vortex theory underestimates the sea-level anomalies above a deep vortex; 4.3

Surface signature of deep anticyclones: dynamic case, CLAM-3DQG model

We show the development of a dipolar signature generated by a deep anticyclone drifting by means of the deep current expressed by Equation (6). The QG simulations are run in a doubly periodic oceanic basin. We work in rescaled coordinates, considering to deal with a basin measuring 600 km × 600 km × 2 km in the zonal, meridional and vertical directions respectively and in a f -plane configuration. We choose to initialize the deep anticyclone using Equation (5), assuring the study of a deep anticyclone whose initial surface signature is zero. This operation also allows one to observe the sea-surface response in a parametric way, for instance, variating the anticyclone characteristics introduced in Section 2.2 (q0 , αR , H,z0 ). The reference simulation consists of a deep anticyclone in the submesoscale range, having radius R=15 km, thickness H=100 m and depth z0 =300 m. Every parameter of the vortex is then incremented of a varying amount (from 10 to 100% of the original value and up to 160% for the radius) and the maximum stream function at the sea surface (a proxy of sea-level anomaly in this context) is extracted and observed. The case of a mesoscale vortex (whose radius R is R = R + 0.7R) is presented in Fig. 14; it shows four snapshots of the surface stream function exhibiting the sea-level bumps and troughs induced by the drifting deep anticyclone.

Daniele Ciani et al. / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 281–311 Ertel PV anomaly (1e11) Vertical Structure

−1 −1

Theoretical Surface Signature of a R−Vortex

(m s ) 1

−500

301

0

−1000 −1 −2

−2000 −2500

−3

−3000

−4

−3500 a 200

250

300 x (km)

350

y

z (m)

−1500

b

−5

400

x

1.6

3 REALISTIC VORTEX POINT VORTEX

1.4

Depth=400m Thickness=300m Radius=40km

−1

2

1 0.8

Ro = 0.7

1.5 1

0.6 0.4

Rotation=0.79rads Thickness=400m Radius=40km

Ro = 1

SLA (cm)

SLA (cm)

1.2

REALISTIC VORTEX POINT VORTEX

2.5

Ro = 0.4

0.5 d

c

0.2 0.5

0.6

0.7 0.8 0.9 Initial Angular Velocity (rad s−1)

1

1.1

0 200

300

400

500

600 700 Depth (m)

800

900

1000

Fig. 13 Parametric study of the sea-level anomaly (SLA) induced by a deep anticyclone. Dependence is studied as a function of the vortex initial angular velocity (c) and depth (d). The monopolar sea-level anomaly is shown in b.

The pattern of the surface signature, in agreement with theoretical predictions, is dipolar and grows in intensity and dimension for increasing times of integration. The positive sea-level anomaly, considering an eastward drift (green arrow), lays in the leading half of the vortex, whose projection at the sea surface is indicated by the green point. The parametric study shows that surface stream function exhibits: • quadratic dependence with respect to the vortex radius; • linear dependence with respect to the vortex initial potential vorticity; • inversely quadratic dependence with respect to the vortex depth z0 ; • linear dependence with respect to the vortex thickness. These results are shown in Fig. 15 and, for every run, refer to the maximum value of the surface stream function observed in the late stages of the simulation (t > 70, over 80 time steps). 4.4

Surface signature of deep anticyclones: dynamic case, ROMS model

To characterize the surface signature of deep anticyclones in a more realistic context, ROMS experiments have been run in the β − plane. The surface expressions of intrathermocline vortices are analyzed and the vortex parameters were varied to account for different oceanic eddies (see Table 1). All the simulations have been initialized with zero steady altimetric signature of the vortices. The reference experiment consists of the evolution of a deep anticyclone in the submesoscale range recently observed in the Arabian Sea [28]. Its characteristics have been reproduced in an idealized context and are shown in Fig. 3. During time integration, the vortex moves

302


time=2/80

Surface Stream Function anomaly

time=20/80 0.4

y

y

0.3

0.2

0.1 x

x VORTEX POSITION AT DEPTH

time=50/80

DRIFT

time=80/80

0

−0.1

y

y

−0.2

−0.3

x

x

−0.4

Fig. 14 Dynamics of a dipolar signature for a mesoscale deep anticyclone (CLAM-3DQG model). X and Y are the zonal and meridional boundaries of the domain. All the quantities are expressed in dimensionless units (du).

southwestward in the basin as predicted by planetary β -effect [25] (Fig. 16 shows the vortex displacement over 300 days). Introduction of planetary β -effect makes the ROMS simulations substantially different from the QG case (CLAM-3DQG model). Here, the displacement of the vortex is not due to a deep current but to selfadvection ( [24], [25]). In this case, the vortex is characterized by a radius of 15 km , a thickness of 100 m , an initial angular velocity field of 0.53 rad s−1 (corresponding to a maximum linear velocity of 0.2 ms−1 ) and a depth of 300 m . The simulated sea-level anomaly is shown in Fig. 17. The maximum signal is reached after about 200 days and, considering the dipolar structure (lower-right snapshot), the difference between positive and negative anomalies (ΔSSH hereafter) does not go beyond the 3 mm value. The maximum horizontal extension of the dipolar signature (after the vortex has been drifting for about one month) is almost 6 times the vortex diameter. In addition, one can recognize the structure predicted by the simple model illustrated in Section 3.2. Furthermore, the initial dipolar signal, triggers a Rossby wave (allowed in β -plane approximation) visible in the trailing half of the anticyclone surface projection. 4.5

Parametric study of dynamic altimetric signatures

The variety of deep anticyclones justifies a parametric study of the dipolar signature (see the parameters of Table 2). Simulations have been run with the prescription of zero sea level anomaly at time t = 0. To provide a representative study, the sea-surface response is analyzed with respect to vortex radius, thickness and depth for three different angular velocity regimes. The chosen velocities are 0.53, 0.79 and 1.06rads−1 (0.2, 0.3 and 0.4 m/s in terms of maximum rotational speed). Particular attention was given to the maximum intensity of the signature and on the time needed to reach the maximum observed value.


a

Surface Stream Function (1e3) (du)

ψ ∝ R2

4

1.4

3

1.2

2

1

1

0.8

0

50 100 150 Radius Increment (%)

0.8

c

ψ ∝ z−2 0

0

ψ∝q

303

b

50 100 Potential Vorticity Increment (%)

1.6

ψ∝H

d

1.4 0.6 1.2 0.4

1 0.8

0.2 0

50 Depth Increment (%)

100

0

50 Thickness Increment (%)

100

Fig. 15 Parametric study on the surface stream function induced by a deep anticyclone (CLAM-3DQG model). a) Dependence on the vortex Radius, b) Dependence on initial potential vorticity, c) Dependence on vortex depth and d) Dependence on vortex thickness. The Stream Function is in dimensionless units and is presented as a function of the parameter increment with respect to the reference simulation. Vortex Displacement

(days)

500 300 450 250

y (km)

400

200

350

150

300

100

250 200 200

50

250

300

350 x (km)

400

450

500

Fig. 16 Displacement of a submesoscale deep anticyclone under planetary β effect (v = 0.2 m/s, R = 15 km, H = 100 m, z0 = 300 m).

4.5.1

Dependence on the vortex radius

Fig. 18 shows the dependence on vortex radius. In the first line, ΔSSH values are computed as the difference between the maximum positive and negative sea-level anomalies in the whole basin; note that the spatial distribution of the signal is similar to the one shown in Fig. 17. The vertical lines in the plots indicate the time step after which the simulation is not physically acceptable, as the vortex crashes against the southern wall of the domain. The interpretation of the results is though possible, because the maximum sea level anomaly always occurs before the vortex crash. The second and third lines show the dependence of the maximum surface signa-

304

Daniele Ciani et al. / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 281–311 Sea Level Anomaly 20 days

50 days

500

500

400

400

300

300

200

200

100

100

(mm) 1.5

1

100

200

300

400

500

600

0

100

200

300

400

500

600

y (km)

0

0.5

0 100 days

200 days

500

500

400

400

300

300

200

200

100

100

0

100

200

300

400

500

600

−0.5

−1

0

100

200

300

400

500

600

−1.5

x (km)

Fig. 17 Sea-level anomaly induced by the displacement of a deep anticyclone in the submesoscale range (v = 0.2 m/s, R = 15 km, H = 100 m, z0 = 300 m). The green spot represents the surface projection of the vortex position at depth.

ture on vortex radius and the time necessary for the maximum signature to occur. The intensity of the altimetric signature has a clear proportionality to the vortex radius, growing from a few millimeters up to more than 2cm for a radius of 40km; the time necessary for the development of the signature reduces almost 4 times from the submesoscale up to the mesoscale range (considering the 15 − 40 km radii range in our case), namely from 400 down to 100 days. Another effect to note is the impact of the vortex radius on the efficiency of the β -drift. Larger radii cause a more rapid southwestward displacement [29], and thus an earlier crash of the vortex against the southern wall of the domain. 4.5.2

Dependence on the vortex thickness

The dipolar signature dependence on thickness shows similar trends with respect to the previous case. Though the signal variation looks very slight, an increase with increasing thickness is observed and the correspondent times of development of the maximum signature becomes smaller for an increment of the parameter. The only exception to this behavior is constituted by the lower-left panel of Fig. 19. The very slight differences between the signals (considering one velocity range at a time) can be principally due to two factors, such as the limited choice of the parameters and the development of the Rossby wave train. The second factor is actually imposed by the physics, while the first expresses the requirements for initialization of the simulations. In fact, not only initialization requires initial sea-level anomalies to be zero, but it needs vortices to be statically stable as well (i.e., the vertical density profile across them must be a monotonic function of the z coordinate). This imposes a restriction on the parameters’ choice. Regarding this series of experiments, the maximum intensity of the dipolar signature (defined as ΔSSH) never crosses the 1.5 cm threshold.


305

Depth=700m − Uθ=0.53s−1 − Thick.= 350m Depth=700m − Uθ=0.79s−1 − Thick.= 350m Depth=700m − Uθ=1.06s−1 − Thick.= 350m

days

400

20 10 0 0

600

200

days

400

30

10

25

25

20

20

20

10 5 0 10

500 400

ΔSSHmax vs Radius 20 30 Vortex Radius (km)

100 0 10

5

400 Day

200

10

500

τ(ΔSSHmax) vs Radius

300

15

0 10

40

ΔSSH (mm)

25

15


300 200 100

20 30 Vortex Radius (km)

40

0 10

days

400

600

10 5

500


200

15

0 10

40

R=15km R=30km R=40km

20

0 0

600

400 Day

ΔSSH (mm)

200


ΔSSH (mm)

10

30

ΔSSH (mm)

20

0 0

Day


ΔSSH (mm)

ΔSSH (mm)

30


40


300 200 100


40

0 10


40

Fig. 18 Dependence of the dipolar signature on vortex radius for three different angular velocity regimes. Line 1: ΔSSH vs time, Line 2: ΔSSHmax vs radius, Line 3: τ (ΔSSHmax ) vs radius.

4.5.3

Dependence on the vortex depth

Dependence of the dipolar signature on vortex depth exhibits a less clear behavior in terms of the response intensity. The results seem to be in contrast with theory, as one gets an increase of the signal (though very slight) for increasing depths, even though the trend is not marked. This happened even if we let vary the depth parameter of almost 100% with respect to its initial value (from 500 to 900 m). Effects of the stratification are probably to investigate, as for example a non substantial change in Brunt-Väisälä frequency in this depth range. The time of development of the signature grows up significantly with depth, exhibiting an increase around 100% in the chosen depth range. 4.5.4

Dependence on Vortex Rotational Velocity

Among the whole set of simulations, we choose the ones exhibiting the maximum values in terms of ΔSSH and characterize the dependence of the signal with respect to the vortex initial rotational velocity. Fig. 21 shows that the altimetric signature exhibits a linear dependence on the vortex angular velocity. Furthermore, for increasing velocities (i.e., relative vorticities), a decreasing time for the onset of maximum surface signature is observed. In this section, the steady signature of deep anticyclones has been investigated numerically by means of a QG model (CLAM-3DQG) and of a primitive equations model (ROMS) in a steady and a dynamic case. In the steady case, we could confirm that the surface signature is a monopolar positive anomaly in sea-surface elevation. Furthermore, a parametric study on the vortex parameters suggested that this feature depends linearly on the vortex radius, thickness and rotational velocity, while it decreases with the vortex square depth.


ΔSSH (mm)

5 0 0 12

ΔSSH (mm)

Thick=250m Thick=350m Thick=450m

200

days

400

Thick=250m Thick=350m Thick=450m

5

200

days

400

10 8

10 5 0 0

600

12

ΔSSHmax vs Thickness

10 8

300 350 400 Vortex Thickness (m)

6 250

450

Thick=250m Thick=350m Thick=450m 200

days

400

600

12 10 8

ΔSSHmax vs Thickness 6 250

θ

15

10

0 0

600

Depth=900m − U =1.06s−1 − R=25km

θ

15

ΔSSH (mm)

ΔSSH (mm)

15 10

Depth=900m − U =0.79s−1 − R=25km

θ

ΔSSH (mm)

Depth=900m − U =0.53s−1 − R=25km

ΔSSH (mm)

306


ΔSSHmax vs Thickness 6 250

450

400

400

400

350

350

350

300 350 400 Vortex Thickness (m) τ(ΔSSH

450

) vs Thickness

300 250 200 250

τ(ΔSSHmax) vs Thickness 300 350 400 Vortex Thickness (m)

450

Day

Day

Day

max

300 250 200 250

τ(ΔSSHmax) vs Thickness 300 350 400 Vortex Thickness (m)

450

300 250 200 250


450

Fig. 19 Dependence of the dipolar signature on vortex thickness for three different angular velocity regimes. Line 1: ΔSSH vs time, Line 2: ΔSSHmax vs thickness, Line 3: τ (ΔSSHmax ) vs thickness.

This behavior can also be compared to the predictions of analytical point vortices, if structures with low Rossby numbers are dealt with and if their potential vorticity structure do not intersect the ocean surface. The mechanism of development of a dipolar surface signature for a deep anticyclone has been confirmed by both the CLAM-3DQG and the ROMS model. For ROMS, the limit of short times has to be considered, because, the vortex displacement in the oceanic basin (in β -plane approximation) allows the formation of a Rossby wave train at the sea surface. The parametric dependence of the dipolar signature on the deep anticyclone parameters is different in a QG (CLAM-3DQG) and Primitive Equations (ROMS) context. In the first case, the modeled anticyclone, living in a f -plane configuration and advected by a deep current, maintains a dipolar signature all along the simulation and its intensity depends linearly on the vortex potential vorticity and thickness, quadratically on the vortex radius and it diminishes with the vortex square depth. In the second case, the dependence is linear on the vortex radius, thickness and rotational velocity while no significant trend can be observed as a function of the vortex depth. We believe that the formation of the Rossby wave train and effects of the ocean background stratification can be responsible of this different behavior, but further analyses are necessary to confirm this.


10 5

ΔSSH(mm)

12

200

days

400

15

Depth=500m Depth=700m Depth=900m

10 5 0 0

600

200

days

Thick.=250m − Uθ=1.06s−1 − R=25km

20

400

12

ΔSSHmax vs Depth

10 8

15 10

10 8

600 700 800 Vortex Depth (mm)

6 500

900

600 700 800 Vortex Depth (m)

900


900

Day

300

Day

300

Day

300

200

200

τ(ΔSSHmax) vs Depth 900

600

ΔSSHmax vs Depth 6 500 400


400

8

400

100 500

days

10

400

200

200

12

ΔSSHmax vs Depth 6 500


5 0 0

600

ΔSSH(mm)

0 0

20 ΔSSH (mm)

15


ΔSSH(mm)

ΔSSH (mm)

20

Thick.=250m − Uθ=0.79s−1 − R=25km

ΔSSH (mm)

Thick.=2 50m − Uθ=0.53s−1 − R=25km

307

τ(ΔSSHmax) vs Depth 100 500


τ(ΔSSHmax) vs Depth

900

100 500


900

Fig. 20 Dependence of the dipolar signature on vortex depth for three different angular velocity regimes. Line 1: ΔSSH vs time, Line 2: ΔSSHmax vs depth, Line 3: τ (ΔSSHmax ) vs depth.

2.4 2.2

320

R=25km, Depth=500m, Thick=250m R=25km, Depth=900m, Thick.=450m R=40km, Depth=700m, Thick.=350m

300

2

280

τ(ΔSSHmax) (days)

ΔSSHmax (cm)

1.8 1.6 1.4 1.2 1

260 240 220 200 180

0.8

160

0.6 0.4 0.2

R=25km, Depth=500m, Thick=250m R=25km, Depth=900m, Thick.=450m R=40km, Depth=700m, Thick.=350m

0.25 0.3 0.35 Initial rotational velocity (m/s)

0.4

140 0.2

0.25 0.3 0.35 Initial rotational velocity (m/s)

0.4

Fig. 21 Dependence of the altimetric signature on vortex angular velocity (left). Time for onset of maximum ΔSSH signal (right).

308


5 Discussion, conclusions and openings In this paper, the signature of deep anticyclones at the sea-surface has been investigated by means of analytical and numerical models. The impact of such deep structures on the sea-surface is a key element for their detection by means of satellite altimetry, which provides global and synoptic information on the world ocean. The capabilities of satellite altimetry will improve in years to come with the SWOT mission [19]. This provides global altimetric measurements of the sea-surface at submesoscale horizontal resolutions ( 15 km) and with accuracy better than 2 cm (which is presently achieved only locally). The detection of deep vortices is an important task in oceanography. In fact deep vortices, together with more easily detectable structures (i.e. surface-intensified vortices), contribute to the global oceanic energy, heat and salinity budget. In recent years, many studies have investigated the surface signature of subsurface anticyclones in realistic and idealized contexts ( [16], [17], [24]) and concluded that, if deep anticyclones are dealt with, the expected influence on the sea surface results in a positive altimetric anomaly, mostly due to their three-dimensional density structure. A recent theoretical approach, based on QG theory and point vortex model, stated that the surface altimetric signature of a Meddy (a subsurface anticyclone) depends linearly on the vortex thickness and potential vorticity, quadratically on its radius and linearly on the inverse of the vortex depth [24]. In this work, analytical models based on the QG framework, as well as numerical experiments in the QG and Primitive Equation context, have allowed us to characterize the surface signature of deep anticyclones, leading to the following main results: • If analytical point vortex models are used in a realistic context, i.e., structures like the R-vortices or Svortices ( [27]) are taken into account, the sea-level anomaly above deep steady anticyclones still depends linearly on the vortex potential vorticity q, the thickness H and quadratically on the vortex radius R. In fact, these parameters are included in the quantity Γ = qHR2 of Equations (19) and (27). Despite this, contrarily to recent theoretical results [24], we found that the surface signature can depend on the inverse of the cubic depth. This is due to a screening effect of the opposite sign potential vorticity anomalies surrounding the central point vortex, which cannot be taken into account if a single point vortex is used. Furthermore, Section 4.2 showed that the point vortex models we proposed are in agreement with the results obtained using realistic vortices. This is true if the anticyclone potential vorticity isosurfaces do not intersect the sea-surface (i.e., if the vortex motion is fully immersed in the ocean) and if the vortex can be described by the QG approximation (Rossby number 0; H(w) = 0, w < 0), the source can be written as M = c p (γa − γm )wH(w),

Q = α (γa − γm )wH(w),

(3)

where the moist adiabatic gradient γm is given by expression γm = γa + (L/c p )(ds/dz), L is the specific heat of condensation and s is the mass fraction of saturated vapor (a function of temperature and pressure) [23-25]. As mentioned above, representation (3) implies that water vapor condensation occurs in ascending air masses, while all the condensate precipitates as rain. Here in the considered case of nonprecipitation convection for condensation heat source we have: Q = α (γa − γm )w,

|x| ≤ x∗ ;

Q = α (γa − γm )wH(w),

|x| > x∗ .

(4)

The physical meaning of such heat source presentation was explained in the Introduction. The moist adiabatic gradient γm depends in general on the temperature and pressure of the equilibrium state. This dependence is neglected in this paper, and γm is treated as constant. System (1) with heat sources (3) or (4) belongs to the class of systems with non-analytic (piecewise-linear or “jumping”) nonlinearities. This nonlinearity does not allow linearization, even in principle. The presence of a nonlinear source is a distinctive mathematical feature of this problem. As it was mentioned above, hereafter the corresponding problem will be considered in the simplest twodimensional arrangement when there is no dependence upon one of the horizontal coordinates (coordinate y) and the analogues of localized atmospheric convective vortices are localized convective rolls or localized systems of convective rolls. Besides that we seek for a solution of problem (1), (2), (4) in the quasistatic approximation. In this approximation, the vertical projection of the momentum equation reduces to the hydrostatic balance pz = gθ [23-25]. We note that the quasistatic approximation does not interfere with the main mathematical aspect of the problem related to the source nonlinearity. This approximation is also frequently adopted in atmospheric numerical models; it is valid under conditions of strong anisotropic exchange μ ν . At last for the sake of simplicity we will consider the problem without rotation assuming that f = 0. While solving the formulated problem we will follow the work [10] where it is possible to find all necessary details. The system (1), (4) with boundary conditions (2) allows a separation of variables: both the solution and the source can be expanded in series in eigenfunctions cos(π nz/h) and sin(π nz/h) of the operator d 2 /dz2 . We consider the case n = 1 (the first vertical mode) and assume that ˜ sin(π z/h)}eκ t , ˜ v, ˜ p) ˜ cos(πz/h), (w, ˜ θ˜ , Q) (u, v, p), (w, θ , Q) = {(u,

(5)

where the parentheses combine amplitudes that are independent of z,t. After nondimensionalization in accor dance with [10] (in particular coordinate x is nondimensionalized on the scale μ /ν (h/π)), we obtain the system of equations for amplitudes which contains three dimensionless parameters: R=

gα (γa − γ )h4 , π4 μν

Rm =

gα (γa − γm )h4 , π4 μν

σ = κ h2 /(π2 ν ).

(6)

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Boris Shmerlin, Maxim Kalashnik, Mikhail Shmerlin / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 313–321

Here, Rm is the parameter characterizing the intensity of latent heat release, R is an analog of the Rayleigh number, and σ is the dimensionless growth rate. We note that the form of the Rayleigh number used here differs from the traditional one [3, 4] by its sign and the factor π4 in the denominator. 3.2

Auxiliary problem

We will consider localized perturbations for which the amplitude of vertical velocity w(x) ˜ is symmetric with respect to the origin of coordinate system x = 0 and changes its sign in localization domain around x = 0 finite number of times and is negative for all |x| ≥ x0 , where x0 corresponds to the point of the last change of sign of w(x) ˜ in the process of growth of coordinate x. Fig. 3 illustrates the dependence of w(x) ˜ on x for such perturbations.

