on the energetics of strati ed turbulent mixing ...sws04rgt/publications/thctheory_23dec08.pdf ·...

Under consideration for publication in J. Fluid Mech. 1

On the energetics of stratified turbulentmixing, irreversible thermodynamics,

Boussinesq models, and the ocean heatengine controversy

REMI TAILLEUX1†1Walker Institute for Climate System Research, University of Reading, Earley Gate, PO Box

243, Reading, RG6 6BB, UK

(Received 22 August 2008 and in revised form ??)

A key issue in stratified turbulence theory is to understand the nature of the link be-tween D(APE), the dissipation rate of available potential energy APE, and Wr,turbulent,the turbulent rate of change of background gravitational potential energy GPEr, whichare the two main measures of turbulent diffusive mixing in stratified fluids. So far,the main understanding about this issue has come from the study of Boussinesq flu-ids with a linear equation of state, in which this link is the deceptively simple equalityWr,turbulent = D(APE), which has been widely interpreted as implying for both termsto be of the same kind, and associated with the irreversible conversion of APE intoGPEr. In contrast, the GPEr laminar rate of change Wr,laminar is traditionally inter-preted as the conversion of internal energy (IE) into GPEr. In this paper, a new viewof energetics is developed that contradicts the traditional view. It relies on regardingIE as the sum of three distinct subcomponents: available internal energy (AIE), exergy(IEexergy), and dead internal energy (IE0). In this new view, D(APE) represents thedisspation rate of APE into IE0, while both Wr,laminar and Wr,turbulent correspond tothe conversion of IEexergy into GPEr. The equality Wr,turbulent = D(APE) is thereforere-interpreted as stating that the conversion rate between IEexergy into GPEr is thesame as the dissipation rate of APE into IE0.

The analysis of non-Boussinesq fluids reveals that the equality D(APE) = Wr,turbulent

is a serendipitous feature of a Boussinesq fluid with a linear equation of state, which is atbest a good approximation, because for real fluids Wr,turbulent is generally smaller thanD(APE), and sometimes even negative for a strongly nonlinear equation of state. In asecond step, the links between stirring and mixing is examined in the case of a wind-andbuoyancy-driven thermally stratified ocean to determine whether these pose any con-straints on the mechanical sources of stirring, as advocated by Munk & Wunsch (1998),and whether they provide additional insights into how the oceans are forced and dissi-pated. The present analysis corrects a certain number of errors in previous approaches tothe issue, and establishes that the coupling between stirring and coupling cannot refutethe traditional buoyancy-driven view of the so-called meridional overturning circulation(MOC), in contrast to previous claims to the contrary. This is not to say, however, thatsome forms of mechanical control of the MOC does not exist, owing to the particularnature of the available potential energy production term. Finally, the present analysissupports the idea that the buoyancy forcing is as important as the mechanical forcing instirring and driving the large-scale ocean circulation, in contrast to recent claims.

† Present address: Walker Institute for Climate System Research, University of Reading,Earley Gate, PO Box 243, Reading RG6 6BB, United Kingdom.

2 R. Tailleux

1. Introduction

1.1. Stirring versus mixing in turbulent stratified fluids

As is well known, one of the most fundamental characteristics of turbulent stratified fluidsis the presence of active stirring that results in significantly elevated values of molecularviscous and diffusive mixing rates as compared with laminar rates. Although stirringand mixing are often used synonimously, they are now recognized as being thermody-namically very different, for being associated with reversible adiabatic and irreversiblediabatic processes respectively, e.g. Eckart (1948); Tseng & Ferziger (2001). While thecoupling between reversible adiabatic stirring and irreversible diabatic mixing in tur-bulent stratified fluids is well established, it is difficult, on the other hand, to find asatisfactory theoretical description of how the two processes are quantitatively related.Part of the problem, obviously, is to provide satisfactory definitions for the processes ofstirring and mixing themselves, as well as to identify what measures their coupling. Toachieve that objective, one of the most promising theoretical framework developed overthe past decade is the available potential energy framework of Winters & al. (1995),which builds upon the pioneering work of Lorenz (1955) developed in the context ofatmospheric energetics. In such a framework, the potential energy PE (i.e., the sum ofthe gravitational potential energy GPE and internal energy IE) is decomposed into itsavailable (APE = AGPE +AIE) and non-available (PEr = GPEr + IEr) components.The interest of such a decomposition is that by construction, the reference state associ-ated with the un-available PE can be affected only by irreversible and diabatic effects,so that measuring its evolution in the absence of external forcing provides a measure ofthe mixing taking place in the fluid.

In the literature, turbulent stratified fluids at low Mach numbers are most often studiedin the context of the Boussinesq approximation, which usually assumes the equationof state to be linear in temperature. In that case, the internal energy component ofthe potential energy is generally neglected, and the focus is on the evolution of thevolume-integrated KE, APE = AGPE, and GPEr. In the case of a mechanically- andthermodynamically-isolated stratified fluid, Winters & al. (1995) show that the evolutionequations for the latter quantities are given by:

d(KE)

dt= −C(KE, APE) − D(KE), (1.1)

d(APE)

dt= C(KE, APE) − D(APE), (1.2)

d(GPEr)

dt= Wr,mixing , (1.3)

where C(KE, APE) = −C(APE, KE) is the reversible conversion between KE andAPE = AGPE, usually called the “density flux”, D(KE) is the viscous dissipation ofKE, D(APE) is the diffusive dissipation of APE, and Wr,mixing is the GPEr rate ofchange due to molecular diffusion. Explicit formula for all these terms are given in Ap-pendix A for a Boussinesq fluid with a linear equation of state, called the L-Boussinesqmodel, as well as for one whose thermal expansion is an increasing function of tempera-ture, called the NL-Boussinesq model. Appendix B generalizes the above expressions tothe case of the fully compressible Navier-Stokes equations (CNSE) with an arbitrary non-

Energetics of turbulent mixing and the ocean heat engine controversy 3

linear equation of state. Both appendices also include the modifications brought aboutby the presence of external mechanical and thermodynamical forcing.

In Eqs. (1.1-1.3), there is only one type of reversible conversion, the “density flux”associated with the APE/KE conversion, which is therefore the only candidate one mayrelate to the concept of stirring. In contrast, there are three different measures of irre-versible mixing, viz., D(KE), D(APE), and Wr,mixing , the first one caused by molecularviscous processes, and the latter two caused by molecular diffusive processes. The overalissue of understanding how the stirring and mixing are inter-related amounts thereforeto understanding how C(APE, KE), D(KE), D(APE), and Wr,mixing are all related.In this paper, the focus will be on turbulent diffusive mixing, for the understanding ofviscous dissipation constitutes somehow a separate issue with its own problematic, e.g.Gregg (1987). It is useful to discuss the links between stirring and mixing from the viewpoint of a turbulent mixing event, defined here as a period of intense mixing preceded andfollowed by laminar conditions, for turbulent properties are usually best understood ina statistical sense requiring some form of averaging. Moreover, there is a huge literatureon such turbulent mixing events, which have been studied numerically, theoretically, andin the laboratory. Appropriate budget equations are obtained by integrating the aboveenergy equations over the duration of the event, which yields:

∆KE = −C(KE, APE) − D(KE), (1.4)

∆APE = C(KE, APE) − D(APE), (1.5)

∆GPEr = W r,mixing , (1.6)

where ∆(.) denotes a difference between the end and beginning of the mixing event, whilethe overbar denotes a time integral over the duration of the mixing event. An importantequation is the “available” mechanical energy equation, obtained by summing the KEand APE equation:

∆KE + ∆APE = −[D(KE) + D(APE)] < 0, (1.7)

which states that the net change in the total “available” mechanical energy KE + APEinitially present before the mixing event is always negative, and entirely due to the viscousand diffusive dissipation of KE and APE respectively. With regard to understandingturbulent diffusive mixing, the key quantity is the total amount of APE diffusivelydissipated by molecular diffusion, which appears as a certain fraction γmixing — oftencalled the “mixing efficiency” — of the total “available” mechanical energy lost over theturbulent mixing event |∆KE| + |∆APE|, i.e.,

γmixing =D(APE)

|∆KE| + |∆APE|=

D(APE)

D(APE) + D(KE), (1.8)

e.g. Peltier & Caulfield (2003) (see also Staquet (2000) for a more detailed discussion ofthe origin of this definition, and of its link with other measures of mixing efficiency, suchas the Bulk Richardson number). There are two particular “archetypical” turbulent mix-ing events, which are such that the available mechanical energy for stirring is initially allin the form of either KE or APE, and which can be regarded as “mechanically-driven”and “buoyancy-driven” respectively. The mixing efficiencies for these two types of mix-ing events are therfore γmixing = D(APE)/|∆KE| and γmixing = D(APE)/|∆APE|respectively. Mechanically-driven mixing is usually studied in connection with shear flowinstability e.g. Winters & al. (1995); Caulfield & Peltier (2000); Staquet (2000); Peltier& Caulfield (2003), while buoyancy-driven mixing is usually studied in connection with

4 R. Tailleux

Rayleigh-Taylor instability, e.g. Linden and Redondo (1991); Dalziel & al (2008). Animportant empirical result is that buoyancy-driven mixing appears to be more efficientthan mechanically-driven mixing, with typical values γmixing = 0.2− 0.3 being reportedfor the latter (with large error bars), compared to γmixing = 0.3− 0.5 being reported forthe former. The result is important, because it suggests that the mixing efficiency of ahybrid mechanically-and buoyancy-driven stratified fluid is likely to adjust to a value in-termediate between purely mechanically-driven and buoyancy-driven mixing efficiencies,a point to keep in mind when seeking to understand the nature of the constraint thatmay exist between stirring and mixing in mechanically- and buoyancy-driven systemssuch as the oceans, a topic taken up in Section 3.

In freely decaying turbulent stratified flows, the coupling between stirring and irre-versible mixing is directly apparent in the APE budget, which shows that: D(APE) =C(KE, APE) + |∆APE|, which states that the diffusively dissipated APE is propor-tional to the time-averaged reversible density flux (as well as to the net amount oflost APE in buoyancy-driven mixing events). Much less straightforward is the link be-tween D(APE) and Wr,mixing , however, whose nature strongly depends on the ther-modynamic equation of state, as will be demonstrated in this paper. That this is so,however, is not widely recognized, for most studies of turbulent diffusive mixing areusually based on the L-Boussinesq model, for which the link between D(APE) andWr,mixing is in contrast deceptively simple. In that case, indeed, it can be shown that ifWr,mixing = Wr,laminar + Wr,laminar is written as the sum of a laminar and enhancedturbulent rate, then one has:

Wr,turbulent = D(APE), (1.9)

e.g., see Winters & al. (1995). This equality, however, is just a serendipitous feature ofthe L-Boussinesq model, which in reality is at best only a good approximation. For realfluids, indeed, a more accurate description of the link between Wr,turbulent and D(APE),based on the results of the present paper, is that Wr,turbulent = ξD(APE), where −∞ <ξ < 1 may be sometimes regarded as approximately constant depending on the particularreference stratification and thermodynamic equation of state considered. Since the caseWr,turbulent < 0 has been discussed almost exclusively only by Nick Fofonoff, e.g. Fofonoff(1962, 1998, 2001) it will be referred to as the Fofonoff regime in this paper, whilethe case Wr,turbulent > 0 will be referred to as the classical regime. Regardless of theregime considered, however, the fact that Wr,turbulent is correlated to D(APE) establishesthat elevated values of Wr,turbulent cannot exist without: 1) finite values of APE, sinceD(APE) = 0 when APE = 0, 2) an APE cascade able to transfer the spectral energyof the temperature (density) field to the small scales at which molecular diffusion isthe most efficient at smoothing out temperature gradients. The discussion of the APEand its cascade appears to be relative new, with the dual KE/APE cascade being onlyrecently discussed by Lindborg (2006). Note that because APE is a globally definedscalar quantity, speaking of APE cascades requires the introduction of the so-called APEdensity, noted Φa(x, t) here, for which a spectral description is possible, e.g. Holliday &McIntyre (1981); Molemaker & McWilliams (2008).

1.2. Nature of energy conversions in turbulent stratified fluid flows and thermodynamiccritique of the irreversible APE/GPEr conversion

In the fluid energetics literature, two terms having opposite signs in two different equa-tions are usually interpreted as an energy conversion between the two forms of energyinvolved. There is no guarantee, however, that the “energy conversion” thus identified is


E=KE+GPE+IEAPE=AGPE+AIE=0

PE=PEr=GPEr+IEr

APE=AGPE+AIE=KE APE=0PEr = KE

=0PE = APEPEr = 0 PE = PEr = KE

(I) Initial laminarstate

(II) KE conversioninto APE and actionof lateral diffusion

(III) Completeconversion of APE

into PEr

Figure 1. Idealised depiction of the diffusive route for kinetic energy dissipation. (I) representsthe laminar state possessing initially no AGPE and AIE, but some amount of KE. (II) representsthe state obtained by the reversible adiabatic conversion of some kinetic energy into APE,which increases APE but leaves the background GPEr and IEr unchanged; (III) represents thestate obtained by letting the horizontal part of molecular diffusion smoothes out the isothermalsurfaces until all the APE in (II) has been converted in background PEr = GPEr + IEr.

actually occurring in reality. This is because energy conversions viewed in this mannerare somehow arbitrary and non-unique, since new energy conversions can be created atwill by adding one term to one equation and subtracting it from a different equation.Physical energy conversions, however, are necessarily unique, for in Nature, it seemsobvious that energy must flow throughout the different existing energy reservoirs in adefinite and reproducible manner. From this, it follows that identifying two terms withopposite signs in two different equations is a necessary, but not sufficient, requirementto identify a genuine physical energy conversion, and that additional physical considera-tions are necessary to establish that such a conversion is a genuine physical one and notjust a mathematical device. Such a point is important to keep in mind when seeking tointerpret the nature of the conversions D(APE) and Wr,turbulent, as is discussed below.

