kinematic versus thermodynamic incompressibility: which...

Under consideration for publication in J. Fluid Mech. 1

Kinematic versus thermodynamicincompressibility: Which does matter for

Boussinesq/Anelastic energetics?

R E M I T A I L L E U X1†1Department of Meteorology, University of Reading,Earley Gate, PO Box 243, Reading, RG6 6BB, UK

(Received 30 April 2014 and in revised form ??)

Although the concept of an “incompressible fluid” as used in the Boussinesq approx-imation takes its origin in a thermodynamic definition of incompressibility based onassuming an infinite speed of sound, the modern acceptation of the term has becomemore generally associated with the kinematic definition of incompressibility associatedwith the use of a divergence-free velocity field. While thermodynamic and kinematic in-compressibility coincide for an equation of state function of entropy alone, in absence ofdiabatic effects, they do not otherwise; yet, most of the literature on Boussinesq energeticsseems to assume that kinematic incompressibility implies thermodynamic incompressibil-ity, and hence that conversions with internal energy are irrelevant for understanding theenergetics of turbulent stratified mixing for instance. This paper shows that the key tounderstanding the issue is to recognise that the energetics of Boussinesq (and anelastic)fluids possesses a term that can be identified as the approximation to the compressiblework of expansion/contraction, which appears in the guise of apparent changes in gravi-tational potential energy due to apparent changes of mass. Consistent approximations tothe “heat” can then be constructed by requiring that the Maxwell relationships be sat-isfied, which ultimately leads to well-defined approximations to the internal energy andother thermodynamic potentials. This approach is illustrated by constructing explicitexpressions for the internal energy and enthalpy for a Boussinesq fluid with a linearequation of state, and by showing how to correct existing models to make them fullyenergetically and thermodynamically consistent. In such models, the conservative energyquantity is the sum of kinetic energy and enthalpy, and gravitational potential energy isa purely thermodynamic quantity that is the difference between enthalpy and internalenergy. Moreover, such models treat potential and in-situ temperatures as fundamentallydistinct quantities. The overall conclusion is that kinematically incompressible Boussi-nesq models can support large compressible effects and leading order conversions betweeninternal energy and mechanical energy, in contrast to what is usually assumed, which isimportant to correctly interpret the energetics of most turbulent diabatic phenomena.

1. Introduction

1.1. What is an incompressible fluid?

Many fluid flows of interest in engineering and geophysical fluid dynamics are charac-terised by low Mach number (M = U0/cs) and small relative density variations (ρ −

† Present address: Department of Meteorology, University of Reading, Earley Gate, PO Box243, Reading RG6 6BB, United Kingdom. E-mail: [email protected]

2 R. Tailleux

ρ0)/ρ0, where U0 is a typical velocity scale, cs is the speed of sound, ρ is the densityand ρ0 a reference density. It has been common practice over the past century to regardsuch flows as incompressible or weakly compressible. There appears to be some confu-sion, however, about what an “incompressible fluid” actually refers to. Historically, theconcept was originally introduced based on a thermodynamic definition of incompress-ibility. Indeed, according to Batchelor (1967) “a fluid is said to be incompressible whenthe density of an element of fluid is not affected by changes in the pressure”, see alsoLilly (1996). In absence of adiabatic effects, the implication is that the density ρ mustbe constant following fluid parcels, i.e., satisfy Dρ/Dt = 0. From mass conservationDρ/Dt+ ρ∇ · v = 0, this implies in turn that ∇ · v = 0 and hence that the fluid veloc-ity field must be divergentless. From a kinematic viewpoint, a divergence-free velocityfield implies that the volume element dV is conserved following fluid parcels, which rep-resents a kinematic definition of incompressibilty. Nowadays, however, the terminology“incompressible approximation” is commonly employed to refer to system of equationsdescribing fluids affected both by diabatic effects and having a pressure-dependent den-sity, as in the case of the seawater Boussinesq approximation used by ocean modellers,e.g., Young (2010), and is therefore entirely based on the kinematic definition of in-compressibility. Clearly, the original meaning of “incompressible” as based on Batchelor(1967) and Lilly (1996) thermodynamic definition seems to have been lost over time tobecome exclusively associated to a fluid with a divergentless velocity field. One objectiveof this paper is to convince the reader that this shift in meaning complicates the correctinterpretation of the energetics of “incompressible” approximations, given that compress-ible effects turn out to be often large and to occur at leading order when diabatic effectsand the density dependence on pressure are retained. The simple solution proposed inthis paper is to show that the changes in gravitational potential energy caused by thechanges in mass due to diabatic effects and the pressure dependence of density can beviewed as an approximation to the compressible work of expansion/contraction.

1.2. Brief historical review of concepts

Over the years, many investigators have sought to take advantage of the smallness of theMach number and buoyancy parameter, among others, to develop sound-proof sets ofequations that offer a number of practical advantages, computational or analytical, overthe fully compressible Navier-Stokes equations. Two particular classes of approximationshave been particularly influential and key to simplifying the numerical and theoreticalanalysis of low Mach number fluid flows, and will be under focus in this paper. Thefirst one is the Oberbeck-Boussinesq approximation (after Oberbeck (1879) and Boussi-nesq (1903)), which in its most common form retains only the rotational divergence freecomponent of the velocity field, and treats the density as constant everywhere exceptwhere it multiplies the acceleration of gravity. Spiegel & Veronis (1960) show that undersome circumstances, the pressure dependence of a compressible fluid can neverthelessbe neglected relative to the temperature dependence, so that formally the Boussinesqapproximation formally reduces to that for an incompressible fluid. The second one isthe Anelastic approximation, e.g., Ogura & Phillips (1962); Lipps & Hemler (1982);Bannon (1996); Durran (1989); Ingersoll & Pollard (1982); Ingersoll (2005); Pauluis(2008). Many other sets can be constructed, which are beyond the scope of this paper, forinstance by using low Mach number asymptotics and multi-scale expansion techniques,e.g., Muller (1998); Klein (2009, 2010). Davies & et al. (2003) reviews of a num-ber of commonly employed reduced sound-proof sets of equations, and show how theyrespectively represent normal modes on the sphere.

Energetics and thermodynamics of the Boussinesq/Anelastic approximations 3

1.3. Fate of thermodynamic/dynamic coupling in Boussinesq/anelastic approximations

The main objective of this paper is to clarify the nature of the compressible effects and ofthe coupling between mechanical energy and internal energy (the dynamics/thermodynamicscoupling) in the Boussinesq/anelastic approximations. For adiabatic motions and a lin-earised equation of state, such approximations are known to decouple either fully oralmost fully the thermodynamics from the dynamics, i.e., Spiegel & Veronis (1960);Ogura & Phillips (1962), and usually admit a well-defined conservative energy quantity,e.g., Lilly (1996). Some form of thermodynamics/dynamics coupling becomes unavoid-able, however, when diabatic and/or a nonlinear equation of state are retained, so thatthe issue of the energetic and thermodynamic consistency of the Boussinesq/anelasticapproximations become less clear in that case. The oceanographic case is instructive inthis respect, as oceanographers have used the Boussinesq hydrostatic approximation inconjunction with a realistic nonlinear equation of state (including the pressure depen-dence) for many decades as the basis for numerical ocean general circulation models ofthe kind used in climate studies, without any apparent obvious drawbacks apart fromthe lack of a well-defined and closed energy budget, see Tailleux (2010). Since Boussi-nesq ocean models appear to work well with a “compressible” equation of state, one maywonder how essential the assumptions of “incompressibility” or “weak compressibility”are in the construction of the Boussinesq and anelastic approximations in the first place.

