Dynamical Potential Energy: A New Approach to Ocean Energetics

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Dynamical Potential Energy: A New Approach to Ocean Energetics FABIEN ROQUET Meteorological Institute of the Stockholm University, Stockholm, Sweden (Manuscript received 6 June 2012, in final form 15 October 2012) ABSTRACT The concept of available potential energy is supposed to indicate which part of the potential energy is available to transform into kinetic energy. Yet it is impossible to obtain a unique definition of available potential energy for the real ocean because of nonlinearities of the equation of state, rendering its usefulness largely hypothetical. In this paper, the conservation of energy is first reformulated in terms of horizontal anomalies of density and pressure for a simplified ocean model using the Boussinesq and hydrostatic approximations. This framework introduces the concept of ‘‘dynamical potential energy,’’ defined as the horizontal anomaly of potential energy, to replace available potential energy. Modified conservation equations are derived that make it much simpler to identify oceanic power input by buoyancy and mechanical forces. Closed budgets of energy are presented for idealized circulations obtained with a general circulation model, comparing spatial patterns of power inputs generated by wind and thermal forcings. Finally, a generalization of the framework to compressible fluids is presented, opening the way to applications in atmosphere energetics.

1. Introduction The concept of energy has historically played a major role in the development of science, as it is one of the most fundamental, general, and significant principles of physical theory (Callen 1985). Applied to the study of physical oceanography, the two fundamental laws of thermodynamics (respectively, conservation of energy and maximization of entropy under specific conditions) have the potential to provide a deep understanding of what shapes the observed large-scale circulation for a given distribution of mechanical and thermohaline forcings. In particular, energy considerations should help to determine if ocean circulation is primarily forced thermally (i.e., by imposed density contrasts) or mechanically (from external sources of momentum). In practice, the relative contribution of wind, buoyancy fluxes, and internal mixing in setting the general circulation remains difficult to assess objectively. The wind forcing is an essential driver of the circulation through Ekman pumping, while buoyancy forcing is a condition sine qua non for the existence of an interior stratification (e.g., Gnanadesikan 1999). However,

Corresponding author address: Fabien Roquet, Meteorological Institute of the Stockholm University, Stockholm, Sweden. E-mail: [email protected] DOI: 10.1175/JPO-D-12-098.1 Ó 2013 American Meteorological Society

buoyancy forcing (i.e., heat and freshwater fluxes) alone is not efficient at driving an interior circulation (Sandstro¨m 1908), thus many authors have argued that a mechanical source of energy was needed to sustain the observed deep circulation, most likely related to wind and tidal mixing (Munk and Wunsch 1998; Huang 1999; Wunsch and Ferrari 2004). Ekman pumping is not very efficient either as follows: of roughly 60 TW of wind power input, only about 1 TW powers the interior circulation by Ekman pumping, with the rest of it going into waves and turbulence (e.g., Wang and Huang 2004). To make further progress, a refined understanding of ocean energetics is needed, preferentially at the local scale. A major difficulty in ocean energetics lies in the definition of the gravitational potential energy (PE), EP [ rgz,

(1)

where r is the in situ density, g is the local gravity, and z is the height relative to an arbitrary reference level. This definition fails to take into consideration that (i) the relative variations of density are very small in the ocean and (ii) mass conservation imposes global constraints on rates of change of PE: any ascending mass flux is always largely compensated by a descending mass flux elsewhere (when using the Boussinesq approximation, this

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latter constraint becomes one about volume conservation and vertical velocities as will be seen later). The concept of available potential energy (APE) was introduced precisely so as to overcome these limitations in the PE definition. APE is defined as the difference between the total potential energy and the minimum total potential energy, which could result from any adiabatic redistribution of mass (Lorenz 1955), corresponding to the fraction of PE that would be released in the ocean circulation while reaching adiabatically a resting hydrostatic state. APE is nowadays a widely used concept in the study of ocean energetics (e.g., Oort et al. 1989; Winters et al. 1995; Toggweiler and Samuels 1998; Huang 1998; Tailleux 2009; Hughes et al. 2009). However, APE is not uniquely defined for a realistic equation of state because of the nonlinear dependence of seawater density on both temperature and salinity (see Tailleux 2012, for a review). This limitation is a serious one as nonlinear effects of the equation of state are important in the ocean (e.g., Gnanadesikan et al. 2005; Iudicone et al. 2008; Klocker and McDougall 2010; Hieronymus and Nycander 2013). Algorithms have been proposed to diagnose APE in numerical simulations with a realistic equation of state (Huang 2005; Ilicak et al. 2012); however, it is unclear whether the APE reference state used in these methods is actually a state of minimum PE. Here a modified framework is proposed for the discussion of ocean energetics, based on what will be called dynamical potential energy (DPE). For a model using Boussinesq and hydrostatic assumptions with a linear equation of state, DPE is simply defined as the horizontal anomaly of PE, that is, the difference between PE and its horizontal average vertical profile. While the justification of DPE relies heavily on the applicability of the hydrostatic approximation, Boussinesq approximations and linearity of the equation of state can be relaxed provided that definitions of PE and DPE are suitably modified. After introducing the concept of DPE in section 2, a quick comparison with APE will be proposed in section 3. The modified framework will be derived in section 4. To demonstrate the utility of this new energy framework, local balances of kinetic energy (KE) and DPE will be presented in section 5, comparing the energetics of three simulated circulations differing only in their surface forcings—wind forcing alone, thermal forcing alone, and wind and thermal forcings together. A discussion of the origin and nature of the power converted from PE to KE will be proposed in section 6. Finally, methods to generalize the DPE framework to cases with nonlinear equation of state and compressible fluids are described in section 7.

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2. Introduction of dynamical potential energy Roquet et al. (2011), discussed the wind-provided input of KE to the large-scale circulation using Ekman theory. Two different representations were used, mapping the following. (i) The direct rate of wind work on the geostrophic circulation, obtained from scalar product of the wind stress and surface geostrophic velocity (Stern 1975), t o ug,o, suitably time and space averaged. (ii) The input of pressure work into the ocean interior, as the product of (minus) the surface pressure elevation times the vertical velocity induced by Ekman pumping, 2po*wEk , again suitably averaged. The surface pressure elevation is proportional to the sea surface height defined relative to the global mean sea level, po* 5 ro gh, and the Ekman pumping velocity, wEk, is related to the curl of the wind stress. Although the two representations have markedly different spatial patterns, their integrals over the global ocean are strictly equal for a steady-state ocean—provided that the horizontal Ekman transport vanishes across coastlines. The direct input of wind work to the geostrophic circulation (first view) is redistributed laterally into the Ekman layer before entering the geostrophic interior as an input of pressure work (second view). Because this input of pressure work is proportional to the horizontal anomalies of sea surface height (i.e., sea surface height relative to global mean sea level), it can be reinterpreted as the power locally generated by Ekman pumping to maintain the local anomaly of sea surface height relative to its equilibrium—which is global mean sea level. The flux of pressure work is in fact simply expressed as a flux of pressure, highlighting the role of pressure in transporting energy in the ocean interior. Pressure has the same dimension as an energy per unit volume: N m22 [ (N m) m23 5 J m23, and a flux of pressure indeed represents a power input per unit area. Pressure, and more specifically its horizontal anomalies, appear as central to understanding oceanic energetics. The framework developed in this paper is in part a generalization of Roquet et al. (2011), exploring the role of horizontal anomalies of pressure and density. For the sake of clarity, we will now focus on a simplified ocean model using the Boussinesq and hydrostatic approximations (e.g., Vallis 2006), with a linear equation of state. Introduce the notation T(x, y, z, t) 5 T(z, t) 1 T*(x, y, z, t)

