Which does matter for Boussinesq/Anelastic energetics?

Under consideration for publication in J. Fluid Mech.
1
Kinematic versus thermodynamic
incompressibility: Which does matter for
Boussinesq/Anelastic energetics?
´ M I T A I L L E U X1 †
RE
1
Department 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 approximation takes its origin in a thermodynamic definition of incompressibility based on
assuming an infinite speed of sound, the modern acceptation of the term has become
more generally associated with the kinematic definition of incompressibility associated
with the use of a divergence-free velocity field. While thermodynamic and kinematic incompressibility coincide for an equation of state function of entropy alone, in absence of
diabatic effects, they do not otherwise; yet, most of the literature on Boussinesq energetics
seems to assume that kinematic incompressibility implies thermodynamic incompressibility, and hence that conversions with internal energy are irrelevant for understanding the
energetics of turbulent stratified mixing for instance. This paper shows that the key to
understanding 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 compressible
work of expansion/contraction, which appears in the guise of apparent changes in gravitational potential energy due to apparent changes of mass. Consistent approximations to
the “heat” can then be constructed by requiring that the Maxwell relationships be satisfied, which ultimately leads to well-defined approximations to the internal energy and
other thermodynamic potentials. This approach is illustrated by constructing explicit
expressions for the internal energy and enthalpy for a Boussinesq fluid with a linear
equation of state, and by showing how to correct existing models to make them fully
energetically and thermodynamically consistent. In such models, the conservative energy
quantity is the sum of kinetic energy and enthalpy, and gravitational potential energy is
a purely thermodynamic quantity that is the difference between enthalpy and internal
energy. Moreover, such models treat potential and in-situ temperatures as fundamentally
distinct quantities. The overall conclusion is that kinematically incompressible Boussinesq models can support large compressible effects and leading order conversions between
internal energy and mechanical energy, in contrast to what is usually assumed, which is
important 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 characterised by low Mach number (M = U0 /cs ) and small relative density variations (ρ −
† Present address: Department of Meteorology, University of Reading, Earley Gate, PO Box
243, 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 density
and ρ0 a reference density. It has been common practice over the past century to regard
such flows as incompressible or weakly compressible. There appears to be some confusion, however, about what an “incompressible fluid” actually refers to. Historically, the
concept was originally introduced based on a thermodynamic definition of incompressibility. Indeed, according to Batchelor (1967) “a fluid is said to be incompressible when
the density of an element of fluid is not affected by changes in the pressure”, see also
Lilly (1996). In absence of adiabatic effects, the implication is that the density ρ must
be constant following fluid parcels, i.e., satisfy Dρ/Dt = 0. From mass conservation
Dρ/Dt + ρ∇ · v = 0, this implies in turn that ∇ · v = 0 and hence that the fluid velocity field must be divergentless. From a kinematic viewpoint, a divergence-free velocity
field implies that the volume element dV is conserved following fluid parcels, which represents a kinematic definition of incompressibilty. Nowadays, however, the terminology
“incompressible approximation” is commonly employed to refer to system of equations
describing fluids affected both by diabatic effects and having a pressure-dependent density, 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 incompressibility. Clearly, the original meaning of “incompressible” as based on Batchelor
(1967) and Lilly (1996) thermodynamic definition seems to have been lost over time to
become exclusively associated to a fluid with a divergentless velocity field. One objective
of this paper is to convince the reader that this shift in meaning complicates the correct
interpretation of the energetics of “incompressible” approximations, given that compressible effects turn out to be often large and to occur at leading order when diabatic effects
and the density dependence on pressure are retained. The simple solution proposed in
this paper is to show that the changes in gravitational potential energy caused by the
changes in mass due to diabatic effects and the pressure dependence of density can be
viewed 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 the
Mach number and buoyancy parameter, among others, to develop sound-proof sets of
equations that offer a number of practical advantages, computational or analytical, over
the fully compressible Navier-Stokes equations. Two particular classes of approximations
have been particularly influential and key to simplifying the numerical and theoretical
analysis of low Mach number fluid flows, and will be under focus in this paper. The
first one is the Oberbeck-Boussinesq approximation (after Oberbeck (1879) and Boussinesq (1903)), which in its most common form retains only the rotational divergence free
component of the velocity field, and treats the density as constant everywhere except
where it multiplies the acceleration of gravity. Spiegel & Veronis (1960) show that under
some circumstances, the pressure dependence of a compressible fluid can nevertheless
be neglected relative to the temperature dependence, so that formally the Boussinesq
approximation formally reduces to that for an incompressible fluid. The second one is
the 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, for
instance by using low Mach number asymptotics and multi-scale expansion techniques,
e.g., M¨
uller (1998); Klein (2009, 2010). Davies & et al. (2003) reviews of a number of commonly employed reduced sound-proof sets of equations, and show how they
respectively 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 of
the coupling between mechanical energy and internal energy (the dynamics/thermodynamics
coupling) in the Boussinesq/anelastic approximations. For adiabatic motions and a linearised equation of state, such approximations are known to decouple either fully or
almost 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 unavoidable, however, when diabatic and/or a nonlinear equation of state are retained, so that
the issue of the energetic and thermodynamic consistency of the Boussinesq/anelastic
approximations become less clear in that case. The oceanographic case is instructive in
this respect, as oceanographers have used the Boussinesq hydrostatic approximation in
conjunction with a realistic nonlinear equation of state (including the pressure dependence) for many decades as the basis for numerical ocean general circulation models of
the kind used in climate studies, without any apparent obvious drawbacks apart from
the lack of a well-defined and closed energy budget, see Tailleux (2010). Since Boussinesq ocean models appear to work well with a “compressible” equation of state, one may
wonder 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 compressible effects in weakly compressible turbulent stratified fluids. The first misconception is
that thermodynamics/dynamics coupling should be small in weakly compressible fluids
irrespective of the particular thermodynamic transformations undergone by fluid parcels,
whereas weak coupling is theoretically justified only for adiabatic motions and fluids with
an equation of state independent of pressure. The second misconception, which is related
to 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 = 0
or ∇ · (ρ0 v) = 0, despite the fact that what physically determines the magnitude of the
compressibility effects and the thermodynamics/dynamics coupling is the nature of the
diabatic effects and nonlinearities of the equation of state.