1.4. Common misconceptions

There appears to be at least two important misconceptions about the role of compress-ible effects in weakly compressible turbulent stratified fluids. The first misconception isthat thermodynamics/dynamics coupling should be small in weakly compressible fluidsirrespective of the particular thermodynamic transformations undergone by fluid parcels,whereas weak coupling is theoretically justified only for adiabatic motions and fluids withan equation of state independent of pressure. The second misconception, which is relatedto the first one, is that it is appropriate to regard a fluid as being “weakly compressible”or “incompressible” whenever its velocity field satisfy the kinematic constraint ∇ ·v = 0or ∇ · (ρ0v) = 0, despite the fact that what physically determines the magnitude of thecompressibility effects and the thermodynamics/dynamics coupling is the nature of thediabatic effects and nonlinearities of the equation of state.

Based on the above arguments, the new view that the present paper aims to promote isthat the Boussinesq and anelastic approximations are capable of supporting potentiallylarge compressible effects and conversions between internal energy and mechanical energy.As this is in stark contrast from the conventional view, this immediately raises severalquestions: 1) Assuming that a Boussinesq/anelastic fluid can indeed describe compress-ible effects and conversions between internal energy and mechanical energy, how do theycompare with that taking place in a fully compressible fluid? Does the answer dependon the nature of the diabatic effects and of the nonlinear equation of state considered?2) Are the Boussinesq and anelastic approximations consistent with the first and secondlaws of thermodynamics? If so, what is the form of the thermodynamic potentials sup-ported by such approximations, and how do they differ from that of a fully compressiblefluid? 3) Is it an intrinsic problem that many Boussinesq and anelastic models fail tobe energetically and thermodynamically consistent, or can such models be modified tocorrect the problem? In the latter case, can this be done without modifying the formalorder of accuracy of the original Boussinesq/anelastic approximations?

4 R. Tailleux

1.5. Organisation of the paper

The main purpose of this paper is to address all the above questions by showing thatthe Boussinesq and anelastic approximations can be endowed with well defined energeticsand thermodynamics that are traceable to the fully compressible Navier-Stokes equations,whether diabatic effects and a nonlinear equation of state are retained or not. There issome precedent to the present ideas. Thus, Young (2010) showed, using ideas previouslydeveloped by Ingersoll (2005), that the seawater Boussinesq equation have a well-definedconserved energy quantity for an arbitrary nonlinear equation of state and for adiabaticmotions, but did not address the issue of conservativeness for diabatic motions. Pauluis(2008) addressed a similar issue for the anelastic approximation for a binary fluid suchas moist air, also with a highly nonlinear equation of state, and discussed energy andthermodynamic consistency issues. None of the above studies, however, clearly addressedthe nature of the conversions between mechanical energy and internal energy, whichare addressed here in significantly more details. Section 2 provides the general theory.Section 3 applies the result to elucidating the thermodynamics of a Boussinesq fluidwith a linear equation of state that has been widely used recently in numerical study ofturbulent mixing, as well as in the context of horizontal convection. Section 4 summarisesand discusses some implications of the results.

2. Thermodynamically consistent and inconsistentBoussinesq/anelastic models

2.1. Definition of a generic Boussinesq/anelastic model

We take as our starting point the anelastic system of equations based on that previouslyconsidered by Ingersoll (2005) and Pauluis (2008), which is sufficiently general to supportboth diabatic effects and an arbitrary nonlinear equation of state, viz,

Dv

Dt+∇ ·

(δP

ρ0

)= −

(ρ− ρ0ρ0

)g0∇Z +

1

ρ0∇ · S, (2.1)

∇ · (ρ0v) = 0, (2.2)

Dη

Dt= η =

q

T= − 1

ρ0∇ · (ρ0Fη) + ηirr, (2.3)

DS

Dt= S = − 1

ρ0∇ · (ρ0FS), (2.4)

ρ = ρ(S, η, P0). (2.5)

Moreover, it is easily verified that the above system reduces to the familiar Boussinesqapproximation in the case where ρ0 = constant. Thereafter, we refer to (2.1-2.5) as theBoussinesq-Anelastic(BA) system, where η is the specific entropy, S is salinity, g0 is aconstant acceleration of gravity, Z is the geopotential height, so that g0Z = Φ is thegeopotential, and S is the stress tensor. The terms η = q/T and S are used as short-handto denote the diabatic effects affecting η and S. Explicit expressions for the latter can beobtained by invoking conservation laws for entropy and salt, as is done further in the text.It is useful at this stage to remark that our definition of buoyancy b = −g0(ρ− ρ0)/ρ0 isthe one that is the most commonly encountered in the literature, but that it differs fromthe definition b = g0(υ − υ0)/υ0 employed by Pauluis (2008) and Young (2010). Animportant result of this paper will be to establish that the latter definition yields a more


accurate momentum and energy budgets, and hence that its use is preferable to achievemaximum energetic and thermodynamic consistency.

Before proceeding, we note that before settling on the above set of equations, Pauluis(2008) initially assumed the reference pressure P0 to be in hydrostatic balance, and henceto satisfy ρ−10 ∂P0/∂z = −g0∂Z/∂z. However, Pauluis found it impossible to simultane-ously achieve energetic consistency and hydrostatic balance, and therefore proposed toalter the latter so that P0 and ρ0 satisfy the following balance instead

∂

∂z

(P0

ρ0

)= −g0

∂Z

∂z. (2.6)

It is easy to see that (2.6) can be immediately integrated to yield P0 = Pa−ρ0gZ, wherePa is the assumed constant and spatially uniform atmospheric pressure at Z = 0. Thereference pressure P0 in Pauluis (2008)’s anelastic model is therefore a hybrid betweenthe Boussinesq reference pressure defined for a constant ρ0, and the actual hydrostaticpressure P0h solution of ∂P0h/∂Z = −ρ0g0. To clarify the link between P0 and P0h, notethat we have

ρ0D

Dt

P0

ρ0=DP0

Dt+ ρ0P0

Dυ0Dt

= ∇ · (P0v) = −ρ0g0DZ

Dt=DP0h

Dt. (2.7)

The apparent impossibility to enforce energetics consistency without altering true hydro-static balance appears to be an important drawback of Pauluis (2008)’s anelastic model,since achieving energy conservation at the expense of momentum balance seems a highprice to pay, warranting a full investigation of its dynamical consequences, perhaps byusing the kind of approach developed by Davies & et al. (2003).

2.2. A brief review of previous approaches to achieve energetics consistency

Prior to discussing the energetics of the BA system in more details, it is useful to reviewsome of the ideas used previously to discuss energetics consistency in Ingersoll (2005),Vallis (2006), Pauluis (2008), Young (2010), Nycander (2010) and McIntyre (2010).A common feature of the latter studies is to take as their starting point the evolutionequation for the kinetic energy, obtained in the usual way by multiplying (2.1) by ρ0v,which after some manipulation can be written as follows:

ρ0D

Dt

v2

2+∇ · (δPv − ρ0Fke) = ρ0b

DZ

Dt− ρ0εK , (2.8)

where the work against the stress tensor is split as usual as the difference between thedivergence of an energy flux minus viscous dissipation v · ∇S = ∇ · (v · S) − ρ0εK =∇ · (ρ0Fke)− ρ0εK . In the special case where ρ−10 ∇ ·S = ν∇2v, then Fke = ν∇v2/2 andεK = ν(‖∇u‖2 + ‖∇v‖2 + ‖∇w‖2). Physically, the key issue here is whether the termρ0bDZ/Dt in the right-hand side of (2.8) can be rationalised as a conversion betweenkinetic energy and potential energy, and if so, what is the nature of the potential energyimplied by such a conversion term. To address this question, the above studies introducedthe following function:

h‡(S, η, Z) = h‡0(S, η)−∫ Z

Z0

b(S, η, Z ′)dZ ′. (2.9)

It is easily shown that the substantial derivative of h‡ is given by

Dh‡

Dt= −bDZ

Dt+ CSS + Cη η, (2.10)

6 R. Tailleux

where CS and Cη are defined by:

CS =∂h‡0∂S−∫ Z

Z0

∂h‡

∂SdZ ′, Cη =

∂h‡0∂η−∫ Z

Z0

∂h‡

∂ηdZ ′. (2.11)

The important result here is that 2.10 admits the same term bDZ/Dt as appears in thekinetic energy equation but with an opposite sign, allowing such a term to disappearupon summing (2.8) and (2.10), which yields

ρ0D

Dt

(v2

2+ h‡

)+∇ · (δPv − ρ0Fke) = ρ0

(CSS + Cη η

)− ρ0εK︸︷︷︸

R

. (2.12)

This result is widely regarded as an important advance in the understanding of theenergetics of the Boussinesq and anelastic approximations, for it establishes that suchapproximations admit a well defined energy quantity, namely ρ0(v2/2 + h‡), that is con-served in absence of diabatic and viscous effects even for an arbitrary nonlinear equationof state. This had long been thought impossible previously.