(2)

for any tracer T. The horizontal mean T is a function of depth and time

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T(z, t) 5

1 A(z)

ðð T(x, y, z, t) dx dy ,

(3)

ÐÐ

where A(z) 5 dx dy is the ocean area at depth z. The hydrostatic balance can be rewritten by separating horizontal mean and horizontal anomalies of both pressure p and density r ›p* ›p 5 2r*g 2 1 rg . ›z ›z

(4)

The two last terms are bracketed, as they nearly cancel (exact cancellation with vertical sidewalls) and are both constant in the horizontal. We will see later how a new pressure variable can be defined, the so-called dynamical pressure, to account for these two horizontal mean contributions. For now, note only that a reduced horizontal hydrostatic balance can be obtained, where horizontal anomalies of the gravity term nearly balance the vertical gradient of pressure anomalies. Define the DPE as the horizontal anomaly of PE EP* [ r*gz .

(5)

Note that this definition applies only for an idealized fluid assuming Boussinesq and hydrostatic approximations and a linear equation of state (or at least an equation of state independent of depth), yet an equivalent definition can be obtained for non-Boussinesq fluids and nonlinear equations of state (see section 7). The rate of conversion of DPE to KE is 2r*gw. Compared to the rate of conversion from PE to KE (2rgw), this expression proves advantageous as follows. (i) For a resting fluid with flat isopycnals, 2rgw increases greatly with depth as it is proportional to the full hydrostatic pressure gradient, while 2r*gw simply vanishes everywhere. (ii) From volume conservation (using Boussinesq assumption), the horizontal mean vertical velocity is always exactly zero, w 5 0. This is why global integrals of both forms of conversion rates are always strictly equal, ððð

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2r*gw dV 5

ððð

2rgw dV .

(6)

In other words, the rate of conversion of DPE to KE is more readily associated with the observed stratification, while representing the same global amount of conversion as does the more conventional definition. This property constitutes the main justification for focusing on DPE to study ocean energetics.

3. Comparison with available potential energy The difference in definition between APE and DPE can be best viewed when comparing their global integrals over a closed ocean basin. The global reservoir of APE is always strictly positive for a stable stratification, except when isopycnals are flat at every depth. In contrast, for DPE, the global reservoir is always exactly zero, ððð (7) EP* dV 5 0. A simple example of DPE and its contrast with APE is shown in Fig. 1a, for a density distribution resembling the idealized pycnocline model proposed by Gnanadesikan (1999). A thermocline separates a shallow, light lens of water from a deeper and denser water mass. The densest water mass is able to reach the surface in the right corner of the domain, creating a horizontal gradient of density where the thermocline is vertical. Reference profiles of density used to calculate both DPE and APE are shown in Fig. 1b, with the resulting density anomalies used to calculate DPE and APE in Figs. 1c and 1d, respectively. Because DPE always integrates to zero globally, a more useful measure of the magnitude of regional anomalies is the integral over the regions of positive DPE, DPE1 [

ððð E*.0 P

EP* dV 5

ððð

jEP*j dV . 2

(8)

The global size of the APE reservoir is ðð

APE 5 (1 2 b)L 5 r0 g

ð0 abH

r0 gz dz 2 bL

ð abH aH

r0 gz dz (9)

b(1 2 b) L(aH)2 , 2

(10)

where a is the ratio of light water-layer depth to fulldepth H, b is the ratio of light-layer length to length of the basin L, and r0 is the difference of density between the two layers. In this example, it is found that the global reservoir of positive DPE values is strictly identical despite qualitative differences in the distribution of density anomalies, DPE1 5 bL

ð0 aH

2(1 2 b)r0 gz dz 5

ðð APE.

Note that equality between global sizes of DPE1 and APE reservoirs will not hold in the general case as, for example, introducing a thermocline tilt destroys it. Yet, the example presented here indicates that they should

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FIG. 1. (a) Comparison of DPE and APE for a simple density distribution is shown. (b) Reference vertical profiles used to compute DPE and APE are shown. Density anomalies obtained as the difference between the density distribution and the reference profile are presented for (c) DPE and (d) APE. The global integral of DPE is exactly zero, while it is strictly positive for APE. However, the global integral of positive values of DPE (DPE1) is strictly equal to the global integral of APE (not true in general).

be comparable for realistic distributions of mass in the ocean (in the model study cases presented later, we found differences of order 20%, see Table 1). Here, it must be emphasized that APE and DPE are very different in nature. APE is fundamentally a thermodynamic concept, making use of an adiabatic rearrangement of fluid parcels to define a state of reference. It relies on the weakness of interior mixing to provide a valuable estimate of the fraction of PE available to the general circulation. In contrast, DPE is a purely dynamical concept finding its justification in the predominance of the hydrostatic equilibrium in the vertical momentum balance, allowing removal of the large static contribution to the hydrostatic pressure in the dynamical equations. Contrary to the situation with APE, no consideration of water mass mixing or adiabatic fluid motion is involved in the definition of the reference state of DPE. APE works well for adiabatic processes, such as wave propagation, baroclinic instability, or the effect of wind pumping on the stratification, since the reference state is then constant through time. However, when mixing is considered, the usefulness of APE becomes questionable as its reference state is constantly changing with

time and rarely corresponds to the resting state that would eventually reach the system by switching off all forcings. With a nonlinear equation of state, the APE reference state is not even well defined. Like in the case

TABLE 1. Size of energy reservoirs and rates of work integrated globally in the WIND, RELAX, and MIXED experiments. Abbreviations for reservoirs are as in the core text. For the total rates of work, WIND is the surface input of frictional power, CONV is the total conversion rate, FRICTION is the total rate of frictional work minus the WIND input, and CORIOLIS is for the Coriolis term, which is slightly working in the model. The total KE trend (sum of the four working rates) is negligible, as circulations are in equilibrium. WIND KE PE DPE1 APE WIND CONV FRICTION CORIOLIS

RELAX

Reservoir size (PJ) 0.77 3.79 27.6 3 108 27.6 3 108 0 8.8 3 103 0 6.3 3 103 Total rate of work (GW) 1.73 0 0 17.99 21.71 220.03 20.02 2.04

MIXED 7.87 27.6 3 108 10.3 3 103 8.0 3 103 25.89 10.42 239.13 2.82

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of APE, the DPE reference state is varying in time in the general case of an unbalanced circulation, yet in practice, DPE is much easier to diagnose and well defined for any equation of state (although its definition has to be slightly changed for a nonlinear equation of state, see section 7). The concept of DPE will now be used to derive a modified framework for the study of energy conservation.