Fig. 3 Illustration of a dependence of w(x) ˜ on x for perturbations within auxiliary problem (in the figure sign “tilde” is omitted).

We assume that for these perturbations the intensity of heat release Q = α (γa − γm )w if |x| ≤ x0 and Q = 0 if ˜ such perturbations satisfies the integral |x| ≥ x0 . By analogy with [10], the amplitude of vertical velocity w(x)of equation: ˆ w(x) ˜ = Rm

|x |≤x0

G(x − x )w(x ˜ )dx ,

where the Green function of the point heat source has the form: 1 (λ1 exp(−λ1 |x|) − λ2 exp(−λ2 |x|)), G(x) = 2 2(λ1 − λ22 )

2 λ1,2

(7)

= 1 + σ + R/2 ± R2 /4 + R(1 + σ )

(8)

In the integral equation (7) x0 is a priori unknown and have to be found in the solution process with the use of self-explanatory condition w(x ˜ 0 ) = 0. Further we will consider only the case R ≥ 0, which corresponds to the Green function localized around the point heat source. The problem has a lot of solutions corresponding to the various values of numbers n, m, called modes: π(n + m) = (n + m)x∗0 , xn,m 0 = √ Rm − R

x∗0 = √

π Rm − R

,

n ≥ 0, m > n.

(9)

In fact these solutions have been obtained in [5, 10], but as it was shown they are not the solutions of the problem formulated in [5, 10]. Now we will make use of these solutions. The modes with numbers m = n + 1 have the greatest growth rate among all the modes with the same value n. With the increase of n the growth rate of such modes increases. Further we will consider only the modes with numbers n, m = n + 1 and designate the corresponding mode with the one number n. Thus xn0 = (2n + 1)x∗0 . For these modes x π π x (10) w˜ n (x) = (−1)n cos[ (2n + 1)( n )] cos[( − ϕ ) n ], |x| ≤ xn0 , 2 x0 2 x0 where 0 < ϕ (n, R, Rm ) < π/2. In domain |x| > xn0 , w˜ n (x) < 0 and is damped exponentially at infinity. As an example dependences w˜ n (x) for the first three modes are shown in Fig. 4(a). Each of these modes exists if Rm ≥


319

Rnm , where Rnm decays with the growth of the mode number n, R0m = 6.19. For the atmosphere Rm 1. For each mode in case of Rm ≥ Rnm there is a curve of neutral stability Rncrit (Rm ), corresponding to the mode dimensionless growth rate σ n = 0. The dimensionless growth rate of the mode changes from zero on the curve of neutral 2 n n 0 stability to the maximum value σmax = Rm /Rm − 1 when R = 0. In case of Rm 1, Rcrit ≈ Rm 1 − (π/Rm ) 3 for n =√0, Rncrit ≈ Rm − [n(n + 1)]−1 for n ≥ 1. Then x∗0 in the instability region changes from minimum value x∗0 min = π/ Rm for all modes when R = 0, to the maximum value at the curve of neutral stability for corresponding 2 −1 mode: x∗0 max = π 3 Rm 6 for the mode n = 0 and x∗0 max = π n(n + 1) for the modes n ≥ 1 (see expression (9)). The curves of neutral stability of various modes are presented in Fig. 4(b). The extreme left line–the limit position of the neutral stability curves when n. tends to infinity. above n as in Microsoft Equation 3.0

Fig. 4 The solutions of the auxiliary problem. left: the dependencies of w(x)/| ˜ w(0)| ˜ on x/x0 for the first three modes n = 0, 1, 2 (in the figure signs “tilde” are omitted); right: the curves of neutral stability of various modes.

3.3

Solutions of the original problem

It is obvious that the mode with number n = 0 is the solution of the original problem if the horizontal dimension of cloud medium is enclosed in the range 0 < x∗ ≤ x∗0 . It is also clear that the mode with number n is the solution of the original problem if the horizontal dimension of cloud medium is enclosed in the range between the next to last and last zeroes of function w˜ n (x). Then from expression (10) we have: [(2n − 1)/(2n + 1)]xn0 ≤ x∗ ≤ xn0 ,

or

(2n − 1)x∗0 ≤ x∗ ≤ (2n + 1)x∗0 .

(11)

Thus values Rm and R define the scale x∗0 . If 0 ≤ R ≤ R0crit (Rm ), then under all values of x∗ the instability will take place and in this case the spatial structure of the unstable perturbation will be defined by value x∗ according ∗ ∗ to the presented above inequality for x∗ (11). If R > Rn−1 crit (Rm ), n ≥ 1, then under x < (2n − 1)x0 the stability ∗ ∗ will take place and under x ≥ (2n − 1)x0 there will be instability and as well as before the spatial structure of the unstable perturbation is defined by value x∗ according to (11). It is obvious that in case of instability two solutions, modes with numbers n and n − 1, correspond to the values x∗ = (2n − 1)x∗0 , n ≥ 1, and in this case the mode with number n has the greater growth rate. It is clear that the mode with number n = 0 is simultaneously the solution of the instability problem in case of precipitation convection, obtained in [5, 10]. The pattern of streamlines of this mode having rotation is presented in Fig. 1. As an example the pattern of streamlines of the mode with number n = 1 is shown in Fig. 5. Constructed solutions are localized spatial systems of convective cloud rolls. In case of rotation the azimuthal velocity component appears additionally. By analogy with [7, 10] in case of axisymmetry analogical solutions

320


Fig. 5 The pattern of streamlines for the mode with number n = 1.

presenting axisymmetric convective vortices with both ascending and descending motions on the axis of symmetry correspond to them. Under the values of the mode number n > 1 they can be interpreted as mesoscale cloud clusters with annular cloud structures. Depending on the anisotropy of turbulent mixing (i.e. ratio μ /ν ) vortices scale changes from the scale of a single convective cloud (tornado) to the scale of a tropical cyclone. Corresponding numerical estimations for the mode n = 0 can be found in [10]. 4 Conclusions The classical Rayleigh theory of convective instability of a viscous and heat conductive rotating atmospheric layer is generalized to the case of phase transitions of water vapor. Earlier for the precipitation convection (PC) we suggested the approach which allowed us to solve the problem of the convective instability of a water vapor saturated atmospheric layer using analytical methods. A brief review of achieved results is presented. In this paper the suggested approach was applied to the analysis of the convective instability of a water vapor saturated atmospheric layer in which there is a finite horizontally domain filled with cloud medium, that is, the theory was extended to the description of the nonprecipitation convection (NPC). A principal difference is stated between moist convection and Rayleigh convection. In particular, the instability region on the plane of model parameters turned out to generally consist of two subregions, in one of which the localized axisymmetric disturbances with a tropical cyclone (hurricane) structure have the highest growth rate. A principal difference between the PC and the NPC is stated too. In case of the PC the ascending motions on the axis of symmetry correspond to such disturbances, in case of the NPC a spontaneous growth of localized vortices both with ascending and descending motions on the axis is possible. Here only two dimensional solutions of the problem were presented. The question of a horizontal spatial structure of three dimensional convective cells with the maximum growth rate remains an open one both for the PC and for the NPC. Apparently, for the PC such cells may by only closed ones [10]. The ideas stated above suggest that for the NPC mesoscale cloud clusters may be composed both from open (with descending motions in the centre) and closed (with ascending motions in the centre) convective cells depending on the cloud spot dimension and model parameters. References [1] Gutman, L.N. (1973), Introduction to the Nonlinear Theory of Mesoscale Meteorological Processes, The Institute for the Promotion of Teaching Science and Technology (IPST): Bangkok, Thailand. [2] Belov, P.N. (1971), Practical Methods in Numerical Weather Forecasting, Defense Technical Information Center: Fort Belvoir, Virginia, United States. [3] Landau, L.D. and Lifshitz, E.M. (1987), Fluid Mechanics, Pergamon Press: Oxford. [4] Gershuni, G.Z. and Zhukhovitskii, E.M. (1976), Convective Stability of Incompressible Fluids, Keter, Jerusalem, Israel. [5] Shmerlin, B.Ya. and Kalashnik, M.V. (1989a), Structure of growing localized modes in a moist convection model, Izv.


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Atmos. Ocean. Phys., 25, 307–312. [6] Shmerlin, B.Ya. and Kalashnik, M.V. (1989b), Structure of amplifying periodic modes in a model of moist convection, Izv. Atmos. Ocean. Phys., 25, 599–604. [7] Kalashnik, M.V. and Shmerlin, B.Ya. (1990a), Spontaneous grows of hurricane-like disturbances in a moist convection model, Izv. Atmos. Ocean. Phys., 26, 577–581. [8] Kalashnik, M.V. and Shmerlin, B.Ya. (1990b), Convective instability for a moist saturated layer, Izv. Atmos. Ocean. Phys., 26, 759–765. [9] Shmerlin, B.Ya., Kalashnik, M.V. and Shmerlin, M.B. (2012), Convective instability of a water-vapor-saturated atmospheric layer. The formation of localized and periodic cloud structures, J. Exp. Theor. Phys., 115, 1111–1127. [10] Shmerlin, B.Ya. and Kalashnik, M.V. (2013), Rayleigh convective instability in the presence of phase transitions of water vapor. The formation of large-scale eddies and cloud structures, Phys.-Uspekhi, 56, 473–485. [11] Yamasaki, M. (1972), Small amplitude convection in a conditionally unstable stratification, J. Met. Soc. Jap., 50, 465–481. [12] Yamasaki, M. (1974), Finite amplitude convection in a conditionally unstable stratification, J. Met. Soc. Jap., 52, 365–379. [13] Asai, T. and Nakasui, I.J. (1977), On the preferred mode of cumulus convection in a conditionally unstable atmosphere, J. Met. Soc. Jap., 55, 151–167. [14] Asai, T. and Nakasui, I.J, (1992), A further study of a preferred mode of cumulus convection in a conditionally unstable atmosphere, J. Met. Soc. Jap., 60, 425–431. [15] Delden, A. (1985), On the preferred mode of cumulus convection, Beitr. Phys. Atmos., 58, 202–219. [16] Huang, X. and Kallen, E. (1986), A low order model for moist convection, Tellus, 38A, 381–396. [17] Chlond, A. (1988), Numerical and analytical studies of diabatic heating effect upon flatness of boundary layer rolls, Beitr. Phys. Atmos., 61, 312–329. [18] Huanq, X.Y. (1990), The organization of moist convection by internal gravity waves, Tellus, 42A, 270–285. [19] Haque, S.M. (1958), The initiation of cyclonic circulation in a vertically unstable air mass, Quart. J. Roy. Met. Soc., 78, 394–406. [20] Lilly, D.K. (1960), On the theory of disturbances in a conditionally unstable atmosphere, Mon. Wea. Rev., 88, 1–17. [21] Kuo, H.L. (1961), Convection in a conditionally unstable atmosphere, Tellus,13, 441–459. [22] Gill, A.E. (1982), in Intense Atmospheric Vortices, (Ed. by Bengtsson, L. and Lighthill, J.), Springer Verlag, Heidelberg. [23] Matveev, L.T. (1967), Fundamentals of General Meteorology: Physics of the Atmosphere, Israel Program for Scientific Translations (Keter): Jerusalem, Israel. [24] Gill, A.E. (1982), Atmosphere-Ocean Dynamics, Academic Press, New York. [25] Emanuel, K.A. (1994), Atmospheric Convection, Oxford University Press, New York.



An Approach to the Modeling of Nonlinear Structures in Systems with a Multi-component Convection Sergey Kozitskiy† Department of Oceanic and Atmospheric Physics, Il’ichev Pacific Oceanological Institute, 43 Baltiyskay str. Vladivostok, 690041, Russia Submission Info Communicated by S.V. Prants Received 1 February 2015 Accepted 2 March 2015 Available online 1 October 2015 Keywords Multidiffusive convection Multiple-scale method Coherent structures Chaos in fluid dynamics

Abstract We consider 3D multi-component convection in a horizontally infinite layer of an uncompressible fluid slowly rotating around a vertical axis. A family of CGLE type amplitude equations is derived by multiple-scaled method in the neighborhood of Hopf bifurcation points. We numerically simulate a case of the three-mode convection at large Rayleigh numbers. It was shown that the convection typically takes a form of hexagonal structures for a localized initial conditions. The rotation of the system prevents the spread of the convective structures on the entire area. The approach to the modeling of the Saturn’s polar hexagon on the basis of amplitude equations is discussed. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction One of the popular methods to investigate the system with a convective instability near the bifurcation points is the method of amplitude equations. For the case of Rayleigh-Benard convection this method was introduced by Newell and Whitehead [1]. It allowed to reduce the original PDE system to nonlinear evolution equations for roll modes. For the 2D roll-type double-diffusive convection the amplitude equations of CGLE type (Complex Ginzburg-Landau Equation) were derived and studied numerically in the work [2]. Three possible types of amplitude equations for the double-diffusive system were obtained in [3] for the different bifurcation points. Also an approach to derivation of amplitude equations for a tall thin slot geometry was formulated in the work [4]. With the use of this approach the family of amplitude equations for 3D double-diffusive convection interacting with a horizontal vorticity field in the neighborhood of Hopf bifurcation points was derived in [5] (in Russian), or in the extended and revised version [6] (in English). In the present article we extend the above approach to the slowly rotating systems with convection and derive the amplitude equations for this case. On the basis of the obtained equations we investigate numerically an important case of three-mode convection with localized initial conditions generating hexagonal nonlinear convective structures. Then we determine an impact of slow rotation on these structures. The applicability of the obtained model for the explanation of the famous Saturn’s polar hexagon phenomenon is briefly discussed. † Corresponding


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Sergey Kozitskiy / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 323–331

2 Formulation of the problem and basic equations Consider 3D double-diffusive convection in a liquid layer of a width h, confined by two plane horizontal boundaries and rotating around a vertical axis. The liquid layer is heated and salted from below. The dimensionless governing equations for momentum and diffusion of temperature and salt are: ut + (uux + vuy + wuz ) = −px + σ Δu + f v − σ (RT θ − RS ξ )K f 2 x , vt + (uvx + vvy + wvz ) = −py + σ Δv − f u − σ (RT θ − RS ξ )K f 2 y , wt + (uwx + vwy + wwz) = −pz + σ Δw + σ (RT θ − RS ξ ) ,

θt + (uθx + vθy + wθz ) − w = Δθ , ξt + (uξx + vξy + wξz ) − w = τ Δξ , ux + vy + wz = 0 .

(1)

Where σ = ν0 /χ is the Prandtl number (σ ≈ 7.0), τ = D/χ is the Lewis number (0 < τ < 1, usually τ = 0.01 − 0.1). RT = (gα h3 /χν )TΔ and RS = (gγ h3 /χν )SΔ are the temperature and the salinity Rayleigh numbers, α and γ are cubic expansion coefficients, f = 2h2 Ω /χ is the dimensionless frequency of rotation and K = χ 2 /4gh3 . Fluid velocity field is represented by the vector v(t, x, y, z) = (u, v, w). Variables θ (t, x, y, z) and ξ (t, x, y, z) denote deviations of temperature and salinity from their stationary linear profiles, so T (t, x, y, z) = T+ + TΔ [θ (t, x, y, z) − z] ,

S(t, x, y, z) = S+ + SΔ [ξ (t, x, y, z) − z] .

T+ and S+ are the temperature and salinity at the lower boundary of the area. Free-slip boundary conditions are used for the dependent variables (the horizontal velocity component is undefined): uz = vz = w = θ = ξ = 0 at z = 0, 1 . It is believed that they are suitable to describe the convection in the inner layers of liquid and do not change significantly the convective instability occurrence criteria for the investigated class of systems [7]. 3 Derivation of amplitude equations In this chapter we briefly present the scheme of derivation of the amplitude equations for the multi-component convection in a slowly rotating system. The details of the derivation are practically the same as in the case of usual double-diffusive convection and can be found in [5, 6]. Consider the equations for double-diffusive convection in the vicinity of a bifurcation point, the temperature and salinity Rayleigh numbers for which are designated as RT c and RSc respectively. In this case the Rayleigh numbers can be represented as follows: RT = RT c (1 + ε 2 rT ),

RS = RSc (1 + ε 2 rS ) .

At least one of the values rT or rS is of unit order, and the small parameter ε shows how far from the bifurcation point the system is. To derive the amplitude equations we use the derivative-expansion method from [8], which is the case of the multiple-scale method. Introduce the slow variables: T1 = ε t ,

T2 = ε 2t ,

X = εx,

Y = εy.

In accordance with the chosen method we assume that the dependent variables now depend on t, T1 , T2 , x, y, z, X , Y , which are considered as independent. Also we replace the derivatives in the equations (1) for the prolonged ones by the rules: ∂t → ∂t + ε∂T1 + ε 2 ∂T2 , ∂x → ∂x + ε∂X , ∂y → ∂y + ε∂Y .


325

According to our assumption of the slow rotation of the system set f = ε f and K = ε K , where f and K are parameters of the order O(1). Then the equations (1) can be written as: 1 (ϕ , ϕ ) − ε N 2(ϕ , ϕ ) . L1 ϕ − ε 2 L2 ϕ − N Lϕ = −ε

(2)

L, L1 , Where we introduce vector of the dependent variables ϕ = (u, v, w, θ , ξ , p), matrix-differential operators 2 L2 and nonlinear operators N1 , N2 . Note that L1 now contains the term describing the Coriolis force, and L contains the term describing the centrifugal force from (1). We seek solutions of equations (2) in the form of asymptotic series in powers of small parameter ε : ∞

ϕ = ∑ ε i ϕi = εϕ1 + ε 2 ϕ2 + ε 3 ϕ3 + · · · .

(3)

i=1

After their substitution in (2) and collection the terms at ε n we obtain the systems of equations to determine the terms of the series (3).

O(ε 2 ) :

ϕ1 = 0 , L ϕ2 = − 1 (ϕ1 , ϕ1 ) , L L 1 ϕ1 − N

O(ε 3 ) :

ϕ3 = − 1 (ϕ1 , ϕ2 ) − N 1(ϕ2 , ϕ1 ) − N 2 (ϕ1 , ϕ1 ) . L L 1 ϕ2 − L 2 ϕ1 − N

O(ε ) :

(4)

The linear system at O(ε 1 ) has a solution in the form of sum of n normal modes (convective rolls): n sin π z λ t ik j ·x 1 + c.c. . +ϕ ϕ1 = ∑ A j (X ,Y, T1 , T2 )ϕˇ 1 j e e cos π z j=1

(5)

The cosine in the braces is selected for variables u1 , v1 , p1 , in another cases the sine is selected. Vectors k j have components k j = (ka j , kb j ). The terms u1 and v1 form the velocity field, against which the convection develops. Parameters of each from n roll-modes λ , ka j , kb j , RT c , RSc are related by the equation: (λ + σ κ 2 )(λ + κ 2 )(λ + τ κ 2) + σ (k2 /κ 2 )[RSc (λ + κ 2 ) − RT c (λ + τ κ 2)] = 0 . Here k2 = ka2 j + kb2 j , and κ 2 = k2 + π 2 . This equation has three roots, two of which can be complex conjugates. In the case of Hopf bifurcation these two roots acquire positive real part at some RT c (ω is a frequency of convective waves): RT c =

κ6 σ +τ RSc + 2 (1 + τ )(τ + σ ) , 1+σ σk

ω2 =

1−τ k2 σ RSc 2 − τ 2 κ 4 > 0 . 1+σ κ

(6)

In this paper we consider double-diffusive convection at Hopf bifurcation points, i.e. in all cases λ = iω . √ Usually the critical wavenumber is kc = π /√2. But for the sufficiently large Rayleigh numbers the characteristic critical wavenumber is of the order 0.23 ω and may reach values of kc ≈ 100 [9] which corresponds to a narrow convective cells. The obtained at O(ε 2 ) and O(ε 3 ) systems in (4) have the following general form: L ϕi = Q i . (1)

(2)

(3)

(1)

Functions Qi include terms, resonating with the left parts of equations, i.e. Qi = Qi + Qi + Qi . Here Qi (2) (3) and Qi generate the secular terms of two types in the solutions, but Qi does’nt generate any secular terms and contains only unimportant terms for the explored case. The conditions of the first type secular terms absence (1) L F j = 0 reduce to demand of orthogonality functions Qi and solutions Fj of the adjoint homogeneous equation

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Sergey Kozitskiy / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 323–331 (2)

and usually take form of amplitude equations. Terms Qi are the constants with respect to quick variables. To (2) prevent the brake of regularity of the asymptotic expansions (5) they should be equal to zero Qi = 0 [4]. These conditions also take form of amplitude equations. To find the amplitude equations derived from the system at O(ε 2 ) write the expressions for components Q2 , omitting zero terms: (1) (0) 1 (ϕ1 , ϕ (0) ) , L 1 ϕ1 − N Q2 = − 1

(2) Q2 = − L1 ϕ1 ,

(3) 1 (ϕ (0) , ϕ (0) ) . Q2 = −N 1 1

(0) (0) (0) Lϕ1 = 0, ϕ1 is the averaged Note that ϕ1 = ϕ1 + ϕ1 , where ϕ1 is the solution of the homogeneous equation fields depending only on slow variables. (2) To exclude secular terms of the second type one should require fulfillment of equalities Q2 = 0. To satisfy these equalities it is sufficient to introduce horizontal stream function Ψ by formulas:

u1 = ΨY ,

v1 = −ΨX ,

ΨT1 = 0 .

(7)

Also it is true p1 = 0 with the accuracy to constants on horizontal variables. The condition of there be no secular terms of the first type in the solutions of the equations at O(ε 2 ) takes a form of the following system: A jT1 + 2α0 (ika j A jX + ikb j A jY ) + (ika j ΨY − ikb j ΨX )A j = 0 ,

j = 1...n.

(8)

where α0 is a certain coefficient calculated in [6, 10]. Note that the term describing the Coriolis force is exactly vanish in these equations. Write the resulting amplitude equations for the system at O(ε 3 ). For this purpose we need the solutions for ϕ1 and ϕ2 , which can be expressed in a general form: (0)

ϕ1 = ϕ1 + ϕ1 ,

(0)

(1)

ϕ2 = ϕ2 + ϕ2 + ϕ2 + ϕ2 .