Based on the above arguments, the new view that the present paper aims to promote is
that the Boussinesq and anelastic approximations are capable of supporting potentially
large compressible effects and conversions between internal energy and mechanical energy.
As this is in stark contrast from the conventional view, this immediately raises several
questions: 1) Assuming that a Boussinesq/anelastic fluid can indeed describe compressible effects and conversions between internal energy and mechanical energy, how do they
compare with that taking place in a fully compressible fluid? Does the answer depend
on 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 second
laws of thermodynamics? If so, what is the form of the thermodynamic potentials supported by such approximations, and how do they differ from that of a fully compressible
fluid? 3) Is it an intrinsic problem that many Boussinesq and anelastic models fail to
be energetically and thermodynamically consistent, or can such models be modified to
correct the problem? In the latter case, can this be done without modifying the formal
order 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 that
the Boussinesq and anelastic approximations can be endowed with well defined energetics
and 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 is
some precedent to the present ideas. Thus, Young (2010) showed, using ideas previously
developed by Ingersoll (2005), that the seawater Boussinesq equation have a well-defined
conserved energy quantity for an arbitrary nonlinear equation of state and for adiabatic
motions, 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 such
as moist air, also with a highly nonlinear equation of state, and discussed energy and
thermodynamic consistency issues. None of the above studies, however, clearly addressed
the nature of the conversions between mechanical energy and internal energy, which
are addressed here in significantly more details. Section 2 provides the general theory.
Section 3 applies the result to elucidating the thermodynamics of a Boussinesq fluid
with a linear equation of state that has been widely used recently in numerical study of
turbulent mixing, as well as in the context of horizontal convection. Section 4 summarises
and discusses some implications of the results.
2. Thermodynamically consistent and inconsistent
Boussinesq/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 previously
considered by Ingersoll (2005) and Pauluis (2008), which is sufficiently general to support
both diabatic effects and an arbitrary nonlinear equation of state, viz,
δP
ρ − ρ0
1
Dv
+∇·
=−
g0 ∇Z + ∇ · S,
(2.1)
Dt
ρ0
ρ0
ρ0
∇ · (ρ0 v) = 0,
(2.2)
Dη
q˙
1
= η˙ =
= − ∇ · (ρ0 Fη ) + η˙ irr ,
Dt
T
ρ0
(2.3)
1
DS
= S˙ = − ∇ · (ρ0 FS ),
Dt
ρ0
(2.4)
ρ = ρ(S, η, P0 ).
(2.5)
Moreover, it is easily verified that the above system reduces to the familiar Boussinesq
approximation in the case where ρ0 = constant. Thereafter, we refer to (2.1-2.5) as the
Boussinesq-Anelastic(BA) system, where η is the specific entropy, S is salinity, g0 is a
constant acceleration of gravity, Z is the geopotential height, so that g0 Z = Φ is the
geopotential, and S is the stress tensor. The terms η˙ = q/T
˙
and S˙ are used as short-hand
to denote the diabatic effects affecting η and S. Explicit expressions for the latter can be
obtained 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 is
the one that is the most commonly encountered in the literature, but that it differs from
the definition b = g0 (υ − υ0 )/υ0 employed by Pauluis (2008) and Young (2010). An
important result of this paper will be to establish that the latter definition yields a more
Energetics and thermodynamics of the Boussinesq/Anelastic approximations
5
accurate momentum and energy budgets, and hence that its use is preferable to achieve
maximum 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 hence
to satisfy ρ−1
0 ∂P0 /∂z = −g0 ∂Z/∂z. However, Pauluis found it impossible to simultaneously achieve energetic consistency and hydrostatic balance, and therefore proposed to
alter the latter so that P0 and ρ0 satisfy the following balance instead
∂Z
∂ P0
= −g0
.
(2.6)
∂z ρ0
∂z
It is easy to see that (2.6) can be immediately integrated to yield P0 = Pa − ρ0 gZ, where
Pa is the assumed constant and spatially uniform atmospheric pressure at Z = 0. The
reference pressure P0 in Pauluis (2008)’s anelastic model is therefore a hybrid between
the Boussinesq reference pressure defined for a constant ρ0 , and the actual hydrostatic
pressure P0h solution of ∂P0h /∂Z = −ρ0 g0 . To clarify the link between P0 and P0h , note
that we have
ρ0
Dυ0
DZ
DP0h
DP0
D P0
=
+ ρ 0 P0
= ∇ · (P0 v) = −ρ0 g0
=
.
Dt ρ0
Dt
Dt
Dt
Dt
(2.7)
The apparent impossibility to enforce energetics consistency without altering true hydrostatic 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 high
price to pay, warranting a full investigation of its dynamical consequences, perhaps by
using 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 review
some 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 evolution
equation for the kinetic energy, obtained in the usual way by multiplying (2.1) by ρ0 v,
which after some manipulation can be written as follows:
ρ0
D v2
DZ
+ ∇ · (δP v − ρ0 Fke ) = ρ0 b
− ρ0 ε K ,
Dt 2
Dt
(2.8)
where the work against the stress tensor is split as usual as the difference between the
divergence of an energy flux minus viscous dissipation v · ∇S = ∇ · (v · S) − ρ0 K =
2
2
∇ · (ρ0 Fke ) − ρ0 εK . In the special case where ρ−1
0 ∇ · S = ν∇ v, then Fke = ν∇v /2 and
2
2
2
εK = ν(k∇uk + k∇vk + k∇wk ). Physically, the key issue here is whether the term
ρ0 bDZ/Dt in the right-hand side of (2.8) can be rationalised as a conversion between
kinetic energy and potential energy, and if so, what is the nature of the potential energy
implied by such a conversion term. To address this question, the above studies introduced
the following function:
Z Z
h‡ (S, η, Z) = h‡0 (S, η) −
b(S, η, Z 0 )dZ 0 .