DEP 5 rgw 1 WD , Dt

(17)

Consider now the conservation of energy for a simplified ocean model using the Boussinesq and hydrostatic approximations (e.g., Vallis 2006), assuming that the equations of state is linear.

where EP 5 rgz is the potential energy and WD 5 Dgz represents the rate of production of PE, that is, sources and sinks of PE related to diabatic processes. The reference level z 5 0 is arbitrary—taken at the surface in the following. Equations (16) and (17) define the energetics for an hydrostatic Boussinesq ocean model. Note that the global constraint of volume conservation has not been used to derive these conventional equations. The continuity equation was only used locally to transform the advection operator into a flux divergence. We turn now to formulating similar equations using DPE.

a. Standard formulation of energetics

b. Modified KE balance

Equations of motion are standard, for momentum conservation, the hydrostatic assumption, continuity, and density conservation

To replace the PE to KE conversion term by a DPE to KE conversion term, expressed as 2r*gw in the equations of energy conservation, the ‘‘dynamical pressure’’ variable is defined as follows:

4. Derivation of the framework

ro

Duh 1 ro f k 3 uh 5 2$h p 1 F , Dt

(11) (12)

$ u 5 0, and

(13) (14)

with D/Dt [ ›/›t 1 u $.

(15)

Here, u 5 (uh, w) is the velocity vector, ro is a constant reference density, f is the Coriolis frequency, and F is the frictional force. Here, D is the local rate of production of density, representing sources and sinks of density due to diabatic processes, such as diffusion, convection, and surface density fluxes. The standard equation for KE is obtained as the scalar product of the momentum equations with the velocity vector DEK 5 2$ ( pu) 2 rgw 1 WF , Dt

ð0 z

›p 5 2rg, ›z

Dr 5 D, Dt

pdyn 5 p* 2

(16)

where EK [ ro juh j2 /2 is the hydrostatic kinetic energy. The term WF 5 uh F represents the rate of work of frictional forces. Note that the pressure divergence term includes both horizontal and vertical pressure fluxes. The equation for PE is obtained from the multiplication of the density Eq. (14) by gz,

›p 1 rg dz, ›z

(18)

where z 5 0 corresponds to the surface level, using a rigid lid assumption for simplicity. The vertical axis is directed upward, so z # 0 in the interior. A method to account for a linear free surface is presented in appendix B. It suffices to note here that it does not fundamentally modify the results presented below. The energetics of a fully nonlinear free surface is non trivial, so it is left aside here. For a basin with vertical sidewalls, dynamical pressure is simply the horizontal anomaly of pressure. In the more general case of sloping bottom topography, the two expressions differ by a function of depth only (which is typically small compared to the magnitude of pressure anomalies). From Eq. (4), dynamical pressure has the same horizontal gradient as the full pressure, and satisfies a reduced hydrostatic balance $h pdyn 5 $h p,

›pdyn 5 2r*g ›z

(19)

producing the modified KE equation DEK 5 2$ (pdyn u) 2 r*gw 1 WF , Dt

(20)

which is very similar to the conventional form in Eq. (16), except that it employs only anomalies about the horizontal average, with the horizontal mean contribution to the hydrostatic gradient suppressed.

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c. DPE balance In the PE Eq. (17), a near cancellation is found between horizontal mean contributions of conversion and the PE advection terms

›r rgw 2 $ (EP u) 5 2gz w 5S. ›z

(21)

The residual S is named the ‘‘stratification term,’’ as it is related to the horizontal mean stratification. The stratification term has neither the form of a conversion nor a flux divergence, and looks like a source or sink of PE, although physically, there is no corresponding local source or sink. Note that S is proportional to w at each depth level, so its horizontal mean always vanishes. This term represents a transfer of power between different areas on the same depth level, redistributing horizontally sources and sinks of PE. Define the rate of production of DPE as the sum of the PE production rate and the stratification term ›r 0 . 5 WD 1 S 5 gz D 2 w WD ›z

(22)

The rate of production of DPE depends on a competition between local rates of production of density by mixing and diffusion processes, and the vertical advection of the horizontal mean density. PE balance becomes ›EP 0 1 $ (EP*u) 5 r*gw 1 WD . ›t

(23)

Note that the time derivative term involves the full PE, but the flux divergence term operates on DPE only, as advection of horizontal mean PE is embedded in the stratification term. The DPE framework is made of Eqs. (20) and (23). The global integral of this system of equations is identical to the global integral of the standard Eqs. (16) and (17). More interestingly, their respective horizontal means are also identical. Differences appear only when comparing their local expressions, with advantage in the modified formulation for mapping and interpreting the energy contributions. The model study in the next section will further illustrate this important point. The case of a hydrostatic Boussinesq ocean model using a nonlinear equation of state will be presented in section 7. The major difference with the linear case is in the definition of PE (Young 2010; Nycander 2011). Otherwise, equations for energy conservation are strictly identical. The case of a compressible hydrostatic model using a nonlinear equation of state will also be treated,

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demonstrating that only the hydrostatic assumption is required to derive this framework.

5. Energy diagnostic of MITgcm simulations To illustrate the use of the DPE framework, a lowresolution idealized configuration of the Massachusetts Institute of Technology general circulation model (MITgcm) (Marshall et al. 1997) is used, consisting of a Northern Hemisphere basin spanning 458 latitude and 608 longitude over 3 km in depth (40 3 48 3 15 grid points). Sidewalls are vertical from the surface to 1200 m, then have a 2% slope until they reach a flat bottom at 3000-m depth—see Hughes and DeCuevas (2001), to see why the slope matters to obtain a realistic quasi-inviscid circulation. Boundary conditions are no-slip on the sidewalls and bottom, with a linear free surface; no mixed-layer scheme is used; simple friction and diffusion schemes (no eddy parameterization mixing scheme) are employed. A linear equation of state is used, that depends only on temperature. Because density and temperature variations are proportional, the numerical model satisfies equations presented in section 4, with r 5 ro (1 2 aT) ,

(24)

F 5 Az ›2z u 1 Ah Dh u, and

(25)

D 5 2ro a(kz ›2z T 1 kh Dh T) ,

(26)

with ro 5 103 kg m23, a 5 2 3 1024 8C21, Ah 5 5 3 103 m2 s21, Az 5 10 m2 s21, kh 5 400 m2 s21, and kz 5 1024 m2 s21, increased to 1 m2 s21 in case of static instability. Three experiments were performed, where the model was run from a resting homogeneous state to near equilibrium (2000 years) using different configurations of forcing fields (see Fig. 2a). In the WIND experiment, a sinusoidal zonal wind stress field was applied at the surface, with a 0.1 N m22 maximum value at 358N. In the RELAX experiment, the surface temperature was relaxed toward a meridional profile of reference ranging from 58C in the north to 208C in the south, with a restoring time scale of 30 days. Wind stress and temperature relaxation were applied together in the MIXED experiment, mimicking North Atlantic conditions.