In the case of Eqs. (1.1-1.3), D(KE) and Wr,laminar have a well-established physicalcharacter, and are well-known to represent the irreversible conversion of KE into IE forthe former, and an irreversible conversion between IE and GPEr for the latter. Thereappears to be some difficulty, however, in properly interpreting the nature of D(APE)and Wr,turbulent. This difficulty arises from the fact that these two terms have the samenumerical value in the L-Boussinesq model, which has been widely interpreted as alsoimplying that the two conversions must also be of the same type, and hence representan irreversible conversion of APE into GPEr, e.g. Winters & al. (1995); Peltier &Caulfield (2003); Munk & Wunsch (1998); Huang (2004); Thorpe (2005) amongothers. From a logical viewpoint, such an interpretation is as flawed as inferring fromthe observation that Peter and Paul are both 50 years old that Peter and Paul musttherefore be the same person. From a thermodynamic viewpoint, such an interpretationconflicts with the assumed diabatic character of the APE dissipation, since one definingsignature of diabatic transformations is to possess a very small ∆GPEr/∆IEr ratio inthe case of nearly incompressible fluids. But clearly, if all diffusively dissipated APE endsup into GPEr, then energy conservation requires that ∆IEr be negligible, resulting inan extremely large ∆GPEr/∆IEr ratio in violation of the general rule about diabatic

6 R. Tailleux

transformations mentioned above. From a thermodynamic viewpoint, the only way toensure that APE dissipation be associated with a small ∆GPEr/∆IEr ratio is that thediffusively dissipated APE be nearly entirely converted into IE, similarly as for viscousdissipation. As to Wr,turbulent, it is easy to convince oneself that it must be of the sametype as Wr,laminar, and hence represent an irreversible conversion between IE and GPEr,since the simple act of decomposing Wr,mixing into a turbulent and laminar part, whichis the way Wr,turbulent was introduced in the first place, is just a mathematical devicethat cannot transform an energy conversion of a given type into a different type. Fromthe above arguments, for which a rigorous thermodynamic basis is given in Appendix B,it follows that the L-Boussinesq model equality Wr,turbulent = D(APE) only means thatthe turbulent dissipation rate of APE into IE is equal to the turbulent conversion rateof IE into GPEr, not that the two conversions are of the same type, and not that thedissipated APE is converted into GPEr . This is important to realize, because as shown inthis paper, the equality between D(APE) and Wr,turbulent is just a serendipitous featureof the L-Boussinesq model, so that it should not be given excessive importance, giventhat the equality is at best only a good approximation in real fluids.

1.3. Internal Energy or Internal Energies?

As recalled above, viscous dissipation converts KE into IE, while molecular diffusionconverts APE into IE, as well as IE into GPEr. As a result, it would seem to be possiblefor all of the dissipated KE and APE to be eventually converted into GPEr through theaction of molecular diffusion. On the other hand, it is also widely agreed that the increaseof GPEr due to the viscous disspation of KE is only a tiny fraction of the total KEdissipated. The only way to reconcile these apparently contradictory results is to admitthe fact that the internal energy into which KE is viscously dissipated must somehowbe different from the internal energy being converted into GPEr. That this is indeedthe case is rigorously established in Appendix B, which shows that the internal energyis best regarded as the sum of at least three distinct subreservoirs: the available internalenergy (AIE), the exergy (IEexergy), and the dead internal energy (IE0). Physically,this decomposition of internal energy is equivalent to the possibility of decomposing thetemperature field as: T (x, y, z, t) = T0(t) + Tr(z, t) + T ′(x, y, z, t), i.e., as the sum of anequivalent thermodynamic equilibrium temperature T0(t), a reference vertical tempera-ture stratification Tr(z, t) linked to Lorenz’s reference state, and a residual, with T0 beingassociated with IE0, Tr with IEexergy , and T ′ with AIE. Physically, AIE is the internalenergy component of Lorenz (1955)’s APE, while IE0 and IEexergy are the internalenergy associated with the equivalent thermodynamic equilibrium temperature T0 andvertical temperature stratification Tr respectively. The idea behind this decompositioncan be traced back to Gibbs (1878), the concept of exergy being common in the ther-modynamic engineering literature, e.g. Bejan, A. (1997). See also Marquet (1991) foran application of exergy in the context of atmospheric available energetics. A full reviewof existing ideas related to the present ones is beyond the scope of this paper, as theengineering literature about available energetics and exergy is considerable. In AppendixB, it is shown that the dissipated KE and APE are essentially converted into IE0, whilethe variations of GPEr due to molecular diffusion are caused at the expenses of IEexergy ,according to the following equations:

d(KE)

dt= C(KE, APE) − D(KE), (1.10)

d(APE)

dt= −C(KE, APE) − D(APE), (1.11)


GPEr+0.2

IEo

+0.8

A) Standard Interpretation of Eq. (1.5)

B) New Interpretation of Eq. (1.5)

IE exergy −0.01IEo−1.0KE

APE 0

+0.8

KE −1.0

APE 0

+0.2

+0.2+0.2

+1.0

GPEr

IE exergy

+0.21

−0.21

+0.21

+0.21

+0.01

Net Change in total IE = +0.79

Net Change in total IE = +0.79

Figure 2. (A) Predicted energy changes for an hypothetical turbulent mixing event under theassumption that the diffusively dissipated APE is irreversibly converted into GPEr; (B) Sameas in A under the assumption that the diffusively dissipated APE is irreversibly converted intoIE0, as the viscously dissipated KE. In both cases, the net energy changes in KE, GPEr, APE,and IE are the same. The only predicted differences concern the subcomponents of the internalenergy IE0 and IEexergy.

d(GPEr)

dt= Wr,mixing , (1.12)

d(IE0)

dt≈ D(KE) + D(APE) = Dtotal, (1.13)

d(IEexergy)

dt≈ −Wr,mixing = −[Wr,laminar + Wr,turbulent]. (1.14)

In this model, the first three equations are just a rewriting of Eqs. (1.1) and (1.3), sothat the main novelty is associated with Eq. (1.13) and (1.14). Physically, these equationsmean that the viscous and diffusive dissipation processes mostly contribute to increasingthe equivalent thermodynamic equilibrium temperature T0 of the system, without affect-ing Tr(z, t). The process by which GPEr increases as the result of molecular diffusion,on the other hand, goes hand in hand with the smoothing out of the vertical temperaturegradient Tr, and stops when Tr = 0. The point made in this paper that APE cannotbe irreversibly converted into GPEr is equivalent to the result that APE does not con-tribute to increase the vertical stratification Tr, which is the only part of the temperatureable to feed the variations of GPEr.

8 R. Tailleux

1.4. Link with the ocean heat engine controversy

The links between stirring and irreversible mixing have received an increased attention inthe oceanic literature over the past decade, following the very influential study by Munk& Wunsch (1998) in which the authors ask the question of whether the energetics ofirreversible mixing can place any constraint on the mechanical sources of stirring, andthereby provide important new insights into how the oceans are forced and dissipated.MW98 start by noting that in the oceans, high-latitudes cooling and its associated deepwater formation lowers the centre of gravity, so that in order to maintain a steady state,a counteracting process is required. MW98 argue that surface heating is too inefficient toaccomplish this purpose, and that the only process large enough to balance the effect ofcooling is turbulent diffusive mixing. Although in their paper, MW98 do not distinguishbetween the available and un-available part of GPE, their argument clearly pertain onlyto the background GPEr, since only the latter is affected by irreversible and diabaticprocesses. In mathematical terms, MW98 argue that the steady-state GPEr balance isgiven by:

Wr,turbulent ≈ −

{d(GPEr)

dt

}

cooling

, (1.15)

which amounts to neglect the laminar contribution of Wr,mixing , as well as to neglect theeffects of surface heating.

In order to link irreversible mixing to the mechanical sources of stirring, MW98’sreasoning amounts to: 1) invoke Eq. (1.9) to link Wr,turbulent and D(APE); 2) invokeOsborn (1980)’s concept of mixing efficiency to link D(APE) to D(KE) by D(APE) =γmixingD(KE); 3) argue that D(KE) is balanced by the mechanical energy input G(KE)due to the winds and tides, leading eventually to MW98’s main result:

γmixingG(KE) = −

{d(GPEr)

dt

}

cooling

. (1.16)

Following MW98, Eq. (1.16) has been widely interpreted in the last decade as imposinga constraint on the mechanical energy input required to sustain a meridional overturingcirculation of about 20 − 30 Sv, and associated maximum meridional heat transport ofabout 2 PW. Thus, estimating the loss of GPEr due to cooling to be O(0.4 TW), andusing the canonical value γmixing = 0.2, MW98 conclude that G(KE) = 0(2 TW) ofmechanical stirring is needed to support the currently observed overturning circulationand poleward heat transport. While this is an interesting and plausible idea, MW98further contend rather provokingly, however, that their result Eq. (1.16) also implies thatthe ocean overturning circulation is not driven by high-latitude cooling, as traditionallyassumed, e.g., Colin de Verdiere (1993), but by the turbulent mixing powered by thewinds and tides, and that as a consequence, the oceans should not be regarded as a heatengine, but rather as a heat transport engine in which the heat transport is achievedpassively.

Although MW98’s arguments have been echoed favorably within the ocean community,e.g. Huang (2004); Paparella & Young (2002); Kuhlbrodt (2007); Nycander & al. (2007),many questions about their validity and actual meaning remain. Indeed, Eq. (1.16) isnot an exact result deduced rigorously from first principles, which furthermore makes theunjustified assumption that the buoyancy forcing does not contribute significantly to theoverall mechanical stirring. Moreover, the fact that Eq. (1.16) appears to pertain only tothe GPEr budget, i.e., the part of the GPE that is not available for conversion into kineticenergy, raises the question of how it can be legitimately used to make any inferences abouthow the kinetic energy of the large-scale circulation is maintained against dissipation,


given that the latter issue has been more traditionally discussed so far in the context ofthe KE/APE budget. Likewise, it is difficult to understand how Eq. (1.16) can tell usanything about the passive or active character of the heat transport. On top of everything,the current confusion about the above issues appears to be exacerbated by terminologyissues, with such important concepts as as ‘heat engine’, ’heat transport engine’, ‘passiveheat transport’, which play such an important role in the present discussion of oceanenergetics, are widely used but rarely, if at all, defined. These issues, and proposedsolutions to clarify them, are taken up in Section 3.

1.5. Purpose and organization of the paper

The main objectives of this paper are twofold, first to convince the reader that D(APE)and Wr,turbulent are two different types of energy conversion, representing two distinctmeasures of irreversible diffusive mixing, and whose coupling in a real fluid is consider-ably more complicated than can be anticipated on the basis of the L-Boussinesq model,and second, to re-examine MW98’s conclusions about the consequences placed by thecoupling between stirring and mixing on our understanding of how the oceans are forcedand dissipated. The backbone of the paper are the theoretical derivations presented inthe appendices A and B, which provide a rigorous theoretical support for all the ar-guments presented in the paper. Appendix A offers a new derivation of Winters & al.(1995)’s framework, which is further extended to the case of a Boussinesq fluid with athermal expansion increasing with temperature. Appendix B further extends Winters &al. (1995)’s results to the case of a fully compressible thermally-stratified fluid, in whichthe decomposition of internal energy into three distinct subreservoirs is presented. Section2 seeks to illustrate the differences between D(APE) and Wr,turbulent using a numberof different viewpoints, and examine some of its consequences, in the context of freelydecaying turbulence. Section 3 extends the analysis to the case of a forced/dissipatedsystem, with the particular case of the oceans in mind. The new framework is used toassess the validity of a number of objections that have been made against the tradi-tional buoyancy-driven view of the so-called thermohaline circulation. Section 4 offers asummary and discussion of the results.

2. A new view of turbulent mixing energetics in freely decayingstratified turbulence

2.1. Boussinesq versus Non-Boussinesq energetics

As mentioned above, a central point of this paper is to argue that irreversible energyconversions in turbulent stratified fluids are best understood if internal energy is notregarded as a single energy reservoir, but as the sum of at least three distinct sub-reservoirs. Obviously, these nuances are lost in the traditional Boussinesq description ofturbulent fluids, since the latter lacks an explicit representation of internal energy, letalone of its three sub-reservoirs. This does not mean that the Boussinesq approximationis necessarily inaccurate of incomplete, but rather that the definitive interpretation ofits energy conversions requires to be checked against the understanding achieved fromthe study of the fully compressible Navier-Stokes equations. Such a study was carriedout, with the results reported in Appendix B. The approach followed consisted in seekingsuccessive refinements of the energy conversions starting from the KE/APE/PEr systemfor which the number of energy conversions is limited and unambiguous. The second stepwas to split PEr into its GPEr and IEr component; the third step further split IEr

into its exergy IEr − IE0 and dead IE0 components. Finally, the last step was to split

10 R. Tailleux

1 2 3

4 5 6

7 8 9

1

2

3

4 6

8

2 4 6

8 1 4

9 5 7

(II) HORIZONTAL MIXING

2 4 6

8 1 4

9 5 7

9 5 7

(I) STIRRING

Figure 3. Idealised depiction of the experimental protocol used to construct Fig. 4, as well asunderlying the method for constructing Figs. 5 and 6. In the top panel, a piece of stratifiedfluid is cut into pieces of equal mass that are numbered from 1 to Ntot, where Ntot is the totalnumber of parcels. A random permutation is generated as a way to shuffle the parcels randomlyand adiabatically, in order to mimic the stirring process. In the bottom panel, all the parcelslying at the same level are homogenised to the same temperature by conserving the total energyof the system, which mimics the horizontal mixing step illustrated in Fig. 1.

