PRL 101, 178501 (2008)
PHYSICAL REVIEW LETTERS
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Nonlinear Waves in Zonostrophic Turbulence Semion Sukoriansky* and Nadejda Dikovskaya Department of Mechanical Engineering/Perlstone Center for Aeronautical Engineering Studies, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Boris Galperin College of Marine Science, University of South Florida, St. Petersburg, Florida, USA (Received 14 May 2008; published 24 October 2008) The Charney-Hasegawa-Mima equation applies to a broad variety of hydrodynamic systems ranging from the large-scale planetary circulations to small-scale processes in magnetically confined plasma. This equation harbors flow regimes that have not yet been fully understood. One of those is the recently discovered regime of zonostrophic turbulence emerging in the case of small-scale forced, barotropic twodimensional turbulence on the surface of a rotating sphere or in its -plane approximation. The commingling of strong nonlinearity, strong anisotropy and Rossby waves underlying this regime is highlighted by the emergence of stable systems of alternating zonal jets and a new class of nonlinear waves, or zonons. This Letter elucidates the physics of the zonons and their relation to the large-scale coherent structures. DOI: 10.1103/PhysRevLett.101.178501
PACS numbers: 92.60.e, 92.10.Hm, 92.10.Lq, 92.10.Ty
The Charney-Hasegawa-Mima equation (CHME) [1,2] is a simple model of planetary and plasma turbulence. Charney [1] showed that for the former, CHME describes the regime of geostrophic turbulence pertinent to the largest planetary scales. Those scales are much larger than the scales at which the flow undergoes two-dimensionalization [3,4]. Even in its simplified, barotropic version (infinite Rossby deformation radius), the commingling of strong nonlinearity, strong anisotropy and Rossby waves gives rise to complicated dynamics. In flows with a small-scale forcing, the inherent anisotropic inverse energy cascade may lead to the development of the regime of zonostrophic turbulence, a subset of geostrophic turbulence [5,6]. This regime is distinguished by an anisotropic spectrum and stable systems of alternating zonal (east-west) jets. In this Letter, we show that another important attribute of zonostrophic turbulence is a new class of nonlinear waves coined zonons. Zonons may form coherent structures observable in physical space. The barotropic vorticity equation on the surface of a rotating sphere (BVES) is an example of the CHME with the infinite Rossby radius. The small scale forced and linearly damped version of this equation is an efficient simple model of the barotropic mode of planetary circulations [5]. Numerical experiments with such a system are at the focus of this Letter. The small-scale forced BVES is given by @ þ Jð c ; þ fÞ ¼ r2p þ ; @t
(1)
where is the vorticity; c is the stream function, r2 c ¼ ; f ¼ 2 sin is the Coriolis parameter (the planetary vorticity); is the angular velocity of the sphere’s rotation; is the latitude, is the longitude; is the hyper0031-9007=08=101(17)=178501(4)
viscosity coefficient; p is the power of the hyperviscous operator (p ¼ 4 in this study), and is the linear friction coefficient. The scale at which the flow frequency is equal to 2 is reciprocal to the wave number of the large-scale friction, nfr . The small-scale forcing, , acting on the scales around n1 , pumps energy into the system at a constant rate. Part of this energy becomes available for the inverse cascade at a rate . The Jacobian, Jð c ; þ fÞ, where JðA; BÞ ¼ ðR2 cosÞ1 ðA B A B Þ and R is the radius of the sphere, represents the nonlinear term. Equation (1) was solved numerically using the decomposition of the stream function in spherical harmonics Ynm ðsin; Þ, m being the zonal index. Conventionally, n and m are nondimensional. Setting R ¼ 1 eliminates the difference between the indices and wave numbers. The setup of the simulations was the same as in [6]. Equation (1) admits a class of linear Rossby-Haurwitz waves (RHWs) [7] with the dispersion relation m !R ðn; mÞ ¼ 2 : (2) nðn þ 1Þ The consistency with the -plane approximation is preserved by setting ¼ =R [8]. The energy spectrum for flows in spherical geometry is P Pn m 2 EðnÞ ¼ nm¼n Eðn; mÞ ¼ nðnþ1Þ where m¼n hj c n j i, 4R2 the modal spectrum, Eðn; mÞ, is the spectral energy density per mode (n, m), and the angular brackets indicate an ensemble or time average [9,10]. The spectrum EðnÞ can be represented as a sum of the zonal and nonzonal, or residual components, EðnÞ ¼ EZ ðnÞ þ ER ðnÞ, where the zonal spectrum is EZ ðnÞ ¼ Eðn; 0Þ. If the characteristic time scales of turbulence and RHWs are t ¼ ½n3 EðnÞ1=2 and R ¼ ½!R ðn; mÞ1 , respectively, then turbulent processes prevail on relatively small