a. Overview of simulated circulations The balanced state of the WIND experiment consists in a classic double-gyre circulation with a western intensification of currents forming a northeast jet between the two gyres (Fig. 3a). In this experiment, the most

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FIG. 2. (a) Meridional distribution of wind stress (gray curve) and temperature relaxation profile (black curve) used in the MITgcm simulations. (b) Global-mean stratification of the balanced state for the three experiments.

notable feature is that the ocean is completely unstratified, permitting simplifications in the energy equations. In particular, r* 5 0 everywhere. Consequently, the dynamical pressure is constant with depth, as expected for a barotropic circulation, and the rate of DPE to KE conversion, as well as all terms in the DPE equation, are identically null. The circulation simulated in the RELAX experiment corresponds to what Stern (1975) called horizontal convection (see review of Hughes and Griffiths 2008), where warm and cold sources are applied on the surface geopotential. A pattern of density anomalies is generated at the surface, requiring the existence of a largescale circulation to establish a stationary state. Although the RELAX circulation features a double-gyre structure like in the WIND experiment (cf. Figs. 3a and 3c), numerous fundamental differences exist between the two circulations. In particular, surface heat fluxes induce a subsurface thermocline separating the surface-intensified

gyres from a slower deep reaching flow (see Fig. 2b and 3d). A strong asymmetry now occurs between the northern and southern gyres. The northern gyre presents a zonally elongated shape that follows closely the northern boundary of the domain. By contrast, the southern gyre is less intense, shallower but more wide-spread, with its western boundary current touching the western wall. Overall, the circulation induced in the RELAX experiment is far more intense than in the WIND experiment: the total reservoir size of KE is 3.8 PJ, 5 times larger than the 0.8 PJ reservoir in the WIND case (see Table 1). This occurs because KE is quadratic in the velocity, so even with a similar depth-integrated transport, a surface intensified circulation is more energetic than a barotropic one. Overall, the MIXED experiment strongly resembles the RELAX experiment, with a similar circulation pattern consisting in two asymmetric gyres. However, MIXED circulation is surface-intensified because of wind forcing, resulting in a KE reservoir with a doubled size compared to RELAX (7.9 PJ). The thermocline is slightly deeper than in the RELAX case (see Fig. 2b and 3f), seen as a 20% increase of the DPE1 reservoir size. This deepening is consistent with wind stress suction in the north, and pumping in the south, thus accentuating the magnitude of horizontal anomalies of density. The size of the APE reservoir can be easily estimated for the three experiments because the equation of state is fully linear. The APE density reference profile is obtained by sorting water parcels by density. In section 3, it was argued that APE and DPE1 reservoirs should generally be of comparable size. Indeed, typical differences are of 20% in the simulations, with a similar increase between the RELAX and the MIXED experiments using either of the definitions.

b. Energetics of the WIND experiment The steady-state balance of KE in the WIND simulation consists of a simple balance between the divergence of dynamical pressure fluxes and the work of frictional forces $ ( pdyn u) 5 WF ,

(27)

where advection of KE is neglected, as it is typically one order of magnitude smaller than other terms. Globally, 1.7 GW of wind stress power input enters at the surface, which is then removed in the interior through friction (Table 1). A small contribution comes from the Coriolis term in the model, as its discretized form is not perfectly orthogonal to the velocity vector. The horizontal mean budget is shown in Fig. 4, including the vertical profile of dynamical pressure flux (Fig. 4a),

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FIG. 3. Distribution of (left) dynamical pressure and (right) horizontal density anomalies for the three experiments. Positive (negative) anomalies are given in red (blue). For each panel, the value of isosurfaces is given in the subtitle (same value used for the surface contour interval). For example, (a) the 60.01 dbar isosurfaces of dynamical pressure for the WIND experiment are shown, and surface contours are superimposed with a 0.01 dbar contour interval.

whose vertical gradient balances the frictional work (Fig. 4b). This budget is exactly the same when considering the standard form of energy equations or the modified one, as pdyn w 5 pw and r*gw 5 rgw (50 in this case). The value of the modified framework appears when considering the local budget. Because the DPE to KE conversion term is zero everywhere, the local description of energetics is greatly simplified. Frictional forces work in two main areas (Fig. 5a). Sources of KE are

found in the first vertical level of the model, and sinks are found at every depth below, mainly along the southern and western boundaries where the currents are strongest. Power is transported between wind-driven sources and dissipative sinks as fluxes of dynamical pressure, moving downward while rotating around the two gyres. Although not shown here, the precise region of dissipation lies close to regions of frictional work as typical distances over which viscous stress fluxes are transported

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FIG. 4. Horizontally integrated budget of energy ÐÐ for (a),(b) WIND, (c),(d) RELAX, and (d),(e) MIXED experÐÐ Horizontally iments. (left) Vertical fluxes of dynamical pressure ( pdyn w)ÐÐand DPE ( Ep* w) are presented. (right) ÐÐ integrated rates of work for (friction) the frictional forces ( W ), (conv) the conversion term 2r*gw, and (PE F ÐÐ prod) ÐÐ the PE production term ( WD ) are shown. Note that the stratification term has no contribution on the horizontal ( S 5 0). Advection of KE is not shown, because it is a small contribution that can safely be neglected here.

before being dissipated are small compared to ocean basin scales, along both vertical and horizontal directions (see appendix A). Figure 6 compares the theoretical distribution of inputs of pressure work induced by Ekman pumping as defined by Roquet et al. (2011) with vertical fluxes of dynamical pressure (i.e., horizontal anomalies of pressure

in this case) taking place below the first vertical level of the model, that is at a 50-m depth. As expected, the two distributions compare favorably, although numerical noise at the grid scale is apparent in the model. Note that the comparison is successful because dynamical pressure is considered instead of full hydrostatic pressure. Otherwise, the large hydrostatic contribution to the pressure

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FIG. 5. Distribution of (left) rates of frictional work and (right) rates of DPE into KE conversion for (a),(b) WIND, (c),(d) RELAX, and (e),(f) MIXED experiments. Regions of power sources are connected to regions of power sinks by large-scale fluxes of dynamical pressure (see Fig. 3), thus following geostrophic contours on the horizontal.

field would indeed hide correlations existing between sea surface height and subsurface isobars.

c. Energetics of the RELAX experiment 1) KE BALANCE In the RELAX experiment, power comes necessarily from the conversion of DPE into KE, as there is no surface wind stress $ ( pdyn u) 5 2r*gw 1 WF ,

