A variational derivation of the thermodynamics of a moist atmosphere with irreversible processes

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by  François Gay-Balmaz

A variational derivation of the thermodynamics of a
moist atmosphere with irreversible processes
Franc¸ois Gay-Balmaz1
7
17 January 2017
1
0
2
Abstract
n
a Irreversible processes play a major role in the description and prediction of atmo-
J spheric dynamics. In this paper, we present a variational derivation of the evolution

4 equations for a moist atmosphere subject to the irreversible processes of molecular
1 viscosity, heat conduction, diffusion, and phase transition. This derivation is based
on the general variational formalism for nonequilibrium thermodynamics established

] inGay-Balmaz and Yoshimura[2017a,b],whichextendsHamilton’sprincipletoincor-
h
porates irreversible processes. It is valid for any state equation and thus also covers
p
thecase of the atmosphere of other planets. In this approach, the second law of ther-
-
h modynamics is understood as a nonlinear constraint formulated with the help of new
t variables, called thermodynamic displacements, whose time derivative coincides with
a
m the thermodynamic force of the irreversible process. The formulation is written both
in the Lagrangian and Eulerian descriptions and can be directly adapted to oceanic
[
dynamics.

1
v
1 1 Introduction
2
9
3 The partial differential equations governing the thermodynamic of the atmosphere are of
0 obviousimportanceforweatherandclimateprediction. These equationsarewell-knownfor
. their extreme complexity,bothfromthe theoreticalandthe computationalside, whichis in
1
0 part due to the many physical processes that they involve, such as phase transition, cloud
7 formation, precipitation, and radiation.
1
: Inabsenceofirreversibleprocesses,theequationsofatmosphericdynamiccanbederived
v
byapplyingHamilton’svariationalprincipletothe Lagrangianfunctionofthe fluid. Thisis
i
X in agreement with a fundamental fact from classical reversible mechanics, namely that the
r motionofthemechanicalsystemisgovernedbytheEuler-Lagrangeequationswhich,inturn,
a
describethecriticalpointsoftheactionfunctionalofthisLagrangianamongallpossibletra-
jectories with prescribed values at the temporal extremities. Hamilton’s principle for fluid
mechanicsintheLagrangiandescriptionhasbeendiscussedatleastsincetheworksofHerivel
[1955], for an incompressible fluid and Serrin [1959] and Eckart [1960] for compressible
flows, see also Truesdell and Toupin [1960] for further references. While in the Lagrangian
description this principle is a straightforward extension of the Hamilton principle of parti-
cles mechanics, in the Eulerian description the variational principle is much more involved
and several approaches have been developed, see Lin [1963], Seliger and Whitham [1968],
Bretherton[1970]. WerefertoSalmon[1983]andSalmon[1988]forfurtherdevelopmentsin
1LMD/IPSL, CNRS, Ecole normale sup´erieure, PSL Research University, Ecole polytechnique, Uni-
versit´e Paris-Saclay, Sorbonne Universit´es, UPMC Univ Paris 06, 24 Rue Lhomond 75005 Paris, France,
[email protected]
1
1 Introduction 2
thecontextofgeophysicalfluids,seealsoMu¨ller[1995]. InHolm, Marsden and Ratiu[2002],
the variational principle in Eulerian description is obtained via the Euler-Poincar´e reduc-
tion theory for several geophysical fluid models, by exploiting the relabelling symmetries.
Physically,the existence ofa variationalHamilton’s principle manifests itself in specific La-
grangianconservationlaws,themostcelebratedbeingtheconservationofpotentialvorticity
which alone allows one to understand many processes taking place in the atmosphere, the
ocean, or in laboratory experiments (e.g. Hoskins, McIntyre, and Robertson [1985]). Con-
servation of potential vorticity is related to Kelvin’s circulation theorem which, thanks to
variational principle, can be interpreted in terms of the general Noether theorem, linking
conservation laws to symmetries. More recently, the variational formulation of geophysi-
cal fluids has been crucially used in Desbrun, Gawlik, Gay-Balmaz, and Zeitlin [2014] and
Bauer and Gay-Balmaz [2016] to derive structure preserving numerical schemes for rotat-
ing stratified fluids in the Boussinesq and pseudo-incompressible approximations,following
Pavlov et al. [2010], Gawlik et al. [2011].