(0) (0) (1) Lϕi = 0, ϕ2 and ϕ2 are linear and nonlinear on Here ϕi are the general solutions of homogeneous equations (1) amplitude terms of the particular solution of the inhomogeneous equation L(ϕ2 + ϕ2 ) = Q2 , ϕi are the averaged fields on the slow horizontal equations, arising as an integrating constants. Write the system at ε 3 in a general form: (1) (2) (3) L ϕ3 = Q 3 + Q 3 + Q 3 . (1)

Then the expression for Q3 take form: (1)

(1l)

(1p)

Q3 = Q3 + Q3 (1l)

(1n)

+ Q3 (1n)

. (1p)

Here we have separately identified linear Q3 and nonlinear Q3 on A j terms, and also terms Q3 , containing (1) ϕ1 . The first equation in (9) is obtained as a result of orthogonality requirement of Q3 and solution Fj of adjoint homogeneous equation. From the condition of there be no secular terms of the second type in the equations at O(ε 3 ) one should (2) require to be true Q3 = 0. Also assume that Δ⊥ Ψ2T1 = 0, as it is true in the case of Ψ. Thus we get second equation in (9). 4 The family of amplitude equations Finally the resulting family of amplitude equations for the system at ε 3 is:

⎧ α4 α1 j (Ψ)A j + J(Ψ, A j ) + N j (A) , ⎪ |k · X| + 2 (k · ∇⊥ )2 A j − α0 Δ⊥ A j + α3 G ∂ A = rA 1 − j ⎨ T2 j k k n j (|A j |2 ) , ⎪ ⎩ (∂T2 − σ Δ⊥)Ω = J(Ψ, Ω) − ∑ G j=1

(9)


327

where Δ⊥ is Laplacian with respect to the slow variables, ∇⊥ is gradient operator, Ω = Δ⊥ Ψ, αi are complex coefficients, D(x) is defined as D(0) = 1 and D(x) = 0 at x = 0. Index j = 1 . . . n denotes the mode number. 2 j ( f ) = π (ka j ∂X + kb j ∂Y )(ka j ∂Y − kb j ∂X ) f . G k4 This family of the systems of amplitude equations depends on the set of n wavevectors which define the shape of in the equations describes an interaction between convection and field of horizontal convective cells. Operator G vorticity, generation of vortex due to convection. Now in the equations we have in the forcing term the parameter proportional to |k · X| and describing the influence of the centrifugal force on the convective pattern. Note that we intentionally choose the modulus in the parameter from the conditions that the convection should be attenuated with the distance from the center of coordinates. The functions N j (A) are the following combination of cubic nonlinear terms:

N j (A) = α2 A j

n

∑ |Aq|2 +

q=1

n

n

n

∑∑ ∑

(1) [D(kq +k p −km −k j )α jmqp A∗m Aq A p

(10)

m=1 q=1 p=q+1

(2) (3) + D(kq −k p −km +k j )α jmqp Am A∗q A p + D(kq −k p +km −k j )α jmqp Am Aq A∗p ] .

coefficients in the article [10]. Coefficients r , α0 , α1 , α2 , α3 in the equations (9) coincide with the same-named √ Therein one can find the expressions for these coefficients at k = π / 2 and graphs of their dependence from (s) frequency ω . Coefficients at the nonlinear terms α jmqp , (s = 1, 2, 3) are presented in [5, 6]. The coefficient α4 describing the influence of centrifugal forces can be expressed as follows:

α4 = −

(iη + σ )(iη + τ )(iη − (1 + τ + σ )) Ω2 h π (1 − τ ) · . · gε 3 k(1 + σ )(τ + σ ) η (iη − 1)

Here η = ω /κ 2 is a parameter having typical value of about η = 20 at large Rayleigh numbers. In this limit we can write: Ω2 h πω (1 − τ ) . α4 ≈ · 2 3 gε kκ (1 + σ )(τ + σ ) By the appropriate choice of the rotation speed you can put this coefficient equal to any convenient quantity. It is this quantity determines the size of the region occupied by the convection. 5 Hexagonal type three-mode convection Consider three-mode convection in the case when the convective rolls are placed at the angles 120 degrees with √ √ respect to each other. The wave vectors are: k1 = (k , 0), k2 = (−k/2 , k 3/2), k3 = (−k/2 , −k 3/2). The system (9) transforms to the following shape: ⎧ ⎪ AT = A(1 − α4 |X |) + α6 AXX − α7 AYY + J(Ψ, A) + α9 AΨXY − iA|A|2 + α11 A(|B|2 + |C|2 ) , ⎪ ⎪ √ ⎪ ⎪ 1 ⎪ = B(1 − α |Y 3 − X |) + α61 BXX + α71 BYY − α72 BXY + J(Ψ, B) B T 4 ⎪ 2 ⎪ √ ⎪ ⎨ 1 − 2 α9 B[ΨXY + 23 (ΨYY − ΨXX )] − iB|B|2 + α11 B(|A|2 + |C|2 ) , √ ⎪ CT = B(1 − 12 α4 |Y 3 + X |) + α61CXX + α71CYY + α72CXY + J(Ψ,C) ⎪ ⎪ √ ⎪ ⎪ 3 1 2 2 + |B|2 ) , ⎪ α C[Ψ − − 9 XY ⎪ 2 2 (ΨYY − ΨXX )] − iC|C| + α11C(|A| ⎪ √ √ ⎪ ⎩ Ω = α Δ Ω + J(Ψ, Ω) − (|A|2 − 1 |B|2 − 1 |C|2 ) + 3 (|B|2 − |C|2 ) − 3 (|B|2 − |C|2 ) . T

8 ⊥

2

2

XY

Here we introduced the coefficients: 1 3 3 1 α61 = α6 − α7 , α71 = α6 − α7 , 4 4 4 4 k2 κ 2 k2 κ 2 α6 = 2 (α1 − α0 ) , α7 = 2 α0 , 4π ω 4π ω

XX

4

√

4

3 (α6 + α7 ) , 2 σ k2 κ 2 α8 = , α9 = ikα3 . 4π 2 ω

α61 =

YY

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Sergey Kozitskiy / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 323–331 Convection amplitude |A+B+C|

Convection amplitude |A+B+C| 6

−10

5.5 −10

5

5

−5

4.5 4

−5 4

3.5 3 0

3

0 2.5 2

2 5

5 1.5 1

1 10

10

−10

−5

0

5

10

0.5

−10

−5

0

5

10

Fig. 1 The numerical solution of the three-mode equations in the area 25 × 25 for T = 2.5 at ω = 20000. The initial conditions are A = B = C = 2 exp(−0.5(X 2 +Y 2 )) and Ω = 0. The left pattern corresponds to the case with α4 = 0 and the right one corresponds to the case with α4 = 0.15 (nonzero centrifugal forces).

We have performed numerical simulations for the case of convection with ω = 20000, σ = 7 and τ = 1/81. The above coefficients for this case have the following values: α6 = 1.043 − 0.42i, α7 = −0.185 + 0.564i, α8 = 9.72, α9 = 0.123 − 0.293i, α11 = 0.000658 − 0.961i. 6 Results of numerical experiments In the calculations we used the numerical scheme developed on the basis of ETD2RK exponential time differencing pseudo-spectral method from [11]. The number of nodes on both horizontal variables was usually 256. The size of the area for calculations as a rule was chosen as 25 × 25, and the calculations were led up to the times about T = 30. We used the periodic boundary conditions natural for the pseudo-spectral methods. As an initial conditions for simulation we choose Gauss bell-like function A = B = C = 2 exp(−0.5(X 2 + Y 2 )). An initial vorticity field was chosen to be zero. The numerical simulation (ω = 20000) shows, that the convection generates various hexagonal patterns (figure 1-3) slowly evolving with the time, and the system never reaches any stationary state. It was noticed that in a time of T = 15 − 30 the condition of diffusive chaos is developed in the system, when the initial state is destroyed and the initially symmetrical convection becomes irregular in both space and time. The simulation shows that the sign of the imaginary part of the coefficient α6 depending on ω is important for the pattern type. Internally the hexagonal convective structures are composed of three roll-type modes nonlinearly interacting with each other. As one roll-type mode has stick-like solution, the three such systems of sticks at the angles of 120 degrees to each other give a case of the hexagonal pattern. Separately we investigated the effect of the slow rotation on the obtained convective patterns. The coefficient α4 was chosen in the range 0.1 − 0.2 to optimally restrict convection on the area of calculation. On the figures 1-3 the left patterns correspond to the case of α4 = 0, and the right patterns correspond to the case of α4 = 0.15. One can easily see that the proper choice of α4 can effectively localize the area of the intensive convection.


Convection amplitude |A+B+C|

329


6

−10

−10 6

5 −5

5

−5

4

0

4 0

3

2

5

3

5

2

1 10

1 10

−10

−5

0

5

10

−10

−5

0

5

10

Fig. 2 The numerical solution of the three-mode equations in the area 25 × 25 for T = 7.5 at ω = 20000. The initial conditions are A = B = C = 2 exp(−0.5(X 2 +Y 2 )) and Ω = 0. The left pattern corresponds to the case with α4 = 0 and the right one corresponds to the case with α4 = 0.15 (nonzero centrifugal forces).


Convection amplitude |A+B+C| 6 5.5

−10

−10 5 5 4.5

−5

4

−5 4

3.5 3

0

0

3

2.5 2 5

2

5 1.5 1

10

0.5

−10

−5

0

5

10

1 10

−10

−5

0

5

10

Fig. 3 The numerical solution of the three-mode equations in the area 25 × 25 for T = 10 at ω = 20000. The initial conditions are A = B = C = 2 exp(−0.5(X 2 +Y 2 )) and Ω = 0. The left pattern corresponds to the case with α4 = 0 and the right one corresponds to the case with α4 = 0.15 (nonzero centrifugal forces).

330


7 An approach to the Saturns’s polar hexagon simulation Saturn’s hexagon is a persisting hexagonal cloud pattern around the north pole of Saturn. The sides of the hexagon are about 13,800 km long. The hexagon does not shift in longitude like other clouds in the visible atmosphere. Saturn’s polar hexagon discovery was made by the Voyager mission in 1981–82, and it was revisited since 2006 by the Cassini mission. It is believed that the hexagon is described by some kind of solitonic solution. Also it is stated that the hexagon forms where there is a steep latitudinal gradient in the speed of the atmospheric winds in Saturn’s atmosphere. And the speed differential and viscosity parameters should be within certain margins. If this is not fulfilled the polygons don’t arise, as at other likely places, such as Saturn’s South pole or the poles of Jupiter. Obviously double-diffusive convection plays in the atmospheres of such planets as Saturn or Jupiter an essential role [12]. Here atmosphere is a mixture of hydrogen with helium, and in the upper atmosphere there exist a vertical negative gradient of temperature due to hot lower layers. Thus we have a diffusive type of doublediffusive convection in a rotation system. As a rule, rotation acts as one more diffusive component, which gives actually a case of triple-diffusive convection and complicates the analysis. Nevertheless preliminary considerations show that at large Rayleigh numbers (as in the case of Hexagon) such system behaves qualitatively as the explored double-diffusive system near the Hopf bifurcation points. So one can expect similar amplitude equations for a slow variations of convective amplitude, but with the different coefficients of such equations. An exact derivation of amplitude equation for the Hexagon’s case is rather cumbersome task, but the obtained in this article results allow to make some hints on possible steps in solving a task of construction the equations having Hexagon as a solution. As we noted, in the case of Hexagon one can expect amplitude equations similar to (9), but with additional terms. Such terms can be incorporated into equations first artificially to provide qualitatively satisfactory results. Then on this basis the more exact model can be derived by the strict methods. As one can see, the solution (Fig. 1-3) for three-mode equations qualitatively resembles Hexagon. But without any additional terms on large time the solution spreads over all area, and its shape becomes more whimsical (Fig. 3, left). The terms taking into account centrifugal forces and possibly the curvature of the surface can effectively localize the area of the convection (Fig. 3, right) and provide the long-living hexagonal patterns. Also one should insert into equations the stabilizing terms, possibly proportional to the fifth order of the amplitude of the convection preventing evolution changes of the Hexagon. The more complex and fundamental problem is the principle of the modes selection when the initial variety of modes turns with the time into the only three modes determining the final form of the convective structure. But these themes are out of the scope of the current article. 8 Conclusion The family of amplitude equations (9) describing 3D multi-component convection in an infinite layer of fluid, interacting with horizontal vorticity field and slowly rotating around a vertical axis is derived. The shape of the convective cells is defined by a finite superposition of roll-type modes. For numerical simulation of the obtained systems of amplitude equations we developed the numerical scheme based on modern ETD (exponential time differencing) pseudospectral methods [11]. The software packages were written for simulation of convection with hexagonal type cells. Numerical simulation has showed that the convection in the system takes the form of hexagonal convective structures slowly varying with time. In the system a state of diffusive chaos is developed (at time T = 15-30), where the initial symmetric state is destroyed and convection becomes irregular both in space and time. Also it was noticed that the centrifugal force can prevent the spread of convection over the all possible area. As a result the hexagonal convective structures stay localized around the center of coordinates. It make possible to use the alike amplitude equation based models to explain such phenomena as Saturn’s polar hexagon. The obtained results induce a deeper understanding of heat and mass transfer processes in the ocean and the


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atmosphere, help to describe more adequately the convective and vortex structures that arise in physical systems with convective instability, and may also be the basis for the construction of more advanced models of systems with multi-component convection. Acknowledgements This work is supported by RFBR grant 14-05-00017. References [1] Newell, A.C. and Whitehead, J.A. (1968), Finite bandwidth, finite amplitude convection. Journal of Fluid Mechanics, 38, 279–303. [2] Bretherton, C.S. and Spiegel, E.A. (1983), Intermittency through modulational instability. Physics Letters, 96A, 152– 156. [3] Kozitskiy, S.B. (2000), Amplitude equations for a system with thermohaline convection, Journal of Applied Mechanics and Technical Physics 41(3), 429–438. [4] Balmforth, N.J. and Biello, J.A. (1998), Double diffusive instability in a tall thin slot, Journal of Fluid Mechanics 375, 203–233. [5] Kozitskiy, S.B. (2012), Model of three dimensional double-diffusive convection with cells of an arbitrary shape, Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki, 4, 43–61. [6] Kozitskiy, S.B. (2014), Structures in 3D double-diffusive convection and possible approach to the Saturn’s polar hexagon modeling, E-print arXiv:1405.3020 [nlin.PS] (http://arxiv.org/abs/1405.3020). [7] Weiss, N.O. (1981), Convection in an imposed magnetic field. part 1. the development of nonlinear convection, Journal of Fluid Mechanics, 108, 247–272. [8] Nayfeh, A.H. (1993), Introduction to perturbation techniques, John Wiley & Sons, New York-Chichester-BrisbaneToronto. [9] Kozitskiy, S.B. (2005), Fine structure generation in double-diffusive system, Physical Review E, 72(5), 056309–1– 056309–6. [10] Kozitskiy, S.B. (2010), Amplitude equations for three-dimensional roll-type double-diffusive convection with an arbitrary cell width in the neighborhood of hopf bifurcation points, Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki, 4, 13–24. [11] Cox, S.M. and Matthews, P.C. (2002), Exponential time differencing for stiff systems, Journal of Computational Physics, 176, 430–455. [12] Leconte, J. and Chabrier, G. (2012), A new vision of giant planet interiors: Impact of double diffusive convection, Astron. Astrophys, 540, A20.



Instability Development in Shear Flow with an Inflection–Free Velocity Profile and Thin Pycnocline S.M. Churilov† Institute of Solar–Terrestrial Physics SB RAS, 126a Lermontov Street, Irkutsk 664033, Russia Submission Info Communicated by S.V. Prants Received 1 February 2015 Accepted 2 March 2015 Available online 1 October 2015 Keywords Inflection-free velocity profile Sharp density stratification Critical layer Nonlinear wave interactions

Abstract Weakly stratified flows of the class under study have a wide 3D spectrum of the most unstable waves with very close growth rates and phase velocities so that their individual critical layers merge into a common one. The analysis of evolution equations for those waves has shown that throughout a weakly nonlinear stage of development their amplitudes grow explosively. During the first (three-wave) phase, the most rapidly growing are low-frequency waves whereas at the next phase, when numerous and diverse higher-order wave interactions come into play, the growth of highfrequency waves is accelerated and they overtake low-frequency waves. The results obtained are illustrated by numerical calculations for some ensembles of waves. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In the theory of hydrodynamic stability of high-Reynolds-number plane-parallel shear flows vx = U (z), an important role is played by so-called critical layer [1, 2] (hereafter referred to as CL), a narrow neighborhood of the critical level z = zc on which the flow velocity is equal to the phase velocity of the disturbance.a Inside CL, fluid particles move together with the wave and interact with the same fragment of the wave during a long time interval, and therefore an intensive exchange of energy and momentum between the individual (fluid particles within CL) and collective (wave) degrees of freedom takes place which is called a wave–flow resonance [3–5]. As a result, even at a small wave amplitude, the motion of fluid particles within CL is rearranged significantly and affects the development of the wave. In particular, it is the wave–flow resonance that is usually responsible for the wave (and, hence, flow) instability in linear approximation. In the linear theory, perturbation can be regarded as a superposition of independent eigenoscillations, each with its own phase velocity and, if the wave–flow resonance condition is fulfilled, its own CL. With the growth of the amplitudes, however, this approximation ceases to be valid, and a weakly nonlinear stage of evolution begins. During this stage, the perturbation can still be considered as an ensemble of waves arranged in almost the same manner as at the linear stage, but it is impossible to neglect their interactions. And it is the CL where the most intensive wave interactions take place because the rearrangement of the flow makes the perturbation magnitude † Corresponding

author. Email address: [email protected] a More precisely, to its real part, U ≡ U(z ) = c = Re c. c c r ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.09.009

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S.M. Churilov / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 333–351

inside CL much greater than outside it. Hence, a significant contribution to the perturbation evolution can be made only by those interactions which involve the waves with a common critical layer, i.e., with nearly the same phase velocities (so-called phase-locked modes, see [6], for example). In this respect, Holmboe waves [7] playing an important role in the dynamics of sharply stratified shear flows are of a great interest. Such flows, with the vertical scale of density variation being much smaller than that of velocity shear, Λ, are widely represented in the astrophysical and geophysical fluid dynamics [8–11] and attract a considerable interest of researchers (see [12] and references therein). Let us consider a class of flows in which Holmboe waves are the only possible waves. In particular, among these are flows of an ideal incompressible fluid which are similar to a boundary layer with an imbedded thin pycnocline. Their velocity U (z) increases monotonically upwards, from zero at the bottom (z = 0) to some U0 as z → ∞ whereas the density ρ0 (z) steadily decreases from ρ1 to ρ2 in a transition layer of thickness Λ, see Fig.1(a). Linear stability of these flows is well-studied [13, 14]. For a brief review of it let us introduce 3

10

Richardson number, J

(z)

Height, z

2

U(z)

1

J0(+)(k) 1

0.1

0.01

J 0( (k) )

zN 0



0

0.001

1

0

Velocity, U, and density, 

1

2

Wave number, k

(a)

(b)

Fig. 1 (a) sketch of the velocity and density profiles, and (b) the instability domain with growth-rate level lines for planar waves in the flow described by Eq.(2); the growth rate maximum, γmax = 0.0480, is marked by cross, and J∗ = (−) min J0 (k) = 3.2 · 10−3.

dimensionless variables so that ρ1 = 1, U0 = 1, U (0) = 1 (i.e., Λ = 1; prime denotes the derivative in z) and write the squared buoyancy frequency as Ω2 (z) ≡ −gρ0 /ρ0 = J n(z) where g is the gravitational acceleration, and J is the bulk Richardson number. The normalized buoyancy profile n(z) is localized in a layer of thickness O() centered at zN = O(1), ˆ ∞ ˆ ∞ dz z n(z), dz n(z) = 1, (1) zN = 0

0

and has a single maximum. Also, we set UN = U (zN ) and presume that U < 0 everywhere. In essence, the medium over pycnocline is homogeneous and opaque to waves. Therefore, the flow can be regarded as a horizontal waveguide, and its eigenoscillations as guided modes. Let us begin with twodimensional (planar) waves, with wave vectors k = (k, 0, 0) parallel to the flow. Waves outrunning the flow are neutrally stable (ci ≡ Im c = 0) and are characterized by the wavelength and the number m = 0, 1, 2, . . . of the eigenfunction nodes in z. The waveguide width is O(), hence, modes with m ≥ 1 (i.e., having nodes inside the pycnocline) do exist when J = O(−1 ) 1 or greater. In what follows, we shall study only the m = 0 mode. Its


335

2

Spanwise wave-vector component, |ky|

Spanwise wave-vector component, |ky|

dispersion equation can be written as J = J0 (k; c) where J0 increases monotonically with both k and cr . (+) As cr reduces (together with J) at fixed k and crosses the c = 1 boundary (at J = J0 (k) ≡ J0 (k; 1)), the wave–flow resonance interaction comes into play and CL is formed. The sign of the vorticity gradient U is such that the mass of fluid particles moving slightly faster than the wave, i.e., having velocity in the range of Uc to Uc + δ U , is greater than that of particles with velocities in the range of Uc − δ U to Uc . For this reason, the wave is unstable as long as its CL is at the periphery of the flow where stratification is negligible. With further reducing of J, the critical level is dropped lower and lower, and when it approaches the pycnocline periphery or even falls inside, the stratification stabilizes the wave. In such a way we reach the lower boundary (−) J = J0 (k) > 0 of the domain of flow instability with respect to planar (independent of y) perturbations [13]. At k = 0, the upper and lower boundaries of the instability domain start from the same point of the (k, J)–diagram, (+) (−) (+) (−) J0 (0) = J0 (0) = O(1) but run in different directions. Whereas J0 increases monotonically with k, J0 initially decreases abruptly, attains its minimal value J∗ = O(2 ), and then increases very slowly (see Fig.1(b)). (−) In the strip 0 ≤ J < J0 (k), planar eigenoscillations do not apparently exist. Properties of three-dimensional (oblique) disturbances can be easily found by application of Squire’s theorem [15] generalized for stratified flows [16, 17]. It states that if U (z) and n(z) profiles are given, complex phase velocity of the oblique wave with the wave vector k3D = (k cos θ , k sin θ , 0) in the flow with the stratification level J is equal to the phase velocity of the planar wave with the wave vector k2D = (k, 0, 0) in the flow with the stratification level J = J/ cos2 θ . As a consequence, the total instability domain of the flow extends from (+) the upper stability boundary J = J0 (k) to the abscissa axis (J = 0), and only oblique waves do exist and are (−) unstable in the strip 0 < J < J0 (k). Moreover, above the strip, up to J = O(3/2 ), oblique waves grow faster than planar as well. Thus, flows belonging to the class under study lose their stability at arbitrarily weak stratification and become unstable with respect to perturbations of all wavelengths (0 < k < ∞). However, the most interesting property is that weakly-supercritical flows, with 2 < J < 3/2 , have a wide three-dimensional spectrum of unstable waves with growth rates γ = O() slowly depending on k (Fig.2(a)) and phase velocities lying in a very narrow (O()) range (Fig.2(b)), see [14] for details.

1

0

0

1

2

Streamwise wave-vector component, kx

(a)

2

1

0

0

1

2

Streamwise wave-vector component, kx

(b)

Fig. 2 The k–dependence of (a) growth rate and (b) deviation (cr − UN )/ of the phase velocity from UN in the flow described by Eq.(2). The instability domain boundaries are drawn in bold.