(28)

where advection of KE is again neglected. The resulting circulation is more vigorous than in WIND, both in terms of KE reservoir size and rates of work. The total conversion rate (18.0 GW, see Table 1) is 10 times larger than the wind-power input in the WIND circulation. The horizontally integrated budget shows a similar balance to the WIND case, except that the source of KE is now provided through the DPE to KE conversion rate (see Fig. 4). Horizontal mean conversion is positive at all depths (Fig. 4d), while the frictional power is negative. The vertical flux of dynamical pressure, which transports

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FIG. 6. Comparison of spatial patterns of (left) upward Ekman-induced inputs of pressure work as defined by Roquet et al. (2011) (po*wEk , see section 2), and (right) dynamical pressure flux below the first vertical level for the three model experiments ([pdynw]50m). The two patterns compare favorably for (top) the WIND case. (middle) In the RELAX case, the buoyancy-driven flux of dynamical pressure is mainly localized along coastal boundaries. (bottom) The MIXED case pattern is the superposition of an Ekman-induced input with a buoyancy-driven pattern as in the RELAX case. Bathymetric contours are superimposed (doted lines).

power between sources and sinks, reverses direction at about 500-m depth, from upward above to downward below (Fig. 4c). Spatial distributions of frictional work and DPE to KE conversion rates are shown in Figs. 5c and 5d, respectively. Areas of large DPE to KE conversions are

confined to thin regions along western, northern, and eastern boundaries, extending deep in the water column, up to 1000 m in the weakly stratified northern gyre (see Fig. 5d). DPE to KE conversion rates are locally of both signs, as the vertical velocity shows. Two surfaceintensified regions of dissipation are found, the first south

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of the western boundary and the second north of the eastern boundary. Another less active region of dissipation is found at depth along the northern flank of the basin, associated with the deep circulation of the northern gyre.

2) DPE BALANCE The DPE to KE conversion rate results from a balance involving the divergence of DPE fluxes and the DPE production rate 0 . $ (EP*u) 5 r*gw 1 WD

(29)

Taking its global integral, it appears that the total conversion rate is powered by the total DPE production rate, equal to the total PE production rate (the global integral of the stratification term is always zero). The horizontally integrated budget indicates that DPE is produced in the interior (see PE production in Fig. 4d), and is then advected upward before being converted (see Fig. 4c). As for KE, the horizontally integrated DPE balance is strictly identical to the standard PE one (EP w 5 EP*w and WD 5 WD0 ). Consider now the local expression of DPE conservation. The spatial distribution of zonally integrated contributions to the DPE balance are presented in Fig. 7 (left panels). The PE production term is positive wherever diffusion acted to increase the local temperature, mainly within the southern thermocline. In contrast, it is negative below the convectively mixed layer in the northern part of the basin. The stratification term is locally of the same order of magnitude as the PE production term, nearly balancing the positive production rates of PE in the thermocline. Recall that the horizontal mean of the stratification term is always zero, thus in effect, this term induces a large horizontal redistribution of the PE produced by diffusion and mixing, concentrating most of it along the northern boundary. Distributions of conversion and DPE production terms are strikingly similar, especially when compared to the distribution of PE production areas. A large part of the conversion occurs in the northernmost sector of the basin. Positive conversion rates are also found along the western boundary (see Fig. 5d), powering the weak southern gyre. This somewhat counterintuitive result illustrates the importance of the stratification term in redistributing horizontally PE production before its actual conversion. Although PE is largely produced in stratified areas, where diffusion and mixing erode efficiently local density gradients, most of the conversion occurs in weakly stratified areas mainly because the largest vertical velocities are observed there.

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d. Energetics of the MIXED experiment One striking feature of the MIXED experiment is that, although it is forced with the same wind stress distribution as the WIND experiment, it has an input of wind power 15 times larger (see Table 1). Added to the power input from DPE to KE conversion, the MIXED circulation receives twice as much power input as does the RELAX experiment. This illustrates the complexity of energy diagnostics, where wind and buoyancy forcings strongly interact to generate the observed circulation. To first-order, the heat forcing seems to control the large-scale distribution of density, as the MIXED stratification is very similar to the RELAX one. The wind forcing controls the intensity of the surface circulation. Because of the stratification, the wind forcing acts on a shallower layer than in the baroclinic case, thus able to produce faster currents, hence the larger wind-power inputs and increased size of the KE reservoir. The DPE to KE conversion rate is modulated by the presence of wind (cf. Figs. 5d and 5f) as Ekman pumping modifies the depth of isopycnals. Bands of negative conversion rates are seen at the surface in both gyres as pumping is directed upward in the northern gyre, and downward in the southern one. As a result, the horizontalmean conversion rate is negative near the surface (Fig. 4f), and the total conversion rate is almost halved compared to the RELAX case (see also Fig. 7, right panels). Using a depth-density streamfunction to analyze the global overturning circulation, Nycander et al. (2007) found that the subsurface circulation was on average thermally indirect, that is, powered by wind, and the deeper circulation was thermally indirect (i.e., driven by heat conduction and small-scale mixing) in a similar simulation. This is consistent with our findings, showing how Ekman pumping has two separate contributions on the KE balance: a direct frictional power input, and an indirect effect on conversion rates through the vertical displacement of isopycnals. In the MIXED experiment, these two effects are acting against each other, yet this is not necessary the case. This simulation illustrates why the common practice of describing separately a thermohaline circulation and a wind-driven one in the ocean has some justification, although this separation is not exact. Wind and heat forcings act differently on the ocean, producing distinct patterns of circulation. This is clearly seen in the distribution of vertical fluxes of dynamical pressure (see Fig. 6). The MIXED pattern is made of the superposition of a wind-driven Ekman pattern and a buoyancy-driven pattern similar to the one in the RELAX case. Again, the wind-driven pattern of pressure flux compares well

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FIG. 7. Zonally integrated distributions of the different contributions to the DPE balance diagnosed for (left) the RELAX and (right) MIXED experiments: (top to bottom) the PE production rate WD , the stratification term S, the DPE production rate WD0 5 WD 1 S, and the conversion rate 2r*gw.

with the theoretical expression discussed in Roquet et al. (2011). However, this pattern is significantly different from the one in the WIND case, although the wind forcing is the same. This is another illustration of the leading role of stratification on the circulation.

6. Conversion and mixing energy supply The total conversion rate is the only contribution to the PE balance that does not depend on the reference level

used to define PE, but it is not directly related to the distribution of mixing and surface buoyancy fluxes. On the other hand, the rate of production of PE has a straightforward relation to mixing and diffusion, but its expression does depend on the reference level. However, there is a strong relationship between the conversion rates and PE production rates that will be discussed here. For a balanced state, the total conversion and total PE production are strictly equal

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FIG. 8. Vertical distribution of sources and sinks of density for (a) RELAX and (b) MIXED experiments. In the Bousinesq approximation, density sources/sinks are included to represent contraction/expansion of water parcels. As the circulation is balanced, the total amount of production and destruction of density are matching in absolute value. The difference in their respective mean depths is the source of conversion power, consistent with the classical Sandstro¨m (1908) argument.