Irreversibleprocessessuchasphasetransition,cloudformation,precipitation,andradia-
tion,playamajorroleinthedescriptionandpredictionofatmosphericdynamicsandithas
been a longstanding questionwhether ornot the variationalformulationsmentionedabove
canbe extendedto include allthese irreversibleprocesses. Inthis paper, weshallpositively
answerthisquestionbypresentingavariationalderivationforthethermodynamicofamoist
atmosphere that includes the irreversible processes of molecular viscosity, heat conduction,
diffusion, and phase transition. We shall use the variational formulation of nonequilibrium
thermodynamics recently developed in Gay-Balmaz and Yoshimura [2017a,b]. The main
aspect of this approach is the introduction, for each of the irreversible process, of a new
variable, called the thermodynamic displacement, whose time derivative coincides with the
thermodynamicforceoftheprocess. Thisnewvariableallowsustoformulatethesecondlaw
ofthermodynamicsasanonlinearconstrainttobeusedinthevariationalformulation,both
fortheLagrangianandEuleriandescriptions. Thisconstraintinvolvesthephenomenological
expressionof eachof the thermodynamic fluxes in terms of the thermodynamic forces. It is
hence called the phenomenologicalconstraint. From the point of view of atmospheric mod-
elling,thisconstraintencodesthevariousparametrizationsofsubgridscaleeffectsingeneral
circulation models of the atmosphere. In absence of irreversible processes, the constraint
disappears and our variational formulation consistently recovers the classical variational
principles in Lagrangianand Eulerian descriptions.
The atmosphereof the Earthis composed of dry air, watersubstance in any of its three
phases, and atmospheric aerosols. For practical use in meteorology, dry air, whose main
components are Nitrogen and Oxygen, can be regarded as an ideal gas. Unlike the compo-
nents of dry air, the proportion of the gas phase of water,i.e., water vapor, is very variable
and plays a major role in the thermodynamics of the atmosphere because of its ability to
condense under atmospheric conditions. The condensed phases of water consist of cloud
particles(waterdropletoricecrystal)thatfollowthemotionofdryairandofhydrometeors
(such as rainand snow)that are falling throughthe air. The atmospheric aerosolsaresolid
and liquid particles in suspension (other than that of water substance) whose study is very
important for atmospheric chemistry, cloud and precipitation physics, and for atmospheric
radiation and optics. It is not significant for atmospheric thermodynamics, and shall not
be considered here. In the first part of the paper (Section 2), we restrict our approach
to the case of dry air. In this case, the only irreversible processes are viscosity and heat
conduction. This situation allows us to introduce the main ideas of our approach in a sim-
plified context. In the second part of the paper (Section 3) we consider the case of a moist
atmospherebyincludingwatersubstanceinitsdifferentphases,namely,watervapor,liquid
water,andsolidwater. The irreversibleprocessesconsideredareviscosity,heatconduction,
2 Variational derivation of the thermodynamic of a dry atmosphere 3
vapordiffusion,andphasechanges. Wealsocomputetheimpactoftheirreversibleprocesses
on the potential vorticity equation and on Kelvin’s circulation theorem, by staying in the
general Lagrangian framework, which provides a useful unified treatment of potential vor-
ticity and circulation theorems for various approximations of the equations of atmospheric
dynamics. Finally, in Section 4, we quickly mention that our approach directly applies to
oceanic dynamics with the irreversible process of molecular viscosity, heat conduction, and
salt diffusion included.
2 Variational derivation of the thermodynamic of a dry
atmosphere
In this section we consider the dynamics of a dry atmosphere subject to the irreversible
processes of molecular viscosity and of heat exchange by molecular conduction.
The equation of state of dry air is the ideal gas law
R T
d
v =v(p,T)= , (2.1)
p
where v is the specific volume, p is the pressure, T is the temperature, R =R∗/m is the
d d
gas constant for dry air written in terms of the universal gas constant R∗ and the mean
molecular weight of dry air m . The expression of all other thermodynamic variables for
d
dry air in terms of p and T can be derived by using the equation of state (2.1) and the
fact that the specific heat C at constant pressure can be assumed to be constant in the
p
atmosphere, which is the hypothesis for a perfect gas. For example, it is deduced that the
specific internal energy and the specific entropy are
u=C T, η =C lnT −R lnp+η ,
v p d 0
where C is the specific heat at constant volume, η is a constant, and we assume that the
v 0
internal energy at T =0 K is zero.