(2.9)
Z0
It is easily shown that the substantial derivative of h‡ is given by
Dh‡
DZ
= −b
+ CS S˙ + Cη η,
˙
Dt
Dt
(2.10)
6
R. Tailleux
where CS and Cη are defined by:
Z Z
∂h‡ 0
∂h‡0
−
dZ ,
CS =
∂S
Z0 ∂S
Cη =
∂h‡0
−
∂η
Z
Z
Z0
∂h‡ 0
dZ .
∂η
(2.11)
The important result here is that 2.10 admits the same term bDZ/Dt as appears in the
kinetic energy equation but with an opposite sign, allowing such a term to disappear
upon summing (2.8) and (2.10), which yields
D v2
ρ0
+ h‡ + ∇ · (δP v − ρ0 Fke ) = ρ0 CS S˙ + Cη η˙ − ρ0 εK .
(2.12)
Dt 2
|
{z
}
R
This result is widely regarded as an important advance in the understanding of the
energetics of the Boussinesq and anelastic approximations, for it establishes that such
approximations admit a well defined energy quantity, namely ρ0 (v2 /2 + h‡ ), that is conserved in absence of diabatic and viscous effects even for an arbitrary nonlinear equation
of state. This had long been thought impossible previously.
Nevertheless, the above result only partially address the energetics and thermodynamic
consistency 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 potential
energy or internal energy, making its link to standard energetics unclear. This issue is
partly remedied in Pauluis (2008) and Young (2010) by using the non-standard buoyancy
b = −g0 (υ −υ0 )/υ0 , in which case h‡ can be linked to the usual specific enthalpy, which in
hydrostatic compressible fluids plays an equivalent role as the total potential energy (as
for 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, when
the traditional buoyancy is used instead.
(b) Establishing the existence of a conserved energy quantity in absence of diabatic and
viscous effects is not sufficient to establish the energetic and thermodynamic consistency
of 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, and
hence that it is created/destroyed at the expenses of an unknown energy reservoir to be
identified. Moreover, diabatic and viscous effects act as sources/sinks of entropy, so that
an additional key issue is whether the BA system is consistent with the second law of
thermodynamics.
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 system
by remarking that the approximated forms of kinetic energy and gravitational potential
energy are in general sufficiently close to their exact counterparts that the only real
issue is about understanding how the BA system approximates internal energy and other
thermodynamic potentials. Therefore, instead of focusing on the kinetic energy budget as
discussed above, it seems more natural to construct an evolution equation for the sum of
the 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 linking
the 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” ρ0 dV rather
than the actual mass ρdV , where dV is the elementary volume of the fluid parcel considered. As a result, it is not entirely clear whether the GPE of a fluid parcel should
be defined as ρdV g0 Z or as ρ0 dV g0 Z. Based on the calculations that follow, we find
Energetics and thermodynamics of the Boussinesq/Anelastic approximations
7
that the most relevant definition of mechanical energy Em (per unit volume) for the
BA system is ρ0 Em = ρ0 [v2 /2 − b(Z − Z0 )] = ρ0 v2 /2 + (ρ − ρ0 )g0 (Z − Z0 ) (for some
reference geopotential height Z0 ). Using the previously obtained evolution equation for
kinetic energy (2.8), the evolution equation for Em is easily verified to be
D v2
Db
ρ0
− b(Z − Z0 ) + ∇ · (δP v − ρ0 Fke ) = −ρ0 (Z − Z0 )
− ρ0 ε K ,
(2.13)
Dt 2
Dt
which can be equivalently written in conservative form:
∂(ρ0 Em )
δP
Db
+ ∇ · ρ0 Em +
v − ρ0 Fke = −ρ0 (Z − Z0 )
− ρ0 ε K .
∂t
ρ0
Dt
(2.14)
For comparison, the corresponding evolution equation for the mechanical energy in a
fully compressible fluid would take the form:
D v2
Dυ
ρ
+ g0 (Z − Z0 ) + ∇ · (P v − ρFke ) = ρP
− ρεK .
(2.15)
Dt 2
Dt
In a compressible fluid, mechanical energy and internal energy are coupled in mainly two
ways: (i) via a reversible conversion between internal energy and kinetic energy accomplished by the work of expansion/contraction P Dυ/Dt; (ii) the irreversible dissipation
of kinetic energy into heat by viscous processes εK , as seen in the r.h.s. of (2.15). Since
viscous dissipation is obviously present in both (2.13) and (2.15), the only question that
needs addressing is whether the following equivalence can be established?
Db
Dυ
↔
ρP
?
(2.16)
Dt
Dt
To see that this is indeed the case, we expand the pressure and specific volume as follows:
−ρ0 (Z − Z0 )
P = Pa − ρ0 g0 Z + · · · ,
υ=
1
ρ − ρ0
+ ···
−
ρ0
ρ20
(2.17)
which uses the underlying assumption in the Boussinesq and anelastic approximations
that (ρ − ρ0 )/ρ0 1. This implies in turn that at leading order, the work of expansion/contraction can be approximated by:
Dυ
D
1
ρ − ρ0
ρP
≈ ρ0 (Pa − ρ0 g0 Z)
−
Dt
Dt ρ0
ρ20
Db
Dυ0
+ ρ0 (Pa − ρ0 g0 Z)
.
(2.18)
Dt
Dt
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 has
been previously noted by Winters & al (1995), Wang & Huang (2005) in the context of
the non-averaged Boussinesq model with a linear equation of state (3.1-3.4), and in the
context of Boussinesq-averaged models by Nycander & al. (2007) and Young (2010).
= −ρ0 (Z − Z0 )
2.4. Thermodynamics of Boussinesq and Anelastic binary fluids
Classical thermodynamics defines the specific internal energy e = e(η, S, υ) as a function
of state, whose value is independent of the thermodynamic path followed. This implies
that the reversible work transfer δW = −P dυ and generalised heat transfer δQ = T dη +
µdS entering the total differential of e, viz.,
de = δQ + δW = T dη + µdS − P dυ,
(2.19)
8
R. Tailleux
are not independent of each other. This interdependence is commonly expressed through
the so-called Maxwell relationships, which merely state that the cross-derivatives for
twice continuously differentiable functions must be equal. In the present case, the interdependence between work and heat variables is thus expressed via the following relations
T =
∂e
,
∂η
µ=
∂e
,
∂S
P =−
∂e
,
∂υ
(2.20)
∂T
∂2e
∂P
∂T
∂2e
∂µ
∂µ
∂2e
∂P
=
=−
,
=
=
,
=
=−
.