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PRL 101, 178501 (2008)
scales where t < R while RHWs are dominant on large scales. The transitional wave number n ¼ 0:5ð3 =Þ1=5 at which t R marks the threshold of the effect induced anisotropization of the inverse energy cascade [6,11]. Another useful flow characteristic is the Rhines wave number, nR ¼ ð=2UÞ1=2 , U being the rms velocity, that can be related to the wave number of the large-scale friction, nfr [6,12]. If the interval (nR , n ) is wide enough, the processes in it can be expected not to depend on friction and be governed solely by the inverse energy cascade and effect. This interval will be referred to as the zonostrophic inertial range. If, on the other hand, nR * n , the drag would be a dominant factor on large scales where n < n . As shown in [5,6], the system (1) indeed attains two basic steady state regimes, friction-dominated and zonostrophic. These regimes differ by the degree of anisotropy and the nature of the wave-turbulence interaction. The width of the zonostrophic inertial range can be measured by the ratio R n =nR . The regime of zonostrophic turbulence emerges for R * 2 while the frictiondominated regime corresponds to R & 1:5. Figure 1 shows the energy spectra in the frictiondominated regime. The total spectrum, EðnÞ, closely follows the classical Kolmogorov-Batchelor-Kraichnan (KBK) scaling [8,13], EðnÞ ¼ CK 2=3 n5=3 ;
CK ’ 6:
(3)
For n > n , the modal spectrum is E ðm; nÞ ’ ð1=2ÞCK 2=3 n8=3 :
nðn þ 1Þ 2 hj c m n ð!Þj i; 4
(5)
where c m n ð!Þ is the time Fourier transform of the spectral coefficient c m n ðtÞ in Eq. (1). In pure 2D turbulence with no
waves, Uð!; n; mÞ has a symmetric bell shape around the zero frequency. In the case at hand, waves manifest themselves as spikes in Uð!; n; mÞ at frequencies !ðn; mÞ. Figure 2 shows Uð!; n; mÞ in the friction-dominated regime. The large-scale modes are populated by narrow spikes corresponding to linear RHWs. A strong RHW signature is present even on scales with n=n > 2. On the smallest scales, the RHW peaks become significantly broadened by turbulence. Even though the effect of rotation is significant and the flow dynamics is dominated by strong nonlinearity punctuated by the inverse energy cascade and the energy spectrum of well developed turbulence, the flow features linear RHWs. This result is
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FIG. 1. The kinetic energy spectra for a friction-dominated regime (nR ¼ 9:2, n ¼ 14:3). The thick and thin solid lines show EðnÞ and Eðn; mÞ, respectively. The dashed lines correspond to the spectra (3) and (4).
strikingly counter-intuitive. Its manifestation can be found in the large-scale terrestrial atmospheric circulation that exhibits a nonlinear, friction-dominated regime [6] but features linear RHWs [7]. Note that in a truly linear system, the excitation of every RHW would require a specific source. In the nonlinear framework, however, all possible RHWs emerge from the random noise of diverse nature. Figure 3 shows spectral anisotropization in the zonostrophic regime where EZ ðnÞ and ER ðnÞ attain the slopes given in [8,13],
(4)
Figure 1 reflects -effect caused anisotropization of the inverse energy cascade and preferential energy flux into low-m, n < n modes [11,12,14,15]. RHWs are a solution of the linearized Eq. (1). Are these waves present in the fully nonlinear equation and, if the answer is yes, how are they affected by the nonlinearity? To answer these questions, consider the velocity correlator, Uð!; n; mÞ ¼
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PHYSICAL REVIEW LETTERS
EZ ðnÞ ¼ CZ 2 n5 ; ER ðnÞ ¼ CK
CZ ’ 0:5;
2=3 n5=3 ;
(6a)
CK ’ 4 to 6:
(6b)
The transitional wave number, n , corresponds to the intersection of (6a) and (6b) [6]. The n5 scaling of the n n/nβ
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FIG. 2. The velocity correlator, Uð!; n; mÞ 107 , for the friction-dominated regime; nR ¼ 9:2, n ¼ 12:3. The filled triangles correspond to the RHWs dispersion relation (2).