Our variational formulation is more naturally expressed in terms of density variables
rather than specific variables, hence we will write the Lagrangian in terms of the mass
density ρ = 1 and entropy density s = η. In addition, the variational derivation has a
v v
simpler form in Lagrangian description, this is why we shall first write it in Lagrangian
variables and then later deduce from it the Eulerian form.
Letusdenotebyx=ϕ(t,X)theLagrangiantrajectoryofdryairparticles. Thevariable
X refers to the label of the particle and x is its current location. The Lagrangian and
Eulerian wind velocities ϕ˙ = dϕ and v are related as
dt
d
ϕ(t,X)=v(t,ϕ(t,X)).
dt
We shall denote by S(t,X) the entropy density in Lagrangian representation, related to
s(t,x) as S(t,X) = s(t,ϕ(t,X))|∇ϕ(t,X)|, where |∇ϕ| denotes the Jacobian of ϕ. The
mass density in Lagrangian representation is ̺ (X) = ρ(t,ϕ(t,X))|∇ϕ(t,X)|. Note that
0
because of mass conservation, the mass density ̺ in Lagrangian representation is time
0
independent. This is in contrastwith the time dependence of S, due to the presence of the
irreversible processes.
The Lagrangian of the dry atmosphere consists of the sum of the kinetic energy and of
thecontributionofEarthrotation,towhichissubstractedthegravitationalpotentialΦand
the internal energy u. In terms of the Lagrangianvariables ϕ, ϕ˙, and S, it reads
1 S |∇ϕ|
L(ϕ,ϕ˙,S)= ̺ |ϕ˙|2+R(ϕ)·ϕ˙ −Φ(ϕ)−u , dX=: LdX. (2.2)
0
2 ̺ ̺
ZD (cid:20) (cid:18) 0 0 (cid:19)(cid:21) ZD
2.1 Variational formulation in Lagrangian description 4
Here the vector field R is the vector potential for the Coriolis parameter, i.e., curlR=2Ω,
whereΩ is the angularvelocityofthe Earth. Onthe righthandside of(2.2)we definedthe
Lagrangiandensity L as the integrand of the LagrangianL.
2.1 Variational formulation in Lagrangian description
In absence of irreversible processes and in absence of heat or matter exchange with the
exterior, the entropy is conserved. This means that in Lagrangian variables the entropy is
time independent, S(t,X) = S (X). In this case, the equations of motion follow from the
0
Hamilton principle:
t2
δ LdXdt=0, (2.3)
Zt1 ZD
wherethecriticalconditionistakenwithrespecttoarbitraryvariationsδϕoftheLagrangian
trajectory ϕ, with δϕ(t )=δϕ(t )=0.
1 2
WeshallnowextendtheHamiltonprinciple(2.3)inordertoincorporatetheirreversible
processes of viscosity and heat conduction. Following Gay-Balmaz and Yoshimura [2017b],
we consider the variational condition
t2
δ L+(S−Σ)Γ˙ dXdt=0, (2.4)
Zt1 ZD
(cid:0) (cid:1)
subject to the phenomenological constraint
∂L
Σ˙ =−Pfr :∇ϕ˙ +J ·∇Γ˙ (2.5)
S
∂S
and with respect to variations δϕ, δS, δΣ, δΓ subject to the variational constraint
∂L
δΣ=−Pfr :∇δϕ+J ·∇δΓ (2.6)
S
∂S
and with δϕ(t )=0, δΓ(t )=0, i=1,2.
i i
This variationalformulationintroduces two new functions, Σ and Γ, whose time deriva-
tive will be ultimately identified with the entropygenerationrate density and the tempera-
ture, respectively. The tensor Pfr is the Piola-Kirchhofffriction stress tensor and J is the
S
entropyfluxdensityinLagrangianrepresentation. Thelinkwiththecorrespondingfamiliar
Eulerian objects will be explained below. The notation “:” indicates the contraction of
tensors with respect to two indices.