336


And finally, if viscosity is small, at the linear stage of development the thickness L of an individual (i.e., belonging to a single wave) CL is equal to the unsteady scale [2] L = Lt = O(|B|−1 |dB/dt|) where B and t are wave amplitude and time respectively. Therefore, as early as at the linear stage of perturbation development L = O(γ ) = O(), and individual critical layers of a broad spectrum of waves merge into a common CL. The importance of this fact is difficult to overestimate. First, all necessary conditions are fulfilled for intensive weakly nonlinear interactions of instability waves belonging to a wide spectral band. Second, using the properties of solutions of the Rayleigh equation [18] that describes the perturbation out of CL (with the pycnocline inside), as well as smallness of the CL thickness, and applying the method of matched asymptotic expansions, we are able to construct not only linear but also weakly nonlinear theory of instability development for the entire class of flows under study, without specifying the velocity and buoyancy profiles. This opens a way toward understanding the overall picture of appearance, self-organization and initial development of the structures and features of the wave field which will be crucial later, during a strongly nonlinear stage of perturbation evolution. Up to the present, three-wave interaction of Holmboe waves [19] and weakly nonlinear evolution of a monochromatic wave [20] are studied. These studies have shown that transition to weakly nonlinear stage occurs at such small wave amplitudes that fluid particles trapped by waves have no time to mix. As a result, nonlinear interactions accelerate the growth of instability waves to an explosive one. In the present paper, we take into account wave interactions of higher orders and complete a weakly nonlinear description of Holmboe waves development up to amplitudes of the order unity, i.e., up to transition to a strongly nonlinear stage. The results obtained will be illustrated by numerical calculations for the model flow with the halocline, i.e., with the stratification due to height-dependent water salinity: U (z) = tanh z,

n(z) =

1 + exp(−2zN /) ; zN = 0.5, = 0.04, J = 0.01. 2 cosh2 [(z − zN )/]

(2)

The paper is organized as follows. In Section 2, basic equations are written and their solutions outside CL are found. In Section 3, equations describing the perturbation inside CL are derived, the scheme of solving them is given, and the linear problem solution is obtained. Section 4 is devoted to nonlinear evolution equations. We analyze their structure and solutions, and construct scenarios of perturbation development. In Section 5, numerical solutions of evolution equations for some wave ensembles are presented, and results obtained are discussed in Section 6. And some auxiliary results are given in Appendix. 2 Basic equations and outer solution Let the bulk Richardson number J = 2 J˜ = O(2 ), and Reynolds number is so large (Re −3 ) that the dissipation is negligible. We consider the weakly nonlinear development of an ensemble of most unstable waves. Since the phase velocities of those waves are close to UN , in the frame of reference moving downstream with velocity UN , both propagation of waves and their growth in amplitude can be described in terms of slow time τ = Lt where L is the thickness of their common CL. It is assumed that the disturbance evolves from an infinitesimal one due to instability and therefore vanishes as τ → −∞. We shall employ matched asymptotic expansions to construct the solution, i.e., we search for solutions both inside CL and in outer regions of the flow, and then, matching them in intermediate domains (L |z − zN | 1), we obtain the evolution equations as matching conditions. Eliminating pressure, one can write the hydrodynamic equations in the Boussinesq approximation [18], in the frame of reference moving with velocity UN , as

∂w ∂ 2ρ ∂ 2ρ ∂w ∂w ∂w D Δw −U + J( 2 + 2 ) = −Δ(u +v +w ) Dt ∂x ∂x ∂y ∂x ∂y ∂z +

∂ ∂u 2 ∂v ∂w 2 ∂u ∂v ∂u ∂w ∂v ∂w [( ) + ( )2 + ( ) + 2( + + )], ∂z ∂x ∂y ∂z ∂y ∂x ∂z ∂x ∂z ∂y

(3 a)


D ∂ρ ∂ρ ∂ρ +v +w ), ρ − n(z)w = −(u Dt ∂x ∂y ∂z

337

(3 b)

∂w ∂ ∂v ∂v ∂v ∂ ∂u ∂u ∂u D G +U = (u + v + w ) − (u + v + w ), Dt ∂y ∂x ∂x ∂y ∂z ∂y ∂x ∂y ∂z

(3 c)

∂u ∂v ∂w ∂u ∂v + =− , G= − , (3 d, e) ∂x ∂y ∂z ∂y ∂x where D/Dt = L ∂ /∂ τ + (U −UN ) ∂ /∂ x, v = (u, v, w) and ρ are the velocity and density disturbances, and Δ is Laplacian. In outer regions of the flow (|z − zN | L), stratification and nonlinearity can be neglected and one may keep only the linear part of the solution, w = ∑[wn (τ , z) ei(kn x+qn y) + c.c.], n

kn = (kn , qn , 0), kn > 0,

(4)

where summation is over all the waves of the ensemble, and c.c. denotes the complex conjugated term. Since amplitudes of interacting waves can, in general, have (or acquire in the process of evolution) different orders of magnitude, it is convenient to assign them different amplitude parameters 0 < εn 1 and to seek the solution for Eqs.(3) in the form (1) (2) wn (τ ) = εn [wn + L wn + . . . ]. (1)

Outside the CL, the main contribution to the n-th wave can be written as wn± = −ikn An · (τ )g± (z; Kn ) where the +/− subscript corresponds to the region over/under zN , Kn2 = kn2 + q2n , and g± (z; K) satisfy the Rayleigh equation with the boundary conditions U (z) d2 g± − ( + K 2 )g± = 0; dz2 U (z) −UN

(5)

g− (0) = 0, |g+ (∞)| < ∞. It should be emphasized that, by virtue of the Rayleigh theorem, g+ and g− are different solutions of the Rayleigh equation at any K. And it is their matching through CL (with the pycnocline inside) that provides the desired evolution equations. In the problem being studied, stratification is weak and passing through the pycnocline changes not so much the eigenfunction itself as its derivative. Hence, to the leading order, g+ (zN ) = g− (zN ) = g∗ . Assuming g∗ = 1, we obtain the expansion when |z − zN | ≡ L|Z| 1 (see [18]) (1)

wn± = −ikn An (τ )[1+L(

UN ln |LZ|+an±)Z + O(L2 |Z| ln |LZ|)]. UN

(6)

Coefficients a+ (Kn ) and a− (Kn ) are real, and can be found only by solving the problem (5). The functions gn±(z) ≡ g± (z; Kn ), when analytically continued into the lower half-plane of the complex z, are linearly independent, and their Wronskian a∗ (Kn ) = gn+ (z) gn− (z) − gn+ (z) gn− (z) = an− − an+ + iπUN /UN = 0

(7)

has the real part proportional to the eigenfunction derivative jump over CL. For each n, equating the wn derivative jump over CL with the O(Lεn ) change in the derivative of the Wn component of the inner solution provides the matching condition ˆ ˆ ˆ R i − dZWn ZZ , − dZ(. . . ) = lim εn [a+ (Kn ) − a− (Kn )]An (τ ) = dZ(. . . ), (8) R→∞ −R kn L

338


which yields the nonlinear evolution equation for the corresponding wave. Finally, solving Eqs.(3 c–e) yields horizontal velocity, (1)

un± = (

q2n U gn± kn2 dgn± + )An Kn2 dz Kn2 (U −UN )

U gn± ikn qn dgn± (1) − ( )An vn± = Kn2 dz U −UN

−→ z→zN

−→ z→zN

[

q2n + O(ln |LZ|)]An , LKn2 Z

(9)

−kn qn + O(ln |LZ|)]An . [ LKn2 Z

3 Inner problem and the linear stage of evolution Inside the critical layer, it is convenient to introduce new variables, Z = (z − zN )/L, N(Z) = L n(z);

w = W, (u, v, G) = L−1 (U , V, Γ), ρ = L−2 P/UN ,

˜ N 2 . Then, accurate as well as the time τ = UN τ (in what follows, the prime for τ will be omitted) and R˜ = J/U to exponentially small terms (compare with Eq.(1)), ˆ ∞ ˆ ∞ dZ N(Z) = 1, dZ Z N(Z) = 0, −∞

−∞

and Eqs.(3) take the form (we retain only principal terms necessary for further analysis; derivatives in τ , x, y, Z are denoted by corresponding subscripts, fN ≡ f (zN )): (

2 ˜ LUN ∂ ∂ + Z )WZZ = − R(P + P ) + (Wx − 12 Z 2WZZx ) xx yy ∂τ ∂x L UN +

( (

1 L2UN

∂ ∂ [Ux2 +Vy2 +WZ2 + 2UyVx + 2UZWx + 2VZ Wy − (U Wx +VWy +WWZ )], ∂Z ∂Z (10 a)

LU 1 ∂ ∂ + Z )P − N(Z)W = − N Z 2 Px − 2 (U Px +V Py +W PZ ), ∂τ ∂x 2UN L UN

(10 b)

LU ∂ ∂ + Z )Γ +Wy = − N (ZWy + 12 Z 2 Γx ) ∂τ ∂x UN

(10 c)

∂ ∂ 1 + 2 [ (U Vx +VVy +WVZ ) − (U Ux +V Uy +W UZ )], L UN ∂ x ∂y Ux +Vy = −WZ ,

Γ = Uy −Vx .

(10 d, e)

Let us begin with a linear part of the inner solution. In line with Eq.(4), we represent it as a sum of waves, and each wave as an expansion (1)

(1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

(2)

(Un ,Vn ,Wn , Pn , Γn ) = εn [(Un ,Vn ,Wn , Pn , Γn ) + L(Un ,Vn ,Wn , Pn , Γn ) + . . . ]. (1)

1. O(εn ). In this order, Eq.(10 a) yields homogeneous equations LnWn ZZ = 0 where Ln = L (kn ) and L (k) = (1) ∂ /∂ τ + ikZ. Integrating them with the ‘initial’ conditions Wn (τ , Z) → 0 as τ → −∞ we obtain Wn ZZ = 0 and, after matching to the outer solution (Eq.(6)), (1)

Wn

= −ikn An (τ ).

(11)


339

Other contributions to the inner solution, both linear and nonlinear, obey the equations of the form L (k) f (τ , Z) = R(τ , Z) with the solution ˆ ∞

f (τ , Z) =

0

dsR(τ − s, Z) exp(−ikZs)

(12)

satisfying f (τ , Z) → 0 as τ → −∞. Substituting Eq.(11) into Eq.(10 b) and using Eq.(12) one obtains (1)

(1)

(1)

Ln Pn = N(Z)Wn = −ikn N(Z)An (τ ) and Pn = −ikn N(Z) An (τ , Z) where (m) An (τ , Z) =

ˆ

∞ 0

(0)

dssm An (τ − s) exp(−ikn Zs), An (τ , Z) ≡ An (τ , Z).

(13)

(m)

Notice that An (τ , Z) decrease as |Z|−(m+1) in the lower half-plane and increase indefinitely in the upper one when |Z| → ∞, and that complex conjugation reverses this property. (1) Further, we employ Eqs.(10 c–e) to find Γn and horizontal components of velocity, (1)

Γn = −kn qn An ,

(1)

Un

=

ikn q2n An , Kn2

(1)

Vn

=−

ikn2 qn An , Kn2

(14)

which, as can be easily verified, are automatically matched to Eqs.(9). (2) (2) 2. O(L εn ). In this order, we obtain the equation LnWn ZZ = kn2 (UN /UN )An (τ ), and its solution is Wn ZZ = kn2 (UN /UN )An or, after integrating by parts, (2)

Wn ZZ = −

iknUN ∂ An (An (τ ) − ). UN Z ∂τ

The right-hand side is regular in Z = 0. Therefore, its integral in Z does not change if we bypass this point from below. Since the term containing ∂ An /∂ τ is regular and decreases as Z −2 (when |Z| → ∞) in the lower half-plane of complex Z, its integral in Z vanishes, and ˆ iπU (2) − dZWn ZZ = −ikn ( N An (τ )). UN 3. O(εn 2 /L). In this order, we calculate the correction for stratification, (3)

(1)

(3)

(1)

˜ n = −ikn Kn2 R˜ N(Z) An (τ , Z), Wn ZZ = −ikn Kn2 RN(Z) ˜ An LnWn ZZ = Kn2 RP and its contribution to Eq.(8), ˆ ˆ (3) 2˜ − dZWn ZZ = −ikn Kn R

∞

ˆ dZN(Z)

−∞

0

∞

dssAn (τ − s) exp(−ikn Zs).

Now, we have calculated all the major linear contributions to Eq.(8) due to the inner solution. Rearranging them to the left-hand side and keeping nonlinear terms (that have yet to be calculated) on the right, we obtain the system of equations (with a∗ (Kn ) defined by Eq.(7)) ˆ ˆ ∞ ˆ ∞ 2 i (N) − dZWn ZZ . dZN(Z) dt tAn (τ − t) exp(−ikn Zt) = − (15) a∗ (Kn )An (τ ) + 2 Kn2 R˜ L k ε L n n −∞ 0 During the linear stage of perturbation development, their right-hand sides are negligible and wave amplitudes grow exponentially, An (τ ) = An0 exp[−ikn (Zn + iΔn )τ ], with growth rates γn = kn Δn which satisfy the dispersion equations ˆ ∞ Kn2 R˜ dZN(Z) (Im (X ) > 0). (16) a∗n ≡ a∗ (Kn ) = 2 N (Zn + iΔn ), N (X ) = 2 kn −∞ (Z − X )

340


4 Nonlinear stage of evolution Let us begin our analysis of nonlinear contributions to evolution equations with a surprising fact. Although in the problem at hand the pycnocline is inside CL, i.e., just there where major nonlinear interactions occur, stratification has no significant impact on nonlinearity because the contribution of density perturbation to W is reduced by the factor 2 /L 1 (see Eq.(10a)). In view of this, the main contribution is due to nonlinear terms constructed of velocity components (see Eqs.(10 a,c)). As is well known [21], weakly nonlinear wave interaction produces pulsations with wave vectors and frequencies which are every possible sums and differences of wave vectors and frequencies of waves belonging to the ensemble studied (recall that wave parameters are chosen in such a manner that x-components kn of wave vectors kn and frequencies ωn ≈ UN kn are positive, see Eq.(4)). If the wave vector of the beating of (r − 1) waves is equal to the wave vector of one of the ensemble waves, kn , the r–wave resonance (or interaction) takes place. We shall denote it by the symbolic formula kn = k1 ± k2 ± · · · ± kr−1 with proper choice of signs.b The contribution of such an interaction to the right-hand side of Eq.(15) is obviously proportional to the product of the amplitudes of the waves whose wave vectors enter into the symbolic formula with positive sign, and the complex conjugated amplitudes of the other waves that produce the beating. There are two fundamental consequences of the fact that the nonlinearity is formed inside CL. Firstly, nonlinear wave interactions are ‘amplified’ inside CL. Indeed, nonlinear terms in the right-hand sides of Eqs.(10 a,c) are not simply the products of the perturbation components but the products divided by L2 , and owing to this the outcome of a r-wave interaction becomes L−2(r−2) 1 times greater. Secondly, successful interactions follow a kind of ‘selection rule’. Namely, (N) for the integral of Wn ZZ over CL be nonzero, the integrand should not be a rapidly decreasing analytic function neither in the lower nor in the upper half-plane of complex Z. Hence, in view of the remark made after Eq.(13), only that wave interaction can contribute to the right-hand side of Eq.(15) which has at least one minus in its symbolic formula. As already mentioned, we study the evolution of an ensemble of the most unstable waves with growth rates γ (k) = O() close to maximum value and slowly-dependent on k (see Fig.2(a)). By the early nonlinear stage a wide spectrum of these waves is formed, with amplitudes of the same order, and it is correct to put the CL thickness L = and all εn = ε during not only the linear stage of development but also early nonlinear one. The contribution to the Eq.(15) from a r-wave interaction is of the order O(L−1 (ε /L2 )r−2 ), and it is easily seen that three-wave interactions come first into play as soon as the wave amplitudes achieve the level

ε = L 3 = 3 . 4.1

(17)

Three-wave stage of evolution

Three-wave interaction of Holmboe waves is studied in sufficient detail [19] and we present here only the results that are needed for demonstration of some ideas underlying the further analysis. In the triad k1 = k2 + k3 , the wave k1 has a maximal frequency. In view of the selection rule, only ‘de(N) cay’ processes (k1 − k3,2 −→ k2,3 , W2,3 ∼ A1 A¯3,2 ; the bar denotes complex conjugate) contribute to Eqs.(15) (N)

whereas ‘fusion’ processes (k2 + k3 −→ k1 , W1 ∼ A2 A3 ) do not, in contrast with conventional (i.e., without CL) three-wave interactions in which the decay and fusion processes compete on equal terms (see, for example, [21]). As a result, in three-wave interaction the highest-frequency wave (k1 ) has an impact on the development of lower-frequency waves (k2 and k3 ), ˆ ∞ dZN(Z) dt tA2,3 (τ − t) exp(−ik2,3 Zt) −∞ 0 ˆ ∞ k1 dt t 2 A1 (τ − t)A¯3,2 τ − t , = Vk2,3 k1 k3,2 k3,2 0

ˆ 2 ˜ R a∗ (K2,3 ) A2,3 (τ ) + K2,3

b Because

∞

phase velocities of all waves are practically the same, corresponding frequency detuning is sufficiently small.

(18)


Vk k1 k2 =

2π k12 q2 (k k1 )[k k1 ]z , K12 K22 UN

341

(k k1 ) = k k1 + q q1 , [k k1 ]z = k q1 − k1 q,

but does not experience their reverse effect. During this stage, the behaviour of wave amplitudes turns out to be dependent on the configuration of wave ensemble [19]. In an isolated triad of interacting waves, the wave k1 rises with its linear growth rate γ1 and has a catalyzing effect [6] on the other two waves parametrically amplifying their growth up to a super-exponential one (∼ exp(aeγ1 τ /3 )). If, however, the ensemble contains many waves, the most high-frequency of them does also rise with the linear growth rate and play only a catalyzing role. And any of the other waves can be parametrically amplified in some triads and serve as a catalyst in the others, stimulating the waves coupled with it to grow faster. In order to describe them, we extend Eq.(18) and construct the nonlinear evolution equation (hereafter referred to as NEE) for some Ak (τ ) by inclusion into its right-hand side the sum of contributions of all the triads containing k, ˆ ∞ ˆ ∞ dZN(Z) dt tAk (τ − t) exp(−ikZt) a∗ (K)Ak (τ ) + K 2 R˜ −∞ 0 ˆ ∞ k dt t 2 Ak+k2 (τ − t)A¯k2 τ − t − t . (19) = ∑ Vk k+k2 k2 k2 0 k2 ,k2 >0 Solutions of these equations grow much faster, according to an explosive law, Ak (τ ) = Ck (τ∗ − τ )−α (k)+iβ (k),

α (k) > 0,

(20)

with the growth index α (k). The main consequence of the transition to an explosive growth is that the CL thickness becomes to be a rapidly growing dynamic variable rather than a parameter, L = |Ak |−1 d|Ak |/dτ ∼ /(τ∗ − τ ).

(21)

Let us substitute Eq.(20) into Eq.(19), calculate each of its terms separately and compare them. It is seen that the growth index is reduced by one with each integration in t and each t in the integrand. As a result, the first term on the left-hand side is growing with the index α (k), whereas the second term which is due to stratification increases much more slowly, with the index (α (k) − 2), and therefore becomes uncompetitive at the explosive stage and provides only small corrections to the law (20). Terms on the right-hand side of Eq.(19) have growth indexes (α (k + k2 ) + α (k2 ) − 3). In view of Eq.(21), integration with respect to t can be related to the L−1 factor in front of the integrals on the right-hand sides of Eq.(8) and Eq.(15), and t μ in the integrand can be related to the L−μ factor (μ = 1, 2) in front of the corresponding term on the right-hand side of Eq.(10 a). Since the growth rate of the sum of ‘explosive’ terms is determined by the highest index, we equate α (k) to the maximal growth index of terms on the right-hand side of Eq.(19) and obtain the equation

α (k) = max[α (k + k2 ) + α (k2 )] − 3. k2

(22)

Unfortunately, its obvious solution, α (k) ≡ 3, should be rejected because the most high-frequency wave has α = 0 (its amplitude growth is as slow as exponential). For further analysis let us impose the (commonly accepted in numerical simulation) condition of disturbance periodicity, with periods 2π /k in x and 2π /q in y. Then the ensemble should consist of waves with wave vectors of the form (mk, pq, 0), with integer 0 < m ≤ M and p. In particular (hereafter, zero z-components will be omitted), the wave vector of the highest-frequency wave kM = (Mk, pM q) (M ≥ 3 because M = 2 corresponds to isolated harmonic-subharmonic triad(s) that grows super-exponentially rather than explosively). In what follows, such an ensemble will be called the M-level ensemble, and the levels will be numbered by m. Waves belonging to the same level have nearly the same frequencies and differ in p (i.e., in the y-component of the

342


wave vector). We assume that, asymptotically, the wave amplitudes at all levels grow explosively with indexes αm depending on m only, and formally assign αM = 0 to the catalyzing wave. Then Eq.(22) yields

α1 = 3

3 M−1 = 3+ ≡ 3 + Δα , α2 = 3 and M−2 M−2

αm = 3

M−m = (M − m)Δα . M−2

(23)

Thus, the first-level waves have the maximum growth index, and the index decreases with m (with the step Δα ) to zero at m = M. Numerical calculations of the three-wave stage of development in ensembles with M = 3, 4 and 5 [19] have demonstrated that wave amplitudes grow explosively, with growth indexes prescribed by Eqs.(23). In particular, in ensembles with M = 3 the growth index of the first-level waves is twice as large as that on the second level, α1 = 6 and α2 = 3. 4.2

Post-three-wave stage: a qualitative analysis of interactions

Upon transiting to the explosive stage, wave amplitudes at different levels begin to grow with different (decreasing with m) rates, and the scaling (17) should be replaced by relations

εm = 3 (L/)αm , 0 < m < M.

(24)

Using them, one can estimate the contributions of nonlinear interactions of different orders to the right-hand side of NEE for a wave belonging to the m-th level. But first recall that CL gain for a r-wave interaction is equal to L−2(r−2) , and only that process can contribute to NEE which has at least one minus in its symbolic formula. An estimate of the maximum contribution of three-wave interactions to Eq.(15) is quite obvious, m =1

2 O(1), max O[εm1 εm2 /(εm L3 )] = max[(L/)(M−2m2 )Δα −3 ] =

m=m1 −m2

m2

where the levels of interacting waves are numbered by mi . Further, it is easily seen that only two types of fourwave interactions can contribute to Eq.(15), m = m1 + m2 − m3 and m = m1 − m2 − m3 . In the process of the first type, to the same m2 and m3 a wave with a smaller m1 corresponds, and therefore the contribution of such a process is greater. A simple estimate, max

6 L L L ε m1 ε m2 ε m3 m3 =1 = max( )(2M+m−m1 −m2 −m3 )Δα = max L( )2(M−m3 )Δα −6 = L( )2Δα , m3 (εm L) L4 L 5 m3

shows that this contribution becomes of the order unity when 2Δα

6

L = L4 = O( 1+2Δα ) = O( M+4 ), or, expressing εm in terms of L,

εm = O(

3(m+4) M+4

),

εm = O(L2+m/2 ).