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PHYSICAL REVIEW LETTERS
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FIG. 5. The correlator Uð!; n; 1Þ (solid lines) for a zonostrophic regime (nR ¼ 5:5, n ¼ 16:2). The filled triangles at the top of the panel correspond to the RHWs dispersion relation (2) with values of n shown near the triangles. To ease tracing the lines at different n, they are shown at different gray scales. FIG. 3. The kinetic energy spectra for a zonostrophic regime (nR ¼ 5:5, n ¼ 16:2). The thick gray and black solid lines show EZ ðnÞ and ER ðnÞ, respectively. The thin lines show Eðn; mÞ. The dashed lines correspond to the spectra (6a), (6b), and (4).
zonal mode extends beyond n up to the intersection with the spectrum (4) [13]. Figure 4 shows Uð!; n; mÞ for the zonostrophic regime. Along with the linear RHWs emerge nonlinear waves referred to as zonons; their frequency is denoted as !z ðn; mÞ. Most intriguingly, zonal zonons appear in the zonal mode where RHWs do not exist. With increasing n, both zonal and nonzonal modes may acquire multiple zonons. What is the physical nature of the zonons? To answer this question, Fig. 5 presents the correlator Uð!; n; 1Þ for n ¼ 3 through 15 viewed through a ‘‘magnifying glass’’ of the logarithmic scale. Up to n ¼ 6, each line Uð!; n; 1Þ exhibits the dominant peak at ! corresponding to its respective RHW frequency (2). The secondary peaks are associated with the zonons. The zonons are forced oscillations excited by RHWs in modes with the same m and
FIG. 4. The velocity correlator, Uð!; n; mÞ, for the regime of zonostrophic turbulence; nR ¼ 5:5, n ¼ 16:2. The filled triangles correspond to the RHWs dispersion relation (2) while the filled circles mark the zonons.
practically all other n. The frequencies of these forced waves are equal to the frequencies of the corresponding ‘‘master’’ RHWs and thus are independent of n. Figure 6 compares zonons with RHWs by presenting !ðmÞ for different n. The RHWs are clearly evident for all modes including those with n > n . Along with RHWs, one distinguishes two groups of zonons identifiable with RHWs with n ¼ 4 and 5. Figure 3 indicates these RHWs are most energetic. Furthermore, !z ðn; mÞ is proportional to m and virtually independent of n for both groups of zonons such that they form wave packets whose zonal phase speeds, cz ¼ !z ðn; mÞ=m, are equal to the zonal phase speeds of the corresponding master RHWs. Note the presence of the zonal zonons, !z ðn; 0Þ 0, whereas !R ðn; 0Þ ¼ 0 for all n. Figure 7 clarifies the nature of the zonal zonons. The total energy of zonal modes, Etot Z , oscillates in time as a superposition of waves with two frequencies corresponding to those of zonal zonons. Nonlinear interactions between zonal and nonzonal modes cause energy oscillations
FIG. 6. Frequencies of RHWs and zonons as functions of m for different n; nR ¼ 5:5, n ¼ 16:2. The empty circles and dashed lines correspond to the RHW dispersion relation (2); the filled circles and the solid black and gray lines show the dispersion relations for the zonons excited by RHWs with n ¼ 4 and 5, respectively.
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FIG. 7.