APE into its AIE and AGPE components. These successive refinements are illustratedin Fig. 10 in Appendix B. An important outcome of the analysis is that the structureand form of the KE/APE/GPEr equations Eqs. (1.1-1.3) obtained for the L-Boussinesqmodel turn out to more generally valid for a fully compressible thermally-stratified fluid,so that one still has:

d(KE)

dt= −C(KE, APE) − D(KE), (2.1)

d(APE)

dt= C(KE, APE) − D(APE), (2.2)

d(GPEr)

dt= Wr,mixing . (2.3)

It can be shown, however, that the explicit expressions for C(KE, APE), D(KE), andD(APE) differ between the two sets of equations, see Appendices A and B for thedetails of these differences. Based on the numerical simulations detailed in the following,the most important point is probably that D(APE) appears to be relatively unaffectedby the details of the equation of state, in contrast to Wr,mixing , which suggests thatthe L-Boussinesq model is able to accurately represent the irreversible diffusive mixingassociated with D(APE). Moreover, since the internal energy contribution to APE isusually small for a nearly incompressible fluid, it also follows that the L-Boussinesq modelshould also be able to capture the time-averaged properties of C(KE, APE), since the


Figure 4. (a) The increase of AGPE as a function of the stirring energy SE (see text for details).Each point represents a different stratification shuffled by a different random permutation. Thecontinuous line represents the straight-line of equation ∆AGPE = SE which would describe theenergetics of the stirring process if AIE were zero, i.e., if the fluid were perfectly incompressible;(b) the change of GPEr as a function of the stirring energy SE dissipated by diffusive mixing;the dotted line is the straight line of equation ∆GPEr = Diffusive dissipated SE which would de-scribe the energetics of turbulent mixing if the irreversible conversion AGPE −→ GPEr existed;(c) The change in dead internal energy IE0 as a function of the diffusively dissipated stirringenergy SE. The dashed line is the straight-line of equation ∆IE0 = diffusively dissipated SE; thefigure shows a near perfect correlation; (d) The change in GPEr as a function of the exergychange. The dashed line is the straight line of equation ∆GPEr = −∆IEexergy. The figureshows again a near perfect correlation.

latter is the difference of two terms expected to be accurately represented by the L-Boussinesq model based on the APE equation. The L-Boussinesq model, however, will ingeneral fail to correctly capture the behaviour of GPEr, unless the approximation of alinear equation of state is accurate enough, as seems to be the case for compositionally-stratified flows for instance, e.g. Dalziel & al (2008). The above properties help torationalize why the L-Boussinesq model appears to perform as well as it often do.

Being re-assured that there are no fundamental structual differences between the en-ergetics of the KE/APE/GPEr system in the Boussinesq and compressible NSE, thenext step is to clarify the link with internal energy. One of the main results of this paper,derived in Appendix B, are the following evolution equations for the dead and exergy

12 R. Tailleux

components of internal energy:

d(IE0)

dt≈ D(KE) + D(APE), (2.4)

d(IEexergy)

dt≈ −Wr,mixing , (2.5)

which were obtained by neglecting terms scaling as O(αP/(ρCp)), for some values of α, P ,ρ, and Cp typical of the domain considered, where α is the thermal expansion coefficient,P is the pressure, ρ is the density, and Cp is the heat capacity at constant pressure.The important point is that such a parameter is very small for nearly incompressiblefluids. For seawater, for instance, typical values encountered in laboratory experimentsdone at atmospheric pressure are α = 2.10−4 K−1, P = 105 Pa, Cp = 4.103 J.K−1.kg−1,ρ = 103 m3.kg−1, which yield αP/(ρCp) = 5.10−6. In the deep oceans, this value canincrease up to O(10−3), but this is still very small. Eqs. (2.4) and (2.5) confirm thatD(APE) and D(KE) are fundamentally similar dissipative processes, in that they bothconvert APE and KE into dead internal energy, while also confirming that Wr,mixing

represent a conversion between IEexergy and GPEr both in the laminar and turbulentcases.

2.2. Analysis of idealized turbulent mixing events

As mentioned above, a central objective of this paper is to convince the reader thatD(APE) and Wr,turbulent are fundamentally two very different measures of irreversiblediffusive mixing, with the former being always positive and representing the dissipationof APE into ’dead internal energy’ IE0, and the latter a conversion between ‘exergy’IEexergy and GPEr , whose sign depends on the particular vertical stratification andequation of state considered. For this reason, the deceptively simple equality Wr,mixing =D(APE) predicted by the L-Boussinesq model is necessarily a source of confusion, since itis easily misinterpreted as implying the conversion of APE into GPEr, an interpretationthat perhaps would make sense if there was indeed a strict equality between Wr,mixing

and D(APE) in reality, which is not the case however. In order to illustrate these newidea, the energy budget of an hypothetical turbulent mixing event associated with shearflow instability is examined in the light of Eqs. (1.10-1.14). Typically, a turbulent mixingevent develops from laminar conditions for which APE = 0, during which APE increasesand oscillates, until the instability subsides and the fluid re-laminarize, at which pointAPE goes back to zero again. As the result of the mixing event, the shear flow has losta certain amount of kinetic energy |∆KE|, the mean vertical temperature gradient hasbeen smoothed out, and GPEr has increased by a certain amount ∆GPEr. A quantitiveanalysis is obtained by time-integrating Eqs. (1.10-1.14) over the duration of the mixingevent, which yields:

∆KE = −Dviscous − Ddiffusive, (2.6)

∆APE = 0, (2.7)

∆GPEr = W r,turbulent + W r,laminar, (2.8)

∆IE0 = Ddiffusive + Dviscous, (2.9)

∆IEexergy = −[W r,laminar + W r,turbulent], (2.10)

where ∆(.) denotes a difference between the end and beginning of the event, while anoverbar denotes a time-integration over the event. From an observational viewpoint, itis important to realize that energy conversions cannot be observed or measured directly,


but are to be inferred from the measured changes of the observable energy reservoirs. Asan illustration, Fig. 2 illustrates the expected changes in a hypothetical turbulent mixingevent as predicted by the standard (top panel) and new interpretation. Both interpre-tations are consistent with overall energy conservation, but differ in their predictionsfor the changes in IE0 and IEexergy †. From an observational viewpoint, therefore, theonly way to discriminate between the traditional and new interpretation of turbulentmixing energetics would require the ability to measure the changes in IE0 and IEexergy

separately.

2.3. An idealized experimental protocol to test the new and old theories

In the absence of any direct measurements of IE0 and IEexergy , which may becomeavailable in the future, the only way to assess the validity of the above new ideas aboutturbulent mixing energetics is probably by means of numerical methods. The followingis concerned with assessing the accuracy of the two key formula Eqs. (2.4) and (2.5) nu-merically, obtained by neglecting terms proportional to the parameter αP/(ρCp), whichis expected to be very small for liquids such as water or seawater. Physically, the purposeis to demonstrate that the diffusively dissipated APE is nearly entirely converted intoIE0, as predicted by Eq. (2.4), and that GPEr variations are nearly entirely accountedfor by corresponding variations in IEexergy , as predicted by Eq. (2.5). To that end, ide-alized energetically consistent mixing events are constructed, and studying numerically.The procedure consists in considering a piece of thermally stratified fluid initially lyingin its Lorenz (1955)’s reference state in a two-dimensional container with a flat bottom,vertical walls, and a free surface exposed at constant atmospheric pressure at its top.The fluid is then discretized on a rectangular array of dimension Nx × Nz into discretefluid elements having all the same mass ∆m = ρ∆x∆z, where x and z are the horizontaland vertical coordinates respectively, as illustrated in Fig. 3. The initial stratification isa vertically-dependent temperature profile T (x, P ) = Tr(P ) regarded as a function ofhorizontal position x and pressure P . Thousands idealized mixing experiments are thengenerated according to the following procedure:

(a) Initialization of the reference stratification. The initial stratification is discretizedas Ti,k = T (xi, Pk) = Tr(Pk), with xi = (i − 1)∆x, i = 0, . . .Nx and Pk = Pmin +kg∆m, k = 1, . . .Nz, where Tr(Pk), k = 1, . . .Nz are Nz random generated numberssuch that Tmin 6 Tr(Pk) 6 Tmax that have been re-ordered in the vertical to create astatically stable stratification, for randomly generated Tmin, Tmax, and Pmin.

(b) Random stirring of the fluid parcels. The fluid parcels are then numbered from 1to N = (Nx + 1)(Nz + 1), and randomly shuffled by generating a random perturbationof N elements, such that each parcel conserves its entropy in the re-arrangement. Sucha step is intended to mimic the adiabatic stirring of the parcels associated with theKE −→ APE conversion. The random stirring of the fluid parcels requires an externalamount of energy — called the stirring energy SE — which is diagnosed by computingthe difference in potential energy between the shuffled state and initial state, i.e.,

SE = (GPE + IE)shuffled − (GPE + IE)initial. (2.11)

The latter computation requires a knowledge of the thermodynamic properties of thefluid parcels. In this paper, such properties were estimated from the Gibbs function forseawater of Feistel (2003) by specifying a constant value of salinity. Thermodynamic

† Note that what we call the predicted changes for IE0 and IEexergy in the standard in-terpretation are simply those required to be consistent with the assumption of an irreversibleconversion of APE into GPEr. Internal energy changes are usually not explicitly discussed inthe literature, which does not mean that such changes are not implied.

14 R. Tailleux

properties such as internal energy, enthalpy, density, entropy, chemical potential, speedof sound, thermal expansion, haline contraction, and several others, are easily estimatedby computing partial derivatives with respect to temperature, pressure, salinity, or anycombination thereof, of the Gibbs function. ‡ The stirring energy SE is none other thanLorenz (1955)’s APE of the shuffled state. Since the stirring leaves the backgroundpotential energy unaffected, the energetics of the random shuffling is given by:

∆APE = ∆AGPE + ∆AIE = SE, (2.12)

∆GPEr = ∆IEr = 0, (2.13)

where Eq. (2.12) states that the stirring energy SE is entirely converted into APE,while Eq. (2.13) expresses the result that being a purely adiabatic process, the stirringleaves the background reference quantities unaltered. Fig. 4 (a) depicts ∆AGPE as afunction of SE for thousands of experiments, all appearing as one particular point onthe plot. According to this figure, ∆AGPE approximates ∆APE within about 10%.This illustrates the point that for adiabatic processes, APE is well approximated by itsgravitational potential energy component.

(c) Isobaric irreversible mixing of the fluid parcels. The last step consists in forcingeach isobaric layer to mix completely according to an isobaric process conserving thetotal enthalpy of each isobaric layer, so that after the mixing step, all temperature withthe same pressure also have the same temperature. Such a process converts the fractionqSE of the APE into background PEr, according to:

∆APE = ∆AGPE + ∆AIE = −qSE (2.14)

∆GPEr + ∆IEr = qSE (2.15)

where 0 < q 6 1. The presence of the factor q is explained by the fact that the horizontalmixing is not necessarily able to dissipate all of the stirring energy SE, which occurswhen the mixing leaves the mixed density profile statically unstable. In that case, thelatter still has a nonzero APE = (1−q)SE, meaning that only the fraction qSE has beendissipated by diffusive mixing. As a result, q = 1 only when the mixed density profileappears to be statically stable. The consequences of the irreversible mixing on GPEr

are illustrated in the panel (b) of Fig. (4), which depicts the GPEr change ∆GPEr

as a function of the fraction of the diffusively dissipated stirring energy qSE. If thestirring energy was entirely dissipated into GPEr, as is usually assumed, one wouldexpect ∆GPEr = qSE, which is plotted as the dashed line. Although the figure showsseveral instances for which this relation works relatively well, the large majority of theexperiments show that ∆GPEr is generally smaller than qSE, and even often negativeas is expected in the Fofonoff regime discussed above. By contrast, panels (c) and (d)of Fig. (4) depict the change in dead internal energy IE0 as a function of the fractionof the diffusively dissipated stirring energy qSE, and change in GPEr as a function ofthe change in exergy IEexergy = IEr − IE0. In both cases, a nearly visually perfectcorrelation is found, which are consistent with the following relations:

∆IE0 ≈ qSE, (2.16)

∆GPEr ≈ −∆(IEr − IE0), (2.17)

and hence in agreement with the approximate Eqs. (2.4) and (2.5). In our opinion, the re-sults of Fig. 4) clearly demonstrate the different nature of the APE dissipation D(APE),and GPEr rate of change Wr,mixing .

‡ Computer programs are available from Rainer Feistel or from the author.


Figure 5. (Top panels) The work of expansion/contraction B normalised by its laminar value(obtained for APE=0) as a function of a normalised APE for a particular stratification corre-sponding to the classical regime, with the right panel being a blow-up of the left panel. (Bottompanels) Same as above figure, for the same temperature stratification, but taken at a meanpressure of 50 dbar instead of atmospheric pressure, which is sufficient to put the system in theFofonoff regime. The figures show that although B is usually negative in every case, it is never-theless positive for small values of APE in the classical regime, as expected from L-Boussinesqtheory. The normalization constant APEmax corresponds to the overall maximum of APE forall experiments.

2.4. Numerical estimates of B, Wr,mixing and D(APE) as a function of APE

Having clarified the nature of the net energy conversions occurring in idealized mixingevents, the following turns to the estimation of the turbulent rates of the three importantconversion terms B, Wr,mixing , and D(APE), which are the main three terms affectedby molecular diffusion in the fully compressible Navier-Stokes equations. As pointed outin the introduction, enhanced rates fundamentally arise from turbulent fluids possessinglarge amounts of small-scale APE. For this reason, this paragraph seeks to understandhow the values of B, Wr,mixing and D(APE) are controlled by the magnitude of APE.