(2.21)
∂υ
∂η∂υ
∂η
∂S
∂η∂S
∂η
∂υ
∂S∂υ
∂S
Assuming that it is justified to regard δWba = (Z − Z0 )db as the BA approximation to
the work transfer δW = −P dυ, similar relations must link δWba and the BA approximation to the heat transfer, denoted δQba , for the BA equations to be thermodynamically
consistent. The following shows that δQba may be written similarly as in the fully compressible case in the form δQba = Tba dηba + µba dSba , where the suffix ba refers to the BA
approximated quantities, and hence that the BA counterpart of the fundamental relation
of thermodynamics (2.19)) may be written
deba = Tba dηba + µba dSba + (Z − Z0 )db.
(2.22)
The specific enthalpy h = e + P υ and specific Gibbs function g = e + P υ − T η are two
other important thermodynamic potentials, which are traditionally obtained from one
another by means of the Legendre transform, e.g., Alberty (2001). It is easily shown
that the corresponding BA approximations to h and g are given by hba = eba − b(Z − Z0 )
and gba = eba − b(Z − Z0 ) − Tba ηba , with the following total differentials:
dhba = d [eba − b(Z − Z0 )] = Tba dηba + µba dSba − bdZ,
dgba = d [eba − b(Z − Z0 ) − Tba ηba ] = −ηba dTba + µba dSba − bdZ.
(2.23)
(2.24)
The above relations establish that constructing self-consistent thermodynamic potentials
for the BA equations is a priori feasible. Such relations do not tell, however, how accurate
the BA thermodynamic potentials should be. To address this issue, it is essential to
construct 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 methods
to construct the BA thermodynamic potentials explicitly. The first method consists in
integrating the Maxwell relationships to obtain unknown and difficult to measure thermodynamic potentials from various measurable quantities, such as the speed of sound
or heat capacity, as discussed in Callen (1985). For the present purposes, this method
is really only suited for constructing the BA thermodynamic potentials in idealised situations for which the equation of state and other thermodynamic quantities are known
in closed analytic form, as will be illustrated later in the text. The second method consists in deducing the BA thermodynamic potentials from the knowledge of the exact
thermodynamic potentials, and is illustrated next.
To proceed, let us assume that the specific Gibbs function g = g(T, S, P ) is known as
a 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
Energetics and thermodynamics of the Boussinesq/Anelastic approximations
9
in balance with ρ0 rather than the anelastic reference pressure P0 , while the tilde indicate
that the exact pressure is replaced by P0h .
To check that this is the case, let us take the total differential of gba , and verify that
it is consistent with (2.24). Using the fact that dg = −ηdT + µdS + υdP , it follows that
dgba = −˜
η dT + µ
˜dS + g0 (1 − ρ0 υ˜)dZ,
(2.26)
where we used the result that dP0h = −ρ0 g0 dZ by definition of P0h . From the above definition 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
ρ˜ − ρ0
,
(2.27)
ηba = η˜,
µba = µ
˜,
b = −g0 (1 − ρ0 υ˜) = −g0
ρ˜
Tba = T,
Sba = S.
(2.28)
Eqs. (2.27) and (2.28) are important, for they establish that the BA specific entropy
ηba and relative chemical potential µba can be simply obtained by replacing the total
pressure 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 and
salinity, do not need to be approximated, allowing a straightforward comparison between
the simulated T/S fields and observations.
Surprisingly, however, (2.27) shows that ρ 6= ρ˜(T, S, P0h ), in contrast to what is usually
assumed in the oceanographic literature, since from the result
ρ − ρ0
ρ˜ − ρ0
= −g0
,
(2.29)
b = −g0
ρ˜
ρ0
it follows that ρ is actually linked to ρ˜ by the relation
ρ˜ − ρ0
ρ = ρ0 1 +
= ρ0 (2 − ρ0 υ˜) .
ρ˜
(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 additional physical justification for the latter. The result is interesting, because it shows
that thermodynamic consistency considerations alone are able to discriminate between
two a priori acceptable definitions of buoyancy, and to reveal that one definition is far
superior than the other. The above results also demonstrate that density should be estimated by using the hydrostatic pressure P0h rather than the full pressure, as advocated
by 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 standard
buoyancy rather than the true buoyancy degrades the accuracy of the horizontal momentum equations in the Boussinesq primitive equations forming the basis of most current
ocean climate models, which we plan on quantifying in a subsequent paper.
As mentioned above, the knowledge of one thermodynamic potential is sufficient to
recover all other thermodynamic potentials via Legendre transforms. It is easily shown
that the relevant BA approximations to the specific internal energy and enthalpy can be
deduced from that for the specific Gibbs function gba as follows:
eba (S, T, Z) = gba + b(Z − Z0 ) + Tba ηba = g˜ + (g0 + b)(Z − Z0 ) + T η˜,
hba (S, T, Z) = eba − b(Z − Z0 ) = g˜ + g0 (Z − Z0 ) + T η˜.