(25) (26)

Assuming that such amplitudes are reached, we are able to evaluate the contributions of further orders. As earlier, we can conclude that the maximal r-wave contribution is due to the interactions of the form k = k1 + k2 + · · · + kr−2 − kr−1. Taking the x-component of this equality, we find m = m1 + m2 + · · · + mr−2 − mr−1 ,

(27)

and, using Eq.(26), obtain max

mr−1 =1 εm1 εm2 . . . εmr−1 L2(r−1)+(m1+m2 +···+mr−1 )/2 = max = max Lmr−1 −1 = 1. 2(r−2) 2+m/2+2r−3 mr−1 (εm L) L L

(28)

Thus, upon reaching the amplitudes described by Eq.(25), many interactions of various orders come simultaneously into play. Namely, all the interaction from the three– to the (m + 3)–wave inclusive begin to contribute


343

to the development of each wave belonging to the m-th level. In general, the exceptions are only the most highfrequency waves, i.e., waves of the M-th level, because their evolution is not affected by three-wave interaction. According to Eq.(28), the interactions of different orders no longer compete with each other and equally affect the development of disturbances. Therefore, the scaling (26) is maintained until the end of the weakly nonlinear stage of development, i.e., up to L = O(1) when all the wave amplitudes on all levels also become of the order unity. It is interesting to note that the amplitude level described by Eq.(25) is a threshold one for isolated triads of interacting waves as well (for them, M = 2). As soon as the amplitudes of two lower-frequency (or the ‘first-level’) waves, k2 and k3 , which increase super-exponentially, become O(5/2 ) they begin to affect the development of the third (higher-frequency, m = M = 2) wave k1 by four- (k1 = k1 + k2,3 − k2,3 ) and five-wave (k1 = k2 + k3 + k2,3 − k2,3 ) interactions, and they all begin to grow explosively, with indexes 3 and 5/2, see Eq.(26). 4.3

Post-three-wave stage: the structure of evolution equations

Since a large number of nonlinear interactions of diverse orders take part in the perturbation development, the deriving the evolution equations is extremely time-consuming, and the equations themselves are highly complicated. Based on the above qualitative considerations and an analysis of the process of pulsation generation inside CL, we try to recognize what the composition and structure of NEEs should be. These evolution equations should provide a correct description for all the stages of weakly nonlinear perturbation development including the linear stage, coming into play of nonlinear interactions which implies transition to an explosive growth, and further growth of wave amplitudes up to the order unity. For this purpose, NEEs should contain, like Eqs.(19), all the linear and three-wave contributions and, in addition, the major contributions due to higher-order interactions. Qualitative analysis of wave interactions made in Section 4.2 has shown that, after transiting to an explosive growth, a distinctive hierarchy of amplitudes is set up (see Eqs.(24) and (26)). Namely, the higher is the frequency of the wave (i.e., the level number m), the less is its amplitude, and only to the end of the weakly nonlinear stage all the amplitudes become of the same order of magnitude. Therefore, only that r-wave interaction is efficient which has one minus and mr−1 = 1 on the right-hand side of its formula (27), and the remaining processes are uncompetitive. And now the main objective is to establish the functional form of contributions to Eqs.(15) made by efficient higher-order interactions. To calculate these contributions, we retain left-hand sides and only nonlinear terms on the right-hand sides in Eqs.(10 a,c) because all the terms omitted have no significant impact on the nonlinearity. By way of illustration, quadratic and cubic in amplitude iterations are given in Appendix. (1) In the first (linear) iteration (see Eqs.(11) and (14)), vertical velocity Wn is independent of´ Z and pro∞ portional to the amplitude An (τ ) whereas the horizontal velocity is proportional to An (τ , Z) = 0 ds An (τ − s) exp(−ikn Zs), i.e., to the integral over all the history of perturbation development which contains Z only in the exponent. Step by step, we substitute calculated iterations into the right-hand sides of Eqs.(10 a,c) and, using Eq.(12), calculate the next iteration of the solution integrating over the history of perturbation development. It is easy to understand that the p-th iteration has the form of an (p + n − 1)–fold integral (n ≥ 0) over the time delays si or a sum of similar integrals. The integrand contains a product of p amplitude functions (each with its own time delay) multiplied by exp(−iΣZ) where Σ is a linear combination of si and by the kernel which is a homogeneous function of si as will be seen later. If the wave vector stands with minus in the symbolic formula of the beating the corresponding amplitude function is complex conjugated (see Appendix). Note that the differentiation with respect to Z does not break the kernel homogeneity because it is equivalent to multiplication by an degree one uniform polynomial of si . To continue the study, let us introduce the notations similar to those used in the dimensional analysis (see [22], for example) choosing amplitude (A) and time delay (s) as ‘units of dimension’ and putting [W (1) ] = A and

344


[Γ(1) ] = [U (1) ] = [V (1) ] = s A, in accordance with Eqs.(11) and (14).c Substituting into the right-hand sides of Eqs.(10 a,c) shows that they are homogeneous functions of s and A with ‘dimensions’ s3 A2 and s2 A2 respectively (2) so that [WZZ ] = s4 A2 , [W (2) ] = s2 A2 , and [Γ(2) ] = [U (2) ] = [V (2) ] = s3 A2 . Hence, passage to the next iteration adds s2 A to the ‘dimension’ and retains the uniformity of kernels. It is easily seen that this is true for each (p-th) iteration so that [W (p) ] = s2(p−1) A p and [Γ(p) ] = [U (p) ] = [V (p) ] = s2p−1 A p . (N) For a wave of the m-th level, we are now able to write the functional form of the contribution to WZZ due . It is a sum of terms to efficient r-wave interaction (see Eq.(27), mr−1 = 1). We denote this contribution by Wmr that differ in structure but all are homogeneous functions of si and amplitudes with ‘dimension’ s2(r−1) Ar−1 . A single term is a (r + n − 1)-fold integral (0 ≤ n < r − 1) of the form ˆ

∞ 0

ˆ

ds1 . . . dsr−1

0

∞

r−2

dsa . . . dsn K (s1 , . . . , sr−1 ; sa , . . . , sn ) e−ikσ Z ∏ A j (τ −S j )A¯r−1 (τ −Sr−1 ),

(29)

j=1

r−2

where σ = ∑ m j S j − Sr−1 , the kernel K is a homogeneous function of degree (r − 1 − n) of its arguments, j=1

each S j is a sum of some of the delays si (i = 1, 2, . . . , r − 1, a, b, . . . , n), and all S j are different (see Appendix, Eqs.(A1) – (A3)). Such a structure is typical for all contributions due to wave interactions, from three– to (m + 3)-wave (in the last case m1 = m2 = · · · = mm+2 = 1). into the right-hand side of the corresponding Eq.(15), integrating it over Z, and using the Substituting Wmr ´∞ relation −∞ dZe−iσ Z = 2πδ (σ ), we see that the integral (29) is nonzero if and only if σ can take on zero within the integration domain. In this case, the delta function allows us to perform integration with respect to one of si , d and therefore the ‘dimension’ of the contribution to Eq.(15) becomes s2r−3 Ar−1 . Let now find the τ -dependence of this contribution when the amplitudes grow explosively, Ai (τ ) ∼ (τ∗ − τ )−αmi , with αm = 2+m/2, see Eq.(26). Introducing new variables ti = si /(τ∗ − τ ) we see that the integral is proportional to (τ∗ − τ ) to the power −(2r − 3) + [2(r − 1) + (m1 + m2 + · · · + mr−2 + 1)/2] = [1 + (m + 2)/2] = αm that is independent of r, in accordance with qualitative analysis carried out in Section 4.2. 5 Numerical calculations In order to illustrate the above analysis, we have calculated a weakly nonlinear evolution of some ‘minimal’ three-level (M = 3) ensembles containing two by two waves at the levels m = 1 and m = 2, and one wave at the level m = 3. Let denote their wave vectors by kmn , where n = 1, 2 enumerates waves at the same (m-th) level in order of increasing ky . For wave interaction to be efficient the equalities k31 = k21 + k12 = k22 + k11 should be held and the waves of the first two levels should obey either k21 = k11 + k12 , k22 = k12 + k12 or k21 = k11 + k11 , k22 = k11 + k12 . In both cases results of calculation are similar. In what follows, we consider the ensemble of the latter kind with k11 = (k, −q), k12 = (k, 3q), k21 = (2k, −2q), k22 = (2k, 2q), k31 = (3k, q).

(30)

In addition, we have calculated the evolution of isolated wave triad k1 = k2 + k3 in two versions, with a frequency ratio of 1:2:3 (k1 = k31 , k2 = k22 , k3 = k11 ) and with an ‘oblique’ subharmonic (k1 = (0.9, 0.5), k2 = (0.45, −0.3), k3 = (0.45, 0.8)), and have obtained very similar results as well. Drawing up the schemes of interactions for these ensembles, we include in them all the three-wave interactions with a non-zero contributions, and only efficient higher-order processes. Also, it is taken into account c Superscript

indicates the order of iteration, and the power of s is the sum of the kernel’s power and the number of integrations. because the boundaries of new domain are defined by homogeneous equation σ = 0. d The change in integration domain due to such an integration does not break the homogeneity with respect to s i


345

that the interactions involving only the waves with collinear wave vectors are uncompetitive (for discussion, see [20, 23]), and that k11 and k21 are collinear. The number of interacting waves is indicated under horizontal braces. For isolated triad, we obtain the scheme k1 = k1 + k2 − k2 = k1 + k3 − k3 = k2 + k2 + k3 − k2 = k2 + k3 + k3 − k3 , 4

5

k2 = k1 − k3 = k2 + k3 − k3 , 3

4

k3 = k1 − k2 = k2 + k3 − k2 , 3

4

whereas for the ensemble (30) k11 = k31 − k22 = k22 − k12 = k11 + k12 − k12 , 4

3

k12 = k31 − k21 = k22 − k11 = k11 + k12 − k11 , 4

3

k21 = k31 − k12 = k21 + k12 − k12 = k22 + k11 − k12 = k11 + k11 + k12 − k12 , 4

3

5

k22 = k31 − k11 = k21 + k12 − k11 = k22 + k11 − k11 = k22 + k12 − k12 4

3

= k11 + k11 + k12 − k11 = k11 + k12 + k12 − k12 , 5

k31 = k31 + k11 − k11 = k31 + k12 − k12 = k21 + k22 − k11 = k22 + k22 − k12 4

= k21 + k11 + k12 − k11 = k21 + k12 + k12 − k12 = k22 + k11 + k11 − k11 5

= k22 + k11 + k12 − k12 = k11 + k11 + k11 + k12 − k11 = k11 + k11 + k12 + k12 − k12 . 5

6

It can be seen that even for such a small wave ensemble the number of interactions that should be taken into account is immense (for comparison, throughout the three-wave stage each wave is involved only in one or two interactions, see the scheme above and [19]). For this reason we do not write NEEs in the full form. We have found all the interaction coefficients for both isolated triad and ensemble (30) and calculated their development at k = 0.3 and q = 0.5. The numerical procedure is the same as described in [19]. Fig.3 shows the position of the ensemble waves inside the instability domain, and Fig.4 demonstrates the evolution of the isolated triad k1 = k31 , k2 = k22 and k3 = k11 with τ . In view of the above analysis, as soon as amplitudes attain the O(3 ) level, the linear stage of development (corresponding to rectilinear parts of amplitude traces) of waves k2 and k3 changes to the super-exponential growth whereas the wave k1 continues to grow exponentially. And when |A2 | and |A3 | become as large as 5/2 all the three waves begin to grow explosively. All these stages are clearly seen in Fig.4 and growth indexes are close to predicted. In Fig.5, the development with time of the ensemble (30) is shown, and Fig.6 demonstrates two successive stages of its explosive growth, three-wave and post-three-wave. All the regimes predicted can be easily identified and growth indexes have the proper values.e In conclusion, it should be noted that the three-wave and post-three-wave stages have different times of ‘explosion’, τA and τB respectively. e For

comparison, in Fig.4(b) and Fig.6 straight dashed lines are drawn, with inclinations corresponding to theoretical values of the growth index.

346


2 12

Spanwise wave number, ky

1

22 31

0 11 21

-1

-2

0

1 Streamwise wave number, kx

2

Fig. 3 Ensemble (30) in the instability domain.

6 Discussion The flows of the class under study (i.e., with an inflection–free velocity profile and sharply stratified) have a remarkable distinction (as compared with a homogeneous boundary layer, for example) giving a better insight into the mechanisms operating on the way from linear instability to a strongly nonlinear stage of perturbation development, i.e., into the mechanisms of transition to turbulence. Flows of this kind have a single mode of eigenoscillations known as Holmboe waves. At a fairly low stratification (J = O(2 )), the most unstable of them have not only very close growth rates but also very close phase velocities (and therefore, a common CL) and those waves form a significant part of the spectrum. As a result, the primary instability amplifies a wide threedimensional spectrum of Holmboe waves with amplitudes of the same order and common critical layer and, after transiting to a weakly nonlinear stage of development, these waves take part in diverse and intensive interactions. Due to stratification, the nonlinearity threshold (Eq.(17)) is low, ε L2 . Hence, by the early nonlinear stage, the mixing of fluid particles trapped by waves has actually no time to even begin, and perturbation growth does not slow down in the process of the nonlinear evolution but is accelerated to an explosive one [2], with growth indexes changing from one phase of development to another (see. Eqs.(23) and (26)). We have described all these processes and traced the evolution of perturbations from the linear stage up to the beginning of strongly nonlinear. Particular attention has been given to the stage of explosive growth. We have found that in the course of its second (post-three-wave) phase a rich variety of wave interactions comes into play and therefore should be taken into consideration. As a result, corresponding NEEs become too cumbersome, and their deriving in the

S.M. Churilov / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 333–351 28

20 16 12

2

8

1 4

3

0

24

 

 

3

20

16

2

  

24

Logarithmic wave amplitudes, ln |An|


28

347

12

-4

8

1

4 -4

-2

0

2

4

Time, 

6

1

2

3

4

5

6

7

Logarithmic time, -ln[

(a)

8

(b)

Fig. 4 Evolution of isolated triad k1 = k2 + k3 : (a) as a whole, and (b) at the explosive stage; curves are marked by ‘numbers’ of waves, τ∗ = 5.124.


40

30

20

11 10

31 22

0

21

12

-10 -3

-2

-1

0

1

Time, 

2

3

4

5

Fig. 5 Development of the ensemble (30).

regular way (as in the case of three-wave interactions [19], for example) seems almost impossible to be done. Therefore, in this study we relied on a qualitative analysis of the problem and, owing to it, were able to found the structure and key properties of NEEs and to analyze the behavior of their solutions. In the problem at hand, the pycnocline turns out to be immersed into the unsteady critical layer with a rapidly changing flow structure, and therefore both the dispersion and nonlinear terms in evolution equations are dependent on the whole history of the disturbance development, and the equations themselves have the form of a system of integral equations, see Eqs.(18) and (19), for example. The objective was to establish the functional form (or, more precisely, scaling properties) of the major nonlinear contributions to NEEs made by higher-order

348


40

8

2

=

22

3

α=

3

11

1 0 -1

21

-2

12

-3

12

30

11

20

α

22

=

3

21

10

7/ 2

6

31

4

α

/2 =5

31

=

5

n

α

Logarithmic wave amplitudes, ln |A |

6

α

Logarithmic wave amplitudes, ln |A |

7 n

-4 -5

0

0.0

0.5

1.0

Logarithmic time, -ln[(τΑ−τ)/τΑ]

(a)

1.5

0

2

4

6

8

10

Logarithmic time, -ln[(τΒ−τ)/τΒ]

(b)

Fig. 6 Three-wave (a) and post-three-wave (b) stages of an explosive growth of the ensemble (30); τA = 3.516, τB = 4.502.

wave interactions, without making bulky calculations. It was shown first that, despite the pycnocline and CL merge together, stratification has no significant impact on the nonlinearity, moreover, after transiting to explosive stage, it ceases to affect the perturbation development as well. Then, CL gain for wave interactions was found, and interactions which can contribute to NEEs were selected (see the second paragraph of Section 4). This allowed to show that the main contribution to NEEs due to r-wave interactions should be a homogeneous function of not only the wave amplitudes (of degree (r − 1), evidently) but, more importantly, also time delays si (of degree (2r − 3)). Thus, we were able, firstly, to make sure that the structure of NEEs is compatible with the (presumed) regime of explosive growth and, secondly, to find the growth index of each wave and to ascertain that during the post-three-wave stage wave interactions of all orders equally affect the perturbation development. Finally, for a few very small wave ensembles we have derived, in a regular manner, all the nonlinear interaction terms and numerically calculated the evolution of wave amplitudes (see Section 5). The numerical and theoretical results are in good agreement and this fact lends support to analysis made. The waves being studied have very close phase velocities, i.e., a weak dispersion. By analogy with the acoustics, one could expect that the rapid growth of high-frequency harmonics accompanied by formation of fronts and structures similar to weak shock waves [24, 25] should take place. However, during the initial (threewave) stage of nonlinear development the scenario is exactly the opposite. Namely, the fastest-growing is a low-frequency component of the spectrum (waves with small m, see Eq.(23)), and the disturbance becomes even more smooth. The reason for this is in a catalytic character of three-wave interactions within the CL where the wave fusion (k2 + k3 −→ k1 ) is forbidden. And only on the next (post-three-wave) stage, when a lot of interactions of diverse orders come into play, the growth of high-frequency part of the spectrum is accelerated. Nevertheless, wave amplitudes in low- and high-frequency parts of the spectrum become of the same order of magnitude only to the end of weakly nonlinear stage of perturbation development. Unfortunately, we can not yet say what exactly should be the amplitude spectrum of Holmboe waves towards the end of the weakly nonlinear stage of development. More specifically, it is hard to tell whether the spectrum is determined mainly by an accidental choice of wave amplitudes and phases at the initial (i.e., linear) stage, or


349

it tends to some universal form. On the one hand, the growth of the waves is very fast (explosive) and there is no sufficient time for phase mixing and ‘forgetting’ the initial conditions. But on the other hand, as is shown above, each wave is involved in a plethora of diverse interactions with other waves, and it may be that scaling properties which help to find growth indexes of all waves enable us to find their amplitude spectrum as well. At least some features of a self-organization of perturbation development seem to be observed (see Discussion section in [19]). Acknowledgements The work was supported in part by RFBR Grants No. 10-05-00094 and No. 14-05-00080. APPENDIX

Beats of two and three waves (N)

Here we write out contributions to WZZ due to beats of two and three waves in order to illustrate some points of the above analysis. Let us start with quadratic in amplitude contributions corresponding to the three-wave interactions. Substituting Eqs.(11) and (14) into the right-hand sides of Eqs.(10 a,c) and using Eq.(12) as well as the notations introduced in Eqs.(18), we find the contribution at the sum frequency (k = k1 + k2 , ω = ω1 + ω2 ),

k1 +k2 = WZZ

ˆ ∞ ik1 k2 k2 q1 k1 k2 (k1 k2 ) k2 q2 [k , k ] ( ds1 ds2 [ 1 2 s21 − (q1 − q2 )s1 s2 − 2 2 s22 ] 1 2 z 2 2 UN K1 K1 K2 K2 0

(1)

×A1 (τ −s1 ) A2 (τ −s2 ) exp(−i[k1 s1 + k2 s2 ]Z) − ˆ ·

k1 k2 (k1 k2 ) (q1 + q2 ) K12 K22 ∞ 0

dsds1 ds2 (s1 − s2 ) A1 (τ −s−s1 ) A2 (τ −s−s2 ) exp(−i[k1 (s + s1 ) + k2 (s + s2 )]Z)),

and that at the difference frequency (k = k1 − k2 , ω = ω1 − ω2 ),

k1 −k2 = WZZ

ˆ ∞ ik1 k2 k2 q1 k1 k2 (k1 k2 ) k2 q2 [k , k ] ( ds1 ds2 [ 1 2 s21 − (q1 + q2 )s1 s2 + 2 2 s22 ] 1 2 z 2 2 UN K1 K1 K2 K2 0

(2)

×A1 (τ −s1 ) A¯2 (τ −s2 ) exp(−i[k1 s1 − k2 s2 )]Z) ˆ ∞ k1 k2 (k1 k2 ) (q1 − q2 ) dsds1 ds2 (s1 − s2 ) A1 (τ −s−s1 ) A¯2 (τ −s−s2 ) exp(−i[k1 (s + s1 ) − k2 (s + s2 )]Z) − K12 K22 0 (N)

Beats of three waves produce cubic in amplitude contributions to WZZ looking much more cumbersome. Let us take, for example, an efficient interaction k = k1 + k2 − k3 . Since Eqs.(10) have quadratic nonlinearity, the contribution of this process is the sum of three, in accordance with the representation k = (k1 + k2 ) − k3 = (k1 − k3 ) + k2 = (k2 − k3 ) + k1 . We write out only the first of them:

350


(k +k2 )−k3

WZZ1

= −

ˆ ∞ ik1 k2 k3 q3 [k +k , k ] ( ds ds1 ds2 ds3 (k1 s1 + k2 s2 − k3 s3 ) 1 2 3 z K32 U 2N 0

×{[k1 , k2 ]z [1 − −

(3)

2((k1 +k2 )k3 ) k3 s3 k1 q1 k1 +k2 k1 k2 s1 s2 + ][ 2 s1 + 2 2 [k1 , k2 ]z 2 (k1 +k2 ) k1 s1 +k2 s2 K1 K1 K2 k1 s1 +k2 s2

k2 q2 2[k1 +k2 , k3 ]z k1 q1 s1 2 k2 q2 s2 2 s2 ] − [ (K1 + (k1 k2 )) + (K2 + (k1 k2 ))]} 2 2 2 (k1 +k2 ) K2 K1 K22

×A1 (τ −s −s1 ) A2 (τ −s −s2 ) A¯3 (τ −s −s3 ) exp(−i[k1 (s+s1 )+k2 (s+s2 )−k3 (s+s3 )]Z) ˆ ∞ k1 k2 (k1 k2 )[k1 , k2 ]z (s1 −s2 ) ds dsa ds1 ds2 ds3 [k1 (s1 +sa )+k2 (s2 +sa )−k3 s3 ]{ 2 2 −(q1 + q2 ) K1 K2 [k1 (s1 +sa ) + k2 (s2 +sa )] 0 ×[1 − ×( −