We first focus on the work of expansion/contraction B, which takes the following form:

B =

∫

V

αP

ρCp

∇ · (κρCp∇T ) dV +

∫

V

αP

ρCp

ρε dV −

∫

V

P

ρc2s

DP

DtdV, (2.18)

obtained by regarding ρ as a function of temperature and pressure. The part of B affectedby molecular diffusion is the first term in the right-hand side of Eq. (2.18), and the oneunder focus here. The second and third term in the r.h.s. of Eq. (2.18) are respectivelycaused by the work of expansion due to the viscous dissipation Joule heating, and to thework of expansion/contraction due to acoustic waves. The study of these two terms isbeyond the scope of this paper.

16 R. Tailleux

The diffusive part of B was estimated numerically for thousands of randomly generatedstratifications, similarly as in the previous paragraph, using a standard finite differencediscretization of the molecular diffusion operator. Unlike in the previous paragraph, how-ever, all the randomly generated stratifications were computed from only two differentreference states pertainting to the classical and Fofonoff regimes respectively, the resultsbeing depicted in the top and bottom panels of Fig. 5 (with the right panels providing ablow-up of the left panels). The main result here is that finite values of APE can makethe diffusive part of B negative and considerably larger by several orders of magnitudethan in the laminar APE = 0 case. This result is important, because it is in stark con-trast to what is usually assumed for nearly incompressible fluids at low Mach numbers.From Fig. 5, it is tempting to conclude that there exists a well-defined relationship be-tween the diffusive part of B and APE, but in fact, the curve B = B(APE) is morelikely to represent the maximum value achievable by B for a given value of APE. Indeed,it is important to realize that a given value of APE can correspond to widely differentspectral distributions of the temperature field. In the present case, it turns out thatthe random generator used tend to generate temperature fields with maximum powerat small scales, which in turns tend to maximise the value of B. It follows that theremust exist smoother stratifications with values of B liying in between the x-axis andthe empirical curve B = B(APE), and that the latter curve must depend on the gridresolution employed in the numerical discretization. Nevertheless, Fig. 5 raises the inter-esting question of whether the empirical curve B = B(APE) could in fact describe thebehaviour of the fully developed turbulent regime, an issue that could be explored usingdirect numerical simulations of turbulence.

The two remaining quantities of interest are Wr,mixing and D(APE), which were nu-merically estimated from the following expressions:

Wr,mixing =

∫

V

αrPr

ρrCpr

∇ · (κρCp∇T ) dV (2.19)

D(APE) =

∫

V

Tr − T

T∇ · (κρCp∇T ) dV =

∫

V

κρCp∇T ·

(T − Tr

T

)

dV. (2.20)

As for B, these two quantities were evaluated for thousands of randomly generated strat-ifications as functions of APE, starting from the same reference states as before. Theresults for Wr,mixing are depicted in the left panels of Fig. (6), and the results for D(APE)in the right panels of the same Figure, with the top and bottom panels correspondingto the classical and Fofonoff regimes respectively. The purpose of the comparison is todemonstrate that whereas there exist stratifications for which the two rates D(APE)and Wr,mixing are nearly identical (top panel, classical regime), as is expected from theclassical literature about turbulent stratified mixing, it is also very easy to constructspecific cases occurring in the oceans for which the two rates become of different signs(bottom panel, Fofonoff regime). The other important result is the relative insensitivity ofD(APE) to the nonlinear character of the equation of state compared to Wr,mixing , sug-gesting that the use of the L-Boussinesq model can still accurately describe the KE/APEinteractions even for strongly nonlinear equations of states, although it would fail to do agood job of simulating the evolution of GPEr outside the linear equation of state regime.This also suggests that the L-Boussinesq should be adequate enough to study the mixingefficiency of turbulent mixing events over a wide range of circumstances, provided that bymixing efficiency one means the quantity γmixing = D(APE)/[D(APE) + D(KE)], andnot γmixing = Wr,mixing/[D(APE) + D(KE)]. Finally, we also experimentally verified


Figure 6. (Left panels) The rate of change of GPEr normalised by its laminar value as a functionof normalised APE, in the classical regime (top panel) as well as in the Fofonoff regime (bottompanel). The stratification is identical to that of Fig. 5. (Right panels) The rate of diffusivedissipation of APE normalised by Wr,mixing laminar value, as a function of a normalised APE,in the classical regime (top panel), as well as for the Fofonoff regime (bottom panel). The figureillustrates the fact that if former can be regarded as a good proxy for the latter in the classicalregime, as is usually assumed, this is clearly not the case in the Fofonoff regime. The two figuresalso illustrate the fact that the former always underestimate the latter for a thermally stratifiedfluid, so that observed values of mixing-efficiencies obtained from measuring GPEr variationsare necessarily lower-bounds for actual mixing efficiencies.

(not shown) that D(APE) is well approximated by the quantity Wr,mixing − B, as isexpected when AIE is only a small fraction of APE.

2.5. Synthesis

An attempt at summarizing the energetics of freely decaying turbulence is depicted inFig. 7 for the classical (top panel) and Fofonoff (middle panel) regimes. The bottompanel seeks to unite the two regimes into a single diagram, by furthermore combiningAIE and AGPE into a single reservoir for APE. Doing so makes the bottom panelbasically identical to the Boussinesq energy flowchart in the new interpretation proposedin this paper that is depicted in the bottom panel of Fig. 2. The middle panel showsthat the energetics of turbulent mixing in the Fofonoff regime appears to differ in fun-damental ways to that of the extensively studied classical regime. Indeed, whereas bothW and B act as net sinks of KE in the classical regime, it appears possible in the Fo-fonoff regime to have a fraction of the KE dissipated into AIE to be converted back toKE. Fofonoff (1998, 2001) suggested this could act as a positive feedback on turbulent

18 R. Tailleux

KE

AGPE AIE

IEo IEr−IEo

GPEr

W −B

AGPE

KE

AIE

IEo

−B

IEr−IEo

−W

GPEr

KE IEo IEr−IEo

GPErAPE

+/− WrW−B

D(KE)

D(KE)

D(KE)

D(APE)

D(APE)

D(APE),mixing

SYNTHESIS

−Wr,mixing

Wr,mixing

CLASSICALREGIME

Wr,mixing

FOFONOFFREGIME

−Wr,mixing

Figure 7. The energetics of freely decaying turbulence for the classical regime (top panel), theFofonoff regime (middle) panel, and a synthesis of both regimes obtained by subsuming AGPEand AIE into APE alone. Note the similarity of the energetics in the lower panel and that ofthe re-interpreted Boussinesq energetics of the lower panels of Fig. 2.

kinetic energy, thereby enhancing turbulent mixing and speeding up the return to theclassical regime that would be eventually achieved after enough reduction in the verticaltemperature gradient. In Fofonoff’s view, the positive feedback implicitly requires theconversion of GPEr into AGPE, which conflicts with the present findings that GPEr

can only be converted into IEexergy , which also supports the conclusions of McDougall(2003) obtained using different arguments. This does not necessarily rule out the idea ofpositive feedback, but if so, it cannot exactly occur as envisioned by Fofonoff. Such anissue appears important, owing to many places in the oceans being in Fofonoff’s regime,and should therefore receive more attention in the future, as it could be important forthe parameterization of turbulent diffusive mixing in numerical models of the oceans.

3. Forced/dissipated balances in the oceans

3.1. A new approach to the mechanical energy balance in the oceans

As an example of application of the above ideas, the following examines the links betweenstirring and irreversible mixing from a number of viewpoints, in the case of a thermally-


IEIEoKE

APE GPEr

exergy

G(KE)

D(KE)

G(APE)

(1−Yo)Qnet Yo Qnet

D(APE)

C(KE,APE)Wr,mixing

Wr,forcing

Figure 8. Energy flowchart for a mechanically- and buoyancy-driven thermally-stratified fluid,where Qnet = Qheating − Qcooling. At leading order, the “dynamics” (the KE/APE/IE0

system) is decoupled from the “thermodynamics” (the IEexergy/GPEr system). The dynam-ics/thermodynamic coupling occurs through the correlation between D(APE) and Wr,mixing .

stratified ocean forced mechanically by a surface wind stress, and thermodynamicallyby surface heat fluxes. Some of the objectives are to clarify the role of buoyancy-forcingin driving the ocean circulation, and to determine if, as proposed by Munk & Wun-sch (1998), the consideration of mixing requirements in the oceans really impose anyconstraints on the mechanical sources of stirring.

3.1.1. Steady-state balance for the ‘available’ mechanical energy ME = KE + APE

As established in Appendices A and B, the energetics of a mechanically- and buoyancy-driven thermally stratified ocean can be described in terms of the following evolutionequations for the volume-integrated kinetic energy KE, available potential energy APE,and background potential energy PEr:

d(KE)

dt= −C(KE, APE) − D(KE) + G(KE), (3.1)

d(APE)

dt= C(KE, APE) − D(APE) + G(APE), (3.2)

d(PEr)

dt= D(KE) + D(APE) − G(APE) + Qnet. (3.3)

A further decomposition of PEr in terms of exergy IEr − IE0, dead internal energyIE0, and GPEr is detailed in Appendix B, and the resulting energy flow paths schemat-ically depicted in Fig. 8. This figure shows the decoupling at leading order between the“dynamics” associated with the KE/APE/IE0 reservoirs, and the “thermodynamics”associated with the IEexergy and GPEr reservoirs. Coupling between the dynamics andthermodynamics occurs at the level of the transfer rates between APE and IE0 on theone hand, and between IEexergy and GPEr on the other hand, which, in the simplestcase are just Wr,mixing ≈ D(APE), and Wr,forcing ≈ G(APE) when the approximationof a linear equation of state is accurate enough. In Fig. 8, the buoyancy forcing enters thesystem through the dead internal energy, so that the APE production actually appearsas a conversion between IE0 and APE in the present case. The remaining part of thebuoyancy forcing is responsible for creating IEexergy necessary to feed the variations ofGPEr.

20 R. Tailleux

The steady-state balance for the mechanical energy ME = KE + APE is discussedfirst, with that for the background PEr being discussed in the next paragraph. To thatend, the steady-state versions of Eqs. (3.1) and (3.2) are summed to yield:

G(KE) + G(APE) = D(APE) + D(KE), (3.4)

which simply states that in a steady-state, the production of mechanical energy by thewind forcing G(KE) and buoyancy forcing G(APE) is balanced by the dissipation of KEand APE by molecular viscous and diffusive processes respectively. Physically, APE canbe created both mechanically and thermodynamically, as illustrated in Fig. 9. AlthoughEq. (3.4) appears to be written down explicitly for the first time here, the result followsnaturally from those obtained by Winters & al. (1995), or from the classical theoryof atmospheric energetics as described by Peixoto & Oort (1992) for instance, in thecontext of the Boussinesq approximation. The main novelty compared to Winters & al.(1995), however, is the new interpretation of D(APE) as a ‘true’ dissipation mechanism,similar to D(KE), not as an energy conversion between APE and GPEr, while thenovelty compared to classical atmospheric energetics is the presence of the mechanicalenergy production term G(KE), and the presence of the diffusive dissipation of APE.This advance was obtained by clarifying the links with the different subcomponents ofinternal energy.

Eq. (3.4) represents a significant departure from the way energetics has been discussedin the oceanographic literature. Indeed, most studies of energetics over the past decadehave so far tended to subsume the production and dissipation of APE into a single termB = G(APE) − D(APE), e.g. Wang & Huang (2005); Paparella & Young (2002);Nycander & al. (2007), in which case Eq. (3.4) reduces to:

G(KE) + B = D(KE). (3.5)

Such an approach appears to have been prompted to a large extent by the desire to makeenergetics theory ‘fits’ with the contention by Munk & Wunsch (1998) that the work ratedone by the surface buoyancy fluxes should be small according to Sandstrom (1908)’stheorem †, in contradiction with the findings by Oort & al. (1994) that the latter is asimportant as the work rate done by the wind. Because D(APE) is usually nearly as largeas G(APE), calling G(APE) − D(APE) the work rate done by the surface buoyancyfluxes obviously does the trick of producing a small number, however, this is physicallyno more justified than calling the quantity G(KE) − D(KE) the work rate done bythe wind, and should be discouraged. The error, obviously, comes from the failure ofrecognizing D(APE) as a dissipation process, which should clearly not included as partof the definition of the work rate done by the surface buoyancy fluxes. A back-of-the-envelope estimate for G(APE), presented below, confirms the idea that G(APE) is aslarge as the mechanical energy input due to the wind, suggesting that the buoyancyforcing has to be considered as an important stirrer of the oceans.

3.1.2. Back-of-the-enveloppe estimate of G(APE) for the world oceans

Assessing the relative importance of the surface buoyancy forcing versus that of thewind forcing in driving the large-scale ocean circulation requires estimating the APEproduction rate G(APE). Previously, Oort & al. (1994) inferred G(APE) = 1.2 ±0.7 TW from observations, on the basis of using the so-called Lorenz approximationfor estimating the APE, and concluded that the surface buoyancy forcing was equallyimportant as the wind forcing in driving the large-scale ocean circulation. However, the

† See Kuhlbrodt (2008) for a translation Sandstrom’s paper.


ρ

ρ

ρ

ρ

1

2

1

2

WindCooling

I) Production of APE by wind forcing II) Production of APE by surface cooling Conversion C(APE,KE) > 0 Conversion C(KE,APE) > 0

Figure 9. Idealized depictions of mechanically-driven (left panel) and buoyancy-driven (rightpanel) creation of APE. I) A wind blowing at the surface of a two-layer fluid causes the tiltof the layer interface, resulting in a net C(KE, APE) > 0 conversion. II) Localized cooling athigh-latitudes sets the density of a fraction of the upper layer to that of the bottom layer, alsoinducing a tilt in the layer interface. The return of the interface to equilibrium conditions (flatinterface) results in a net C(APE,KE) > 0 conversion.