(2.31)
(2.32)
Taking the total differential of the above expressions thus yields
deba = T d˜
η+µ
˜dS + (Z − Z0 )db,
(2.33)
10
R. Tailleux
dhba = T d˜
η+µ
˜dS − 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 thermodynamic potentials from the knowledge of the specific enthalpy h = h(η, S, P ), whose
natural variables are specific entropy, salinity, and pressure. To that end, let us introduce
˜ + g0 (Z − Z0 ),
hba = h(η, S, P0h (Z)) + g0 (Z − Z0 ) = h
(2.35)
and verify that it is the relevant Boussinesq/anelastic approximation to the specific enthalpy. This can be checked by taking that the total differential of hba , viz.,
dhba = T˜dη + µ
˜dS + g0 (1 − ρ0 υ˜)dZ = T˜dη + µ
˜dS − bdZ,
(2.36)
agrees with (2.23). QED. The corresponding expression for the internal energy eba =
hba + b(Z − Z0 ) can be written as
˜ + (g0 + b)(Z − Z0 ).
eba = h
(2.37)
Its total differential is:
deba = T˜dη + µ
˜dS −ρ0 g0 υ˜dZ +(b+g0 )dZ +(Z −Z0 )db = T˜dη + µ
˜dS +(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 thermodynamic
potentials, the gravitational potential energy ep = b(Z − Z0 ) can be written as the
difference 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 thermodynamic quantity entirely determined from the knowledge of temperature, salinity,
and Z. It follows that the BA enthalpy represents the total potential energy of the BA
equations, and hence that the sum v2 /2 + hba must represent the relevant conservative
energy quantity for the BA system, which is demonstrated in the following. The BA
equations therefore behave as the hydrostatic compressible equations, for which the sum
v2 /2 + h is also the relevant conserved energy quantity, as is well known. This exact
correspondence can be viewed as an illustration of the duality between the Boussinesq
and 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 the
specific internal energy and enthalpy. Thus, combining (2.33) and (2.34) with the evolution equations (2.3) and (2.4) for η and S respectively, one obtains
Deba
˙ + ρ0 (Z − Z0 ) Db ,
= ρ0 [q˙ + µ
˜S]
(2.40)
Dt
Dt
h
i
Dhba
DZ
ρ0
= ρ0 q˙ + µ
˜S˙ − ρ0 b
.
(2.41)
Dt
Dt
From the above argument, we anticipate that hba is the relevant total potential energy
for the BA equations, and hence that v2 /2 + hba is the corresponding total energy. By
ρ0
Energetics and thermodynamics of the Boussinesq/Anelastic approximations
11
combining either one of (2.40) or (2.41) with the mechanical energy equation (2.13), the
two following equivalent evolution equations for the total energy Et = v2 /2 + eba − b(Z −
Z0 ) = v2 + hba are obtained:
h
i
D v2
− b(Z − Z0 ) + eba + ∇ · [δP v − ρ0 Fke ] = ρ0 q˙ + µ
˜S˙ − εK ,
(2.42)
ρ0
Dt 2
h
i
D v2
ρ0
+ hba + ∇ · [δP v − ρ0 Fke ] = ρ0 q˙ + µ
˜S˙ − εK .
(2.43)
Dt 2
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 easily
shown to be the case only if the right-hand side of (2.42) is the divergence of some energy
flux Fq , i.e.,
ρ0 (q˙ + µS˙ − εK ) = −∇ · (ρ0 Fq ).
(2.44)
As we show now, this imposes a constraint on the form of the entropy flux Fη and
irreversible entropy production η˙ irr in (2.3). Assuming salt to be conservative quantity,
and hence such that ρ0 S˙ = −∇ · (ρ0 FS ) for some salt flux FS , implies for the evolution
equation of specific entropy:
ρ0 q˙
ρ0 εK + µ∇ · (ρ0 FS ) − ∇ · (ρ0 Fq )
Dηba
=
=
Dt
T
T
ρ0 (Fq − µ
˜ FS )
ρ0 ε K
µ
˜
1
−∇·
=
+ ρ0 Fq · ∇
− FS · ∇
.
(2.45)
T
T
T
T
The latter expression establishes that the entropy flux Fη and irreversible entropy production η˙ irr in (2.3) must be related to the salt flux FS and enthalpy/internal energy
flux Fq by:
Fq − µ
˜ FS
Fη =
,
(2.46)
T
1
µ
˜
εK
+ Fq · ∇
− FS · ∇
.
(2.47)
η˙ irr =
T
T
T
In non-equilibrium thermodynamics, this is usually the point at which the second law of
thermodynamics is invoked to further constrain Fq and FS , in order to guarantee that
η˙ irr > 0, e.g., see de Groot & Mazur (1962). Going back to the evolution equation for
internal energy and enthalpy, note that (2.44) implies:
ρ0
ρ0
Deba
Db
= −∇ · (ρ0 Fq ) + ρ0 εK + ρ0 (Z − Z0 )
,
Dt
Dt
(2.48)
Dhba
DZ
= −∇ · (ρ0 Fq ) + ρ0 εK − ρ0 b
,
(2.49)
Dt
Dt
so that Fq appears as the diffusive flux of internal energy or enthalpy. Both equations
can be regarded as equivalent forms of the first law of thermodynamics.
ρ0
2.9. Alternative forms of the first law of thermodynamics
Although entropy is perhaps the most natural measure of heat, the fact that it increases
irreversibly upon mixing makes it strongly nonconservative. In contrast, temperature or
enthalpy mix more linearly, hence are more conservative, and therefore of more practical
interest. This motivates examining the implication of the above results for the derivation
of 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 = −˜
η dT + µ
˜dS − bdZ,
the Maxwell relationships provide the following partial derivatives for the entropy:
∂ η˜
∂µ
˜
∂ η˜
∂b
∂ υ˜
=−
,
=
= ρ0 g0
= g0 α
˜,
∂S
∂T
∂Z
∂T
∂T
by defining α
˜ = ρ0 ∂ υ˜/∂T as the relevant definition of the thermal expansion coefficient.
Using the well known result that ∂η/∂T = cp dT /T , this implies that we can write:
c˜p
∂µ
˜
dT −
dS + g0 α
˜ dZ
T
∂T
As a result, it follows that the temperature equation can be written as:
d˜
η=
DT
T D˜
η
T ∂µ
˜ DS
α
˜ g0 T DZ
=
+
−
.
Dt
c˜p Dt
c˜p ∂T Dt
c˜p Dt
(2.50)
(2.51)
2.9.1. Simplifications in absence of salinity
Some of the general principles affecting temperature variables are most easily understood by first discarding temporarily the effects of salinity, in which case the above
equation simplifies to
DT
∇ · (κ˜
cp ∇T ) εK
αg0 T DZ
=
.