2((k1 +k2 )k3 ) 2[k1 +k2 , k3 ]z k3 s3 [k1 , k2 ]z ]− + [k1 +k2 − 2 2 (k1 + k2 ) k1 (s1 +sa ) + k2 (s2 +sa ) (k1 + k2 ) k1 s1 +k2 s2

k1 q2 k2 q1 k1 k2 (k1 k2 )[k1 , k2 ]z (q1 +q2 )sa (s1 −s2 ) s1 − 2 s2 ) + 2 2 2 K2 K1 K1 K2 (k1 s1 +k2 s2 )[k1 (s1 +sa ) + k2 (s2 +sa )]

k1 +k2 k1 k2 s1 s2 [k1 , k2 ]2z ]} A1 (τ −s −s1 −sa ) A2 (τ −s −s2 −sa ) A¯3 (τ −s−s3 ) K12 K22 (k1 s1 +k2 s2 )2

× exp(−i[k1 (s+s1 +sa )+k2 (s+s2 +sa )−k3 (s+s3 )]Z)) ˆ ∞ k12 q1 2 ik1 k2 k3 ( ds ds ds (k s +k s ){((k +k −k )(k +k ))[k , k ] [ s − 1 2 3 1 1 2 2 1 2 3 1 2 1 2 z K12 1 (k1 +k2 )2U 2N 0 −

k22 q2 2 k1 k2 (k1 k2 ) k1 q1 s1 2 s2 − (q1 −q2 )s1 s2 ] − [k1 +k2 , k3 ]z (k1 s1 +k2 s2 )[ (K1 + (k1 k2 )) 2 2 2 K2 K1 K2 K12

+

k2 q2 s2 2 (K2 + (k1 k2 ))]} A1 (τ −s1 −s3 ) A2 (τ −s2 −s3 ) A¯3 (τ −s3 ) K22

× exp{−i[k1 (s1 + s3 ) + k2 (s2 + s3 ) − k3 s3 ]} ˆ ∞ k1 k2 (k1 k2 ) ds ds1 ds2 ds3 [k1 (s+s1 )+k2 (s+s2 )]{ ((k1 +k2 −k3 )(k1 +k2 ))[k1 , k2 ]z −(q1 +q2 ) K12 K22 0 ×(s1 −s2 ) − [k1 +k2 , k3 ]z [k1 (s+s1 )+k2 (s+s2 )][k1 +k2 − −

[k1 , k2 ]z k1 q2 k2 q1 ( 2 s1 − 2 s2 ) k1 s1 +k2 s2 K2 K1

k1 k2 (k1 k2 )[k1 , k2 ]z (q1 +q2 )s(s1 −s2 ) k1 +k2 k1 k2 s1 s2 ]} [k1 , k2 ]2z + 2 2 2 2 2 K1 K2 (k1 s1 +k2 s2 ) K1 K2 (k1 s1 +k2 s2 )[k1 (s+s1 ) + k2 (s+s2 )]

×A1 (τ −s −s1 −s3 ) A2 (τ −s −s2 −s3 ) A¯3 (τ −s3 ) × exp(−i[k1 (s+s1 +s3 )+k2 (s+s2 +s3 )−k3 s3 ]Z)). As would be expected, the right-hand sides of both Eqs.(A1) and (A2) are homogeneous functions of degree k1 +k2 decreases as |Z|−4 in the lower half-plane, it does not 4 in si whereas that of Eq.(A3) has degree 6. Since WZZ


351

k1 −k2 contribute to the right-hand side of the corresponding Eq.(15) whereas WZZ does (compare with the nonlinear term in Eq.(19)): ˆ ∞ ˆ 2iπ k12 q2 (k1 − k2 ) 2 k1 k1 −k2 2 ¯ = − (k k ) [k , k ] ds s A ( τ − s) A τ − s . K − dZWZZ 1 2 1 2 z 1 2 1 k2 K12 K22UN 0 (k +k )−k

For the same reason, WZZ 1 2 3 does contribute to Eq.(15) as well. And finally, it should be emphasized that the exponents in Eqs.(A1) – (A3) are constructed from the time delays S j of wave amplitudes exactly in that manner as described after Eq.(29). References [1] Maslowe, S.A. (1986), Critical layers in shear flows, Annual Reviews in Fluid Mechanics, 18, 405–432. [2] Churilov, S.M. and Shukhman, I.G. (1996), Critical layer and nonlinear evolution of disturbances in weakly supercritical shear layer, Izvestiya, Atmospheric and Oceanic Physics, 31, 534–546. [3] Timofeev, A.V. (1971), Oscillations of inhomogeneous flows of plasma and liquids, Sov. Phys. Uspekhi, 13, 632-646. [4] Andronov, A.A. and Fabrikant, A.L. (1979) Landau damping, wind waves and whistle. In: Nonlinear Waves (edited by A.V. Gaponov-Grekhov), 68–104, Nauka, Moscow (in Russian). [5] Fabrikant, A. (2002), Plasma-hydrodynamic analogy for waves and vortices in shear flows. Sound-Flow Interactions. Lecture Notes in Physics, Vol. 586 (edited by Aurégan Y. et al.), 192–209, Springer-Verlag, New York. [6] Wu, X. and Stewart, P.A. (1996), Interaction of phase-locked modes: a new mechanism for the rapid growth of threedimensional disturbances, Journal of Fluid Mechanics, 316, 335–372. [7] Holmboe, J. (1962), On the behaviour of symmetric waves in stratified shear layers. Geofys. Publik., 24, 67–112. [8] Garaud, P. (2001), Latitudinal shear instability in the solar tachocline, Monthly Notices of the Royal Astronomical Society, 324, 68–76. [9] Gavrilov, N.M., Fukao, S., Hashiguchi, H., Kita, K., Sato, K., Tomikawa, Y. and Fujiwara, M. (2006), Combined MU radar and ozonesonde measurements of turbulence and ozone fluxes in the tropo-stratosphere over Shigaraki, Japan, Geophysical Research Letters, 33, L09803. [10] Yoshida, S., Ohtani, M., Nishida, S. and, Linden, P.F. (1998), Mixing processes in a highly stratified river. In: Physical Processes in Lakes and Oceans (edited by J. Imberger), 389-400. AGU, Washington, DC. [11] Tedford, E. W., Carpenter, J. R., Pawlowicz, R., Pieters, R. and Lawrence G. A. (2009), Observation and analysis of shear instability in the Fraser River estuary, Journal of Geophysical Research, 114, C11006. [12] Carpenter, J.R, Tedford, E. W., Rahmani, M. and Lawrence, G. A. (2010), Holmboe wave fields in simulation and experiment, Journal of Fluid Mechanics, 648, 205–223. [13] Churilov, S.M. (2008), Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points. Part 2. Continuous density variation, Journal of Fluid Mechanics, 617, 301–326. [14] Churilov, S.M. (2010), Three-dimensional instability of shear flows with inflection-free velocity profiles in stratified media with a high Prandtl number, Izvestiya, Atmospheric and Oceanic Physics, 46, 159–168. [15] Squire, H.B. (1933), On the stability of three-dimensional disturbances of viscous flow between parallel walls, Proceedings of the Royal Society London A, 142, 621–628. [16] Yih, C.-S. (1955), Stability of two-dimensional parallel flows for three-dimensional disturbances, Quarterly Applied Mathematics, 12, 434–435. [17] Smyth, W. D. and Peltier, W. R. (1990), Three-dimensional primary instabilities of a stratified, dissipative, parallel flow, Geophysical and Astrophysical Fluid Dynamics, 52, 249–261. [18] Drazin, P.G. and Reid, W.H. (2004), Hydrodynamic Stability, 2nd edition, Cambridge University Press, Cambridge. [19] Churilov, S.M. (2011), Resonant three-wave interaction of Holmboe waves in a sharply stratified shear flow with an inflection-free velocity profile. Physics of Fluids, 23, 114101. [20] Churilov, S.M. (2009), Nonlinear stage of instability development in a stratified shear flow with an inflection-free velocity profile. Physics of Fluids, 21, 074101. [21] Craik, A.D.D. (1985), Wave Interactions and Fluid Flows, Cambridge University Press, Cambridge. [22] Bridgman, P.W. (1932), Dimensional Analysis, Yale University Press, New Haven. [23] Churilov, S.M. and Shukhman I.G. (1994), Nonlinear spatial evolution of helical disturbances to an axial jet, Journal of Fluid Mechanics, 281, 371–402. [24] Landau, L.D. and Lifshitz, E.M. (1987), Fluid Dynamics, Pergamon Press, Oxford. [25] Moiseev, S.S., Sagdeev, R.Z., Tur, A.V. and Yanovsky, V.V. (1979), Effect of vortices on acoustic turbulence spectrum. In: Nonlinear Waves (edited by A.V. Gaponov-Grekhov), 105–115, Nauka: Moscow (in Russian).



Transient Free Surface Flow Past a Two-dimensional Flat Stern Osama Ogilat1,2, Yury Stepanyants2† 1,2 Jerash

University, Irbid international Street, Jerash, Amman, 26150, Jordan;

2 University

of Southern Queensland, West St., Toowoomba, QLD, 4350, Australia Submission Info

Communicated by S.V. Prants Received 16 November 2014 Accepted 2 March 2015 Available online 1 October 2015 Keywords Surface wave Stern model Transient flow Steady state

Abstract A transient free surface flow past a two-dimensional semi-infinite flat plate in the fluid of a finite depth is considered in the linear approximation. It is assumed that the fluid is inviscid and incompressible and the flow is irrotational. The plate is suddenly submerged at relatively small depth below the free surface into the fluid uniformly moving with a constant velocity. The linearized problem is solved for relatively small Froude numbers F < 1 using the Laplace and Fourier transforms, as well as the Wiener– Hopf technique. It is shown that eventually at large time, the transient solution approaches asymptotically the steady-state solution. Peculiarities of the solution obtained are discussed and illustrated graphically. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The analytical model for transient two-dimensional free surface flow past a semi-infinite flat plate in a fluid of finite depth is considered in this paper. It is shown that the solution obtained in the linear approximation asymptotically approaches the steady state solution at large times, t → ∞. In the near-wake zone close to the separation point where the fluid detaches from the plate, the problem considered can be treated as the model of a flow behind a ship stern. Such model can be used to gain insight into the physical phenomena occurring in water behind a real ship having a wide blunt stern. The results obtained in this paper can be treated also as the proof of stability of an earlier derived steady-state solution [1]. These results demonstrate that the surface disturbance converges to the steady state after some transient period following the sudden submerging of a plate into water. The steady state is exactly that which has been earlier calculated for the plane stern in the paper by Ogilat et al. [1]. Two-dimensional free surface flows past a semi-infinite plate has been a subject of a big interest for the last three decades beginning with the steady-state problem considered by Vanden-Broeck [2], Schmidt [3], Zhu & Zhang [4]. In his paper, Vanden-Broeck [2] solved the fully nonlinear problem numerically and generated waves downstream where the flow separates tangentially from the flat stern. Schmidt [3] studied the steadystate problem analytically in the linear approximation case and showed that for certain special plate shapes the † Corresponding


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O. Ogilat, Y. Stepanyants / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 353–369

downstream waves can be completely eliminated. Zhu & Zhang [4] considered analytically the time dependent linear problem and showed that the steady state is asymptotically achievable in the long-time limit. Solving the steady problem by means of the Wiener–Hopf technique the authors showed that the closed form solution in the far field zone can be obtained for an arbitrary hull shape. In the further numerical studies of the fully nonlinear problem Madurasinghe & Tuck [5] and Farrow & Tuck [6] showed that generated waves cannot be completely eliminated if the free surface detaches from the stern at a stagnation point. Haussling [7] treated transient linear and √ nonlinear problems numerically for the ship model moving with draught-defined Froude numbers Fr ≡ V / gd ∈ [1, 4], where V is ship velocity, g is acceleration due to gravity, and d is the draught of the ship. √ (Notice that this definition of the Froude number differs from the depth-defined Froude number F ≡ V /p gh used here, where h is the fluid depth. The relationship between these two definitions is given as F = Fr d/h. Therefore in our notation, the results obtained by Haussling [7] pertain to small Froude numbers since usually d/h ≪ 1). It was shown that for Fr > 3 the nonlinear effects are negligibly small for most practical purposes, whereas for Fr < 3 these effects may be significant. The studies mentioned above were all performed for a fluid of infinite depth. For a fluid of finite depth, McCue & Stump [8] considered linear problem using the Wiener–Hopf technique for a special case of a flat plate, McCue & Forbes [9] and Maleewong & Grimshaw [10] treated the particular case of a flat plate using the boundary integral equation method, whereas Binder [11] and Ogilat et al. [1] considered the more general problem of curved semi-infinite plate, using the Wiener–Hopf technique, boundary integral method and weakly nonlinear approximation. These works deal with a steady forced Korteweg–de Vries (KdV) equation to gain an insight into what types of plate shapes can eliminate waves. In this paper we consider a linearized problem of surface wave generation in the fluid of a finite depth behind a flat semi-infinite plate. It is shown that the resulting mixed boundary-value problem can be solved analytically by means of Laplace transform on time and a Fourier transform on horizontal coordinate x. We also apply the Wiener–Hopf technique and show that the transient flow arising, when the plate is suddenly submerged into small depth d under water surface, asymptotically approaches to steady-state flow early derived by Ogilat et al. [1]. It is assumed that the fluid is inviscid and incompressible, surface tension is neglected, and fluid flow is irrotational.

2 Linear analysis 2.1

Mathematical formulation

Unsteady two-dimensional free surface flow past a semi-infinite plate in a fluid of finite depth is considered, as shown in Figure 1(a). Cartesian coordinates are introduced such that the x-axis ˜ lies on the bottom of the fluid domain and the y-axis ˜ is directed vertically upwards. For x˜ < 0, t˜ = 0, the fluid is bounded above by a semiinfinite flat plate, and for x˜ > 0, t˜ = 0, the upper boundary consists of the free surface. It is assumed that the flow for x˜ < 0 is that of a uniform stream of depth H and speed c, as shown in Figure 1(a). For the dimensional variables we use tilde reserving the same notations free of tilde for the corresponding dimensionless variables. Suppose now that the plate is suddenly submerged into the fluid at a depth d ≪ H below the undisturbed free surface at t˜ > 0+ , as shown in Figure 1(b). The distance between the horizontal bottom and the plate is h so that H = h + d. Assuming that the speed of upstream fluid is uniform at the constant depth h and equal to V , we can obtain the downstream fluid velocity from the mass conservation principle: Vdown = V h/(h + d). As the flow is irrotational, the x˜ and y˜ components of the velocity field can be presented through the velocity potential Φ: u˜ =

∂Φ , ∂ x˜

v˜ =

∂Φ . ∂ y˜

(1)

Substituting these expressions into the continuity equation, ∇ · v = 0, where v = (u, ˜ v), ˜ we obtain as usual the


355

(a)

(b) Fig. 1 Sketch of free surface flow past a semi-infinite flat plate in a fluid of finite depth. In frame (a) the plate is located at the level of undisturbed free surface y = H, whereas in frame (b) the plate is suddenly submerged into the fluid for t˜ > 0+ .

Laplace equation for the velocity potential Φ:

∂ 2Φ ∂ 2Φ + 2 = 0, ∂ x˜2 ∂ y˜ in the domain D˜ :

  −∞ < x˜ < 0, 

0 ≤ x˜ < ∞,

for ∀t˜.

(2)

0 < y˜ < h; 0 < y˜ < h + η˜ (x, ˜ t˜).

Two boundary conditions are required at the free surface, as the surface position, η˜ , and Φ have to be determined. We obtain one boundary condition by requiring that the fluid velocity is tangential to the free surface, namely, ∂ η˜ ∂ Φ ∂ η˜ ∂Φ + = : x˜ > 0, y˜ = h + η˜ (x, ˜ t˜), ∀t˜. (3) ˜ ∂t ∂ x˜ ∂ x˜ ∂ y˜ To obtain the second boundary condition, we apply Bernoulli equation along the free surface; for x˜ > 0, y˜ = h + η˜ (x, ˜ t˜), and ∀t˜ we have

∂Φ 1 2 1 P˜ + (u˜ + v˜2 ) + gy˜ = V 2 + gh + : ∂ t˜ 2 2 ρ

(4)

where ρ is the density, g is the acceleration due to gravity, P˜ = P− − Pa , P− is the constant pressure at x˜ → −∞, and Pa is the atmospheric pressure on the free surface.

356


At the solid boundaries the normal component of the fluid velocity must be zero. This gives:

∂Φ = 0 : −∞ < x˜ < 0, y˜ = h, ∀t˜ at the plate, ∂ y˜

(5)

∂Φ = 0 : −∞ < x˜ < ∞, y˜ = 0, ∀t˜ at the bottom. ∂ y˜

(6)

To complete the formulation of the unsteady problem, two initial conditions are needed. Assuming that the fluid is incompressible and plate submerging d is small in comparison with the total depth H we can consider that the flow is uniform upstream and downstream at the initial instant of time so that the velocity potential is Φu = V x, ˜ (−∞ < x˜ < 0);

Φd =

V h x˜ , (0 ≤ x˜ < ∞). h+d

Combining these two equations and neglecting small-amplitude evanescent modes arising at x˜ = 0, we find that the initial condition for the velocity potential approximately is given by Φ(x, ˜ y, ˜ 0) = [(

Vh ˆ x) −V )H( ˜ +V ]x, ˜ h+d

(7)

ˆ x) where H( ˜ is the Heaviside function. The initial position of the free surface at y = H can be considered as a perturbation of a height d ≪ H relative the level of the plate y = h. Then we obtain the following initial condition for the free surface: ˆ x). η˜ (x, ˜ 0) = d H( ˜

(8)

Equations (2)–(8) form the mixed boundary-value problem with given initial conditions for the unknown functions, free surface η˜ (x, ˜ y, ˜ t˜) and velocity potential Φ(x, ˜ y, ˜ t˜). To solve this problem, it is convenient firstly to reduce it to the dimensionless form. To this end, let us scale all spatial coordinates by the height h, and all velocities – by the upstream speed V ; that is we let x = x/h, ˜ y = y/h, ˜ η = η˜ /h, v = v/V ˜ , u = u/V ˜ , φ = Φ/V h, ˜ ρ gh. Then, the Laplace equation (2) in the domain t = V t˜/h, and also P = P/   −∞ < x < 0, 0 < y < 1; D:  0 ≤ x < ∞, 0 < y < 1 + η (x,t) reads

∂ 2φ ∂ 2φ + 2 = 0, ∂ x2 ∂y

for ∀t,

(9)

and the boundary conditions (3)–(4) at the free surface x > 0, y = 1 + η (x,t) for ∀t become:

∂η ∂φ ∂η ∂φ + = ; ∂t ∂x ∂x ∂y

(10)

∂φ 1 2 1+η 1 P+1 + (u + v2 ) + = + 2 . ∂t 2 F2 2 F

(11)

The boundary conditions at the rigid plate (5) and bottom (6) reduce to:

∂φ = 0 : −∞ < x < 0, y = 1, ∀t at the plate, ∂y

(12)


357

∂φ = 0 : −∞ < x < ∞, y = 0, ∀t at the bottom. ∂y

(13)

The initial conditions (7)–(8) in the dimensionless variables read

φ (x, y, 0) = [1 −

ε ˆ ˆ H(x)]x ≈ [1 − ε H(x)]x, 1+ε

ˆ η (x, 0) = ε H(x),

(14) (15)

where ε = d/h ≪ 1 is the small parameter. The other two dimensionless parameters, the Froude number F and the applied pressure P, were introduced as V F=√ , gh

P=

P− − Pa . ρ gh

(16)

In this paper we are only interested in subcritical flows, for F < 1. The problem (9)–(13) posted here naturally reduces to the steady state problem studied in [1] if we omit all temporal derivatives letting ∂ t = 0. In the following section we linearize the problem (9)–(13) and solve it by means of Laplace transform and Wiener–Hopf technique. The formal solution can be presented in quadrature in the form of definite integrals. To calculate the integrals we use numerical approximation of the contour integrals in the complex plane exploiting the best rational approximation method suggested by Trefethen et al. [12]. The results obtained are discussed them in concluding section. 2.2

Linearized stern flow problem

In this section we linearize the problem (9)–(13) under the assumption that P/(1 − F 2 ) ≪ 1 and solve the resultant linear equations exactly using the Wiener–Hopf technique. In the present study we follow the working and notations used in the paper [1] for the steady-state problem. The analysis of the finite-depth problem performed here for the flat plate is similar to that earlier undertaken for a curved plate in a fluid of finite depth [1] or infinite depth [3, 4] respectively. Note then that without the external pressure exerting on the plate (i.e., when P = 0) the problem posted above has the trivial unperturbed solution which represent the uniform stream:

φ = x,

η = 0,

(17)

which is formally valid for all Froude numbers F leaving apart the problem of flow stability. When a small external pressure is applied on the plate, the solution of the linearized problem (9)–(13) can be considered as weakly perturbed stationary solution (17): P P φ1 (x, y,t) + o( ), 2 1−F 1 − F2 P P η (x,t) = [1 + η1 (x,t)] + o( ), 2 1−F 1 − F2

φ (x, y,t) = x +

(18) (19)

where the small parameter P/(1 − F 2 ) ≈ ε ≪ 1 is chosen such that the following steady linear problem coincides with McCue & Stump [8] for the special case of a flat plate (recall however that we use here different normalization similar to that used by Binder [11], rather than by McCue & Stump [8]). By applying the linearization (18)–(19) to equations (9)–(13), we find the function φ1 (x, y,t) satisfies the Laplace equation in the fluid domain D : −∞ < x < ∞, 0 < y < 1, ∀t:

∂ 2 φ1 ∂ 2 φ1 + = 0, ∂ x2 ∂ y2

(20)

358


and boundary condition at the bottom:

∂ φ1 =0: ∂y

−∞ < x < ∞,

y = 0,

∀t.

(21)

The boundary conditions on the free surface and rigid flat plate in the linear approximation can be presented as (−∞ < x < ∞ and y = 1 for ∀t):

∂ φ1 ∂ 2 φ1 ∂ ∂ φ1 ∂ 2 φ1 + F 2[ 2 + 2 ( )− ] = 0, ∂y ∂t ∂t ∂x ∂ y2

(22)

∂ φ1 = m(x,t), ∂y

(23)

where at the flat plate, x < 0, function m(x,t) is: m(x,t) ≡ 0,

x < 0; ∂ η1 ∂ η1 + ), m(x,t) = −( ∂t ∂x

(24) x > 0.

(25)

The initial conditions (14)–(15) in the linear approximation become: ˆ φ1 (x, y, 0) = −xH(x), ˆ η1 (x, 0) = −1 + H(x).