Nevertheless, the above result only partially address the energetics and thermodynamicconsistency of the BA system, as it fails to resolve the following key issues:

(a) At first sight, the energy quantity h‡ looks nothing like gravitational potentialenergy or internal energy, making its link to standard energetics unclear. This issue ispartly remedied in Pauluis (2008) and Young (2010) by using the non-standard buoyancyb = −g0(υ−υ0)/υ0, in which case h‡ can be linked to the usual specific enthalpy, which inhydrostatic compressible fluids plays an equivalent role as the total potential energy (asfor instance in studies of available potential energy, e.g., Huang (2005), Pauluis (2007)).The link between h‡ and the standard enthalpy remains to be clarified, however, whenthe traditional buoyancy is used instead.

(b) Establishing the existence of a conserved energy quantity in absence of diabatic andviscous effects is not sufficient to establish the energetic and thermodynamic consistencyof the BA system in the most general case. When diabatic and viscous effects are retained,the presence of R in (2.12) clearly shows that v2/2 + h‡ is not conserved in general, andhence that it is created/destroyed at the expenses of an unknown energy reservoir to beidentified. Moreover, diabatic and viscous effects act as sources/sinks of entropy, so thatan additional key issue is whether the BA system is consistent with the second law ofthermodynamics.

2.3. Alternative approach to the energetics of the BA system

In this paper, we propose an alternative approach to the energetics of the BA systemby remarking that the approximated forms of kinetic energy and gravitational potentialenergy are in general sufficiently close to their exact counterparts that the only realissue is about understanding how the BA system approximates internal energy and otherthermodynamic potentials. Therefore, instead of focusing on the kinetic energy budget asdiscussed above, it seems more natural to construct an evolution equation for the sum ofthe kinetic energy and gravitational potential energy (i.e., the total mechanical energy),and examine the nature of the terms going into its right-hand side with the aim of linkingthe latter to internal energy.

There is some ambiguity in defining gravitational potential energy in the BA system,since the quantity that is conserved following fluid parcels is the “BA mass” ρ0dV ratherthan the actual mass ρdV , where dV is the elementary volume of the fluid parcel con-sidered. As a result, it is not entirely clear whether the GPE of a fluid parcel shouldbe defined as ρdV g0Z or as ρ0dV g0Z. Based on the calculations that follow, we find


that the most relevant definition of mechanical energy Em (per unit volume) for theBA system is ρ0Em = ρ0[v2/2 − b(Z − Z0)] = ρ0v

2/2 + (ρ − ρ0)g0(Z − Z0) (for somereference geopotential height Z0). Using the previously obtained evolution equation forkinetic energy (2.8), the evolution equation for Em is easily verified to be

ρ0D

Dt

[v2

2− b(Z − Z0)

]+∇ · (δPv − ρ0Fke) = −ρ0(Z − Z0)

Db

Dt− ρ0εK , (2.13)

which can be equivalently written in conservative form:

∂(ρ0Em)

∂t+∇ ·

[ρ0

(Em +

δP

ρ0

)v − ρ0Fke

]= −ρ0(Z − Z0)

Db

Dt− ρ0εK . (2.14)

For comparison, the corresponding evolution equation for the mechanical energy in afully compressible fluid would take the form:

ρD

Dt

[v2

2+ g0(Z − Z0)

]+∇ · (Pv − ρFke) = ρP

Dυ

Dt− ρεK . (2.15)

In a compressible fluid, mechanical energy and internal energy are coupled in mainly twoways: (i) via a reversible conversion between internal energy and kinetic energy accom-plished by the work of expansion/contraction PDυ/Dt; (ii) the irreversible dissipationof kinetic energy into heat by viscous processes εK , as seen in the r.h.s. of (2.15). Sinceviscous dissipation is obviously present in both (2.13) and (2.15), the only question thatneeds addressing is whether the following equivalence can be established?

−ρ0(Z − Z0)Db

Dt↔ ρP

Dυ

Dt? (2.16)

To see that this is indeed the case, we expand the pressure and specific volume as follows:

P = Pa − ρ0g0Z + · · · , υ =1

ρ0− ρ− ρ0

ρ20+ · · · (2.17)

which uses the underlying assumption in the Boussinesq and anelastic approximationsthat (ρ − ρ0)/ρ0 � 1. This implies in turn that at leading order, the work of expan-sion/contraction can be approximated by:

ρPDυ

Dt≈ ρ0(Pa − ρ0g0Z)

D

Dt

(1

ρ0− ρ− ρ0

ρ20

)

= −ρ0(Z − Z0)Db

Dt+ ρ0(Pa − ρ0g0Z)

Dυ0Dt

. (2.18)

Eq. (2.18) shows that the equivalence (2.16) is exact in the Boussinesq case ρ0 =constant, but only approximate in the anelastic case. The above correspondence hasbeen previously noted by Winters & al (1995), Wang & Huang (2005) in the context ofthe non-averaged Boussinesq model with a linear equation of state (3.1-3.4), and in thecontext of Boussinesq-averaged models by Nycander & al. (2007) and Young (2010).

2.4. Thermodynamics of Boussinesq and Anelastic binary fluids

Classical thermodynamics defines the specific internal energy e = e(η, S, υ) as a functionof state, whose value is independent of the thermodynamic path followed. This impliesthat the reversible work transfer δW = −Pdυ and generalised heat transfer δQ = Tdη+µdS entering the total differential of e, viz.,

de = δQ+ δW = Tdη + µdS − Pdυ, (2.19)

8 R. Tailleux

are not independent of each other. This interdependence is commonly expressed throughthe so-called Maxwell relationships, which merely state that the cross-derivatives fortwice continuously differentiable functions must be equal. In the present case, the inter-dependence between work and heat variables is thus expressed via the following relations

T =∂e

∂η, µ =

∂e

∂S, P = − ∂e

∂υ, (2.20)

∂T

∂υ=

∂2e

∂η∂υ= −∂P

∂η,

∂T

∂S=

∂2e

∂η∂S=∂µ

∂η,

∂µ

∂υ=

∂2e

∂S∂υ= −∂P

∂S. (2.21)

Assuming that it is justified to regard δWba = (Z−Z0)db as the BA approximation tothe work transfer δW = −Pdυ, similar relations must link δWba and the BA approxima-tion to the heat transfer, denoted δQba, for the BA equations to be thermodynamicallyconsistent. The following shows that δQba may be written similarly as in the fully com-pressible case in the form δQba = Tbadηba+µbadSba, where the suffix ba refers to the BAapproximated quantities, and hence that the BA counterpart of the fundamental relationof thermodynamics (2.19)) may be written

deba = Tbadηba + µbadSba + (Z − Z0)db. (2.22)

The specific enthalpy h = e + Pυ and specific Gibbs function g = e + Pυ − Tη are twoother important thermodynamic potentials, which are traditionally obtained from oneanother by means of the Legendre transform, e.g., Alberty (2001). It is easily shownthat the corresponding BA approximations to h and g are given by hba = eba−b(Z−Z0)and gba = eba − b(Z − Z0)− Tbaηba, with the following total differentials:

dhba = d [eba − b(Z − Z0)] = Tbadηba + µbadSba − bdZ, (2.23)

dgba = d [eba − b(Z − Z0)− Tbaηba] = −ηbadTba + µbadSba − bdZ. (2.24)

The above relations establish that constructing self-consistent thermodynamic potentialsfor the BA equations is a priori feasible. Such relations do not tell, however, how accuratethe BA thermodynamic potentials should be. To address this issue, it is essential toconstruct the BA thermodynamic potentials explicitly.