+
−
Dt
c˜p
c˜p
c˜p Dt
(2.52)
Here, we assumed that the diabatic effects modifying entropy exclusive arise from standard molecular diffusion given by the classical Fourier law ∇ · (ρ0 Fq ) = −∇ · (κρ0 c˜p ∇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 cannot
be 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 possesses
a term related to change in pressure, which some authors, e.g. Pons & Le Qu´er´e (2005,
2007), call the “piston effect”, and the resultant Boussinesq equations, the thermodynamic Boussinesq equations.
2.9.2. Accounting for compressibility effects
In the atmospheric and oceanic practice, compressibility effects are traditionally eliminated by constructing evolution equations for potential temperature θ or conservative
temperature Θ respectively. By definition, θ is the temperature that a parcel with temperature T at a given geopotential height Z would have if lifted adiabatically to a reference
level Z = Z0 , usually taken at sea level. In the ocean, potential temperature is thus
defined 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 two
levels Z and Z0 . Differentiating (2.53) yields
c˜rp
c˜p
∂µ
˜
∂µ
˜r
dT −
dS + α
˜ g0 dZ = dθ −
dS,
(2.54)
T
∂T
θ
∂θ
where we defined c˜rp = cp (θ, S, Z0 ) and µ
˜r = µ
˜(θ, S, Z0 ). This expression shows that it is
possible to rewrite the evolution equation for entropy as follows:
dηba =
c˜rp Dθ
∂µ
˜ DS
Dηba
=
+
θ Dt
∂θ Dt
Dt
(2.55)
Energetics and thermodynamics of the Boussinesq/Anelastic approximations
13
In absence of salinity, we can write:
c˜rp Dθ
∇ · (κ˜
cp ∇T ) εK
=
+
.
θ Dt
T
T
This can be transformed in the following equation for θ,
Dθ
κ˜
cp θ∇T
=∇·
+ θ˙irr ,
Dt
c˜rP T
where the nonconservative production/destruction of θ is given by:
θεK
θ
+ r .
θ˙irr = −κ˜
cp ∇T · ∇ r
c˜p T
c˜p T
(2.56)
(2.57)
(2.58)
In the literature, θ is often treated as a conservative variable, which consists in neglecting
the nonconservative term θ˙irr . Alternatively, one may remark that the equation for θ can
also be written as:
˜θ
Dh
Dθ
=
,
(2.59)
c˜rp
Dt
Dt
˜ θ (θ) = hb (η, 0) is the enthalpy a parcel would have if moved adiabatically from
where h
˜ θ was called “potential enthalpy” by McDougall (2003).
Z to Z = 0. For this reason, h
This allows one to defined a new temperature variable Θ, also conserved for adiabatic
˜ θ = c0 dΘ, where c0 is an arbitrarily defined specific heat capacity
motions, such that: dh
p
p
representative of cp at Z = 0. McDougall (2003) discusses a possible choice for c0p in the
oceanic context. As a result, it is possible to write the above equation as:
˜θ
DΘ
1 Dh
κ˜
cp θ∇T
˙ irr ,
+Θ
(2.60)
= 0
=∇·
Dt
cp Dt
c0p T
where this time, the nonconservative production/destruction of Θ is given by:
κ˜
cp
θ
θεK
˙
Θirr = − 0 ∇T · ∇
.
+
cp
T
T
(2.61)
˙ irr is sigMcDougall (2003) shows that the globally averaged nonconservative term Θ
nificantly smaller than θ˙irr for the present-day distribution of temperature and salinity
˙ irr is not necessarily smaller than θ˙irr when
fields. It is important to note, however, that Θ
estimated locally, as the relative ordering of the two terms depends on the particular fluid
considered and on the degree of turbulence experienced by the fluid.
3. Thermodynamics and energetics of a Boussinesq fluid with a
linear 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 idealised
linear equation of state, which is governed by the following standard equations:
Dv
1
g0 (ρ − ρ0 )
+ ∇δP = −
k + ν∇2 v,
Dt
ρ0
ρ0
(3.1)
∇ · v = 0,
(3.2)
DT
= κ∇2 T,
Dt
(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 pressure
anomaly defined relative to the reference Boussinesq pressure P0 = −ρ0 g0 z, ρ is the
density, 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 g0
is a nominal value of the acceleration of gravity. An implicit assumption in obtaining the
temperature equation is that the heat capacity cp0 is constant.
As should be clear from the arguments developed above, the above Boussinesq model is
neither energetically nor thermodynamically consistent when molecular diffusive effects
are retained. Yet, it has nevertheless been extensively used in recent theoretical discussion
of 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 numerical
simulations of stratified turbulence, in the sense that in such studies, both κ and ν
are usually interpreted as representing the molecular values of diffusivity and viscosity
respectively. The following seeks to clarify the nature of the difficulties involved, and
discuss 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 should
interpret the temperature variable T in (3.4) as in-situ temperature or potential temperature. On the one hand, the fact that T is materially conserved in absence of diffusive
effects suggests the latter interpretation, whereas the fact that molecular diffusion acts
in homogenising T suggests the former interpretation. In the following, the thermodynamic implications of each interpretation are considered in turn, in order to establish
whether apparent innocuous differences in interpretation matter. We first discuss the
interpretation of T as in-situ temperature, in which case the expression for buoyancy
becomes:
b = αg0 (T − T0 ).
(3.5)
Physically, assuming an equation of state linear in in-situ temperature is equivalent to
assume 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 the
evolution of pressure, which from a mathematical viewpoint is determined by a nonlocal
elliptic operator.
The interpretation of T as in-situ temperature suggests describing the thermodynamic
properties of the fluid in terms of the specific Gibbs function gba , whose natural variables
are T and Z. The partial differential relations determining gba are
∂gba
= −ηba ,
∂T
cp0 = −T
∂gba
= −b,
∂Z
∂ 2 gba
,
∂T 2
(3.6)
(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
Energetics and thermodynamics of the Boussinesq/Anelastic approximations
15
specific entropy is obtained:
ηba
∂gba
=−
= cp0 ln
∂T
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 /T 0) +
αg0 (Z − Z0 ) = cp0 ln (θ/T0 ) − αg0 Z0 , yielding:
αgZ
θ = T exp
(3.10)
cp0
This makes it clear that in order for a model to be thermodynamically consistent, potential temperature is always different from the in-situ temperature. Now, we previously established that the expressions for internal energy and enthalpy were given by:
eba = gba + b(Z − Z0 ) + T ηba and hba = eba − b(Z − Z0 ) = gba + T ηba , yielding therefore
the following expressions:
hba = cp0 T + αg0 T0 (Z − Z0 ),
(3.11)
αg0 (Z − Z0 )
T.