(26) (27)

Finally, given the solution to equations (20)–(27), the shape of the free surface can be recovered for x > 0, y = 1 and ∀t > 0 via the equation PF 2 ∂ φ1 ∂ φ1 η (x,t) = P − + . (28) 1 − F2 ∂t ∂x In the next section, the linearized problem (20)–(27) is solved using the Laplace transform method together with the Wiener–Hopf technique. 2.3

Laplace and Fourier transforms

Following paper [1], we solve Eqs. (20)–(27) by means of the Wiener–Hopf technique. Before doing that, we apply the Laplace transform to functions φ1 , n and m: φ¯ (x, y, s); n(x, ¯ y, s); m(x, ¯ y, s) ˆ∞ = {φ1 (x, y,t); n(x, y,t); m(x, y,t)} e−st dt. (29) 0

It is assumed that there is a real number α such that these transforms exist for Re(s) > α . Application of the Laplace transform to Eqs. (20)–(21) gives ∇2 φ¯ = 0,

φ¯y = 0,

−∞ < x < ∞, −∞ < x < ∞,

0 < y < 1, y = 0.

(30) (31)

And Laplace transform of Eqs. (22)–(23) gives at y = 1 and −∞ < x < ∞: ˆ φ¯y + F 2 [s2 φ¯ + 2sφ¯x − φ¯yy + 2xδ (x) + (sx + 2)H(x)] = n(x, ¯ s),

(32)


359

φ¯y = m(x, ¯ s),

(33)

where m(x, s) = 0, and n(x, s) = const for x ≤ 0, since the stern is supposed to be flat. Let us denote the Fourier transform on the variable x of the function φ¯ (x, y, s) as: ˆ ∞ φˆ¯ (k, y, s) = φ¯ (x, y, s)eikx dx.

(34)

−∞

Similar notations will be used for other dependent variables. As will be seen further, to avoid singularities in the transformed functions [see, e.g., Eq. (43) below for function G(k, s)] and insure convergence of the integral in (34), we consider complex values of k with Im(k) > 0. Moreover, as will be shown later, variation of this parameter is restrict by the strip in the complex k-plane, 0 < Im(k) < τ+ , parallel to the axis Re(k) and bounded from the top by some real value τ+ . The choice of τ+ will be explained at the end of Appendix B. After application of the Laplace and Fourier transforms to Eqs. (28) and (30) we obtain:

ηˆ¯ (k, s) =

P−1 PF 2 − (s − ik)φˆ¯ , s 1 − F2

d 2 φˆ¯ (k, y, s) − k2 φˆ¯ (k, y, s) = 0. dy2

(35) (36)

The general solution to the former equation is

φˆ¯ (k, y, s) = A(k, s) cosh ky + B(k, s) sinh ky.

(37)

The Fourier transform of the boundary condition (31) dictates that the constant B(k, s) ≡ 0, therefore solution (37) reduces to φˆ¯ (k, y, s) = A(k, s) cosh ky. (38) The Fourier transform of boundary conditions (32) and (33) yield: 2ik − s ˆ¯ s), φˆ¯y − F 2 [φˆ¯yy − (s2 + 2iks)φˆ¯ + ] = n(k, k2 ∂ φˆ¯1 ˆ¯ s). = m(k, ∂y

(39) (40)

Substituting (38) into Eqs. (39) and (40), we obtain: ˆ¯ s) = kA(k, s)G(k, s) sinh k − F 2 n(k,

2ik − s , k2

ˆ¯ s) = kA(k, s) sinh k, m(k,

(41) (42)

where

(s + ik)2 . (43) k tanh k ˆ¯ s) and m(k, ˆ¯ s) are the Fourier transforms of functions n(x, Here functions n(k, ¯ s) and m(k, ¯ s), respectively. Equations (42) and (41) can be reduced to the corresponding equations for steady-state problem [1] for the flat stern, if we put s = 0 and omit the last term in Eq. (41). These two terms appear in the result of application of Laplace transform to non-stationary equations containing time derivative. According to the well-known property ˆ f ′ ) = sL( ˆ f ) − f (0+ ), where Lˆ stands for the Laplace transform, and prime denotes of the Laplace transform, L( time derivative. ˆ¯ s) and n(k, ˆ¯ s): Eliminating A(k, s) from Eqs. (41) and (42), we obtain the relationship between functions m(k, G(k, s) = 1 + F 2

ˆ¯ s) = m(k,

ˆ¯ s) + (2ik − s)(F /k)2 n(k, . G(k, s)

(44)

360


And from Eqs. (42) and (38) we find cosh ky ˆ¯ s). m(k, φˆ¯ (k, y, s) = k sinh k

(45)

ˆ¯ s) is determined, we can restore the unknown function η (x,t) describing the free surface through Eqs. Once m(k, (45) and (35). 2.4

Wiener–Hopf technique

ˆ¯ s), we need to apply the inverse Laplace and Fourier transforms to Eq. (44). To restore the original function m(k, This can be done by means of the Wiener–Hopf technique. The key step of this technique is factorization of the function G(k, s) such that 1 G(k, s) = 2 G+ (k, s)G− (k, s), (46) k where G+ (k, s) is analytic and non-zero in the upper-half plane Im(k) > 0, and G− (k, s) is analytic and non-zero in the lower-half plane Im(k) < τ+ of the complex plane k (the details are presented in Appendix B. ˆ¯ s) and n(k, ˆ¯ s) can be presented as Functions m(k, m(k, ¯ˆ s) = mˆ¯ − (k, s) + m¯ˆ + (k, s), ˆ¯ s) = nˆ¯− (k, s) + nˆ¯+ (k, s), n(k, where ˆ0

mˆ¯ − (k, s), nˆ¯− (k, s) =

−∞

{m(x, ¯ s), n(x, ¯ s)} eikx dx

are analytic functions in the lower-half plane, and mˆ¯ + (k, s), nˆ¯+ (k, s) =

ˆ∞

{m(x, ¯ s), n(x, ¯ s)} eikx dx

0

are analytic functions in the upper-half plane. Since the plate is flat, so that m(x,t) ≡ 0 for x < 0, then mˆ¯ − (x, s) ≡ 0. Assuming that the atmospheric pressure is constant at x > 0, we obtain that nˆ¯+ (x, s) ≡ 0. Taking into account these properties, we can present Eq. (44) in the form mˆ¯ + (k, s) =

k2 nˆ¯− (k, s) + F 2 (2ik − s) . k2 G(k, s)

(47)

Substituting now Eq. (46) into Eq. (47), we obtain the Wiener–Hopf equation: mˆ¯ + (k, s)G+ (k, s) =

k2 nˆ¯− (k, s) + F 2 (2ik − s) , G− (k, s)

(48)

where the left-hand side represents the analytic function in the upper half-plane Im(k) > 0, whereas the righthand side is the analytic function in the lower half-plane, Im(k) < τ+ . Because both sides of Eq. (48) are equal in the strip 0 < Im(k) < τ+ , then assuming that they are independent of k (which is the case for the steady problem, when s = 0 [1]), we can conclude that the left-hand side is the analytic continuation of the right-hand side and vice-versa. Following then the same procedure as has been described in papers [1, 8] and denoting left-hand sides of Eq. (48) by E(s), we obtain E(s) mˆ¯ + (k, s) = . (49) G+ (k, s)


361

Substituting Eq. (49) into Eq. (45) and applying then the inverse Fourier and Laplace transforms to Eq. (45), we can find the integral representation of exact solution to the linearized problem for the normalized velocity potential: ˆ ˆ −i γ +i∞ ∞+iδ e−ikx dk φ1 (x, y,t) = 2 [ ]E(s)est ds, (50) 4π γ −i∞ −∞+iδ G+ (k, s)k tanh k where γ is a vertical contour for the inverse Laplace transform that is located to the right of all singularities; δ is a real constant in the range 0 < δ < τ+ , forcing the path of integration to lie in the strip 0 < Im(k) < τ+ as required [13]. Equation (50) together with Eq. (28) provides the integral representation for the free surface η (x,t) in terms of inverse Fourier and Laplace transforms: ˆ P 1 γ +i∞ I(x, s)est ds, 1 − F 2 2π i γ −i∞

η (x,t) = P − 1 + where I(x, s) =

E(s)F 2 2π

ˆ

∞+iδ

−∞+iδ

(51)

ik − s e−ikx dk. G+ (k, s)k tanh k

(52)

Due to complicated nature of the function G+ (k, s), it is too difficult and, perhaps, even impossible to calculate the inverse transforms in Eq. (51) analytically. But this can be done numerically by means of the approximate method for the inverse Laplace transform proposed by Trefethen et al. [12]. In the next section we will calculate the integral by that method. 2.5

Numerical approximation of the contour integral

Following the method developed by Trefethen et al. and briefly described in Appendix A, we let s = z/t. Using the rational approximation of the integral as described in Appendix A, we can present solution (51) in the form

η (x,t) = P − 1 −

P 1 2 1 − F 2π it

N

∑ C j I(x, z j /t).

(53)

j =1

From the numerical investigation performed in this study, we found that ten values (N = 10) of poles and residues from Table 1 in Appendix A provides quite good approximation of the inverse Laplace transform of function I(x, z/t). At small times even a bit less number of poles and residues can be used to achieve the appropriate accuracy, but this is insignificant from the computational point of view. To find function I(x, z/t) in Eq. (53) we need to determine function G+ (k, z/t), which depends upon the roots of function f1 (k, z/t) for different values of z as shown in Appendix B. Function G+ (k, z/t) can be presented in the factorized form ∞ ik z k G+ (k, z/t) = Γ(1 − )(F )2 e2itk/z ∏ (1 − )ek/ζ j , (54) π t ζj j =1 where Γ(x) is a special gamma-function [14, 15]. Using properties of gamma-function (see, e.g., Sect. 6.1.31 in ref. [15]), Γ(1 −

ik k ik )Γ(1 + ) = , π π sinh k

(55)

we can express the gamma-function Γ (1 − ik/π ) in Eq. (54) in terms of the gamma-function Γ (1 + ik/π ). This allows us to present the denominator in the integral of Eq. (52) in the following form: G+ (k, s)k tanh k =

(kFzeikt/z /t)2 Γ(1 + ik/π ) cosh k

∞

k

∏ (1 − ζ j )ek/ζ . j =1

j

(56)

362


Substitution of Eq. (56) into Eq. (52), allows one to calculate the integral by closing the contour in the lower half-plane of k, where zeros of function G+ (k, z/t) are at the points k = 0 and k = ζ j ( j = 1, 2, 3, . . .). Using then the residue theorem, we obtain I(x, z/t) = −E(z/t)[ where

∞

∏= ∏

3 + 0.183755(z/t) i − ∏], F 2 (z/t)2

(iζ j − z/t)Γ(1 + iζ j /π ) cosh ζ j e−iζ j x

ζ j (z/t)2 e1+2it ζ j /z

j =1

(57)

.

Substituting into function I(x, z/t) consecutively N values of z j from Table 1 in Appendix A instead of z, we obtain from Eq. (53) the approximate solution for free surface η (x,t):

η (x,t) = P − 1 −

P 1 1 − F 2 2π t

10

∑ C j E (z j /t) (A j + iB j ) ,

(58)

j =1

where Aj =

3 + 0.183755 (z j /t) F 2 (z j /t)2 ∞

Bj =

∏

j =1

,

(iζ j − z j /t)Γ(1 + iζ j /π ) cosh ζ j e−iζ j x

ζ j (z j /t)2 e1+2it ζ j /z j

.

To find function E (z/t), we note first that the first term in the sum (58), A j ∼ 1/F 2 , becomes predominant when the Froude number decreases, so that the term B j can be omitted. To obtain then the expected steadystate solution at any fixed value of x when t → ∞, we have to assume that E (z j /t) ≡ E0 z j /t, where E0 is a constant which can be determined from the condition η (0,t) = 1. To clarify this statement we note that when the Froude-independent term B j is omitted, then each term in the sum (58) contains the product E (z j /t) A j = E (z j /t)

3 + 0.183755 (z j /t) (z j /t)2

.

From this expression one can see that the denominator has a singularity ∼ t −2 when t → 0, whereas the numerator has a weaker singularity, ∼ t −1 . Thus, if we chose function E (z j /t) ∼ z j /t, then the ratio remains finite when t → 0. 3 Results and Discussions Thus, with the help of Laplace transform and the Wiener–Hopf technique we derived a formal solution to the problem of transient two-dimensional flow past a semi-infinite flat plate in a fluid of a finite depth in the subcritical case, F < 1. The free surface profile η (x,t) is given by Eq. (58). It can be computed for any instant of time with the appropriate accuracy using, for instance, ten values of poles z j and residues C j from Table 1 in Appendix A (the accuracy can be raised if required; then more data should be added into the Table using the Matlab code provided by Trefethen et al. [12]). To use this formula for different time moments, function G(k, z/t) should be factorized anew every time and roots of function f1 (k, z/t) should be found as shown in Appendix B. We studied the dependence of accuracy of calculation of a free surface on the number of poles taken from Table 1. It is expected that after the transient process, when t → ∞, the free surface in the near-field zone behind


363

1.05 1.04

y

1.03 1.02 1.01 1 0.99 −1

0

1

2x

3

4

5

Fig. 2 (Color online) Free surface profile at y = 1 + η , where η is given by Eq. (58). Different lines correspond to different numbers of poles (see the text). The plot was generated for P = 0.01, F = 0.5 and with 60 complex roots of function G+ (k, z/t).

1.05 1.04

y

1.03 1.02 1.01 1 0.99 −1

0

1

2x

3

4

5

Fig. 3 (Color online) Free surface profile at y = 1 + η , where η is given by Eq. (58), with fixed number of poles N = 10 from Table 1, but with different numbers of complex roots of function G+ (k, z/t). Dashed line pertains to 40 roots, whereas solid line pertains to 60 roots; dots represent steady-state solution [1]. The plot was generated for P = 0.01, F = 0.5 ant t = 1000.

the stern should be stationary. Hence, its shape can be compared with the steady state solution obtained in [1] for the same parameters. Figure 2 illustrates the comparison of free surface shape obtained in this study at t = 1000 with the steady-state solution. As one can see, solution obtained with only 8 poles (see dashed line in the figure) does not match with the steady-state solution, whereas solution obtained with 10 poles (solid line) does match and is practically indistinguishable from the steady-state solution presented by dots. Next figure illustrates the influence of the number of roots of function G+ (k, s) on the accuracy of solution. In this case N = 10 poles were chosen but number of roots were taken 40 and 60. As follows from Fig. 3, 40 roots is not enough to obtain an accurate solution. Only the first period of wave can be satisfactorily described with the help of 40 roots. Wave amplitude is reproduced with fairly good accuracy, but not the wave period. Thus, it was concluded that the optimal number of poles is N = 10 and number of roots is 60. In further studies these numbers were used. The transition to the steady-state regime is illustrated by Fig. 4. As one can see from that figure, in the near-field zone, the transient solution approaches stationary solution by t = 1000. At early times, the maximal splash may be significantly greater than the stationary wave amplitude.

364


1.1

t=1 t=5 t = 10

y

1.05

1

0.95 −1

0

1

2 x

3

4

5

(a)

1.05

t=100,

t=1000,

Steady

1.04

y

1.03 1.02 1.01 1 0.99 −1

0

1

2x

3

4

5

(b) Fig. 4 (Color online) Free surface profile at y = 1 + η , where η is given by Eq. (58) in different instants of time. The plot was generated for P = 0.01, F = 0.5, N = 10 and 60 complex roots of function G+ (k, z/t).

Fig. 5 (Color online) Maximal wave amplitude A = ηmax against time as per Eq. (58). The plot was generated for P = 0.01, F = 0.5.


365

As one can see from Fig. 4(a), the waves are undeveloped in relatively short times, and the free surface approaches the unperturbed horizontal level at large x. After elapsing some time, generated waves become stationary, at least, at the near-field zone just behind the stern. In the transient period large amplitude splash is generated, as one can see in Fig. 4. The dependence of maximal wave amplitude on time is shown in Fig. 5. As follows from this figure, after the first splash, maximal wave amplitude drops down, increases again and only after that it gradually decreases to the steady-state value. For the considered parameters (see figure caption), maximal amplitude of the splash is 3.4 times greater than the amplitude of stationary wave.

4 Conclusion In this paper we have investigated the transient problem of wave generation past the flat stern which is suddenly submerged at the initial instant of time into the moving water at small depth d which is much less than the total fluid depth h. The main purpose of this study was to demonstrate how the transient problem with such initial perturbation converges to the steady problem studied earlier [1]. In general, a nonlinear problem may have different steady state solutions, therefore if one of them has been found or constructed numerically, there is no guarantee that it can be realized in the process of evolution of any reasonable initial perturbation. Here we have shown that the steady state solution for the plane stern can be realized in the process of long-term evolution of the initial-value problem described above. From this point of view the steady-state solution can be treated as asymptotically stable on the Lyapunov sense. We have found, however, that the onset of steady-state regime is not monotonic as can be bee seen from Fig. 4. In particular, maximal wave amplitude behind the stern increases first and attains the value ≈ 1.1 at t = 10, but then it decreases to ≈ 1.02 at t = 100, increases again almost to 1.1 and only after that it gradually decreases and approaches asymptotically the value ≈ 1.03 by t = 1000 (see Fig. 5). By that time the transient solution becomes indistinguishable from the steady-state solution – see Fig. 4b). √ In this study it is assumed that the Froude number is less than the critical one, F ≡ V / gh < 1. Solution to the linearized problem has been obtained in the analytic form, Eq. (58), and then calculated numerically. The traditional methods to this class of mixed boundary problems were used: Laplace transform on time, Fourier transform on x and Wiener–Hopf technique. It was shown that in the transient period high amplitude splash is generated, then the amplitude of highest wave drops down and quickly increases again. After that the wave amplitude gradually approaches the steady state which is characterized by a constant amplitude quasi-sinusoidal trailing wave. The time dependence of the free-surface perturbation behind the plate was found numerically and presented graphically. The problem of the stern shape optimization to reduce generated waves was not considered in this paper, whereas for the steady-state problem it has been shown [1] that optimal shapes do exist such that the trailing waves can be almost completely eliminated. We leave this problem for future study.

Acknowledgments One of the authors (O.O.) is very grateful to Ian Turner, Scott McCue and John Belward for their support and valuable advices.

366


APPENDIX

A. Numerical calculation of contour integrals Here we briefly describe the method of numerical calculation of contour integrals suggested in the paper by Trefethen et al. [12] in 2006. The method is applicable to integrals of the form ̥=

1 2π i

ˆ

Γ

f (z) ez dz,

(1)

where f (z) is the analytic function of z, where z 6∈ (−∞ 0], and Γ denotes a Hankel contour that lies in the region of analyticity of function f (z). Suppose the integral (1) is approximated by the quadrature method ̥=

1 2π i

N

ˆ

Γ

f (z) ez dz ≈

∑ Ck ez

k

f (zk ),

(2)

k =1

The sum in Eq. (2) can be interpreted according to residue calculus as 1 ̥= 2π i

ˆ

r(z) f (z)dz.

(3)

c

It is assumed here that | f (z)| → 0 as |z| → ∞, c is a closed contour in the region of analyticity of function f (z) that winds clockwise around each point of singularity zk , and r(z) is the rational function N

r(z) =

Ck ezk

∑ z − zk .

k =1

Letting Γ′ be a contour for the integral in Eq. (1) like Γ, except lying between R−, where R− : z ∈ (−∞ 0], and the point zk , then Γ′ is equivalent to Γ for the integral (1) of ez f (z). Now if we define Γ to be the union of Γ′ with a large circular arc of a radius R → ∞, then the integral on such closed contour Γ′ is equivalent to the integral on the contour c in the integral (3) involving r(z) f (z) (see [12]) (see Fig. 6). Therefore, Eq. (3) can be replaced by ˆ 1 ̥N = r(z) f (z) dz. (4) 2π i Γ′ To obtain the higher accuracy result, many values of N are needed, in general. However Trefethen et al. [12] have shown that much less number of values can be used by exploiting the best rational approximation. According to these authors, if one supposes that the poles and residues of function r(z) are z1 , z2 , . . . , zN and C1 ,C2 , . . . ,CN , respectively, then the integral in (3) is equal to the sum in Eq. (2). This is a rational function deviation from ez which decreases at the optimal rate as (9.28903)−N when N → ∞. The Matlab code has been provided by Trefethen et al. [12] to compute the best approximation of the function ez . In the result of this the integral in Eq. (3) can be approximated by the following sum: 1 2π i

ˆ

Γ

N

r(z) ez dz ≈

∑ C j f (z j ),

(5)

j =1

where z j and C j are the poles and residues, respectively, shown in Table 1. These quantities were used in calculation of the free surface shape in Eq. (58).


367

Fig. 6 (Color online) The contour of integration consisting of the path Γ′ and path c of infinitely large radius R. Red dots show the positions of poles in Eq. (2).

B. Factorization of G(k, s) Here we present the details of factorisation of function G(k, s). Let G(k, s) = f1 (k, s) f2 (k, s)/k2 , where functions f1 (k, s) and f2 (k, s) are f1 (k, s) = k sinh k + F 2 (s + ik)2 cosh k, k f2 (k, s) = . sinh k

(6) (7)

The splitting of the Wiener–Hopf equation depends on the roots of functions f1 (k, s) and f2 (k, s). Assuming s = 0, we obtain the same functions for steady-state problem that has been presented in paper [1]. For the unsteady problem we put s = z/t where values of z have been found by Trefethen et al. [12] for the inverse Laplace transform (see Appendix A). Equation (6) with s = z/t becomes f1 (k, z/t) = k sinh k + F 2 (z/t + ik)2 cosh k.

(8)

Table 1 The first ten values of poles z j and residues C j provided by Trefethen et al. [12] through the Matlab program with F = 0.5. j

zj

Cj

1

4.0277 + 1.1939i

2

4.0277 − 1.1939i

−4.8184 − 21.0546i

3 4

3.2838 + 3.5944i 3.2838 − 3.5944i

5

1.7154 + 6.0389i

6

1.7154 − 6.0389i

7 8 9 10

−0.8944 + 8.5828i

−4.8184 + 21.0546i 7.1172 + 8.8195i 7.1172 − 8.8195i

−2.5656 − 1.2164i −2.5656 + 1.2164i 0.2726 + 0.0142i

−0.8944 − 8.5828i

0.2726 − 0.0142i

−5.1612 − 11.3752i

−0.0058 − 0.0007i

−5.1612 + 11.3752i

0.0058 + 0.0007i

368


6Im(k) s s s s s s

s s s

s s s s s s s s

Re(k)

Fig. 7 (Color online) Zero isolines of functions Re f1 (k, z/t) (blue lines) and Im f1 (k, z/t) (red lines). Dots indicate intersection points of isolines. The plot was generated for F = 0.5, t = 1 and z1 = 4.0277 + 1.1939i.