PHYSICAL REVIEW LETTERS
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Total energy of zonal modes as a function of time.
around the saturated value which can be calculated from the spectrum (6a) [8]. Clearly, all zonons are ‘‘slave’’ waves excited by RHWs. They have dispersion relations different from (2) and should be recognized as an entity completely different from RHWs. How do the zonons appear in the physical space? The RHWs with n ¼ 4 (denoted as nE ) are the most energetic, their respective packets of zonons are dominant in physical space and easiest to observe. The phase speed of these packets is !R ðnE ; mÞ=m cRE . In physical space, these zonon packets are expected to form westward propagating eddies detectable in the Hovmo¨ller diagrams (longitude-time plots of the stream function). The slope of the demeaned diagrams yields a velocity of the zonally propagating eddies relative to local zonal flows. If eddies are really comprised of zonons, their zonal phase speed should be equal to cz ¼ cRE . The Hovmo¨ller diagrams shown in Fig. 8 reveal westward propagating eddies at three different off-equatorial latitudes at which the zonal jets have their maximum, minimum, and zero velocity. Figure 9 further demonstrates that cz ¼ cRE at all three latitudes. The westward propagating eddies can be identified with the energetic zonon packets moving largely independently of the zonal flows. A new class of nonlinear waves in 2D turbulence with a -effect, zonons, is documented here for the first time. Zonons are forced oscillations excited by RHWs in other modes via nonlinear interactions. Zonons are an integral part of the zonostrophic regime. They emerge in the process of energy accumulation in the large-scale modes and
FIG. 8. The Hovmo¨ller diagrams of the stream function at three different latitudes in the zonostrophic regime. The white lines show the slope used to calculate the angular velocities.
FIG. 9. Comparison of zonal phase speeds computed from the Hovmo¨ller diagrams and the dispersion relation (2) for most energetic RHWs for a number of different simulations.
formation of the steep spectrum (6a). The mechanism of zonon generation is accommodated neither in conventional second-moment closure theories nor in theories of wave turbulence and weakly-nonlinear wave interactions. Future research should clarify zonons’ roles in planetary circulations and their relation to large oceanic eddies detected in satellite altimetry [16] provided that the oceanic circulation is marginally zonostrophic [6,17]. Partial support of this research by the ARO Grant No. W911NF-05-1-0055, ONR Grant No. N00014-07-11065, and the Israel Science Foundation Grant No. 134/03 is gratefully acknowledged.
*
[email protected] [1] J. Charney, J. Atmos. Sci. 28, 1087 (1971). [2] A. Hasegawa and K. Mima, Phys. Fluids 21, 87 (1978). [3] G. Vallis, Atmospheric and Oceanic Fluid Dynamics (Cambridge University Press, Cambridge, England, 2006). [4] C. Cambon, N. Mansour, and F. Godeferd, J. Fluid Mech. 337, 303 (1997). [5] B. Galperin, S. Sukoriansky, N. Dikovskaya, P. Read, Y. Yamazaki, and R. Wordsworth, Nonlin. Proc. Geophys. 13, 83 (2006). [6] S. Sukoriansky, N. Dikovskaya, and B. Galperin, J. Atmos. Sci. 64, 3312 (2007). [7] J. Holton, An Introduction to Dynamic Meteorology (Elsevier Academic Press, San Diego, CA, 2004), 4th ed. [8] H.-P. Huang, B. Galperin, and S. Sukoriansky, Phys. Fluids 13, 225 (2001). [9] G. Boer, J. Atmos. Sci. 40, 154 (1983). [10] G. Boer and T. Shepherd, J. Atmos. Sci. 40, 164 (1983). [11] G. Vallis and M. Maltrud, J. Phys. Oceanogr. 23, 1346 (1993). [12] P. Rhines, J. Fluid Mech. 69, 417 (1975). [13] S. Sukoriansky, B. Galperin, and N. Dikovskaya, Phys. Rev. Lett. 89, 124501 (2002). [14] G. Holloway and M. Hendershott, J. Fluid Mech. 82, 747 (1977). [15] A. Chekhlov, S. Orszag, S. Sukoriansky, B. Galperin, and I. Staroselsky, Physica (Amsterdam) 98D, 321 (1996). [16] D. Chelton, M. Schlax, R. Samelson, and R. de Szoeke, Geophys. Res. Lett. 34, L15606 (2007). [17] B. Galperin, H. Nakano, H.-P. Huang, and S. Sukoriansky, Geophys. Res. Lett. 31, L13303 (2004).
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