ððð

2rgw dV 5

ððð WD dV .

(30)

The total PE production is then independent of the reference level. The total production of density is zero in steady state. Divide the ocean interior into two volumes, one with sources of density (i.e., regions where diffusion and mixing tends to increase the local density), and another one with sinks of density. The two volumes contain the same amount of density variation, with opposite signs D[

ððð

jDj dV 5 2

ððð

ððð D dV 5 2

D.0

D dV . (31)

D,0

Now consider the mean depth of each 1 z [ D 1

ððð

Dz dV,

D.0

1 z [2 D 2

ððð

Dz dV . (32)

D,0

Finally, the total conversion rate, which is equal to the total PE production rate, is proportional to the total amount of density sources and the mean depth difference between density source and sink volumes, ððð (33) 2rgw dV 5 Dg(z1 2 z2 ) .

Hence, the total conversion rate available to power the circulation depends on both the intensity of diffusion and its vertical distribution. To have a positive total conversion, the center of gravity of density sources (cooling) must be above the center of gravity of density sinks (heating) (Sandstro¨m 1908). This result can be reinterpreted in terms of expansion and contraction areas, as the Boussinesq approximation essentially consists in expressing volume expansion as a density sink and contraction as a density source. Although heat flux is applied at ocean surface, regions of contraction and expansion are found at depth, and a buoyancy driven circulation can be generated (Jeffreys 1926). In the RELAX experiment, expansion occurs at a mean depth of 217 m compared to 334 m for contraction, thus Dz 5 117 m (Fig. 8). In comparison, Dz is only 46 m in the MIXED experiment, as Ekman pumping induces a sink of density at the surface, which explains the reduction of total conversion rate. In both cases, the height difference is small compared to the basin depth, consistent with the Sandstro¨m (1908) argument that horizontal convection is inefficient. Yet, this small difference in depth is sufficient to produce a total conversion rate comparable (though smaller) to the wind power input in the MIXED experiment, and far bigger than the power input in the wind only case. In this sense, Ekman pumping is not very efficient either. Note also that the decrease in MIXED total conversion as compared to

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RELAX is mainly due to shoaling of the mean expansion depth, due to large wind-related sinks of density at the surface. This wind-induced effect on conversion rates is thought to be large in the real ocean, due to the open zonal channel in the Southern Ocean, resulting in a negative total conversion rate (Toggweiler and Samuels 1998). The origin of power, which is converted from PE to KE, can now be better understood, as it comes directly from diffusion and mixing. The type of energy supply depends on the nature of the mixing involved. In the real ocean, mixing of density can result of a large variety of processes, including molecular diffusion, eddy stirring, wave–current interaction, double-diffusion or convective adjustments (Wunsch and Ferrari 2004). Each process has its own power supply. For example, convection depends on static stability properties of the water-column and is powered by PE conversion. Internal mixing can be greatly enhanced by turbulence and wave activity, such as internal tides, which are mechanically powered. Nonlinearities of the equation of state also provide large sources and sinks of density associated with the isoneutral mixing. In the model, constant vertical and horizontal diffusivity coefficients were specified. These parameters are unrealistically large compared to molecular diffusivities to ascertain numerical stability and to account for subgridscale turbulence and stirring. The origin of the mixing power source remains unspecified in our model, as the processes involved in this diffusion are not resolved. More sophisticated models generally use subgrid scale parameterization trying to mimic the diffusion induced by these unresolved scales, often using energetic arguments to be as realistic as possible. That subgrid scale mixing is parameterized means that sources and sinks of PE are locally applied directly in the interior of the simulated ocean. Note that knowledge of mixing energy supplies and mixing efficiency is not sufficient to determine the total conversion. One must know the spatial distribution of density sources and sinks, which depend also on geometrical constraints and buoyancy forcings. In the limit case of no buoyancy forcing, the ocean eventually becomes homogeneous, that is, spatial anomalies of density (hence DPE) vanish. No conversion can then happen, regardless of the intensity of mixing.

7. Generalization to seawater Boussinesq and compressible models In earlier sections, a DPE framework has been derived using the simple Boussinesq hydrostatic model, using an assumption of linear equation of state. This assumption will now be relaxed using the so-called seawater Boussinesq approximation (Vallis 2006) where the equation

of state is any given function of conservative temperature (McDougall 2003), salinity, and depth. Young (2010) and Nycander (2011) showed how this model is strictly equivalent to the compressible primitive equations in the limit of small relative density perturbations. The DPE framework is in fact not limited to models using the Boussinesq approximation. An equivalence can be obtained for compressible hydrostatic models, noting that this latter model is isomorphic to the Boussinesq hydrostatic model when formulated in pressure coordinates (e.g., de Szoeke and Samelson 2002; Marshall et al. 2004; Vallis 2006). Derivations given in this section are here to demonstrate that the concept of DPE is potentially very general, requiring only the hydrostatic assumption to be valid.

a. Seawater Boussinesq model The conservation of energy is here stated using the seawater Boussinesq approximation (Vallis 2006; Young 2010; Nycander 2011). Contrary to the simple Boussinesq approximation, the equation of state can be nonlinear. In the seawater Boussinesq model, the density conservation Eq. (14) is replaced by an equation of state, and two equations for temperature and salinity conservation r 5 r~(u, S, z) , Du 5 Du , Dt

(34)

and

(35)

DS 5 DS , Dt

(36)

where diffusion, mixing and surface fluxes are included in the local terms Du and DS. The procedure to obtain the equation of PE is described in Young (2010) and Nycander (2011). Define the Boussinesq potential energy (BPE) as the vertical integral of the equation of state times g fb (u, S, z) [ 2 EbP 5 E P

ð0

r~g dz.

(37)

z

This definition corresponds to a variant of the Boussinesq dynamic enthalpy of Young (2010) and the effective potential energy of Nycander (2011). Note that for an equation of state independent of z, the BPE definition reduces to the standard PE definition EP 5 rgz. Differentiating BPE definition gives fb DEbP ›E P 5 Dt ›z

u,S

fb Dz ›E P 1 Dt ›u

S,z

fb DS Du ›E P 1 . Dt ›S Dt u,z

(38)

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›f 5 2a, ›p

fb /›z) 5 rg [see Eq. (37)], and By definition, (›E u,S P Dz/Dt 5 w, so their product gives (minus) the rate of conversion. Using also Eqs. (35) and (36), the equation for BPE is found to be identical to the PE Eq. (17)

(43)

=p up 5 0,

(44)

a5a ~ (u, S, p) , DEbP Dt

5 rgw 1 WD ,

(39)

Du 5 Du , Dt

where fb ›E P WD [ ›u

S,z

fb ›E u P D 1 DS ›S

DS 5 DS , Dt (40)

u,z

is the new expression for the rate of PE production by diabatic processes. Hence, using BPE instead of the standard definition of PE, the conservation of energy takes the same form as in the case of a linear equation of state. If the standard PE definition was used instead of BPE, the equation for internal energy should be explicitly considered, adding a lot of complexity in the set of energy equations. The DPE-based framework is then obtained using horizontal anomalies of BPE as the definition for DPE. Note that the stratification term ›Eb S 5 w rg 2 P ›z

! (41)

cannot be directly related to ›r/›z anymore because of nonlinearities of the equation of state (and more specifically, because the equation of state depends on z).