2.5. Construction of the BA thermodynamic potentials

Depending on the particular fluid and problem considered, there are mainly two methodsto construct the BA thermodynamic potentials explicitly. The first method consists inintegrating the Maxwell relationships to obtain unknown and difficult to measure ther-modynamic potentials from various measurable quantities, such as the speed of soundor heat capacity, as discussed in Callen (1985). For the present purposes, this methodis really only suited for constructing the BA thermodynamic potentials in idealised sit-uations for which the equation of state and other thermodynamic quantities are knownin closed analytic form, as will be illustrated later in the text. The second method con-sists in deducing the BA thermodynamic potentials from the knowledge of the exactthermodynamic potentials, and is illustrated next.

To proceed, let us assume that the specific Gibbs function g = g(T, S, P ) is known asa function of its natural arguments, as is the case for seawater, e.g. Feistel (2003), IOC(2010). We aim to show that the function gba defined by

gba = gba(T, S, Z) = g(T, S, P0h(Z)) + g0(Z − Z0) = g + g0(Z − Z0), (2.25)

is the corresponding BA approximation of g, where P0h is the actual hydrostatic pressure


in balance with ρ0 rather than the anelastic reference pressure P0, while the tilde indicatethat the exact pressure is replaced by P0h.

To check that this is the case, let us take the total differential of gba, and verify thatit is consistent with (2.24). Using the fact that dg = −ηdT + µdS + υdP , it follows that

dgba = −ηdT + µdS + g0(1− ρ0υ)dZ, (2.26)

where we used the result that dP0h = −ρ0g0dZ by definition of P0h. From the above defi-nition of tilde quantities, η = η(T, S, P0h(Z)), µ = µ(T, S, P0h(Z)) and υ = υ(T, S, P0h(Z)).Clearly, (2.26) is consistent with (2.24) provided that

ηba = η, µba = µ, b = −g0(1− ρ0υ) = −g0(ρ− ρ0ρ

), (2.27)

Tba = T, Sba = S. (2.28)

Eqs. (2.27) and (2.28) are important, for they establish that the BA specific entropyηba and relative chemical potential µba can be simply obtained by replacing the totalpressure P by P0h in the exact expressions for η and µ viewed as functions of T , S,and P . They also state the the two main observable quantities, namely temperature andsalinity, do not need to be approximated, allowing a straightforward comparison betweenthe simulated T/S fields and observations.

Surprisingly, however, (2.27) shows that ρ 6= ρ(T, S, P0h), in contrast to what is usuallyassumed in the oceanographic literature, since from the result

b = −g0(ρ− ρ0ρ

)= −g0

(ρ− ρ0ρ0

), (2.29)

it follows that ρ is actually linked to ρ by the relation

ρ = ρ0

(1 +

ρ− ρ0ρ

)= ρ0 (2− ρ0υ) . (2.30)

Note, however, that the result ρ = ρ is possible if Pauluis (2008) and Young (2010)definition of buoyancy b = −g0(υ − υ0)/υ0 is used instead, which therefore provides ad-ditional physical justification for the latter. The result is interesting, because it showsthat thermodynamic consistency considerations alone are able to discriminate betweentwo a priori acceptable definitions of buoyancy, and to reveal that one definition is farsuperior than the other. The above results also demonstrate that density should be esti-mated by using the hydrostatic pressure P0h rather than the full pressure, as advocatedby Young (2010), to avoid degrading the accuracy and consistency of the BA equations.An important consequence of the above result is to suggest that the use of the standardbuoyancy rather than the true buoyancy degrades the accuracy of the horizontal momen-tum equations in the Boussinesq primitive equations forming the basis of most currentocean climate models, which we plan on quantifying in a subsequent paper.

As mentioned above, the knowledge of one thermodynamic potential is sufficient torecover all other thermodynamic potentials via Legendre transforms. It is easily shownthat the relevant BA approximations to the specific internal energy and enthalpy can bededuced from that for the specific Gibbs function gba as follows:

eba(S, T, Z) = gba + b(Z − Z0) + Tbaηba = g + (g0 + b)(Z − Z0) + T η, (2.31)

hba(S, T, Z) = eba − b(Z − Z0) = g + g0(Z − Z0) + T η. (2.32)

Taking the total differential of the above expressions thus yields

deba = Tdη + µdS + (Z − Z0)db, (2.33)

10 R. Tailleux

dhba = Tdη + µdS − bdZ, (2.34)

which are consistent with (2.22) and (2.23).

2.6. Derivations starting from the knowledge of the specific enthalpy

An alternative approach to that based on the Gibbs function is to construct the ther-modynamic potentials from the knowledge of the specific enthalpy h = h(η, S, P ), whosenatural variables are specific entropy, salinity, and pressure. To that end, let us introduce

hba = h(η, S, P0h(Z)) + g0(Z − Z0) = h+ g0(Z − Z0), (2.35)

and verify that it is the relevant Boussinesq/anelastic approximation to the specific en-thalpy. This can be checked by taking that the total differential of hba, viz.,

dhba = Tdη + µdS + g0(1− ρ0υ)dZ = Tdη + µdS − bdZ, (2.36)

agrees with (2.23). QED. The corresponding expression for the internal energy eba =hba + b(Z − Z0) can be written as

eba = h+ (g0 + b)(Z − Z0). (2.37)

Its total differential is:

deba = Tdη+µdS−ρ0g0υdZ+(b+g0)dZ+(Z−Z0)db = Tdη+µdS+(Z−Z0)db, (2.38)

which is again consistent with (2.22).

2.7. Remark on gravitational potential energy in the BA equations

It is of interest to remark here that from the above expressions of the BA thermodynamicpotentials, the gravitational potential energy ep = b(Z − Z0) can be written as thedifference between the BA specific enthalpy and internal energy, viz.,

b(Z − Z0) = hba − eba. (2.39)

This establishes, therefore that ep in the BA equations can be viewed as a purely ther-modynamic quantity entirely determined from the knowledge of temperature, salinity,and Z. It follows that the BA enthalpy represents the total potential energy of the BAequations, and hence that the sum v2/2 + hba must represent the relevant conservativeenergy quantity for the BA system, which is demonstrated in the following. The BAequations therefore behave as the hydrostatic compressible equations, for which the sumv2/2 + h is also the relevant conserved energy quantity, as is well known. This exactcorrespondence can be viewed as an illustration of the duality between the Boussinesqand non Boussinesq equations established by de Szoeke & Samelson (2002).