The δ-notation in (2.4) indicates that we compute the variation of the functional with
respect to all the fields, namely ϕ,S,Σ,Γ. The constraint (2.5) is referred to as a phe-
nomenological constraint, since the expression of the thermodynamic fluxes Pfr and J are
S
obtained through phenomenological laws, also called parameterizations of the irreversible
processes. It is a nonlinear constraint on the time derivatives of the field variables. We
note that the right hand side of this constraint is of the generic form J Λ˙ , for the
α α α
thermodynamic fluxes J (here Pfr and J ) acting on the rate of thermodynamic displace-
α S
ments Λ˙ (here ϕ˙ and Γ˙). The constraint (2.6) is referred to as a variaPtional constraint,
α
since it is a constraint on the variations to be considered in (2.4). One passes from (2.5)
to (2.6) by formally replacing each occurrence of a time derivative by a δ-variation, i.e.,
J Λ˙ J δΛ . This is reminiscent of what happens in the Lagrange-d’Alembert
α α α α α α
principle in nonholonomic mechanics, see Remark 2.1 below.
P P
2.2 Variational formulation in Eulerian description 5
Taking the variations in (2.4), using the variational constraint (2.6) and collecting the
terms proportional to δϕ, δΓ, and δS, we get the three conditions
d ∂L ∂L ∂L−1 ∂L
+DIV +Γ˙ Pfr − =0
dt∂ϕ˙ ∂∇ϕ ∂S ∂ϕ
(cid:18) (cid:19)
∂L−1 ∂L
S˙ =DIV Γ˙ J +Σ˙, Γ˙ =− ,
S
∂S ∂S
(cid:18) (cid:19)
where DIV denotes the divergence with respect to the labels X. From the last equation,
we have Γ˙ = −∂L = T, the temperature in material representation. This attributes to Γ
∂S
the meaning of temperature displacement, as considered in Green and Naghdi [1991] and
introducedinvon Helmholtz [1884]. The secondequationsimplifies to S˙+DIVJ =Σ˙ and
S
attributes to Σ˙ the meaningofentropy generation rate density. Fromthe firstequationand
the constraint, we thus get the system
d ∂L ∂L ∂L
+DIV −Pfr − =0
dt∂ϕ˙ ∂∇ϕ ∂ϕ
 (cid:18) (cid:19)
 T(S˙ +DIVJ )=Pfr :∇ϕ˙ −J ·∇T,
S S
for the fields ϕ(t,X) and S(t,X). We will discuss the parameterization of the thermo-
dynamic fluxes Pfr and J in terms of the thermodynamic forces below in the Eulerian
S
description.
We leave to the reader the computation of the explicit form of these equations for the
Lagrangian L of the dry atmosphere in (2.2). We shall only present the computation in
Eulerian variables below.
Remark 2.1 (Lagrange-d’Alembert principle in nonholonomic mechanics). We now com-
ment on the analogy between our variational formulation and the Lagrange-d’Alembert
principle used in nonholonomic mechanics. Let us consider a mechanical system with con-
figuration variable q and velocity v = q˙. We assume that the system is subject to a linear
constraint on velocity, i.e., a constraint of the form ω(q)·q˙ = 0, for a q-dependent linear
map ω(q). Typical examples are rolling constraints. The extension of Hamilton’s principle
to nonholonomic systems (called the Lagrange-d’Alembert principle) consists in imposing
that the action functional is critical with respect to variations δq subject to the linear con-
straint ω(q)·δq = 0. It formally follows by replacing the time derivative in the constraint
ω(q)·q˙ =0 by a δ-derivative, see, e.g., Bloch [2003]. In our case, the passage from (2.5) to
(2.6) can be formally seen as a generalization of this approach, to the case of a nonlinear
constraint. We refer to Gay-Balmaz and Yoshimura [2017b] for an extensive discussion of
the principle (2.4)–(2.6).