Using the isomorphism between hydrostatic Boussinesq equations of motion and primitive equations in pressure coordinates (e.g., de Szoeke and Samelson 2002; Marshall et al. 2004; Vallis 2006), it is straightforward to generalize the framework to the study of compressible fluids, paving the way for an application of this framework to the study of atmospheric energetics. Let us express primitive equations of motion in the pressure coordinate system, where pressure is used as the vertical coordinate. We define the velocity vector up 5 (uph , wp ), and the gradient operator =p 5 (=ph , ›/›p), in this coordinate system. Here, wp 5 Dp/Dt is the cross-pressure pseudovertical velocity (often referred as omega), which has dimension of a power per unit volume, J s21 m23. Equations of motion are (42)

(46) (47)

with D/Dt [ ›/›t 1 u $ 5 ›/›tjp 1 up =p .

(48)

Here, f 5 gz is the geopotential, and a 5 1/r is the specific volume. Note that the momentum equations are given for a unit mass parcel of fluid, instead of a unit volume parcel in the Boussinesq formulation. This is a more natural form for compressible fluids, as the conserved quantity is mass, not volume. In the following, the subscript m is used for a quantity given in unit mass, as for the frictional force noted F m. The equation of state is a function of conservative temperature, salinity (or humidity for the atmosphere) and pressure, and gives the specific volume instead of density. For the compressible hydrostatic model, KE per unit mass is given as EK,m [ juph j2 /2, and PE per unit mass is equal to the geopotential value, EP,m 5 f. By analogy with the seawater Boussinesq hydrostatic model, we introduce an effective PE variable, tentatively called compressible potential energy (CPE),

b. Compressible model

Duph p p 1 f k 3 uh 5 2=h f 1 F m , Dt

and

(45)

c (u, S, p) [ EcP,m 5 Eg P,m

ðp

a ~ dp

(49)

po

where Po is an arbitrary chosen pressure of reference. Note that CPE corresponds to the dynamic enthalpy defined by Young (2010). Take the scalar product of momentum equations with the velocity vector for the KE equation, and the total differential of CPE for the PE equation DEK,m Dt

5 2=p (fup ) 2 awp 1 WF ,m , and (50)

DEcP,m 5 awp 1 WD,m . Dt

(51)

The rate of conversion of PE to KE is given by (minus) the product of the specific volume by the cross-pressure velocity 2awp, WF ,m is the rate of frictional work, and

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WD ,m is the rate of CPE production by diabatic processes c ›Eg P,m WD,m [ ›u

c ›Eg P,m D 1 ›S u

S,p

DS .

(52)

u,p

To retrieve analogous modified equations of energy conservation, we modify the definition of the horizontal mean, taking it along isobars instead of along geopotentials. Horizontal (isobaric) means are here functions of pressure and time, using the notation ^ t) 1 T 0 (x, y, p, t) for a tracer T. We T(x, y, p, t) 5 T(p, 0 define DPE as the isobaric anomaly of CPE, EcP,m . A dynamical geopotential variable is also defined, in analogy with the dynamical pressure 1 0 ^ › f @ 1a ^ A dp . fdyn 5 f0 1 po ›p ðp

(53)

Finally, the modified set of energy equations is obtained DEK,m Dt

5 2=p (fdyn up ) 2 a0 wp 1 WF ,m ,

and

›EcP,m 0 1 =p (EcP,m up ) 5 a0 wp 1 WD,m 1 S m , ›t

(54) (55)

where the stratification term per unit mass is Sm 5 w

p

a ^2

›EPc^,m

!

›p

.

(56)

Note that if the equation of state is independent of p, the stratification term simply reduces to S m 5 2(p 2 po )wp

›^ a . ›p

(57)

The DPE to KE conversion rate is given as (minus) the spatial correlation between the isobaric anomaly of specific volume a0 and the cross-isobar vertical velocity wp. In analogy with the Boussinesq case, the global ^ p 5 0 derived from the continuity equation constraint w is implicitly used here to remove the large static contribution 2^ awp from the conversion term.

8. Discussion and conclusions In this paper, a new framework is proposed to study ocean energetics, based on the concept of dynamical potential energy. DPE is defined as the horizontal anomaly

473

of PE (where definitions of horizontal anomaly and PE depends on the considered model). For an incompressible fluid, the standard form of PE to KE conversion rate is replaced by a more dynamically relevant expression, the DPE to KE conversion rate, which is proportional to the spatial correlation between vertical velocities and horizontal anomalies of density. This definition is motivated by the fact that the horizontal mean vertical velocity is then zero, and therefore horizontal means of both expressions for conversion rates are exactly equal. The DPE framework provides modified expressions for KE and PE balance, which are easier to map and interpret, as most of the hydrostatic contributions have been removed. At the same time, this framework simply reduces to standard energy conservation when integrated globally or horizontally. So standard interpretations of ocean energetics are conserved, while gaining a physical understanding of energy balances directly at the local scale. A new pressure variable, tentatively called the dynamical pressure, is introduced so as to satisfy a reduced hydrostatic balance function of the horizontal anomaly of density. Dynamical pressure, which is essentially equal to the horizontal anomaly of pressure, accounts for all dynamical effects embedded in the pressure variable in such a way that the static component of the hydrostatic pressure field is optimally removed. The dynamical pressure proves to be very practical to visualize the large-scale circulation. In a barotropic case, dynamical pressure is by definition vertically constant, thus simply showing the spatial pattern of sea surface height extending from surface to bottom. Fluxes of dynamical pressure represent the main way to transport power on long distances, linking KE sources and sinks. There are mainly two types of sources and sinks of KE, related to the following. (i) The rate of work of frictional forces. This power source locally accelerates the flow when it is positive or acts as a brake (internal and bottom friction, eventually dissipated). The wind-power input is a mechanical source of power, with an overall accelerating effect, while bottom and internal friction are most generally sinks of power, transforming KE into Joule heating. (ii) The rate of conversion of DPE. The flow is locally accelerated when relatively light waters are upwelled or when relatively heavy waters are downwelled, and decelerated otherwise. This power source comes from the PE generated by diffusion and mixing, destabilizing the large-scale distribution of mass, which is eventually converted after being spatially redistributed through PE advection.