2.8. Consequence for energy conservation and “heat”-related quantities

We now turn to the problem of clarifying the nature of the BA approximation to “heat”-related quantities. To that end, let us first write down the evolution equations for thespecific internal energy and enthalpy. Thus, combining (2.33) and (2.34) with the evolu-tion equations (2.3) and (2.4) for η and S respectively, one obtains

ρ0DebaDt

= ρ0[q + µS] + ρ0(Z − Z0)Db

Dt, (2.40)

ρ0DhbaDt

= ρ0

[q + µS

]− ρ0b

DZ

Dt. (2.41)

From the above argument, we anticipate that hba is the relevant total potential energyfor the BA equations, and hence that v2/2 + hba is the corresponding total energy. By


combining either one of (2.40) or (2.41) with the mechanical energy equation (2.13), thetwo following equivalent evolution equations for the total energy Et = v2/2+eba−b(Z−Z0) = v2 + hba are obtained:

ρ0D

Dt

[v2

2− b(Z − Z0) + eba

]+∇ · [δPv − ρ0Fke] = ρ0

[q + µS − εK

], (2.42)

ρ0D

Dt

[v2

2+ hba

]+∇ · [δPv − ρ0Fke] = ρ0

[q + µS − εK

]. (2.43)

Now, in order for the BA system to be energetically consistent, it remains to verify that(2.42) and (2.43) are consistent with the principle of energy conservation. This is easilyshown to be the case only if the right-hand side of (2.42) is the divergence of some energyflux Fq, i.e.,

ρ0(q + µS − εK) = −∇ · (ρ0Fq). (2.44)

As we show now, this imposes a constraint on the form of the entropy flux Fη andirreversible entropy production ηirr in (2.3). Assuming salt to be conservative quantity,and hence such that ρ0S = −∇ · (ρ0FS) for some salt flux FS , implies for the evolutionequation of specific entropy:

ρ0DηbaDt

=ρ0q

T=ρ0εK + µ∇ · (ρ0FS)−∇ · (ρ0Fq)

T

=ρ0εKT−∇ ·

{ρ0(Fq − µFS)

T

}+ ρ0

[Fq · ∇

(1

T

)− FS · ∇

(µ

T

)]. (2.45)

The latter expression establishes that the entropy flux Fη and irreversible entropy pro-duction ηirr in (2.3) must be related to the salt flux FS and enthalpy/internal energyflux Fq by:

Fη =Fq − µFS

T, (2.46)

ηirr =εKT

+ Fq · ∇(

1

T

)− FS · ∇

(µ

T

). (2.47)

In non-equilibrium thermodynamics, this is usually the point at which the second law ofthermodynamics is invoked to further constrain Fq and FS , in order to guarantee thatηirr > 0, e.g., see de Groot & Mazur (1962). Going back to the evolution equation forinternal energy and enthalpy, note that (2.44) implies:

ρ0DebaDt

= −∇ · (ρ0Fq) + ρ0εK + ρ0(Z − Z0)Db

Dt, (2.48)

ρ0DhbaDt

= −∇ · (ρ0Fq) + ρ0εK − ρ0bDZ

Dt, (2.49)

so that Fq appears as the diffusive flux of internal energy or enthalpy. Both equationscan be regarded as equivalent forms of the first law of thermodynamics.

2.9. Alternative forms of the first law of thermodynamics

Although entropy is perhaps the most natural measure of heat, the fact that it increasesirreversibly upon mixing makes it strongly nonconservative. In contrast, temperature orenthalpy mix more linearly, hence are more conservative, and therefore of more practicalinterest. This motivates examining the implication of the above results for the derivationof consistent evolution equations for various temperature variables. To that end, we need

12 R. Tailleux

to relate temperature and entropy. From the differential of the Gibbs function

dgba = −ηdT + µdS − bdZ,

the Maxwell relationships provide the following partial derivatives for the entropy:

∂η

∂S= − ∂µ

∂T,

∂η

∂Z=

∂b

∂T= ρ0g0

∂υ

∂T= g0α,

by defining α = ρ0∂υ/∂T as the relevant definition of the thermal expansion coefficient.Using the well known result that ∂η/∂T = cpdT/T , this implies that we can write:

dη =cpT

dT − ∂µ

∂TdS + g0αdZ (2.50)

As a result, it follows that the temperature equation can be written as:

DT

Dt=T

cp

Dη

Dt+T

cp

∂µ

∂T

DS

Dt− αg0T

cp

DZ

Dt. (2.51)

2.9.1. Simplifications in absence of salinity

Some of the general principles affecting temperature variables are most easily under-stood by first discarding temporarily the effects of salinity, in which case the aboveequation simplifies to

DT

Dt=∇ · (κcp∇T )

cp+εKcp− αg0T

cp

DZ

Dt. (2.52)

Here, we assumed that the diabatic effects modifying entropy exclusive arise from stan-dard molecular diffusion given by the classical Fourier law ∇· (ρ0Fq) = −∇· (κρ0cp∇T ).This shows that the evolution equation for in-situ temperature in general possesses:

(a) a term related to molecular diffusion, that is general not-conservative, i.e., it cannotbe expressed as the divergence of a flux because cp is not constant;

(b) it usually incorporate the Joule heating due to viscous dissipation; c) it possessesa term related to change in pressure, which some authors, e.g. Pons & Le Quere (2005,2007), call the “piston effect”, and the resultant Boussinesq equations, the thermody-namic Boussinesq equations.

2.9.2. Accounting for compressibility effects

In the atmospheric and oceanic practice, compressibility effects are traditionally elim-inated by constructing evolution equations for potential temperature θ or conservativetemperature Θ respectively. By definition, θ is the temperature that a parcel with temper-ature T at a given geopotential height Z would have if lifted adiabatically to a referencelevel Z = Z0, usually taken at sea level. In the ocean, potential temperature is thusdefined as the implicit solution of the following equation

ηba(T, S, Z) = ηba(θ, S, Z0), (2.53)

which simply states the equality of the entropy of the fluid parcel considered at the twolevels Z and Z0. Differentiating (2.53) yields

dηba =cpT

dT − ∂µ

∂TdS + αg0dZ =

crpθ

dθ − ∂µr

∂θdS, (2.54)

where we defined crp = cp(θ, S, Z0) and µr = µ(θ, S, Z0). This expression shows that it ispossible to rewrite the evolution equation for entropy as follows:

crpθ

Dθ

Dt=∂µ

∂θ

DS

Dt+DηbaDt

(2.55)


In absence of salinity, we can write:

crpθ

Dθ

Dt=∇ · (κcp∇T )

T+εKT. (2.56)

This can be transformed in the following equation for θ,

Dθ

Dt= ∇ ·

(κcpθ∇TcrPT

)+ θirr, (2.57)

where the nonconservative production/destruction of θ is given by:

θirr = −κcp∇T · ∇(

θ

crpT

)+θεKcrpT

. (2.58)

In the literature, θ is often treated as a conservative variable, which consists in neglectingthe nonconservative term θirr. Alternatively, one may remark that the equation for θ canalso be written as:

crpDθ

Dt=DhθDt

, (2.59)

where hθ(θ) = hb(η, 0) is the enthalpy a parcel would have if moved adiabatically fromZ to Z = 0. For this reason, hθ was called “potential enthalpy” by McDougall (2003).This allows one to defined a new temperature variable Θ, also conserved for adiabaticmotions, such that: dhθ = c0pdΘ, where c0p is an arbitrarily defined specific heat capacityrepresentative of cp at Z = 0. McDougall (2003) discusses a possible choice for c0p in theoceanic context. As a result, it is possible to write the above equation as:

DΘ

Dt=

1

c0p

DhθDt

= ∇ ·(κcpθ∇Tc0pT

)+ Θirr, (2.60)

where this time, the nonconservative production/destruction of Θ is given by:

Θirr = −κcpc0p∇T · ∇

(θ

T

)+θεKT

. (2.61)

McDougall (2003) shows that the globally averaged nonconservative term Θirr is sig-nificantly smaller than θirr for the present-day distribution of temperature and salinityfields. It is important to note, however, that Θirr is not necessarily smaller than θirr whenestimated locally, as the relative ordering of the two terms depends on the particular fluidconsidered and on the degree of turbulence experienced by the fluid.