The roots of this equation with complex function f1 (k, z/t) of the complex variable k depend on two parameters F and z/t. We investigate further positions of roots in the complex plane k for ten complex values of z j from Table 1. Thus, complex function f1 actually contains two real parameters F and t. To find roots of f1 (k, z j /t), one can plot zero isolines for functions Re f1 (k, z/t) and Im f1 (k, z/t) and find their intersections as shown in Fig. 7. For each value of z j function f1 (k, z j /t) has infinite number of complex roots in the upper half-plane denoted by ξl and infinite number of complex roots in the lower half-plane denoted by ζ j . One can show that asymptotically at large values of indices l and j, the roots can be be presented by the following formulae kl ≈ i(π /2 + l π ) and k j ≈ −i(π /2 + jπ ). Using the Weierstrass infinite product theorem [14, 15], f1 can be factorized in the following way z Fzeikt/z 2 ∞ k f1 (k, ) = ( ) ∏ (1 − )ek/ξl t t ξl l =1

∞

k

∏ (1 − ζ j )ek/ζ ; j

j =1

z k ik ik f2 (k, ) ≡ = Γ(1 + )Γ(1 − ). t sinh(k) π π Thus, function G(k, z/t) can be presented in one of the forms: 1 f1 (k, z/t) f2 (k, z/t) k2 ∞ F z ikt/z 2 ik ik ∞ k k = ( e ) Γ(1 − )Γ(1 + ) ∏ (1 − )ek/ζ j ∏(1 − )ek/ξl kt π π j =1 ζj ξl l =1

G(k, z/t) =

=

1 G− (k, z/t)G+ (k, z/t). k2

(9)


369

From here we derive z ik ∞ k G+ = (F eikt/z )2 Γ(1 − ) ∏(1 − )ek/ζ j , t π j =1 ζj

(10)

ik ∞ k ) ∏(1 − )ek/ξl , π l =1 ξl

(11)

G− = Γ(1 +

where G+ (k, z/t) is the analytic function of k in the upper half-plane and G− (k, z/t) is the analytic function of k in the lower half-plane. Function G+ (k, z/t) can be substituted into Eq. (52) for the inverse Fourier transform providing function I(x, s) which will be used then in Eq. (51) for calculation of the free surface. For the practical implementation the infinite products in Eqs. (10) and (11) are replaced by the finite products from j = 1 to j = J in Eq. (10) and from l = 1 to j = L in Eq. (11), where the numbers J and L are almost equal, and their sum is equal to the total number of roots used in the calculations (in particular, J + L = 60 for calculations shown in Figs. 4 and 5). The number J determines the width of the the strip in the upper half-plane of the complex k-plane – see the text after Eq. (34). The value of τ+ must be such that all J zeros of function G+ are contained in the strip. References [1] Ogilat, O., McCue, S. W., Turner, I. W., Belward, J. A., and Binder, B. J. (2011), Minimising wave drag for free surface flow past a two-dimensional stern, Phys. Fluids, 23, 07210001. [2] Vanden-Broeck, J.-M. (1980), Nonlinear stern waves, J. Fluid Mech., 96, 603. [3] Schmidt, G. H. (1981), Linearized stern flow of a two-dimensional shallow-draft ship, J. Ship Res., 25, 236. [4] Zhu, S. P. and Zhang, Y. (2003), A flat ship theory on bow and stern flows, ANZIAM J., 45, 1. [5] Madurasinghe, M. A. and Tuck, E. O. (1986), Ship bows with continuous and splashless flow attachment, J. Austral. Math. Soc. ser. B, Appl. Math., 27, 442. [6] Farrow, D. E. and Tuck, E. O. (1995), Further studies of stern wavemaking, J. Austral. Math. Soc. Ser. B, 36, 424. [7] Haussling, H. J. (1980), Two-dimensional linear and nonlinear stern waves, J. Fluid Mech., 97, 759. [8] McCue, S. W. and Stump, D. M. (2000), Linear stern waves in finite depth channels, Q. J. Mech. Appl. Math., 53, 629. [9] McCue, S. W. and Forbes, L. K. (2002), Free-surface flows emerging from beneath a semi-infinite plate with constant vorticity, J. Fluid Mech., 461, 387. [10] Maleewong, M. and Grimshaw, R. H. J. (2008), Nonlinear free surface flows past a semi-infinite flat plate, Phys. Fluids, 20, 062102. [11] Binder, B. J. (2010), Steady free-surface flow at the stern of a ship, Phys. Fluids, 22, 012104. [12] Trefethen, L. N., Weideman, J. A. C., and Schmelzer, T. (2006), Talbot quadratures and rational approximation, BIT Numerical Mathematics, 46, 653–670. [13] Noble, B. (1988), Methods Based on the Wiener-Hope Technique, Chelsea Press: New York. [14] Korn, G. A. and Korn, T. M. (1968), Mathematical Handbook, McGraw-Hill Book Company: New York et al. [15] Abramowitz, M. and Stegun, I. A. (1970), Handbook of Mathematical Functions, Dover: New York.



Cavitating Flow between Two Shear Moving Parallel Plates and Its Control Yan Liu1 , Sheng Ren2 , Jiazhong Zhang3† 1 School

of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, P. R. China and Quality Control Research Institute of the Ministry of Water Resources, Hangzhou 310012, P. R. China 3 School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, P. R. China 2 Standard

Submission Info Communicated by S.V. Prants Received 27 February 2015 Accepted 2 March 2015 Available online 1 October 2015 Keywords Cavitation Lattice Boltzmann method Drag reduction Cavitating flow Flow pattern

Abstract Two parallel plates with shear moving velocity in opposite direction is introduced as external excitations to induce cavitating flow between them, and a developed scheme based on Lattice Boltzmann method is used to simulate and analyze the evolution of the cavitation or phase transition. First, the principles and simulation process of Lattice Boltzmann method and potential models for single component multiphase flow are introduced, including a special model to the moving boundary conditions. Then, the numerical simulations of evolution of phase transition, induced by shear motions of two parallel plates, are carried out in detail, and the complicated pattern formation of cavitating flows are analyzed in such micro- and multiphase dynamic system and some new results are obtained. In particular, the influences of main parameters, such as initial density and moving velocity, on the cavitation and flow pattern are studied further. The results show that the shear moving motion of two parallel plates could induce the cavitation, and the cavitation and cavitating flow pattern could be controlled availably and efficiently by the main parameters listed above. Further, the method and analysis could be extended to flowing liquid, and an idea of drag reduction utilizing the cavitation due to phase transition in such liquid is proposed. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The resistance forces acted on the underwater bodies can be classified into friction drag, pressure drag, wave making drag. Among them, the friction drag plays an important role in drag reduction. Indeed, for conventional naval vessels, the friction drag can account for 50% of the total resistance force. However, for the underwater vehicle, such percentage is much higher up to 70%. Therefore, it is important to reduce the friction drag for the underwater vehicle as both high velocity and long voyage are required in modern navigation. Under such background, one strategy for drag reduction, especially for the underwater vehicle, is presented in decades [1]. That is, the body of the vehicle can be covered partially or totally by gas or the mixture of gas † Corresponding

author. Email address: [email protected]


372

Yan Liu, Sheng Ren, Jiazhong Zhang / Discontinuity, Nonlinearity, and Complexity 4(3) (2015) 371–379

and liquid, so that the flow structure around the body is rearranged and the friction drag is reduced consequently, because of the great difference between densities of gas and liquid [1, 2]. However, one of problems arise as the strategy is applied to engineering, that is, how to keep the layer full of gas or liquid-gas mixture be stable and controllable around the body, as the complicated flow conditions are varying instantaneously in practice. To this end, some extra devices or assemblies should be attached to the vehicle [2, 3]. For the underwater bodies, another potential strategy is presented further to induce the phase transition of the liquid around the bodies from liquid phase to gaseous phase via some routes, and the evolution of the phase transition can be controlled efficiently. Such strategy is the cavitation. Following the definition of cavitation, the phase transition from liquid to gas means the appearance of vapor cavities inside an initially homogeneous liquid medium, occurring in very different situations. Indeed, cavitation is a unique phenomenon in the field of hydrodynamics, although it can occur in any hydraulic machinery such as pumps, propellers etc. Normally, the generation of cavitation can lead to severe damage in hydraulic machinery. Therefore, the prevention of cavitation is an important concern. However, there exists great potential to utilize cavitation in various important applications, such as friction drag reduction as mentioned above [4-7]. In fact, the cavitating flow pattern or vapor structures are often unstable, and, they often violently collapse as reaching a region with increased pressure. Another problem arises, that is, how to induce and control the cavitating flow availably, since cavitation bubbles are extremely unstable. More, phase transition is one of the main features in cavitation, and it is a complex process with some micro and dispersed interfaces [8-10]. Mathematically, the system is a dynamic system with discontinuity and non-smoothness. The traditional Navier-Stokes equations are not available for it. In other words, the traditional Computational Fluid Dynamics is difficult to simulate the evolution of phase transition, unless some models are combined to it. To this end, the Lattice Boltzmann method, a mesoscopic method, is introduced to simulate the phase transition [11]. In this paper, a method for inducing cavitation is presented, which is composed by two parallel plates with shear moving velocity in opposite direction. The evolution of the cavitation or phase transition is studied in detail by a developed scheme based on Lattice Boltzmann method. Further, the influences of main parameters, such as initial density and moving velocity, on the cavitation and flow pattern are studied. 2 Model and computational method In this Section, the model for phase transition of liquid between two shear moving parallel plates, with same velocity in opposite direction, is presented. More, Lattice Boltzmann method is developed and used to study the phase transition, from liquid phase to gasous phase. Herein the D2Q9 model is introduced to the lattice model [11]. For the sake of simplicity, the parameters are dimensionless in this study. 2.1

Model

Figure 1 depicts two shear moving parallel plates in 2-dimension. The length of plate is L, the distance between two plates H, and the boundary conditions at two ends are periodic. The two plates move with the same velocity but in opposite direction. The specific values of the parameters are set as L = 1024, H = 128, and numerical computational domain is meshed by 1024 × 128 grids. 2.2

Lattices on walls

For the computational domain shown in Fig. 1, choosing one point on the surface of the plate as example, the distribution functions f2 , f5 , f6 and density ρ are unknown. Following previous paper [11], we have,


373

Fig. 1 Shear flow between two shear moving plates

8

∑ fi = ρ

i=0 8

∑ f i e i = −ρ u −

i=0

(1) Δt F 2

f2 − f2eq = f4 − f4eq

(2) (3)

For convenience, u in above equations denotes u(t). F is the resultant force between liquid particles [11]. Further, yields, ⎧ ρ (1 − uy ) = f0 + f1 + f3 + 2( f4 + f7 + f8 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ f2 = f4 + ρ ueq y ⎨ 3 (4) 1 1 1 eq ⎪ ⎪ ( f f = f + − f ) + ρ (u + u ) − ρ u y 5 7 3 1 x y ⎪ ⎪ 2 2 3 ⎪ ⎪ ⎪ ⎪ 1 1 1 ⎪ ⎩ f6 = f8 + ( f1 − f3 ) − ρ (ux − uy ) − ρ ueq y 2 2 3 ΔtFy eq ΔtFx τ Fx eq τ Fy , uy = − , ux = −u + , uy = . Fx and Fy are the components of resultant where ux = −u − 2ρ 2ρ ρ ρ force F in x and y directions, respectively. It should be noted that both Fx and Fy are the function of ρ , which can be obtained by iterative schemes. 3 Phase transition induced by steady shear motion of plates As an initial step, the motions of the plates are steady. Following the numerical method above, the initial value of density, ρint , is set as 0.693, and u(t) = 0.005, the relaxation time in dimensionless τ = 1.0. By the Lattice Boltzmann method, the evolution of the phase transition is shown in Fig. 2. In the initial stage, it is clear that there exists a region occupied by fluid with low density, near the moving plate. However, in the region far away from the plates, the fluid there with low density is mixed with the fluid with high density. As evolution or development of the phase transition, the region occupied by fluid with low density becomes

374


Fig. 2 Evolution of phase transition with ρint = 0.693 and u(t) = 0.005

broaden, and there exist some distinct interfaces, which are the main factor in the analysis of bubble stability, between fluids with low and high density, as shown in Figs. 2(b) and (c). Herein, fluid with low density means bubbles. As the particles of gaseous and liquid fluid separate and gather continuously, the numbers of liquid drop and bubble become less gradually, and finally the bubbles are separated from the liquid drop, resulting in two liquid drops with different size surrounded by gaseous fluid, as shown in Fig. 2(d). Consequently, the interfaces between drops and bubbles become circular gradually, because of the surface tension of the bubble and moving plates. For such complex dynamic system, it is important to study the dynamic behaviors. The density and pressure are the two main variables in the system. Figure 3 is the density and pressure distributions of the cavitating flow near one liquid drop in x direction at time t = 100000, at center line y = 114. It can be seen that the density around the interface between bubble and drop increases rapidly and monotonously from gaseous fluid to liquid fluid. However, another variable, the pressure, rises firstly with small amplitude, then drops sharply, and finally


375

Fig. 3 Density and pressure distribution in x direction at y = 114 at time t = 100000

Fig. 4 Streamlines near interfaces at t = 100000

rises sharply again to the pressure value of liquid fluid. In a sense, the behavior of pressure is much more active or sensitive. Such complex transient phenomena, which are the key understanding of cavitation, are difficult to be captured by the traditional computational fluid dynamics. At time t = 100000, the streamlines near interfaces are depicted in Fig. 4. It is clear that there are two symmetric vortex-pair in large scale around each of the interfaces, both in gaseous and liquid fluid. In particular, there exist some vortices in small scale near the surfaces of plates.

376



3.1

Influences of initial density on pattern formation of cavitating flow

As the initial density is increased to 0.8, the evolution of phase transition versus initial density is shown in Fig. 5. In comparison with the Case with ρint = 0.693, there are some differences between them. That is, there are some layers in the two-phase or cavitating flow in the tunnel composed by the two shear moving plates, and gaseous fluid is adhered to the moving plates, the liquid fluid is located between the two gaseous layers, as shown in Fig. 5(e). As the initial density is increased further to 0.98 and 1.1, the pattern of cavitating flow is similar with Fig. 5(e). The difference is just the area in size, as shown in Fig. 6. Hence, a conclusion can be drawn that there is a relationship between pattern formation of cavitating flow and initial density, as the two plates move with same velocity in opposite directions. In other words, the cavitaton could be controlled and the initial density can be considered as a control parameter. Specifically, as ρint is set a small value, there will be more liquid drops appearing between plates as the evolution of phase transition reaches a steady state. As ρint is set a large value, there will be some layers, full of gas or liquid, appearing between plates as steady state is reached. More, in comparison between Figs. 5(e) and


377

Fig. 6 Pattern of cavitating flow with different initial density and u(t) = 0.005

6, it can be seen that the area of liquid phase with high ρint is larger than that with low ρint . 3.2

Influences of moving velocity on pattern formaton of cavitating flow

In this Section, the influences of moving velocity of plates on the cavitation will be studied in detail. The initial density in the computational domain is set as ρint = 0.693. As u(t) = 0.015, the evolution of phase transition is shown in Fig. 7. It is clear that the pattern of cavitating flow at initial stage is similar with that described by Fig. 2. That is, there is a layer full of gasous fluid on each surface of plates, and the region far away from plates is occupied by the mixture of gaseous and liquid fluid, refer to Figs. 2(a), (b) and Figs. 7(a), (b). However, it is different from the evolution of phase transition described by Figs. 2, 5 and 6 with u(t) = 0.005. Specifically, as u(t) = 0.015, there are many bubbles adhered to the surfaces of plates as the evolution of phase transition reaches a steady state. Hence, it can be drawn that the velocity of moving plates plays an important role in the cavitation by this way, and the cavitaton could be controlled efficiently. In a sense, the velocity can also be considered as a control parameter. Figure 8 shows the pattern of cavitating flow as the moving plates with different velocity and ρint = 0.693. As u(t) = 0.005, the liquid fluid is separated by gaseous fluid, resulting in two liquid drops with different size, as shown in Fig. 8(a). More, there is a alternative distribution between liquid fluid (L1, L2) and gaseous fluid (V1-1, V1-2, V2) in x direction. Here, gaseous fluid (V1-1) shares same bubble (V1) with gaseous fluid (V1-2), since the boundary conditions at two ends are periodic in the computational domain. As u(t) is increased to 0.006, the steady pattern of cavitating flow is shown in Fig. 8(b). Obviously, it can be seen that the liquid fluid (L) runs through the whole computational domain in x direction. For the gaseous fluid, there are two flatten bubbles with different size, which are adhered to each surface of the plates. Taking bubbles VB1 (including VB1-1 and VB1-2) and VB2 as examples, their lengths in x direction are 441 and 542, respectively. In particular, the bubbles (VT1, VT2) are centrosymmetric with respective to bubbles (VB1 and VB2), respectively. Indeed, this property is the result from the shear motion of plates in opposite direction. Further, as u(t) is set as 0.01 and 0.015, there are four bubbles adhered to the each surface of plates, as shown in Fig. 8(c) and (d). The pattern of cavitating flow is similar with that shown by Fig. 8(b), that is, the bubbles are centrosymmetric with respective to centerline. In comparison between Fig. 8(c) and (d), it can be found that the length of touching surfaces with plates by bubble is different, though the number of bubble is the same. In other words, the cavitation can be intensified as shear velocity is increased further.

378



4 Conclusions A method by two parallel plates with shear moving velocity to induce cavitating flow is presented, and the feasibility is approved numerically, by a developed scheme based on Lattice Boltzmann method. The results show that the shear moving motion of two parallel plates could induce the cavitation, and the cavitation and cavitating flow pattern could be controlled efficiently by the main parameters, namely, initial density and shear moving velocity. The method and analysis could be extended to flowing liquid, and the influences of unsteady shear moving on the cavitaton and its flow pattern will be studied in the next work. Acknowledgements This research is supported by the National Natural Science Foundation of China (No51305355), the China Scholarship (No.201406295017), the National Key Technology R&D Program of China, (2013BAF01B02), and the National Fundamental Research Program of China (973 Program, No.2012CB026002).


379

Fig. 8 Pattern of cavitating flow with different velocity and ρint = 0.693

References [1] Ceccio, S.L. (2010), Friction drag reduction of external flows with bubble and gas injection, Annual Review of Fluid Mechanics, 42, 183–203. [2] Cao, W., Wei, Y., Wang, C., Zou Z. and Huang, W. (2006), Current status, problems and applications of supercavitation technology, Advances in Mechanics, 36(4), 571–579. (in Chinese). [3] Zhang, B., Zhang, Y. and Yuan, X. (2009), Effects of the profile of a supercavitating vehicle’s front-end on supercavity generation, Journal of Marine Science and Application, 8(4), 323–327. [4] Vlasenko, Y.D. (1998), Experimental investigations of high-speed unsteady supercavitating flows, Proc. Third International Symp. on Cavitation, 39–44. [5] Savchenko, Y.N., Vlasenko, Y.D., and Semenenko, V. (1999), Experimental studies of high-speed cavitated flows, International Journal of Fluid Mechanics Research, 26 (3), 365–374. [6] Singhal, A.K., Athavale, M.M., Li, H. and Jiang, Y. (2002), Mathematical basis and validation of the full cavitation model, Journal of Fluids Engineering-Transactions of the ASME, 124 (3), 617–624. [7] Chen, X. and Lu, C.J. (2005), Numerical simulation of ventilated cavitating flow around a 2D foil, Journal of Hydrodynamics, 17 (5), 607–614. [8] Kawakami E. and Arndt R.E.A. (2011), Investigation of the behavior of ventilated supercavities, Journal of Fluids Engineering-Transactions of the ASME, 133 (9), 091305. [9] Li, X.B., Wang G.Y., Zhang, M.D. and Shyy, W. (2008), Structures of supercavitating multiphase flows, International Journal of Thermal Sciences, 47 (10), 1263–1275. [10] Zhang, M.D., Wang, G.Y., Dong, Z.Q., Li, X.B., and Gao, D.M. (2008), Experimental observations of cavitating flows around a hydrofoil, Journal of Beijing Institute of Technology, 17 (3), 274–279. [11] Ren, S, Zhang, J.Z., Zhang, Y.M., and Wei, D. (2014), Phase transition in liquid due to zero-net mass-flux jet and its numerical simulation using lattice Boltzmann method, Acta Physica Sinica, 63(2), 024702. (in Chinese).

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Aims and Scope The interdisciplinary journal publishes original and new results on recent developments, discoveries and progresses on Discontinuity, Nonlinearity and Complexity in physical and social sciences. The aim of the journal is to stimulate more research interest for exploration of discontinuity, complexity, nonlinearity and chaos in complex systems. The manuscripts in dynamical systems with nonlinearity and chaos are solicited, which includes mathematical theories and methods, physical principles and laws, and computational techniques. The journal provides a place to researchers for the rapid exchange of ideas and techniques in discontinuity, complexity, nonlinearity and chaos in physical and social sciences. Topics of interest include but not limited to • • • • • • • • • • • • • •

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Igor Belykh Department of Mathematics & Statistics Georgia State University 30 Pryor Street, Atlanta, GA 30303-3083 USA Email: [email protected]

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Andrey Shilnikov Department of Mathematics and Statistics Georgia State University, 100 Piedmont Ave SE Atlanta GA 30303, USA Fax: +1 404 413 6403 Email: [email protected]

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An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 4, Issue 3

September 2015

Contents Modeling Fluid Dynamics in the Ocean and Atmosphere S.V. Prants.................................................................................................................

219－223

Clustering of a Positive Random Field –What is This? V.I. Klyatskin………...……..…………………………………..…………………..

225－242

Equilibrium Distributions for Hydrodynamic Flows V.I. Klyatskin………...……..…………………………………..…………………..

243－255

Hyperbolicity in the Ocean S.V. Prants, M.V. Budyansky, M.Yu. Uleysky, J. Zhang……………....…………...

257－270

Application of the Hydromechanical Model for a Description of Tropical Cyclones Motion Boris Shmerlin, Mikhail Shmerlin.………………………...…………..…………..

271－279

Influence of Deep Vortices on the Ocean Surface Daniele Ciani, Xavier Carton, Igor Bashmachnikov, Bertrand Chapron, Xavier Perrot…………………………...…………..…………………………..…….……

281－311

The Formation of Localized Atmospheric Vortices of Different Spatial Scales and Ordered Cloud Structures Boris Shmerlin, Maxim Kalashnik, Mikhail Shmerlin……...……..…………….....

313－321

An Approach to the Modeling of Nonlinear Structures in Systems with a Multi-component Convection Sergey Kozitskiy……...……..…………………………………………...……….....

323－331

Instability Development in Shear Flow with an Inflection–Free Velocity Profile and Thin Pycnocline S.M. Churilov……...……..……………………………………………...……….....

333－351

Transient Free Surface Flow Past a Two-dimensional Flat Stern Osama Ogilat, Yury Stepanyants……...…………………………...…………….....

353－369

Cavitating Flow between Two Shear Moving Parallel Plates and Its Control Yan Liu, Sheng Ren, Jiazhong Zhang……...….…………………...…………….....

371－379

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