Lorenz’s approximation for APE is known to overestimate G(APE), e.g. Huang (1998a),so that it seems useful to seek an independent estimate based on the exact formula forG(APE) derived in Appendix B, viz,

G(APE) =

∫

S

(T − Tr

T

)

κρCp∇T · ndS. (3.6)

In order to estimate G(APE), note that Tr represents the temperature that the parcelwould have if lifted adiabatically to its position in Lorenz’s reference state. Intuitively,surface parcels are expected to be closer to their reference position in the warmer regionsof the oceans than in the cold ones, meaning that Eq. (3.6) must be dominated by thepart of the integral corresponding to surface cooling. Keeping only the first term in aTaylor series expansion of Tr yields Tr ≈ T + Γ(Pr − Pa), where Γ = αT/(ρCp) is theso-called adiabatic lapse-rate. The resulting estimate for G(APE) is:

G(APE) ≈

(αrPr

ρrCpr

)

cooling

Qcooling (3.7)

with (αrPr/(ρrCpr))cooling a typical value of the parameter for the regions of cooling.Using the values α = 5.10−5, K−1, Pr = 2000 dbar = 2.107 Pa, ρr = 103 kg, Cpr =4.103 J.kg−1.K−1, and Qcooling = 2 PW yields:

G(APE) =5.10−5 × 2.107

103 × 4.103× 2.1015 W = 0.5 TW

which is consistent with the lower bound of Oort & al. (1994)’s estimate. The presentscaling arguments supports, therefore, Oort & al. (1994)’s conclusions that the surfacebuoyancy forcing is likely to be as important as the wind forcing in driving the large-scalecirculation.

3.2. A new look at the GPEr balance and Munk & Wunsch (1998)’s theory

Physically, the mechanical energy produced by the wind and the surface buoyancy fluxesserves mainly two functions: driving the large-scale ocean circulation, and stirring the

22 R. Tailleux

oceans so as to maintain high rates of turbulent molecular viscous and diffusive mixing.In the present framework, the place where the stirring and mixing processes appear to beconnected is related to the coupling existing between the GPEr turbulent rate of changedue to molecular diffusion and surface forcing on the one hand, and the APE productionand dissipation rates on the other hand, as is further clarified below. To that end, let usmake use of the result derived in Appendix B establishing that the evolution equationfor GPEr in a forced/dissipated ocean takes the following form:

d(GPEr)

dt= Wr = Wr,mixing − Wr,forcing, (3.8)

where the explicit expressions for Wr,mixing and Wr,forcing are given at leading order bythe following expressions:

Wr,mixing ≈

∫

V

κρCp∇T · ∇

(αrTr

ρrCpr

)

dV, (3.9)

Wr,forcing ≈

∫

S

(αrPr

ρrCpr

)

κρCp∇T · ndS, (3.10)

which are valid for a fully compressible thermally-stratified ocean. In a steady-stateocean, the GPEr balance simply becomes:

Wr,mixing = Wr,forcing. (3.11)

Eq. (3.11) states that the lowering of the background center of gravity by the surfacebuoyancy fluxes must be balanced by the lifting of the center of gravity by turbulentmixing, which is exactly the conclusion arrived at by Munk & Wunsch (1998). Theconnection with their results can be made more precise by noting that in the case ofa Boussinesq fluid with a linear equation of state, which is that considered by MW98,Wr,mixing can be rewritten as:

Wr,mixing ≈

∫

V

κ‖∇zr‖2αr

∂Tr

∂zr

dV ≈

∫

V

Kρρ0N2r dV (3.12)

by using the definition of turbulent diapycnal diffusivity of Winters & al. (1995), whichis exactly the same expression as used by MW98 to estimate the increase of GPEr dueto the turbulent diapycnal mixing. With regard to the forcing, it is easily shown thatWr,forcing ≈ G(APE), by comparison with the expression for G(APE) developed in theprevious section. For a Boussinesq fluid with a linear equation of state, this approximationis even exact, as established in Appendix A. In the following, we therefore assume thatWr,forcing and G(APE) have approximately the same value.

3.2.1. Connection with turbulent mixing

In order to connect the irreversible diffusive mixing with the mechanical sources ofstirring, MW98 used the classical Osborn (1980)’s model Kρ = γmixingε/N

2, relating theturbulent diapycnal diffusivity Kρ to the mixing efficiency γmixing , the viscous dissipationrate of kinetic energy ε, and the squared buoyancy frequency N 2. Furthermore assumingthe overall viscous dissipation D(KE) to be balanced by the work rate done by the windsand tides G(KE), MW98 rewrote the GPEr balance as follows:

Wr,mixing = γmixingG(KE) = Wr,forcing. (3.13)


However, as mentioned above, Wr,forcing = G(APE) in a Boussinesq fluid with a linearequation of state, so that the above equation can also be rewritten as:

G(KE) =G(APE)

γmixing

. (3.14)

In their paper, MW98 argue that Eq. (3.14) should be regarded as a constraint on theoverall mechanical energy production G(KE) required to sustain an overturning circu-lation and associated meridional heat transport of the observed strength. Specifically,estimating Wr,forcing ≈ 0.4 TW, and using the canonical value γmixing = 0.2, they arriveat G(KE) = 2 TW.

Both Eq. (3.14) and its interpretation are questionable, however, because in the turbu-lent literature, the mixing efficiency is usually defined as the fraction of the mechanicalenergy available for stirring that is eventually expanded into irreversible diffusive mix-ing. By definition, the former is the sum G(KE) + G(APE), while the latter is given byD(APE), so that

γmixing =D(APE)

G(KE) + G(APE). (3.15)

Now, the connection between stirring and mixing is related to the coupling betweenWr,turbulent and D(APE), which is simply given by Wr,turbulent = D(APE) for a Boussi-nesq fluid, but which is more complex in a real fluid. For lack of sufficient understandingabout the non-Boussinesq case, we stick with the Boussinesq case, for which the GPEr

balance is:

Wr,forcing = G(APE) = Wr,larminar + Wr,turbulent (3.16)

which yields the following constraint:

G(APE) = Wr,laminar + γmixing [G(KE) + G(APE)] ≈ γmixing [G(KE) + G(APE)].(3.17)

Re-arranging yields:

G(KE) ≈G(APE)

1 − γmixing

, (3.18)

or equivalently,

γmixing ≈G(APE)

G(APE) + G(KE). (3.19)

Eq. (3.18) is the exact counterpart (apart from the neglect of the assumed small Wr,laminar

term) of Eq. (3.14) previously derived by MW98. The comparison between the two ap-proaches reveal that the error in MW98 stems from the assumption that the surfacebuoyancy fluxes do not significantly contribute to stirring in the oceans. This shows,therefore, that MW98’s contention that the surface buoyancy forcing is unimportantfor maintaining the overturning circulation in a steady-state is an assumption, not aconsequence of their results, in contrast to what is usally assumed.

It is unclear, however, whether Eq. (3.17), which expresses the coupling between stir-ring and mixing, should be interpreted as a constraint on G(KE), assuming G(APE)and γmixing to be given, as suggested by Eq. (3.18), or more simply as the definition ofmixing efficiency for a dissipative ocean forced both by mechanical and thermodynam-ical forcing. Indeed, it is important to note that γmixing is known to vary significantlydepending on whether turbulent mixing is mechanically-driven or buoyancy-driven. Forinstance, the value of γmixing = 0.2 is typically reported for turbulent mixing driven byshear instability, e.g. Gregg (1987); Peltier & Caulfield (2003), whereas values as high as

24 R. Tailleux

γmixing ≈ 0.35 (Linden and Redondo (1991)) or even γmixing ≈ 0.5 (Dalziel & al (2008))have been reported for buoyancy-driven turbulent mixing associated with the Rayleigh-Taylor instability. Note that if one assumes G(APE) = 0.5 TW, and G(KE) = 1 TW,then this yields σγmixing = 1/3, which is well within the observed range.

3.2.2. Generalization to a non-Boussinesq ocean

Munk & Wunsch (1998)’s result and its corrected versions given above are only validfor a Boussinesq ocean for which Wr,turbulent and D(APE) are equal. In the actualoceans, however, the equation of state is strongly nonlinear, so that the equality is nolonger expected to hold. Although the precise link between Wr,turbulent and D(APE)is not understood in the general case, it is useful to make an attempt at generalizingMunk & Wunsch (1998)’s results in that case by assuming a coupling of the formWr,turbulent = ξD(APE), with 0 < ξ < 1 some positive constant lower than unity, asmotivated by our empirical study of Wr,turbulent and D(APE) presented in the previoussection, while still assuming Wr,forcing ≈ G(APE). In that case, the GPEr budget yields:

Wr,forcing = G(APE) = Wr,mixing = ξD(APE), (3.20)

so that from the definition of mixing efficiency given by Eq. (3.15), one arrives at:

ξγmixing =G(APE)

G(KE) + G(APE)(3.21)

and hence

G(KE) =G(APE)

1 − ξγmixing

(3.22)

showing that the result is the same as previously, with γmixing simply replaced byξγmixing . As a result, the nonlinearities of the equation of state appears to require a largerG(KE) than previously. This is in agreement with the conclusions of Gnanadesikan & al.(2005), which argues that cabelling in the oceans requires additional additional amountof stirring over an ocean without it.

3.3. Implications for the driving mechanisms of the overturning circulation

The issue presently debated in the ocean community is whether the above argumentsseeking to link irreversible mixing to the mechanical sources of stirring imply, as con-tended by Munk & Wunsch (1998), that the meridional overturning circulation shouldbe regarded as mechanically-driven, rather than buoyancy-driven as has long been as-sumed. To clarify this question, note that the issue of how the kinetic energy circulationis maintained against dissipation mechanisms has been traditionally addressed from theviewpoint of the KE/APE in the past, at least in the atmospheric literature, since onlythe available part of potential energy can be converted into kinetic energy. As discussedabove, however, it seems clear that Munk & Wunsch (1998)’s arguments pertain onlyto the background GPEr budget, i.e., the part of the GPE that cannot be convertedinto kinetic energy. It is hard to understand, therefore, how Munk & Wunsch (1998)’sarguments, in themselves, can contradict the traditional buoyancy-driven view of themeridional overturning circulation.

There is one way, however, in which the overturning circulation can be regarded aspartially mechanically-driven, which is related to the fact that the magnitude of G(APE)depends on the thermocline depth, which itself is affected by the overall amount of me-chanical energy available for the stirring. The winds and tides, therefore, can be regardedas exerting an indirect control on the strength of the overturning circulation by modulat-ing the thermocline depth, which in turns with modulate the value of G(APE). Likewise,


it can also be argued that there is some buoyancy control over the mechanical energyinput due to the winds, since G(KE) depends on the surface ocean velocity, which iscontrolled both by buoyancy and mechanical forcing.

4. Summary and conclusions

4.1. A new view of turbulent diffusive mixing in stratified fluids

In this paper, the nature of the link between D(APE) and Wr,turbulent has been revisited,by showing that the two terms represent fundamentally different measures of turbulentdiffusive mixing, which are associated with two different types of energy conversions,namely the irreversible conversion of APE into dead internal energy IE0 for the former,and an irreversible conversion between exergy IEexergy and GPEr for the latter. Themain result is that the widespread according to which D(APE) and Wr,turbulent is aserendipitous feature of the L-Boussinesq model, which is at best a good approxima-tion, because for real fluids, Wr,turbulent will in general be smaller than D(APE), andsometimes even negative in the so-called Fofonoff regime. This is in contrast with the tra-ditional view, which has widely interpreted the above equality as implying for D(APE)and Wr,turbulent to be of the same kind, and associated with the irreversible conversionof APE into GPEr.

4.2. Consequences

An unexpected consequence of the present study is the finding that the work of expan-sion/contraction, and hence the velocity divergence, must be considerably larger thanpreviously thought in turbulent stratified fluids, so that the widespread view that thelatter can be regarded as incompressible at low Mach numbers appear to be technically in-correct. Whether this invalidates the use of the incompressible Boussinesq approximationto describe turbulent stratified fluids is unclear, because it so happens that a large workof expansion/contraction must be largely balanced by a correspondingly large densityflux associated with the divergent velocity, suggesting that the smallness of compressibil-ity effects might in fact not be essential to the validity of the Boussinesq approximation.Although this could possibly question the validity of using the incompressible Boussi-nesq equations to describe turbulent stratified fluids, it so happens that the work ofexpansion/contraction appears to be largely canceled out by the density flux due to thedivergent part of the velocity, suggesting that the smallness of compressibility effects isnot essential to the validity of the Boussinesq approximation. Further work is needed,however, to lay down the rigorous physical foundations demonstrating that this is indeedthe case. Note, however, that the validity of the Boussinesq approximation in that caseis limited to the description of the KE/APE interactions. With regard to the simulationof GPEr variations, Boussinesq appears to be valid only when the approximation of alinear equation of state is good enough, since it cannot otherwise deal with the casewhere GPEr variations are negative, as is conjectured to be case in the Fofonoff regime(Fofonoff (1998, 2001)).

Another consequence relates to the proper way to look at mixing efficiency, whoseclassical definition is as the fraction of the total available mechanical energy (in KE andAPE forms) that are participating in the irreversible mixing of the fluid. Because of thecommon misinterpretation of Eq. (1.9) as implying the irreversible conversion of APEinto GPEr, as well as overlooking the fact that the equality Eq. (1.9) is serendipitousto a Boussinesq fluid with a linear equation of state, the following two definitions are

26 R. Tailleux

usually considered equivalent:

γdef1mixing =

D(APE)

|∆(APE + KE)|, γdef2

mixing =Wr,turbulent

|∆(APE + KE)|(4.1)

As discussed in this paper, however, only the first definition is technically correct, becauseonly D(APE) can be traced back to the mechanical source of energy stirring the fluid,whereas Wr,turbulent is a conversion between IEexergy and GPEr. For this reason, thetwo definitions can only lead similar numbers only in the case of a Boussinesq fluid witha linear equation of state. In general, however, γdef2

mixing < γdef1mixing . This is an important

point to keep in mind to properly interpret published values of mixing efficiencies.