3. Thermodynamics and energetics of a Boussinesq fluid with alinear equation of state

3.1. Standard (energetically inconsistent) Boussinesq model

In this section, we seek to illustrate the above ideas for a Boussinesq fluid with an idealisedlinear equation of state, which is governed by the following standard equations:

Dv

Dt+

1

ρ0∇δP = −g0(ρ− ρ0)

ρ0k + ν∇2v, (3.1)

∇ · v = 0, (3.2)

DT

Dt= κ∇2T, (3.3)

14 R. Tailleux

ρ = ρ0 [1− α(T − T0)] , (3.4)

where v = (u, v, w) is the three-dimensional velocity field, δP = P − P0 is the pressureanomaly defined relative to the reference Boussinesq pressure P0 = −ρ0g0z, ρ is thedensity, T is the temperature, ρ0 and T0 are reference constant density and temperature,k is the unit normal vector pointing upwards in the direction opposite to gravity, and g0is a nominal value of the acceleration of gravity. An implicit assumption in obtaining thetemperature equation is that the heat capacity cp0 is constant.

As should be clear from the arguments developed above, the above Boussinesq model isneither energetically nor thermodynamically consistent when molecular diffusive effectsare retained. Yet, it has nevertheless been extensively used in recent theoretical discussionof the energetics of horizontal convection, e.g., Paparella & Young (2002), Wang &Huang (2005), Winters & Young (2009), Tailleux (2010b); Tailleux & Rouleau (2010),as well as in discussing the energetics of turbulent mixing in stratified fluids by Winters& al (1995). Moreover, such a model also forms the basis for numerous direct numericalsimulations of stratified turbulence, in the sense that in such studies, both κ and νare usually interpreted as representing the molecular values of diffusivity and viscosityrespectively. The following seeks to clarify the nature of the difficulties involved, anddiscuss how to resolve them.

3.2. Equation of state linear in (in-situ) temperature

From a thermodynamic viewpoint, it is not immediately obvious whether one shouldinterpret the temperature variable T in (3.4) as in-situ temperature or potential temper-ature. On the one hand, the fact that T is materially conserved in absence of diffusiveeffects suggests the latter interpretation, whereas the fact that molecular diffusion actsin homogenising T suggests the former interpretation. In the following, the thermody-namic implications of each interpretation are considered in turn, in order to establishwhether apparent innocuous differences in interpretation matter. We first discuss theinterpretation of T as in-situ temperature, in which case the expression for buoyancybecomes:

b = αg0(T − T0). (3.5)

Physically, assuming an equation of state linear in in-situ temperature is equivalent toassume that isothermal compressibility, rather than adiabatic incompressibility, is zero.From Appendix A, this appears to implicitly assume a finite but negative speed of sound,which is unphysical. This negative speed of sound, however, does not appear to control theevolution of pressure, which from a mathematical viewpoint is determined by a nonlocalelliptic operator.

The interpretation of T as in-situ temperature suggests describing the thermodynamicproperties of the fluid in terms of the specific Gibbs function gba, whose natural variablesare T and Z. The partial differential relations determining gba are

∂gba∂T

= −ηba,∂gba∂Z

= −b, (3.6)

cp0 = −T ∂2gba∂T 2

, (3.7)

which can easily be integrated to yield

gba = −cp0 [T ln (T/T0)− T ]− αg0(Z − Z0)(T − T0), (3.8)

up to some arbitrary constant of integration. From (3.6), the following expression for the


specific entropy is obtained:

ηba = −∂gba∂T

= cp0 ln

(T

T0

)+ αg0(Z − Z0). (3.9)

This makes it possible to derive an exact expression for the potential temperature θ,which is the implicit solution of the equation ηba(T,Z) = ηba(θ, 0), i.e., cp0 ln (T/T0) +αg0(Z − Z0) = cp0 ln (θ/T0)− αg0Z0, yielding:

θ = T exp

{αgZ

cp0

}(3.10)

This makes it clear that in order for a model to be thermodynamically consistent, po-tential temperature is always different from the in-situ temperature. Now, we previ-ously established that the expressions for internal energy and enthalpy were given by:eba = gba + b(Z −Z0) + Tηba and hba = eba − b(Z −Z0) = gba + Tηba, yielding thereforethe following expressions:

hba = cp0T + αg0T0(Z − Z0), (3.11)

eba = cp0

[1 +

αg0(Z − Z0)

cp0

]T. (3.12)

These relations can be written in terms of other variables by using the expression for θ, aswell as b. For instance, the enthalpy can be written in terms of the potential temperatureand Z as follows:

hba = cp0θ exp

{−αg0Z

cp0

}+ αg0T0(Z − Z0). (3.13)

eba = cp0

[1− αg0Z0

cp0+ ln

{αg0θ

b+ αg0T0

}](T0 +

b

αg0

)(3.14)

Thus, despite the extreme simplicity of the equation of state, the Boussinesq fluid ap-pears to have a non-trivial underlying thermodynamic structure. As the following shows,interpreting T as potential temperature introduces significant changes in the expressionsfor enthalpy and internal energy.

3.3. Improvement of the model energetic and thermodynamic consistency

Having clarified the form of the thermodynamic potentials, we now show how to modifythe above Boussinesq model to make it energetically and thermodynamically consistent.This requires two main changes, namely modifying the heat equation and retaining thedistinction between potential temperature θ and in-situ temperature T . The modifiedBoussinesq model thus becomes:

Dv

Dt+

1

ρ0∇δP = bk + ν∇2v (3.15)

b = αg0(T − T0) (3.16)

∇ · v = 0 (3.17)

Dθ

Dt=θ

Tκ∇2T +

θεKcp0T

(3.18)

T = θ exp

{−αg0Z

cp0

}(3.19)

16 R. Tailleux

Now, (3.18) shows that the potential temperature θ is materially conserved in absenceof molecular diffusive and viscous effects, while the molecular heat flux is now downgradient of the in-situ temperature, as requested by the theory of non-equilibrium ther-modynamics, e.g., de Groot & Mazur (1962). Alternatively, the temperature equation(3.18) can also be written in the form

Dθ

Dt= ∇ ·

(κθ

T∇T)

+ θirr +θεKcp0T

, (3.20)

obtained by entering the term θ/T within the divergence operator, where

θirr = −κ∇T · ∇(θ

T

)= −καg0

cp0

θ

T

∂T

∂z= −καg0

cp0

∂θ

∂z+κα2g20c2p0

θ. (3.21)

From a practical viewpoint, it is useful to note that the the molecular diffusive term canbe written entirely in terms of the potential temperature, using the result that

κθ

T∇T = κ∇θ − καg0θ

cp0k, (3.22)

assuming for simplicity Z(z) = z, as is always done in practice. Interestingly, theseequations introduce the following length scale: L = cp0/(αg0). For typical values, cp0 =4.103J.kg−1.K−1, α = 10−4 K−1 and g0 = 10 m.s−1, which yields: L = 4.106 m. As Lis huge compared to the typical length scales at which molecular diffusion is important,it follows that the nonconservative term is probably negligible. Nevertheless, the addedterms are simple enough that they can be easily be added to existing numerical models,since there exist a simple analytical expressions for them.