2.2 Variational formulation in Eulerian description
In terms of Eulerian fields, the Lagrangian(2.2) reads
ℓ(v,ρ,s)= L(v,ρ,s)dx,
ZD
for the Lagrangiandensity L given by
1
L(v,ρ,s)=ρ |v|2+R·v−Φ−u(s/ρ,1/ρ) . (2.7)
2
(cid:18) (cid:19)
The Eulerian quantities γ(t,x) and σ(t,x) associated to Γ(t,X) and Σ(t,X) are defined as
Γ(t,X)=γ(t,ϕ(t,X)) and Σ(t,X)=σ(t,ϕ(t,X))|∇ϕ(t,X)|; (2.8)
2.2 Variational formulation in Eulerian description 6
the entropy flux density j in Eulerian representation is related to J as
s S
∇ϕ(t,X)·J (t,X)=|∇ϕ(t,X)|j (t,ϕ(t,X)); (2.9)
S s
finally, the friction stress σfr is related to the Piola-Kirchhofffriction stress Pfr as
Pfr(t,X)·∇ϕ(t,X)=|∇ϕ(t,X)|σfr(t,X). (2.10)
By using these relations,we canrewrite the variationalformalism(2.4)–(2.6) entirely in
terms of Eulerian variables as follows:
t2
δ (L+(s−σ)D γ)dxdt=0, (2.11)
t
Zt1 ZD
subject to the phenomenological constraint
∂L
D¯ σ =−σfr :∇v+j ·∇D γ (2.12)
t s t
∂s
and with respect to variations
δv=∂ ζ+v·∇ζ−ζ·∇v, δρ=−div(ρζ), δs, δσ, and δγ, (2.13)
t
that are subject to the variational constraint
∂L
D¯ σ =−σfr :∇ζ+j ·∇D γ (2.14)
δ s δ
∂s
with ζ(t )=0 and δγ(t )=0, i=1,2.
i i
Thefirsttwoexpressionsin(2.13)areobtainedbytakingthevariationswithrespecttoϕ,
u,andρ,oftherelationsϕ˙(t,X)=u(t,ϕ(t,X))and̺ (X)=ρ(t,ϕ(t,X))|∇ϕ(t,X)|,respec-
0
tively, and defining ζ(t,x) as δϕ(t,X) = ζ(t,ϕ(t,X)). These formulas can be also directly
justified by employing the Euler-Poincar´e reduction theory, see Holm, Marsden and Ratiu
[2002].
In (2.12) and (2.14), we introduced the notations D f := ∂ f +v·∇f, D¯ f := ∂ f +
t t t t
div(fv), D f :=δf +ζ·∇f and D¯ f :=δf +div(fζ) for the Lagrangiantime derivatives
δ δ
and variations of scalar fields and density fields.
By applying (2.11) and using the expression for the variations δv and δρ, we find the
condition
t2 ∂L ∂L
+(s−σ)∇γ ·(∂ ζ+v·∇ζ−ζ·∇v)− div(ρζ)
∂v t ∂ρ
Zt1 ZD(cid:20)(cid:18) (cid:19)
∂L
+ +D γ δs−D¯ (s−σ)δγ−δσD γ dxdt=0.
t t t
∂s
(cid:18) (cid:19) (cid:21)
Using the variational constraint (2.14) and collecting the terms proportional to ζ, δs, and
δγ, we obtain the three conditions
∂L ∂L ∂L−1
(∂t+£v) ∂v +(s−σ)∇γ =ρ∇∂ρ −σ∇Dtγ−div Dtγ∂s σfr
(cid:18) (cid:19) (cid:18) (cid:19)
∂L−1
+div D γ j ∇γ,
t s
∂s
(cid:18) (cid:19)
∂L−1 ∂L
D¯ (s−σ)=div D γ j , +D γ =0,
t t s t
∂s ∂s
(cid:18) (cid:19)
2.2 Variational formulation in Eulerian description 7
where we introduced the Lie derivative notation£vm:=v·∇m+∇vT·m+mdivv for a
one-form density m along a vector field v. Further computations finally yield the system
∂L ∂L ∂L
(∂t+£v)∂v =ρ∇∂ρ +s∇∂s +divσfr

 ∂∂Ls(D¯ts+divjs)=−σfr :∇v−js·∇∂∂Ls (2.15)
D¯ ρ=0,
whoselastequation,themastsconservationequation,followsfromthedefinitionofρinterms
of ̺ . These are the general equations of motion, in Lagrangian form, for fluid dynamics
0
subject to the irreversible processes of viscosity and heat conduction. From one of the
above conditions, we note that D γ = −∂L =: T, which attributes to γ the meaning of
t ∂s
temperature displacement in Eulerian variables. From the above conditions, we also note
that the variable σ verifies
1
D¯ σ =D¯ s+divj = (σfr :∇v−j ·∇T). (2.16)
t t s s
T
Therefore, D¯ σ corresponds to the total entropy generation rate density of the system. At
t
this stage, the expressions of σfr and j are still not specified.
s
In absence of irreversible processes (i.e., if σfr =0 and j =0), equations (2.15) recover
s
thegeneralformoftheEuler-Poincar´eequations derivedinHolm, Marsden and Ratiu[2002]
by Lagrangianreduction.