4.3. Revisiting mechanically-driven theories of the meridional overturning circulation

New insights into how the circulation of a thermally-stratified ocean driven by surfacewind stress and heat fluxes is forced and maintained against dissipation were obtained byconsidering the “available” mechanical energy KE/APE budget. The mechanical powerinput G(KE) due to the wind stress is given by the work rate done by the surface windstress against the ocean surface velocity, as is already well-known, while the mechanicalpower input due to the surface buoyancy fluxes is given by the APE production rateG(APE), as originally suggested by Lorenz (1955) in the context of atmospheric ener-getics. Simple scaling estimates suggest that G(APE) is comparable with G(KE), whichsupports Oort & al. (1994)’s previous conclusions that the buoyancy forcing is equallyimportant as the mechanical forcing in driving and stirring the oceans, but disagreeswith the conclusions of many recent studies that have argued the opposite, either on thebasis of Sandstrom (1908)’s theorem, or on the basis of an erroneous definition of thework rate done by surface buoyancy fluxes that does not distinguish between productionand dissipation of APE. It is argued here that the latter definition makes no more sensethan to regard the quantity G(KE)−D(KE) as the work rate done by the surface windstress. The finding that G(APE) is large and positive means that the oceans are ableto convert a large amount of thermal energy into mechanical energy, and are thereforeclearly a heat engine according to the classical engineering definition for such a device.There appears therefore no physical basis for the concept of “heat transport engine”and “passive heat transport” proposed by MW98, which appears to be related to theerroneous belief in the possibility for APE to be irreversibly converted into GPEr. Thelarge value of G(APE) in the oceans appears to be therefore compatible with the ideathat the circulation of the meridional overturning circulation is maintained against dis-sipation by the conversion of a fraction of G(APE) into kinetic energy, in agreementwith the traditional buoyancy-driven view of the overturning circulation as described byColin de Verdiere (1993) for instance (see also Mullarney & al. (2004)). Nevertheless,there appears to be one particular way by which the meridional overturning circulationcan also be regarded as mechanically-driven in some sense. This is due to the fact thatG(APE) depends on the vertical stratification, and in particular on the thermoclinedepth, in the sense that G(APE) expected to increase with an increasing thermoclinedepth, for a given fixed distribution of surface heat fluxes. In other words, the idea here isthat G(APE) is not independent of G(KE), and that G(APE) is likely to increase withG(KE) through the increase in thermocline depth that would result from the enhancedturbulent diapycnal mixing that would result from an increase in G(KE). As it turnsout, the second mechanical energy budget for GPEr allows us to derive MW98’s mainresult that the GPEr budget is mainly a balance between the decrease of GPEr due tohigh-latitude cooling and the increase of GPEr due to turbulent mixing, from first prin-ciples. The energy constraint on G(KE)MW98 derived by MW98 is corrected, however,


for the effect of buoyancy forcing on the stirring, with the old and new constraints givenby:

G(KE)MW98 =Wr,forcing

γmixing

, G(KE)New =G(APE)

1 − γmixing

(4.2)

Whether such equations truly represent a constraint on G(KE) needs to be furtherdiscussed, however, because the equation for G(KE)new can also be written as:

γmixing =G(APE)

G(KE) + G(APE)=

D(APE)

D(APE) + D(KE), (4.3)

raising the question of whether the result shouldn’t be better interpreted as a constrainton mixing efficiency instead. It is important to realize, indeed, that mixing efficiencycan be quite different from mechanically-driven mixing versus buoyancy-driven mixing,as stated in the introduction, so that the issue is whether γmixing simply adjusts to aninbetween value. For instance, taking G(APE) = 0.5 TW, and G(KE) = 1 TW, yieldsγmixing = 0.333, while taking G(KE) = 2 TW would yield γmixing = 0.2, in both casesyielding values compatible with experimental measurements of γmixing .

4.4. Future work

In some sense, the present paper only touches on the surface of the issues that need to beaddressed to fully understand the precise link between reversible stirring and irreversiblemixing. As a result, considerable further work remains to clarify this link in more generalcircumstances, including the case where the equation of state also depends upon salinity,giving rise to the possibility of double-diffusion effects, and storing energy in chemicalform. Such extensions would be useful to better connect the present results to many lab-oratory experiments based on the use of compositionally-stratified fluids, although thepresent evidence is that the use of a linear equation of state is probably accurate enoughto describe the latter. To that end, many technical difficulties need to overcome. Indeed,salinity complicates the definition of Lorenz (1955)’s reference state to such an extentthat it is not even clear that such a state can be uniquely defined, e.g. Huang (2005),as may be the case in presence of humidity in the atmosphere, e.g. Tailleux & Grand-peix (2004). A potentially important generalization would also to further decompose theinternal energy in order to isolate the available acoustic energy considered by Bannon(2004), which in the present paper is included as part of our definition of APE.

The author gratefully acknowledges discussions, comments, and a number of contraryopinions from J.C. McWilliams, J. Moelemaker, W. R. Young, T.J. McDougall, R.X.Huang, G. Nurser, D. Webb, P. Muller, A. Wirth, D. Straub, M. E. McIntyre, M. Am-baum, J. Whitehead, T. Kuhlbrodt, R. Ferrari, D.P. Marshall, R. M. Samelson, R. A.deSzoeke, G. Vallis, H.L. Johnson, A. Gnanadesikan, J. Nycander, O. Marchal, and J.Gregory which contributed to a much improved version of two earlier drafts. I am par-ticularly indebted to D. Straub and A. Wirth for helping me realize that what I formerlyinterpreted as an artifact of the Boussinesq approximation was in fact an artifact of usinga linear equation of state. I also thank Chantal Staquet and two anonymous referees fortheir helpful and constructive comments. This study was supported by the NERC fundedRAPID programme.

28 R. Tailleux

Appendix A. Energetics of Incompressible Navier-Stokes Equations

A.1. Boussinesq equations with equation of state nonlinear in temperature

The purpose of this appendix is to document the energetics of the Boussinesq system ofequations that form the basis for most inferences about stratified turbulence for fluid flowsat low Mach numbers, and which is commonly used in the theoretical and numerical studyof turbulence, e.g. Winters & al. (1995); Caulfield & Peltier (2000); Staquet (2000);Peltier & Caulfield (2003). In order to go beyond the usual case of a linear equation ofstate, a slight generalization is introduced by allowing the thermal expansion coefficientto vary with temperature. The resulting set of equations is therefore as follows:

Dv

Dt+

1

ρ0∇P = −

gρ

ρ0z + ν∇2v (A 1)

∇ · v = 0 (A 2)

DT

Dt= κ∇2T (A 3)

ρ(T ) = ρ0

[

1 −

∫ T

T0

α(T ′)dT ′

]

(A 4)

where v = (u, v, w) is the three-dimensional velocity field, P is the pressure, ρ the density,T the temperature, ν = µ/ρ the kinematic viscosity, µ the (dynamic) viscosity, κ themolecular diffusivity, g the acceleration of gravity, and ρ0 a reference density. The classicalBoussinesq model, called L-Boussinesq model in this paper, is simply recovered by takingα to be a constant in Eq. (A 4). In that case, Eqs. (A 3) and (A 4) may be combined toobtain the following diffusive model for density:

Dρ

Dt= κ∇2ρ (A 5)

as assumed in many numerical studies of turbulence, e.g. Winters & al. (1995); Caulfield& Peltier (2000); Staquet (2000); Peltier & Caulfield (2003). When the temperaturedependence of α is retained, the resulting model is called here the NL-Boussinesq model.

A.2. Standard energetics

Evolution equations for the KE and GPE are obtained by the standard procedure, e.g.Batchelor (1967); Landau & Lifshitz (1987), assuming that the system is forced me-chanically by an external stress τ , and thermodynamically by external heat fluxes, bothassumed to act at the surface boundary located at z = 0. The first equation is premulti-plied by ρ0v and volume-integrated. After re-organisation, the equation becomes:

d(KE)

dt=

d

dt

∫

V

ρ0v2

2dV =

∫

∂V

τ · usdS

︸︷︷︸

G(KE)

−

∫

V

ρgw dV

︸︷︷︸

W

+

∫

V

ρ0ε dV

︸︷︷︸

D(KE)

(A 6)

where W is the so-called density flux, D(KE) is the viscous dissipation rate of kineticenergy, and G(KE) is the rate of work done by the external stress. The time evolutionof the total gravitational potential energy of the fluid, i.e., the volume integral of ρgz, is:

d(GPE)

dt=

d

dt

∫

V

ρgz dV =

∫

V

ρgw dV

︸︷︷︸

W

−

∫

V

ρ0gzακ∇2T dV

︸︷︷︸

B

, (A 7)


where B is the Boussinesq approximation of the work of expansion/contraction. In thepresent case, it is possible to derive an explicit analytical formula for B:

B = −

∫

V

gzρ0ακ∇2T dV =

∫

V

κgρ0α∂T

∂zdV +

∫

V

κρ0gzdα

dT(T )‖∇T‖2 dV

= κg [〈ρ〉bottom − 〈ρ〉top]︸︷︷︸

BL

+

∫

V

κρ0gzdα

dT(T )‖∇T‖2 dV, (A 8)

by using integration by parts, and using the fact that the surface term vanishes becausethe surface is by assumption located at z = 0, where 〈ρ〉bottom and 〈ρ〉top denote thesurface-integral of the bottom and top value of density. For a linear equation of state,B = BL will in general be small, because of the smallness of the molecular diffusivity κ,and finite top-bottom density difference. When α increases with temperature, however, Bmay become significantly larger than BL in turbulence strong enough as to make ‖∇T‖2

large enough for the second term in Eq. (A 8) to overcome BL, pointing out the possiblycritical role of nonlinearity of the equation of state in strongly turbulent fluids.

A.3. Lorenz (1955)’s available energetics

We now seek evolution equations for the available and un-available parts of the gravita-tional potential energy, as previously done by Winters & al. (1995) in the case of theL-Boussinesq equations. By definition, the expression for the GPEr is:

GPEr =

∫

V

ρrgzrdV, (A 9)

where zr = zr(x, t) and ρr = ρr(zr, t) are the vertical position and density of the parcelsin the Lorenz (1955)’s reference state. In Boussinesq models, fluid parcels are assumedto conserve their in-situ temperature in the reference state, so that Tr(zr, t) = T (x, t).Taking the time derivative of Eq. (A 9) thus yields:

d(GPEr)

dt=

∫

V

gzr

Dρr

DtdV +

∫

V

gρr

Dzr

DtdV

︸︷︷︸

=0

= −

∫

V

gzrρ0ακ∇2TdV

= −

∫

∂V

gzrρ0ακ∇T · n dS +

∫

V

κρ0g∇T · ∇(αzr)dV

= −Wr,forcing +

∫

V

κρ0g‖∇zr‖2 ∂(αzr)

∂zr

∂Tr

∂zr

dV

︸︷︷︸

Wr,mixing

, (A 10)

where Wr,forcing is the rate of change of GPEr due to the external surface heating/cooling.For a Boussinesq fluid, this term is identical to the APE production rate G(APE), asshown in the following, i.e., Wr,forcing = G(APE). The above formula was obtained byusing the following intermediate results:

∇[T (x)] = ∇[Tr(zr(x))] =∂Tr

∂zr

∇zr, (A 11)

∇[α(T )zr] = ∇[α(Tr(zr))zr] =∂(αzr)

∂zr

∇zr, (A 12)

as well as the important result that the integral involving the term Dzr/Dt vanishesidentically established by Winters & al. (1995). In the latter paper, the result was

30 R. Tailleux

established by using an explicit formula for the reference stratification. An alternative wayto recover such a result is achieved by noting that the velocity vr = (Dxr/Dt, Dzr/Dt)of the fluid parcels in the reference state must satisfy the continuity equation:

∇r · vr = 0, (A 13)

where ∇r is the divergence operator in the reference space state (xr , zr), from which itfollows that the surface-integral of Wr,mixing = Dzr/Dt along each constant zr level mustvanish, which implies Winters & al. (1995)’s result. The equation for APE = AGPE =GPE − GPEr becomes:

d(APE)

dt=

d(GPE)

dt−

d(GPEr)

dt

= W − (Wr,mixing − B) + Wr,forcing = G(APE) + W − D(APE), (A 14)

where

D(APE) = Wr,mixing − B

=

∫

V

ρ0gακ‖∇zr‖2 ∂Tr

∂zr

dV − BL

︸︷︷︸

DL(APE)

+

∫

V

κρ0g(zr − z)dα

dT‖∇zr‖

2

(∂Tr

∂zr

)2

dV

︸︷︷︸

DNL(APE)

, (A 15)

by using the results that:∫

V

ρ0gκdα

dT‖∇T‖2dV =

∫

V

ρ0gκdαr

dTr

‖∇zr‖2

(∂Tr

∂zr

)2

dV, (A 16)

∫

V

ρ0gκ‖∇zr‖2 ∂(αzr)

∂zr

∂Tr

∂zr

dV =

∫

V

ρ0gκ‖∇zr‖2

(

1 + zr

dα

dT

∂Tr

∂zr

)∂Tr

∂zr

dV. (A 17)

Empirically, it is usually found that D(APE) > 0, which is not readily apparent fromthe form of D(APE), and for which a rigorous mathematical proof remains to be es-tablished. Interestingly, while Wr,mixing and B appear to be both strongly modified bya temperature-dependent α, this is much less so for their difference D(APE), which isusually found empirically to be well approximated by its “linear” part DL(APE). Thisis important, because it clearly establishes that D(APE) and Wr,mixing may be signifi-cantly different when the temperature dependence of α is retained, in contrast to whatis generally admitted based on the L-Boussinesq model. This suggests that results basedon the study of the L-Boussinesq model are likely to be more robust and accurate for thedescription of the KE/APE dynamics than for the description of GPEr. The conditionfor |DNL(APE)| � |DL(APE)| to be satisfied is that dα/dT |dTr/dzr||zr−z| � 1, whichappears to be satisfied in practice for water or seawater. Whether this is also true forother types of fluids still needs to be established.