3.4. Equation of state linear in potential temperature

In this section, we revisit the above results by now assuming the equation of state tobe linear in potential temperature, rather than in in-situ temperature, and hence thatthe adiabatic compressibility is zero, as is generally implicitly assumed in traditional lowMach number asymptotics. In this case, the buoyancy becomes:

b = αθg0(θ − θ0), (3.23)

where αθ = αT/θ is the isentropic thermal expansion (see Eq. (A11) in Tailleux (2010)assuming constant cp0). The use of θ (which is closely related to entropy by dηba =cp0dθ/θ) suggests deriving the thermodynamic properties of the fluid from the specificenthalpy, whose natural variables are ηba and Z, and whose total differential is:

dhba = Tdηba − bdZ =Tcp0θ

dθ − bdZ. (3.24)

The partial derivatives of hba with respect to θ and Z yields the following system ofpartial differential equations:

∂hba∂θ

=cp0T

θ,

∂hba∂Z

= −b = −αθg0(θ − θ0). (3.25)

It is easily verified that ∂2hba/∂Z∂θ = ∂2hba/∂θ∂Z = −αθg0, and hence that the system(3.25) can be integrated to yield the following expression between T and θ

T

θ= 1− αθg0Z

cp0, (3.26)


using the fact that by definition, T = θ at Z = 0. By inserting this result into theequation for ∂hba/∂θ in (3.25), straightforward integration yields:

hba = cp0

[1− αθg0Z

cp0

](θ − θ0) =

(θ − θ0θ

)cp0T, (3.27)

up to an arbitrary constant of integration. The expression for the internal energy istherefore given by:

eba = hba + b(Z − Z0) = [cp0 − αθg0Z0](θ − θ0). (3.28)

Since Z0 is constant, (3.28) shows that in the present model, internal energy is linearlyproportional to potential temperature, and hence that θ is a direct measure of internalenergy. Moreover, it is useful to note that owing to cp0 being constant, conservativetemperature Θ can be defined to coincide with θ.

The alterations to the heat equation needed to make the Boussinesq model energeticallyand thermodynamically consistent for an equation of state linear in θ are the same as foran equation of state linear in T , so that the only difference with the previous case is thedifferent expression linking T and θ, which is now given by (3.26) in place of (3.19). Insummary,

Dθ

Dt=θ

Tκ∇2T +

θεKcp0T

, (3.29)

T = θ

[1− αθg0Z

cp0

]. (3.30)

As previously, it is possible to manipulate the diffusive term to rewrite the heat equationso as to introduce a nonconservative production/destruction of θ as follows:

Dθ

Dt= ∇ ·

(κθ

T∇T)

+ θirr +θεKcp0T

(3.31)

where

θirr = −κ∇T · ∇(θ

T

)= − καθg0/cp0

[1− αθg0Z/cp0]2

∂T

∂z= −καθg0

cp0

(θ

T

)2∂T

∂z. (3.32)

Moreover, it is also possible to write the temperature equation entirely in terms of poten-tial temperature if desired, at the price of introducing some additional depth-dependentterms. Although the proposed corrections are arguably small in liquids such as wateror seawater, they may start having a detectable effect on temperature fluctuations in astrongly turbulent fluid in the limit of very weak stratification, which is rarely studiedin the literature. In any case, the proposed corrections should be straightforward to im-plement in existing numerical Boussinesq models, as well as easily adapted to the caseof an arbitrary nonlinear equation of state.

4. Summary and discussion

Although the concept of an “incompressible fluid” as used in the Boussinesq approx-imation takes its origin in a thermodynamic definition of incompressibility (associatedwith zero adiabatic compressibility, or equivalently with an infinite speed of sound), themodern acceptation of the term has become more generally associated with the kine-matic view of incompressibility associated with the use of a divergence-free vector field.While the thermodynamic and kinematic definitions of incompressibility coincide for anequation of state that depends only on potential temperature or entropy, in absence of

18 R. Tailleux

diabatic effects, they do not otherwise, raising the question of which is best appropriateto understand the energetics of a Boussinesq fluid in the most general case. In this paper,we showed that the key to understand this issue is to recognise that the term gzDρ/Dtthat enters the energetics of the Boussinesq approximation, which differs from zero if ρdepends on pressure and/or in presence of diabatic effects, should be regarded as theBoussinesq approximation to the thermodynamic work of expansion/contraction. Whilethis has been recognised previously by a number of authors, e.g., Winters & al (1995),Wang & Huang (2005) among others, the novelty here is to show how to use Maxwellrelationships to reconstruct consistent approximations of all thermodynamic propertiesof the fluid, and to determine corrections to the heat equation to ensure global ther-modynamic and energetic consistency. How to apply these ideas was illustrated in theparticular case of the extensively used Boussinesq model, for which we constructed ex-plicit expressions for the internal energy and enthalpy. One key result is to show thatthe conservative energy quantity of the Boussinesq model is the sum of kinetic energyplus the Boussinesq enthalpy, similarly as for the compressible hydrostatic Navier Stokesequations. Another important result is that the gravitational potential energy can beviewed as a thermodynamic property of the fluid, which is equal to the difference be-tween the Boussinesq enthalpy and Boussinesq internal energy. The proposed methoddoes not appear to work for the anelastic system of equations, for which a fully con-sistent formulation that would also retain hydrostatic modes of motion remains to bederived.

In contrast to the compressible Navier-Stokes (CNS) equations, in which the thermo-dynamic work of expansion/contraction represents a conversion between internal energyand kinetic energy, the term gzDρ/Dt is more naturally interpreted as a conversion be-tween internal energy and gravitational potential energy in the Boussinesq model. Forthe Boussinesq energetics to be traceable to that of the CNS, it is therefore necessaryto assume that the term gzDρ/Dt actually represents two successive energy conversions,one that first converts internal energy into kinetic energy, followed by one of equal mag-nitude that converts kinetic energy into gravitational potential energy. In some recentpapers by Tailleux (2009, 2013), we argued on the basis of a rigorous analysis of theenergetics of the CNS that compressibility effects and conversions with internal energyshould be regarded as important as conversions between kinetic energy and gravitationalpotential energy in a turbulent stratified fluid. We argued in particular that the observednet increase in GPE over a turbulent mixing event should be regarded as occurring atthe expense of internal energy, rather than at the expense of kinetic energy, in contrastto what is usually assumed following Winters & al (1995)’s interpretation of Boussinesqenergetics. Physically, this can be understood from the fact that turbulence greatly in-creases local values of the molecular diffusive term κ∇2ρ, and hence of the compressiblework gzDρ/Dt = gzκ∇2ρ, resulting in enhanced local transfers between internal energyand gravitational potential energy.

Our results strongly support the view that Boussinesq models can support large andleading order conversions between internal energy and mechanical energy when molec-ular diffusive effects are retained. As a result, while such models can still be regardedas “incompressible” from a kinematic viewpoint, they need to be regarded as thermo-dynamically compressible from the purposes of correctly understanding and interpretingtheir energetics, which is still largely unappreciated. The hope is that by providing anexplicit way to construct thermodynamic potentials, as well as the corrections neededto make Boussinesq models energetically and thermodynamically consistent, it becomeseasier to understand how to develop interpretations of Boussinesq energetics that aretraceable to that of the compressible Navier-Stokes equations.


Appendix A. Proof that the density dependence upon pressure cannever be fully eliminated for general diabatic motions

The purpose of this appendix is to prove that the original concept of an incompressiblefluid historically defined as one whose density is not affected by changes in pressure (asstated in Batchelor (1967) or Lilly (1996) for instance) is physically meaningful for adi-abatic motions only, but otherwise a thermodynamic impossibility for general diabaticmotions. To see this, recall that the way the density of actual fluids is affected by pressurechanges depends sensitively on the nature of the thermodynamic transformation expe-rienced by the fluid parcels. In thermodynamics, the degree to which density is affectedby pressure changes is generally quantified by means of the coefficient of compressibilityρ−1∂ρ/∂P , where the partial derivative is computed along the particular thermodynamicpathway considered. In general, however, the coefficients of compressibility defined for dif-ferent thermodynamic pathways are not independent of each other. For instance, classicalthermodynamic arguments show that the isothermal compressibility γ = ρ−1∂ρ/∂P |T islinked to the adiabatic compressibility ρ−1∂ρ/∂P |η = 1/(ρc2s) by:

γ =1

ρ

∂P

∂ρ

∣∣∣∣T,S

=1

ρc2s+α2T

ρcp=

1

ρc2s+ αΓ, (A 1)

where Γ = αT/(ρcp) is the adiabatic lapse rate. It follows immediately that setting upthe adiabatic compressibility to zero (by taking the zero Mach number limit cs = +∞),while effectively making the fluid “incompressible” for adiabatic motions, fails to do sofor general diabatic transformations (this assumes that the thermal expansion coefficientα remains finite, as is normally the case for most fluids). Note that for seawater, typicalvalues are: 1/(ρc2s) ≈ 4.10−10 Pa

−1, whereas αΓ = α2T/(ρcp) ≈ 7.5 × 10−13 Pa−1, using

cs = 1500 m.s−1, α = 10−4 K−1, cp = 4.103 J.K−1.kg−1, and ρ = 103 kg.m−3. For thesevalues, the limit cs = +∞ decreases the isothermal compressibility by about two to threeorders of magnitude, so that even if it fails to fully eliminate compressibility effects fornon-adiabatic motions, it appears nevertheless capable of reducing them considerably.This result helps to understand a general conclusion of the present study that any ther-modynamically consistent formula must distinguish between the potential temperatureand in-situ temperature.