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To illustrate the use of the DPE framework, and to demonstrate the technical feasibility of its numerical diagnostics, several simulations using a state-of-the-art ocean general circulation model were analyzed. Using a simple squared basin configuration, the energetics of circulations induced by wind and/or buoyancy forcings was characterized. The wind forcing alone generated a circulation through a net input of power at the surface, transported in the interior as a flux of dynamical pressure where it was eventually dissipated. In contrast, the buoyancy forcing induced a large-scale circulation from global positive conversion rates. When wind and buoyancy forcings were combined, both patterns of energetics were superimposed, with the surface circulation being primarily controlled by the wind, and the deeper circulation being more influenced by the buoyancy fluxes and conversion rates. However, response to both wind and buoyancy forcings was not a simple addition of responses to each forcing taken separately (see also Saenz et al. 2012). The wind power input was greatly enhanced by the presence of a subsurface thermocline, while the total conversion rate was modulated by an indirect effect of wind on isopycnal depths. To better understand how buoyancy forcings power a circulation, a simple relation has been derived to link internal mixing with conversion rates [see Eq. (33)]. It shows that the total conversion rate available to power the circulation not only depends on the intensity of mixing acting to erode density gradients, but also on its vertical distribution. Hence, the presence of strong mixing coefficients is not a sufficient condition for large amounts of conversion. The mean depths where expansion and contraction occur must also be sufficiently different. A negative total conversion rate is even possible, if the mean depth of density sources is pushed below the density sinks depth, for instance due to the wind. The DPE framework should allow a better understanding of what sets the distribution and magnitude of conversion rates, as it provides an objective way to analyze each term of the energy budget directly at the local scale. A straightforward method to generalize this framework to a compressible fluid in hydrostatic balance has been presented, using the well-known isomorphism between hydrostatic Boussinesq equations of motion and primitive equations in pressure coordinates (e.g., de Szoeke and Samelson 2002; Marshall et al. 2004; Vallis 2006). This opens the way for potential applications of the DPE framework to the study of atmosphere energetics. The question of how DPE and APE frameworks compare has been briefly mentioned. APE and DPE definitions are strictly different except for the trivial case of a resting ocean with flat isopycnals. The general

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philosophy leading to their definition is also dramatically different, as APE derives from purely thermodynamic considerations, while DPE comes from dynamical constraints imposed by the hydrostatic balance. Yet, as seen in section 3 and in the model study, the amount of positive DPE values compares well with the total APE reservoir, indicating that both definitions are somehow connected. DPE presents several advantages over APE, namely, DPE has a simpler and more practical definition, and it is perfectly well defined for any equation of state. In comparison, the definition of APE rapidly becomes overly complicated as the determination of the minimum PE state of reference is a difficult and computationally expensive task. Moreover, APE is not uniquely defined for a realistic nonlinear equation of state (i.e., a nonlinear function of both temperature and salinity), making its use for the real ocean circulation highly hypothetical. Nonetheless, APE is very popular as it has provided a nice framework to explain fundamental processes such as baroclinic instability or the effect of wind on the stratification. The question of whether DPE can fully replace APE to understand ocean (and atmosphere) energetics is now open and will require further investigations. Acknowledgments. Many thanks go to Carl Wunsch, Jonas Nycander, and Gurvan Madec for extremely helpful discussions and comments. Jean Michel Campin was also of precious help, in particular to carry out model experiments. This work started at MIT (Cambridge, Massachusetts) during a post-doctoral period supported by the National Ocean Partnership Program (NOPP) through funding of NASA to the ECCO-GODAE Consortium.

APPENDIX A Work of Viscous Forces Frictional forces can be written using the viscous stress tensor, t 5 [t ij], where t ij is ith component of the viscous stress along the jth direction F i 5 2 å ›j t ij .

(A1)

j

Parameterizing the stress tensor using a Fickian diffusion of momentum, t ij 5 2ronj›jui, the work of frictional forces can be expressed as WF 5 ro å ui ›j (nj ›j ui ) i,j

(A2)

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The sink of kinetic energy owing to viscous dissipation is transformed into internal energy by Joule effect (increasing locally the temperature). However its effect is supposedly so small in the ocean that the Joule effect is most often neglected in ocean general circulation models (see e.g., Griffies 2004).

APPENDIX B Energetics of a Linear Free Surface

FIG. A1. Frictional work and dissipation for a one-dimensional normal jet. The velocity profile follows a standard Gaussian distribution, with a standard deviation of 1. Frictional work rate, dissipation rate and meridional flux of KE are normalized, using ronx 5 1.

5

å ›j (nj ›j EK ) 2 ro å nj (›j ui )2 , j

(A3)

i,j

with uh 5 (u1, u2). The first right-hand side term is given as the divergence of a viscous flux, function of the KE gradient, and the second term is a negative-definite quantity corresponding to the viscous dissipation, function of squared velocity shears. The wind stress acts as surface boundary condition for the stress tensor, thus providing energy as a vertical viscous flux. This power input is then transported in the Ekman boundary layer where it is dissipated while generating the velocity shear within Ekman spiral. To better understand the relation between frictional work and viscous dissipation, these quantities are diagnosed for a Gaussian jet (see Fig. A1). Most of the frictional work enters at the core of the jet, where velocity and curvature are both largest. In contrast, dissipation is concentrated on the flank of the jet, where the shear is strongest. Power is transported from the center to the flanks of the jet as a viscous stress flux proportional to the gradient of KE. The typical distance between areas of frictional work, and dissipative areas, is given by the width of the jet. In the real ocean, jet width is typically of the order of 100 km horizontally, and 50 m vertically. These spatial scales are thus negligible compared to the size of ocean basins.

Let us consider a linear free surface, modeled as a layer enclosed between the surface geopotential (z 5 0) and the sea surface height (z 5 h). Variations of density within this layer are neglected, r 5 ro, as well as horizontal velocities, uh 5 0. A first consequence is that the vertical velocity is constant vertically within this layer to satisfy continuity, and equal to the rate of change of sea surface height wh 5

›h . ›t

(B1)

Neglecting horizontal velocities within the layer means that the hydrostatic KE is always equal to zero. Divergence of the pressure flux term can be integrated vertically, yielding ðh

$ ( pu) dz 5

0

ðh 0

›( pw) dz 5 2po wh , ›z

(B2)

where po 5 rogh is the pressure along the surface geopotential. A simple manipulation shows that 2po wh 5

ðh 0

2ro gwh dz 5

› ›t

ðh 0

EP dz .

(B3)

In other words, the pressure divergence term is always exactly compensated by the amount of conversion taking place within the surface layer, which is itself powered by changes in free surface PE content. Finally, in this simple yet physically relevant model, the budget of energy within the free surface is trivial. To treat free surface within the more general problem of full ocean energetics, it suffices to note that the source of the power entering the ocean interior as a flux of pressure along the surface geopotential is generated by a change in PE converted within the free surface. More realistic models of free surface will require greater care in the expression of their energetics, however the linear free surface already gives an excellent first-order picture of the importance of free surface

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