Appendix B. Energetics of compressible Navier-Stokes Equations

B.1. Compressible Navier-Stokes Equations (CNSE)

The purpose of this appendix is to generalise Winters & al. (1995)’s results to the fullycompressible Navier-Stokes equations, which are written here in the following form:

ρDv

Dt+ ∇P = −ρgz + ∇ · S (B 1)

Dρ

Dt+ ρ∇ · v = 0 (B 2)


DΣ

Dt=

Q

T=

ρε −∇ · Fq

ρT(B 3)

I = I(Σ, υ), (B 4)

T = T (Σ, υ) =∂I

∂Σ, P = P (Σ, υ) = −

∂I

∂υ. (B 5)

In the present description, the three-dimensional Eulerian velocity field v = (u, v, w),the specific volume υ = 1/ρ (with ρ the density), and the specific entropy Σ are taken asthe dependent variables, with the thermodynamic pressure P and in-situ temperature Tbeing diagnostic variables as expressed by Eqs. (B 4-B5), where I is the specific internalenergy, regarded as a function of Σ and υ. Further useful notations are: D/Dt = ∂/∂t +(v · ∇) is the substantial derivative, ε is the dissipation rate of kinetic energy, Fq =−kT ρCp∇T is the diffusive heat flux, Cp is the specific heat capacity at constant pressure,kT is the molecular diffusivity for temperature, g is the acceleration of gravity, and z isan normal unit vector pointing upward. Moreover, S is the deviatoric stress tensor:

Sij = µ

(∂ui

∂xj

+∂uj

∂xi

)

+

(

λ −2µ

3

)

δij

∂u`

∂x`

(B 6)

in classical tensorial notation, e.g., Landau & Lifshitz (1987), where Einstein’s summa-tion convention for repeated indices has been adopted, and where δi,j is the Kroneckerdelta. The parameters µ and λ are the shear and bulk (or volume) viscosity respectively.

B.2. Standard energetics

The derivation of evolution equations for the standard forms of energy in the contextof the fully compressible Navier-Stokes equations is a standard exercise, e.g. de Groot& Mazur (1962); Landau & Lifshitz (1987); Griffies (2004), so that only the finalresults are given. In the standard description of energetics, only the volume-integratedkinetic energy (KE), gravitational potential energy (GPE), and internal energy (IE) areconsidered, viz.,

KE =

∫

V

ρv2

2dV, GPE =

∫

V

ρgzdV IE =

∫

V

ρI(Σ, υ) dV, (B 7)

whose standard evolution equations are respectively given by:

d(KE)

dt= −

∫

V

ρgw dV

︸︷︷︸

W

+

∫

V

PDυ

Dt︸︷︷︸

B

dm + G(KE) − D(KE) − Pa

dVol

dt, (B 8)

d(GPE)

dt=

∫

V

ρgw dV

︸︷︷︸

W

, (B 9)

d(IE)

dt=

∫

V

ρQ dV −

∫

V

PDυ

Dtdm

︸︷︷︸

B

= D(KE) + Qheating − Qcooling − B, (B 10)

where G(KE) is the rate of work done by the mechanical sources of energy on thefluid, Qheating (resp. Qcooling) is the surface-integrated rate of heating (resp. cooling)due to the thermodynamic sources of energy, and Vol is the total volume of the fluid;additional definitions and justifications are given further down the text. Summing Eqs.(B 8-B10) yields the following evolution equation for the total energy total energy TE =

32 R. Tailleux

KE + GPE + IE:

d(TE)

dt= G(KE) + Qheating − Qcooling − Pa

dVol

dt, (B 11)

which says that the total energy of the fluid is modified:• by the rate of work done by the mechanical sources of energy;• by the rate of heating/cooling done by the thermodynamic sources of energy;• by the rate of work done by the atmospheric pressure Pa against the volume changes

of the fluid.As these derivations are quite standard, justifications for the above equations are onlybriefly outlined. Thus, the KE equation (B 8) is classically obtained by multiplying themomentum equation by v, and integrating over the volume domain. The term W resultsfrom the product of v with the gravitational force vector, whereas the product v · ∇P =∇·(Pv)−P∇·v = ∇·(Pv)−(P/υ)Dυ/Dt yields the work of expansion/contraction minusthe work fone by the atmospheric pressure against total volume changes. The product ofthe velocity vector by the stress tensor is written as the sum G(KE) − D(KE), whereG(KE) represents the work input due to the external stress, and D(KE) the positivedissipation of kinetic energy. The general expression for the mechanical energy input is:

G(KE) =

∫

∂V

vS · ndS =

∫

∂V

τ · vdS (B 12)

where vS is the vector of component (Sv)j = Sijui, while Sn = τ is the stress appliedalong the surface boundary enclosing the fluid. G(KE) is therefore the work of theapplied stress done against the fluid velocity. If one assumes no-slip boundary conditionon all solid boundaries, then this work is different from zero only on the free surface. Thefunction D(KE) is the dissipation function:

D(KE) =

∫

V

{

µ

(∂ui

∂xj

+∂uj

∂xi

−2

3δij

∂u`

∂x`

)2

+ λ(∇ · v)2

}

dV, (B 13)

where again the summation convention for repeated indices has been used, e.g., Landau& Lifshitz (1987). The equation for GPE (B 9) is simply obtained by taking the timederivative of its definition, using the fact that D(ρgzdV )/Dt = ρgw, since D(ρdV )/Dt =0 from mass conservation. The equation for IE (B 10) results from the fact that thedifferential of internal energy in the entropy/specific volume representation is given bydI = TdΣ − Pdυ. The term Qheating and Qcooling represents the surface-integrated netheating and cooling respectively going through the surface enclosing the domain.

B.3. Available energetics

In this paragraph, we seek to derive separate evolution equations for the available and un-available parts of the total potential energy PE = IE+GPE+PaVol, as initially proposedby Lorenz (1955), building upon ideas going back to Margules (1903). Specifically, PEis decomposed as follows:

PE =

∫

V

ρ [I(Σ, υ) + gz] dV + PaVol

=

∫

V

ρ [I(Σ, υ) + gz] dV −

∫

V

ρ [I(Σ, υr) + gzr] dV + Pa (Vol − Vol,r)

︸︷︷︸

APE


+

∫

V

ρ [I(Σ, υr) + gzr] dV + PaVol,r

︸︷︷︸

PEr

(B 14)

where PEr is the potential energy of Lorenz (1955)’s reference state, and APE =PE − PEr is the available potential energy. As is well known, the reference state is thestate minimizing the total potential energy of the system in an adiabatic re-arrangementof the fluid parcels. From a mathematical viewpoint, Lorenz (1955)’s reference state canbe defined in terms of a mapping taking a parcel located at (x, t) in the given state toits position (xr , t) in the reference state, such that the mapping preserves the specificentropy Σ and mass ρdV of the parcel, viz.,

Σ(x, t) = Σ(xr, t) = Σr(zr, t), (B 15)

ρ(x, t)dV = ρ(xr, t)dVr = ρr(zr, t)dVr, (B 16)

where the second condition can be equivalently formulated in terms of the JacobianJ = ∂(xr)/∂(x) of the mapping between the actual and reference state as follows:

ρ(x, t) = ρ(xr, t)∂(xr)

∂(x)= ρr(zr, t)

∂(xr)

∂(x). (B 17)

Prior to deriving evolution equation for PEr and APE, it is useful to mention threeimportant properties of the reference state, namely:

(a) The density ρr = ρr(zr, t) and pressure Pr = Pr(zr, t) of the Lorenz (1955)’sbackground reference state are functions of zr alone (and time);

(b) The background density ρr and pressure Pr are in hydrostatic balance at all times,i.e., ∂Pr/∂zr = −ρrg (this is a consequence of the reference state being the state min-imising the total potential energy in an adiabatic re-arrangement of the parcels);

(c) The velocity vr = (Dxr/Dt, Dyr/Dt, Dzr/Dt) of the parcels in the reference statesatisfies the usual mass conservation equation:

Dυr

Dt= υr∇r · vr, (B 18)

where ∇r · vr is the velocity divergence expressed coordinates system of the referencestate, which is a consequence of the mass of the fluid parcels being conserved by themapping between the actual and reference state.Eq. (B 18) is important, for it allows an easy demonstration of the following result:

∫

V

ρPr

Dυr

DtdV =

∫

Vr

ρrPr

Dυr

DtdVr =

∫

Vr

Pr∇r · vrdVr

=

∫

∂Vr

Prvr · nrdSr −

∫

Vr

vr · ∇PrdVr = Pa

dVol,r

dt+

∫

Vr

ρrgWr,mixingdVr

︸︷︷︸

Wr

, (B 19)

which establishes the equivalence between the work of expansion and the work againstgravity in the reference state, where nr is a outward pointing unit vector normal to theboundary ∂Vr enclosing the fluid in the reference state. In Eq. (B 19), the first equalitystems from expressing the first integral in the reference state; the second equality usesEq. (B 18); the third equality results from the integration by part; the final equality stemsfrom that Pr depends on zr and t only, and that it is in hydrostatic balance, and fromusing the boundary condition vr ·nr = wr = ∂ηr/∂t at the surface assumed to be locatedat zr = ηr(t).

34 R. Tailleux

B.4. Evolution of the background PEr

We seek an evolution equation for the background PEr by taking the time derivative ofthe expression in Eq. (B 14), which yields:

d(PEr)

dt=

∫

V

ρ

[

Tr

DΣ

Dt− Pr

Dυr

Dt+ gwr

]

dV + Pa

dVol,r

dt+

∫

V

[I(Σ, υr) + gzr]D(ρdV )

Dt︸︷︷︸

=0

=

∫

V

ρTr

Q

TdV =

∫

V

ρQ dV +

∫

V

ρ

(Tr − T

T

)

Q dV

= Qnet + (1 − γε)D(KE) + D(APE) − G(APE), (B 20)

where the final result was arrived at by making use of Eq. (B 19), as well as of thedefinitions:

∫

V

ρQ dV =

∫

V

{∇ · (κρCp∇T ) + ρε} dV = Qnet + D(KE), (B 21)

Qnet =

∫

S

κρCp∇T · n dS = Qheating − Qcooling (B 22)

G(APE) =

∫

S

(T − Tr

T

)

κρCp∇T · n dS (B 23)

D(APE) =

∫

V

κρCp∇T · ∇

(T − Tr

T

)

dV, (B 24)

γεD(KE) =

∫

V

(T − Tr

T

)

ρε dV (B 25)

where n is the unit normal vector pointing outward the domain. Eq. (B 22) expresses thenet diabatic heating Qnet due to the surface heat fluxes as the sum of a purely positiveQheating and negative −Qcooling contributions. Eq. (B 23) defines the rate of availablepotential energy produced by the surface heat fluxes. The term D(APE), as defined byEq. (B 24), is physically expected to represent the rate at which APE is dissipated bymolecular diffusion, so that it is expected to be positive in general, which has been sofar only been established empirically using randomly generated temperature fields, buta rigorous mathematical proof is lacking. Finally, Eq. (B25) states that a tiny fractionof the diabatic heating due to viscous dissipation might be recycled to produce work. Ifγε could be proven to be positive, it could probably be included as part of the G(APE).In the following, it will just be neglected for simplicity.

B.5. Evolution of Available Potential Energy (APE)

In the previous section, we defined the total potential energy as the sum of GPE, IE,and the quantity PaVol, see Eq. (B 14). As a result, using the evolution equations forGPE and IE previously derived, the evolution equation for PE is given by:

d(PE)

dt= W − B + Qnet + Pa

dVol

dt. (B 26)

Now, combining this equation with the one previously derived for PEr allows us to derivethe following equation for the available potential energy APE = PE − PEr, defined as


KE PEr

APE

G(KE) Qnet

D(KE)KE

APE

G(KE) Qnet

IEr GPErD(KE) Wr,mixing

KE

APE GPEr

Wr,forcing

Wr,forcingW

r,m

ixin

g

D(APE

)

G(APE

)

C(KE,APE)

C(KE,APE)

C(KE,APE)

D(APE

)

G(APE

)

D(KE)

G(KE) (1−Yo)Qnet Yo Qnet

IEo IEr−IEoKE IEo IEr−IEo

G(KE) (1−Yo)Qnet Yo Qnet

Wr,

mix

ing W

r,forcing

AGPE AIE GPEr

W−B

D(KE)

Wr,forcing

Wr,mixing

G(A

PE)

D(A

PE)

(III)(IV)

(I) (II)

D(APE

)

G(APE

)

Figure 10. Successive refinements of the energetics of a forced/dissipated stratified fluid. Panel(I): KE/APE/PEr representation. Panel (II): Decomposition of PEr int IEr + GPEr. Panel(III): Decomposition of IEr into a dead part IE0 and exergy part IEexergy = IEr − IE0. Panel(IV): Decomposition of APE into AIE and AGPE, revealing the link between C(KE, APE)to the density flux W and work of expansion/contraction B.

the difference between the potential energy and its background value:

d(APE)

dt≈ W − B + Pa

dVol

dt︸︷︷︸

C(KE,APE)

+Qnet+D(KE)−[

D(KE) + D(APE) + Qnet − G(APE)]

= C(KE, APE) + G(APE) − D(APE), (B 27)

where the final expression neglects the small term γεD(KE). The corresponding energyflowchart for the KE/APE/PEr system is very simple, and is illustrated in Fig. 10 (PanelI). This diagram shows that mechanical energy enters the fluid via the KE reservoir,and that thermal energy enters it via the PEr reservoir. There are two dissipation routesassociated with the viscous dissipation of KE and the diffusive dissipation of APE. Onlya certain part G(APE) of the thermodynamic energy input can be converted into APEand hence into KE, which is processed via the PEr reservoir. The two-headed arrowindicates the reversible conversion between KE and APE.