REFERENCES

Alberty, R. A. 2001 Use of Legendre transforms in chemical thermodynamics. Pure Appl.Chem. 73, 1349–1380.

Bannon, P.R. 1996 On the anelastic approximation for a compressible atmosphere. J. Atm.Sciences 53, 3618–3628.

Batchelor, G.K. 1967 An introduction to Fluid Dynamics. Cambridge University Press. 615pp.

Boussinesq, J. 1903 Theorie analytique de la chaleur. Vol 2. Gauthier-Villars, Paris.

Callen, H.B. 1985 Thermodynamics and an introduction to thermostatistics. Wiley. 493 pp.

Davies, T., Staniforth, A., Wood, N. & Thuburn, J. 2003 Validity of anelastic and otherequation sets as inferred from normal-mode analysis. Q. J. Roy. Meteorol. Soc. 129, 2761–2775.

de Groot, S.R. & Mazur, P. 1962 Non-equilibrium thermodynamics. North Holland Pub-lishers.

de Szoeke, R. A. & Samelson, R. M. 2002 The duality between Boussinesq and non-Boussinesq hydrostatic equations of motion. J. Phys. Oceanogr. 30, 2194–2203.

Durran, D. R. 1989 Improving the anelastic approximation. J. Atmos. Sci. 46, 1453–1461.

20 R. Tailleux

Feistel, R. 2003 A new extended Gibbs thermodynamic potential of seawater. Prog. Oceanogr.58, 43–114.

Huang, R. X. 2005 Available potential energy in the world’s oceans. J. Mar. Res. 63, 141–158.

Ingersoll, A. P. & Pollard, D. 1982 Motions in the interiors and atmospheres of Jupiterand Saturn: Scale analysis, anelastic equations, barotropic stability criterion. Icarus 52,62–80.

Ingersoll, A. 2005 Boussinesq and anelastic approximations revisited: Potential energy releaseduring thermobaric instability. J. Phys. Oceanogr. 35, 1359–1369.

IOC, SCOR and IAPSO 2010 The international thermodynamic equation of seawater - 2010:Calculation and use of thermodynamic properties. Intergovernmental Oceanogrphic Com-mission, Manuals and Guides No. 56, UNESCO (English), 196 pp.

Klein, R. 2009 Asymptotics, structure, and integration of sound-proof atmospheric flow equa-tions. Theor. & Comput. Fluid Dyn. 23, 161–195.

Klein, R. 2010 Scale-dependent models for atmospheric flows. Annu. Rev. Fluid Mech. 42,249–274.

Lipps, F. B. & Hemler, R. S. 1982 A scale analysis of deep moist convection and some relatednumerical calculations. J. Atmos. Sci. 39, 2192–2210.

Lilly, D.K. 1996 A comparison of incompressible, anelastic and Boussinesq dynamics. Atmos.Res., 40, 143–151.

McDougall, T.J. 2003 Potential enthalpy: A conservative oceanic variable for evaluating heatcontent and heat fluxes. J. Phys. Oceanogr. 33, 945–963.

McIntyre, M. E. 2010 On spontaneous imbalance and ocean turbulence: generalizations of thePaparella-young epsilon theorem. In Turbulence in the Atmosphere and Oceans. Proc. In-ternational IUTAM/Newton Inst. Workshop held 8-12 December 2008, ed. D. G. Dritschel,Springer-Verlag.

Muller, B. 1998 Low-Mach number aymptotics of the Navier-Stokes equations J. EngineeringMath. 34, 97–109.

Nycander, J., Nilsson, J., Doos, & Brostrom, G. 2007 Thermodynamic analysis of theocean circulation. J. Phys. Oceanogr. 37, 2038–2052.

Nycander, J.2010 Horizontal convection with a nonlinear equation of state: generalization ofa theorem of Paparella and Young. Tellus 62A, 134–137.

Oberbeck, A. 1879 Uber die Warmeleitung der Flussigkeiten bei Berucksichtigung derStromungen infolge vor Temperaturedifferenzen (On the thermal conduction of liquidstaking account flows due to temperature differences). Ann. Phys. Chem., Neue Folge. 7,271–292.

Ogura, Y. & Phillips, N. A. 1962 Scale analysis of deep and shallow convection in theatmosphere. J. Atmos. Sci. 19, 173–179.

Paparella, F. & Young, W. R. 2002 Horizontal convection is non turbulent. J. Fluid Mech.466, 205-214.

Pauluis, O. 2007 Sources and sinks of available potential energy in a moist atmosphere. J. Atm.Sc., 64, 2627–2641.

Pauluis, 0. 2008 Thermodynamic consistency of the analestic approximation for a moist atmo-sphere. J. Atm. Sc., 65, 2719–2729.

Pons, M. & P. Le Quere 2005 An example of entropy balance in natural convection. Part 2:the thermodynamic Boussinesq equations. C. R. Mecanique 333, 133–138.

Pons, M. & P. Le Quere 2007 Modeling natural convection with the work of pressure-forces,a thermodynamic necessity. Int. J. Numer. Meth. Heat Fluid Flow 17, 322–332.

Spiegel, E. A. & Veronis, G. 1960 On the Boussinesq approximation for a compressible fluid.Astrophys. Journal 131, 442–447.

Tailleux, R. 2009 On the energetics of stratified turbulent mixing, irreversible thermody-namics, Boussinesq models, and the ocean heat engine controversy. J. Fluid Mech. 638,339–382.

Tailleux, R. & Rouleau, L. 2010 The effect of mechanical stirring on horizontal convection.Tellus A 62, 138–153.

Tailleux, R. 2010 Identifying and quantifying nonconservative energy production/destructionterms in hydrostatic Boussinesq primitive equation models. Ocean Modell., 34, 125–136.


Tailleux, R. 2010 On the buoyancy power input in the oceans energy cycle. Geophys. Res.Lett., 37, L22603, doi:10.1029/2010GL044962.

Tailleux, R. 2013 Irreversible thermodynamic work and available potential energy dissipationin turbulent stratified fluids. Physica Scripta, 014033

Vallis, G.2006 Atmospheric and oceanic fluid dynamics. Cambridge University Press.Vanneste, J. & O. Buhler 2011 Streaming by leaky surface acoustic waves. Proc. R. Soc.

Lond. A, 467, 1779–1800.Wang & Huang, R. X.2005 An experimental study on thermal circulation driven by horizontal

differential heating. J. Fluid Mech. 540, 49–73.Winters, K. B., Lombard, P. N., and Riley, J. J., & d’Asaro (1995) Available potential

energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115–228.Winters, K.B. & Young, W.R.2009 Available potential energy and buoyancy variance in

horizontal convection. J. Fluid Mech. 629, 221–230.Young, W. R.2010 Dynamic enthalpy, conservative temperature, and the seawater Boussinesq

approximation. J. Phys. Oceanogr., 40, 394–400.

kinematic versus thermodynamic incompressibility: which...

Documents