(3.12)
eba = cp0 1 +
cp0
These relations can be written in terms of other variables by using the expression for θ, as
well as b. For instance, the enthalpy can be written in terms of the potential temperature
and Z as follows:
αg0 Z
hba = cp0 θ exp −
+ αg0 T0 (Z − Z0 ).
(3.13)
cp0
αg0 Z0
αg0 θ
b
eba = cp0 1 −
+ ln
T0 +
(3.14)
cp0
b + αg0 T0
αg0
Thus, despite the extreme simplicity of the equation of state, the Boussinesq fluid appears to have a non-trivial underlying thermodynamic structure. As the following shows,
interpreting T as potential temperature introduces significant changes in the expressions
for 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 modify
the above Boussinesq model to make it energetically and thermodynamically consistent.
This requires two main changes, namely modifying the heat equation and retaining the
distinction between potential temperature θ and in-situ temperature T . The modified
Boussinesq model thus becomes:
Dv
1
+ ∇δP = bk + ν∇2 v
Dt
ρ0
(3.15)
b = αg0 (T − T0 )
(3.16)
∇·v =0
(3.17)
θ
θεK
Dθ
= κ∇2 T +
Dt
T
cp0 T
αg0 Z
T = θ exp −
cp0
(3.18)
(3.19)
16
R. Tailleux
Now, (3.18) shows that the potential temperature θ is materially conserved in absence
of molecular diffusive and viscous effects, while the molecular heat flux is now down
gradient of the in-situ temperature, as requested by the theory of non-equilibrium thermodynamics, e.g., de Groot & Mazur (1962). Alternatively, the temperature equation
(3.18) can also be written in the form
κθ
θεK
Dθ
=∇·
∇T + θ˙irr +
,
(3.20)
Dt
T
cp0 T
obtained by entering the term θ/T within the divergence operator, where
2 2
˙θirr = −κ∇T · ∇ θ = − καg0 θ ∂T = − καg0 ∂θ + κα g0 θ.
T
cp0 T ∂z
cp0 ∂z
c2p0
(3.21)
From a practical viewpoint, it is useful to note that the the molecular diffusive term can
be written entirely in terms of the potential temperature, using the result that
κθ
καg0 θ
∇T = κ∇θ −
k,
T
cp0
(3.22)
assuming for simplicity Z(z) = z, as is always done in practice. Interestingly, these
equations introduce the following length scale: L = cp0 /(αg0 ). For typical values, cp0 =
4.103 J.kg−1 .K−1 , α = 10−4 K−1 and g0 = 10 m.s−1 , which yields: L = 4.106 m. As L
is huge compared to the typical length scales at which molecular diffusion is important,
it follows that the nonconservative term is probably negligible. Nevertheless, the added
terms 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 to
be linear in potential temperature, rather than in in-situ temperature, and hence that
the adiabatic compressibility is zero, as is generally implicitly assumed in traditional low
Mach number asymptotics. In this case, the buoyancy becomes:
b = αθ g0 (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 =
cp0 dθ/θ) suggests deriving the thermodynamic properties of the fluid from the specific
enthalpy, whose natural variables are ηba and Z, and whose total differential is:
dhba = T dηba − bdZ =
T cp0
dθ − bdZ.
θ
(3.24)
The partial derivatives of hba with respect to θ and Z yields the following system of
partial differential equations:
∂hba
cp0 T
=
,
∂θ
θ
∂hba
= −b = −αθ g0 (θ − θ0 ).
∂Z
(3.25)
It is easily verified that ∂ 2 hba /∂Z∂θ = ∂ 2 hba /∂θ∂Z = −αθ g0 , and hence that the system
(3.25) can be integrated to yield the following expression between T and θ
T
αθ g0 Z
=1−
,
θ
cp0
(3.26)
Energetics and thermodynamics of the Boussinesq/Anelastic approximations
17
using the fact that by definition, T = θ at Z = 0. By inserting this result into the
equation for ∂hba /∂θ in (3.25), straightforward integration yields:
θ − θ0
αθ g0 Z
(θ − θ0 ) =
cp0 T,
(3.27)
hba = cp0 1 −
cp0
θ
up to an arbitrary constant of integration. The expression for the internal energy is
therefore given by:
eba = hba + b(Z − Z0 ) = [cp0 − αθ g0 Z0 ](θ − θ0 ).
(3.28)
Since Z0 is constant, (3.28) shows that in the present model, internal energy is linearly
proportional to potential temperature, and hence that θ is a direct measure of internal
energy. Moreover, it is useful to note that owing to cp0 being constant, conservative
temperature Θ can be defined to coincide with θ.
The alterations to the heat equation needed to make the Boussinesq model energetically
and thermodynamically consistent for an equation of state linear in θ are the same as for
an equation of state linear in T , so that the only difference with the previous case is the
different expression linking T and θ, which is now given by (3.26) in place of (3.19). In
summary,
θ
θεK
Dθ
= κ∇2 T +
,
(3.29)
Dt
T
cp0 T
αθ g0 Z
T =θ 1−
.
(3.30)
cp0
As previously, it is possible to manipulate the diffusive term to rewrite the heat equation
so as to introduce a nonconservative production/destruction of θ as follows:
κθ
θεK
Dθ
=∇·
∇T + θ˙irr +
(3.31)
Dt
T
cp0 T
where
θ˙irr = −κ∇T · ∇
2
θ
∂T
καθ g0 θ
καθ g0 /cp0
∂T
=−
.