B.6. Splitting of PEr into GPEr and IEr

Although the KE/APE/PEr system offers a simple picture of the energetics of a (tur-bulent or not) stratified fluid, it is useful to further decompose the background PEr

reservoir into its GPE and IE component, in order to establish the link with the exist-

36 R. Tailleux

ing literature about turbulent mixing, as well as with Munk & Wunsch (1998)’s theory.The particular questions to be addressed is to understand how much of D(KE) andD(APE) are actually spread over GPEr and IEr. Likewise, to what extent do Qnet andG(APE) affect GPEr compared to IEr, where we have the following definitions:

IEr =

∫

V

ρ(x, t)I(Σ, υr) dV =

∫

Vr

ρr(zr, t)I(Σ, υr) dVr (B 28)

GPEr =

∫

V

ρ(x, t)gzr(x, t)dV =

∫

Vr

ρr(zr, t)gzrdVr, (B 29)

by expressing the integrals in the coordinates system associated either with the actualstate or reference state. By definition,

d(GPEr)

dt=

∫

V

ρgDzr

DtdV +

∫

V

gzr

D(ρdV )

dt︸︷︷︸

=0

=

∫

V

ρgwr dV = Wr (B 30)

so that the evolution equation for IEr + PaVol,r = PEr − GPEr is simply:

d(IEr + PaVol,r)

dt= Qnet + D(KE) + D(APE) − G(APE) − Wr (B 31)

In order to make progress, we need to relate Wr to the different sources and sinks affectingPEr, as identified in Fig. (10). To that end, we use the fact that Wr is related to thework of expansion in the reference state, as shown by Eq. (B 19), and regard υ = υ(Σ, P )as a function of entropy and pressure, for which the total differential is given by:

dυ = ΓdΣ −1

ρ2c2s

dP (B 32)

where Γ = αT/(ρCp) is the so-called adiabatic lapse rate, e.g. Feistel (2003), and c2s =

(∂P/∂ρ)Σ is the squared sound of speed. As a result, the expression for Wr becomes:

Wr =

∫

Vr

Pr

Dυr

Dtρr dVr − Pa

dVol,r

dt=

∫

Vr

P ′

rρr

[

αrTr

ρrCpr

Q

T−

1

ρ2rc

2sr

DPr

Dt

]

dVr (B 33)

where P ′

r = Pr − Pa is the pressure corrected by the atmospheric pressure, by notingthat we have:

dVol,r

dt=

∫

V

Dυr

Dtρ dV =

∫

Vr

Dυr

DtρrdVr . (B 34)

In order to simplify Eq. (B 33), let us recall that mass conservation can be rewritten inhydrostatic pressure coordinates as follows:

∇r · ur +∂

∂Pr

DPr

Dt= 0 (B 35)

e.g. Haltiner & Williams (1980); de Szoeke & Samelson (2002). As a result, it follows thatintegrating Eq. (B 35) from the surface where Pa = cst, and hence where DPr/Dt = 0,to an arbitrary level indicates that the surface integral of DPr/Dt along must vanishalong any level zr = cst. As a consequence, the term depending on DPr/Dt in Eq. (B 33)must vanish. The remaining term can be written as follows:

Wr =

∫

V

P ′

rαrTr

ρrCprT{∇ · (κρCp∇T ) + ρε} dV


=

∫

V

P ′

rαr

ρrCpr

(

1 +Tr − T

T

)

∇ · (κρCp∇T ) dV +

∫

V

P ′

rαrTr

ρrCprTρε dV

∫

V

P ′

rαr

ρrCpr

∇ · (κρCp∇T ) dV

︸︷︷︸

Wr,mixing−Wr,forcing

+

∫

V

P ′

rαr

ρrCpr

(Tr − T

T

)

∇ · (κρCp∇T ) dV

︸︷︷︸

Υr,apeD(APE)

+

∫

V

P ′

rαrTr

ρrCprTρε dV

︸︷︷︸

Υr,keD(KE)

= Wr,mixing − Wr,forcing + Υr,apeD(APE) + Υr,keD(KE) (B 36)

where we defined:

Wr,mixing = −

∫

V

κρCp∇T · ∇

(αrP

′

r

ρrCpr

)

dV (B 37)

Wr,forcing = −

∫

S

αrP′

r

ρrCpr

κρCp∇T · n dS (B 38)

Υr,apeD(APE) =

∫

V

αrP′

r

ρrCpr

(Tr − T

T

)

∇ · (κρCp∇T ) dV (B 39)

Υr,keD(KE) =

∫

V

αrPr

ρrCpr

Tr

Tρε dV (B 40)

Eq. (B 36) shows that the variations of GPEr are affected by:• Turbulent mixing, associated with Wr,mixing . This expression is similar to the one

previously derived for the L-Boussinesq model. The classical Boussinesq expression canbe recovered from using the approximation T ≈ Tr, taking αr, ρr, and Cpr as constant,and using the approximation Pr ≈= −ρ0gzr, which yields:

Wr,mixing ≈

∫

V

κρ0g‖∇zr‖2α

∂Tr

∂zr

dV ;

• The surface forcing, associated with Wr,forcing . Likewise, the L-Boussinesq expres-sion can be recovered by making the same approximation, yielding:

Wr,forcing ≈

∫

V

αgzr

Cp

QsurfdS

Note that in the L-Boussinesq approximation, we have:

Wr,forcing ≈ G(APE)

which is not generally true in the fully compressible Navier-Stokes equation;• The contribution from the viscous and diffusive dissipation of KE and APE respec-

tively associated with D(KE) and D(APE). Note that the coefficient Υr,ape and Υr,ke

are very small for a nearly incompressible fluid such as seawater. For instance, typicalvalues are: α = 10−4 K−1, P = 4.107 Pa, ρ = 103 kg.m−3, and Cp = 4.103 J.kg−1.K−1,which yields:

Υr = O

(10−4 × 4.107

103 × 4.103

)

= O(10−3

).

From this, it follows that at leading order, the direct effects of D(APE) and D(KE) onGPEr can be safely neglected compared to the other two effects, so that:

d(GPEr)

dt= Wr ≈ Wr,mixing − Wr,forcing (B 41)

38 R. Tailleux

The resulting modifications to the energy flowchart are then displayed in Fig. 10 (PanelII). At leading order, the effects of the forcing and mixing on GPEr appear as conversionterms with IEr.

B.7. Further partitioning of internal energy into a “dead” and “exergy” component

As seen previously, the L-Boussinesq model is such that:

D(APE) ≈ Wr,mixing , G(APE) ≈ Wr,forcing, (B 42)

which may gives the impression, based on Fig. 10 (Panel II) that the APE dissipated byD(APE) is actually converted into GPEr, while G(APE) may also appear as originatingfrom GPEr. The purpose of the following is to show that this is actually not the case.To that end, we introduce an equivalent isothermal state having exactly the same energyas Lorenz (1955)’s reference state, i.e., that is defined by:

IEr + GPEr + PaVol,r︸︷︷︸

PEr

= IE0 + GPE0 + PaVol,0︸︷︷︸

PE0

. (B 43)

Because both Lorenz’s reference state and the equivalent thermodynamic equilibriumstate are in hydrostatic balance at all time, PEr and PE0 are just the total enthalpies ofthe two states. This makes it possible to define each parcel by their horizontal coordinates(x, y) and hydrostatic pressure P , and to assume that the dead state can be obtainedfrom Lorenz’s reference state by an isobaric process, so that (x0, y0, P0) = (xr , yr, Pr),which in turns implies (dx0/dt, dy0/dt, dP0/dt) = (dxr/dt, dyr/dt, dPr/dt).

Prior to looking at the evolution of the dead state, let us establish that if the pressureP is in hydrostatic balance at all times, then we have the following result:

∫

V

DP

DtdV =

∫

V

u · ∇hP dV, (B 44)

where u is the horizontal part of the 3D velocity field, and ∇h the horizontal nablaoperator. The proof is:∫

V

(DP

Dt− u · ∇hP

)

dV =

∫

V

(∂P

∂t+ w

∂P

∂z

)

dV =d

dt

∫

V

P dV −Pa

dVol

dt−

d

dt

∫

V

ρgzdV

=d

dt

{

PaVol + MtotgHb +

∫

V

ρgzdV

}

− Pa

dVol

dt−

d

dt

∫

V

ρgzdV = 0

where Vol and Mtot are the total volume and mass of the fluid, whose expressions are:

Vol =

∫

S

(η(x, y, t) + Hb)dxdy, gMtot =

∫

S

(Pb(x, y, t) − Pa)dxdy,

where z = η(x, y, t) is the equation for the free surface, Pb(x, y, t) is the bottom pres-sure, Hb is the total depth of the basin, and where the expression between brackets wasobtained by using the following result:

∫

V

PdV =

∫

S

[Pz]η−Hb

dxdy +

∫

V

ρgzdV =

∫

S

[Paη + PbHb]dxdy +

∫

V

ρgzdV

= Pa

∫

S

(η + H)dxdy

︸︷︷︸

PaVol

+ H

∫

S

(Pb − Pa)dxdy

︸︷︷︸

MtotgH

+

∫

ρgzdV.

The important consequence of Eq. (B 44) is that the volume integral of DP/Dt identically


vanishes when P is independent of the horizontal coordinates, as is the case for Pr andP0. Now, using the expression for the enthalpy I + P/ρ:

d(I + P/ρ) = CpdT +

(

υ − T∂υ

∂T

)

dP = CpdT + υ (1 − αT ) dP,

we can derive the following equation for PE0,

d(PE0)

dt=

∫

V0

ρ0

(

Cp0DT0

Dt+ (1 − α0T0)

DP0

Dt

)

dV0 =dT0

dt

∫

V0

ρ0Cp0dV0, (B 45)

which naturally provides the following equation for T0:

dT0

dt=

D(KE) + D(APE) + Q − G(APE)∫

V0

ρ0Cp0dV0. (B 46)

We can now derive an evolution equation for GPE0, using the relation:

d(GPE0)

dt=

∫

V0

P ′

0

Dυ0

Dtρ0dV0 (B 47)

where P ′

0 = P0 − Pa. Now, expressing dυ = υαdT + υγdP , where γ is the isothermalexpansion coefficient, we arrive at the following expression:

d(GPE0)

dt=

∫

V0

ρ0P′

0

[

υ0DT0

Dt− υ0γ0

DP0

Dt

]

dV0 =dT0

dt

∫

V

α0P′

0dV0ρ0Cp0dV0 (B 48)

noting again that the term proportional to DP0/Dt must vanish from the argumentsdeveloped above, so that we simply have:

d(GPE0)

dt= Υ0

[


(B 49)

where

Υ0 =

∫

V0

P ′

0α0dV0∫

V0

ρ0Cp0dV0. (B 50)

As a result, it follows that:

d(IE0 + PaVol,0)

dt=

d(PE0 − GPE0)

dt

= (1 − Υ0)[


(B 51)

Let us now define the exergy part of the IEr + PaVol,r as

IEexergy = IEr − IE0 + Pa(Vol,r − Vol,0) (B 52)

The equation is:

d(IEexergy)

dt= −Wr + Υ0

[


= (Υ0 − Υr,ke)D(KE) + (Υ0 − Υr,ape)D(APE) + Υ0

[

Qnet − G(APE)]

−Wr,mixing + Wr,forcing. (B 53)

Again, the neglect of the terms proportional to αP/(ρCp) yields the following simplifi-cation:

d(IE0 + PaVol,0)

dt≈ D(APE) + D(KE) + Qnet − G(APE), (B 54)

40 R. Tailleux

d(IEexergy)

dt≈ Wr,mixing − Wr,forcing. (B 55)

The corresponding energy flowchart is this time illustrated in Fig. 10 (Panel III). Thisfigure shows that when IEr is decomposed into its dead and exergy part, a decouplingbetween the KE/APE/IE0 and IEr − IE0/GPEr reservoirs appears at leading order.Note, however, that the rates between the reservoirs remain coupled, owing to the cor-relation between D(APE) and Wr,mixing , as well as between G(APE) and Wr,forcing

discussed in this paper, and which is a central topic of turbulent mixing theory.

B.8. Separate evolution of APE into GPE and IE components

We conclude the evolution equation of energetics by further splitting the APE reservoirinto its GPE and IE components. Using the previous relations, one easily shows that:

d(AGPE)

dt= W − Wr ≈ W − Wr,mixing + Wr,forcing (B 56)

d(AIE)

dt≈ W − B + G(APE) − D(APE) − [W − Wr,mixing + Wr,forcing]

≈ −B + Wr,mixing − D(APE) + G(APE) − Wr,forcing . (B 57)

For seawater, it is generally found that AIE accounts for around 10% of the totalAPE, so that to a good approximation APE ≈ AGPE, which is implicit in the Boussi-nesq approximation. Equating d(AGPE)/dt with d(APE)/dt amounts to require thatd(AIE)/dt ≈ 0. By imposing that the forcing and mixing terms vanish separately, oneobtains:

D(APE) ≈ Wr,mixing − B, (B 58)

G(APE) ≈ Wr,forcing , (B 59)

which are equivalent to those of the L-Boussinesq and NL-Boussinesq model. The cor-responping energy flowchart is depicted in Fig. 10 (Panel IV). The key feature of thisfigure is to reveal that the conversion rates between AGPE and AIE are identical tothose taking place between IEr−IE0 and GPEr, which appears to be where the couplingbetween stirring and mixing fundamentally occurs.

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