=−
2
T
cp0
T
∂z
[1 − αθ g0 Z/cp0 ] ∂z
(3.32)
Moreover, it is also possible to write the temperature equation entirely in terms of potential temperature if desired, at the price of introducing some additional depth-dependent
terms. Although the proposed corrections are arguably small in liquids such as water
or seawater, they may start having a detectable effect on temperature fluctuations in a
strongly turbulent fluid in the limit of very weak stratification, which is rarely studied
in the literature. In any case, the proposed corrections should be straightforward to implement in existing numerical Boussinesq models, as well as easily adapted to the case
of an arbitrary nonlinear equation of state.
4. Summary and discussion
Although the concept of an “incompressible fluid” as used in the Boussinesq approximation takes its origin in a thermodynamic definition of incompressibility (associated
with zero adiabatic compressibility, or equivalently with an infinite speed of sound), the
modern acceptation of the term has become more generally associated with the kinematic view of incompressibility associated with the use of a divergence-free vector field.
While the thermodynamic and kinematic definitions of incompressibility coincide for an
equation 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 appropriate
to 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/Dt
that enters the energetics of the Boussinesq approximation, which differs from zero if ρ
depends on pressure and/or in presence of diabatic effects, should be regarded as the
Boussinesq approximation to the thermodynamic work of expansion/contraction. While
this 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 Maxwell
relationships to reconstruct consistent approximations of all thermodynamic properties
of the fluid, and to determine corrections to the heat equation to ensure global thermodynamic and energetic consistency. How to apply these ideas was illustrated in the
particular case of the extensively used Boussinesq model, for which we constructed explicit expressions for the internal energy and enthalpy. One key result is to show that
the conservative energy quantity of the Boussinesq model is the sum of kinetic energy
plus the Boussinesq enthalpy, similarly as for the compressible hydrostatic Navier Stokes
equations. Another important result is that the gravitational potential energy can be
viewed as a thermodynamic property of the fluid, which is equal to the difference between the Boussinesq enthalpy and Boussinesq internal energy. The proposed method
does not appear to work for the anelastic system of equations, for which a fully consistent formulation that would also retain hydrostatic modes of motion remains to be
derived.
In contrast to the compressible Navier-Stokes (CNS) equations, in which the thermodynamic work of expansion/contraction represents a conversion between internal energy
and kinetic energy, the term gzDρ/Dt is more naturally interpreted as a conversion between internal energy and gravitational potential energy in the Boussinesq model. For
the Boussinesq energetics to be traceable to that of the CNS, it is therefore necessary
to 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 magnitude that converts kinetic energy into gravitational potential energy. In some recent
papers by Tailleux (2009, 2013), we argued on the basis of a rigorous analysis of the
energetics of the CNS that compressibility effects and conversions with internal energy
should be regarded as important as conversions between kinetic energy and gravitational
potential energy in a turbulent stratified fluid. We argued in particular that the observed
net increase in GPE over a turbulent mixing event should be regarded as occurring at
the expense of internal energy, rather than at the expense of kinetic energy, in contrast
to what is usually assumed following Winters & al (1995)’s interpretation of Boussinesq
energetics. Physically, this can be understood from the fact that turbulence greatly increases local values of the molecular diffusive term κ∇2 ρ, and hence of the compressible
work gzDρ/Dt = gzκ∇2 ρ, resulting in enhanced local transfers between internal energy
and gravitational potential energy.
Our results strongly support the view that Boussinesq models can support large and
leading order conversions between internal energy and mechanical energy when molecular diffusive effects are retained. As a result, while such models can still be regarded
as “incompressible” from a kinematic viewpoint, they need to be regarded as thermodynamically compressible from the purposes of correctly understanding and interpreting
their energetics, which is still largely unappreciated. The hope is that by providing an
explicit way to construct thermodynamic potentials, as well as the corrections needed
to make Boussinesq models energetically and thermodynamically consistent, it becomes
easier to understand how to develop interpretations of Boussinesq energetics that are
traceable to that of the compressible Navier-Stokes equations.
Energetics and thermodynamics of the Boussinesq/Anelastic approximations
19
Appendix A. Proof that the density dependence upon pressure can
never be fully eliminated for general diabatic motions
The purpose of this appendix is to prove that the original concept of an incompressible
fluid historically defined as one whose density is not affected by changes in pressure (as
stated in Batchelor (1967) or Lilly (1996) for instance) is physically meaningful for adiabatic motions only, but otherwise a thermodynamic impossibility for general diabatic
motions. To see this, recall that the way the density of actual fluids is affected by pressure
changes depends sensitively on the nature of the thermodynamic transformation experienced by the fluid parcels. In thermodynamics, the degree to which density is affected
by pressure changes is generally quantified by means of the coefficient of compressibility
ρ−1 ∂ρ/∂P , where the partial derivative is computed along the particular thermodynamic
pathway considered. In general, however, the coefficients of compressibility defined for different thermodynamic pathways are not independent of each other. For instance, classical
thermodynamic arguments show that the isothermal compressibility γ = ρ−1 ∂ρ/∂P |T is
linked to the adiabatic compressibility ρ−1 ∂ρ/∂P |η = 1/(ρc2s ) by:
1
α2 T
1
1 ∂P =
+
(A 1)
= 2 + αΓ,
γ=
ρ ∂ρ T,S
ρc2s
ρcp
ρcs
where Γ = αT /(ρcp ) is the adiabatic lapse rate. It follows immediately that setting up
the adiabatic compressibility to zero (by taking the zero Mach number limit cs = +∞),
while effectively making the fluid “incompressible” for adiabatic motions, fails to do so
for general diabatic transformations (this assumes that the thermal expansion coefficient
α remains finite, as is normally the case for most fluids). Note that for seawater, typical
values are: 1/(ρc2s ) ≈ 4.10−10 Pa −1 , whereas αΓ = α2 T /(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 these
values, the limit cs = +∞ decreases the isothermal compressibility by about two to three
orders of magnitude, so that even if it fails to fully eliminate compressibility effects for
non-adiabatic motions, it appears nevertheless capable of reducing them considerably.
This result helps to understand a general conclusion of the present study that any thermodynamically consistent formula must distinguish between the potential temperature
and in-situ temperature.
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