We now write this system in the case of the Lagrangian L of the dry atmosphere given
in (2.7), but keeping a general expression for the internal energy. The partial derivatives
are
∂L ∂L 1 ∂L ∂u
=ρ(v+R), = |v|2+R·v−Φ−u+ηT −vp, =− =−T.
∂v ∂ρ 2 ∂s ∂η
On one hand, we note that
∂L ∂L 1
ρ∇ +s∇ =ρ∇ |v|2+R·v −ρ∇Φ−∇p,
∂ρ ∂s 2
(cid:18) (cid:19)
on the other hand, using D¯ ρ=0, we have
t
T
(∂t+£v)(ρ(v+R))=ρ(∂tv+v·∇(v+R)+∇v ·(v+R)).
From (2.15) we thus get, as desired, the equations of motion for a heat conducting gas
subject to the Coriolis, gravitational, and viscous forces:
ρ(∂ v+v·∇v+2Ω×v)=−ρ∇Φ−∇p+divσfr
t
 T(D¯ s+divj )=σfr :∇v−j ·∇T (2.17)
t s s
 D¯ ρ=0.
t
The second equation in (2.17) can be rewritten in terms of the temperature as
1
D T =−ρc2Γdivv+ σfr :∇v−div(Tj ) , (2.18)
t s ρC s
v
(cid:0) (cid:1)
whereC (v,T)=T ∂η(T,v)isthespecificheatatconstantvolume,c2(v,η)= ∂p(v,η)isthe
v ∂T s ∂ρ
square of the speed of sound, and Γ(p,η)= ∂T(p,η) is the diabatic temperature gradient.
∂p
2.3 The case of dry air and ideal gases 8
For meteorologicalapplications, the potential temperature θ is preferred as the temper-
aturetodescribethethermodynamicequation,sinceitturnsouttobeconservedinabsence
ofviscosityandheatconduction. Ageneraldefinitionofthe potentialtemperaturevalidfor
any state equation is given by
p0
θ(p,T):=T + Γ(p′,η(p,T))dp′, (2.19)
Zp
where p is a given reference pressure. One obtains, from (2.18), the potential temperature
0
equation
1 ∂θ
D θ = σfr :∇v−div(Tj ) . (2.20)
t s
ρC ∂T
p
(cid:0) (cid:1)
2.3 The case of dry air and ideal gases
The above development is applicable to any state equation. For the case of a perfect gas,
suchasthedryair,thereareseveralsimplifications. ThespecificheatcoefficientC in(2.18)
v
is a constant and from (2.1), we have p = ρR T in (2.17) and ρc2 = p in (2.18), where
d s ρCv
R is the gas constantfor dry air. In this case, the potential temperature in (2.19) recovers
d
its usual expression θ = T/π, where π := (p/p0)Rd/Cp is the Exner pressure associated to
p . One obtains, from (2.20), the potential temperature equation
0
θ
D θ = (σfr :∇v−div(Tj )). (2.21)
t s
ρTC
p
It is the system (2.17) with the second equation replaced by (2.21), together with the state
equationfortheidealgasp=ρR T,thatistraditionallyusedtodescribethedynamicsofa
d
dryatmosphere,see,forinstance,Gill[1982]. Ourvariationalapproachishowevernaturally
expressed in terms of the entropy density rather than the potential temperature.
In some atmospheres, the perfect gas hypothesis may no longer be made. For example,
for the atmosphere of Venus, while the state equation of an ideal gas can still be used, the
specific heat depends on the temperature (Seiff et al. [1985]). An analytic approximation
for this temperature dependence is givenby C (T)=C (T/T )ν, where C ,ν,T are con-
p p0 0 p0 0
stants, see Lebonnois et al [2010], which yields a correspondingexpressionfor the potential
temperature,accordingtothegeneraldefinitionabove. Ofcourse,ourvariationalformalism
does apply to this case, as it does for any one component gas.
2.4 Heat exchanges
So far, we have considered the atmosphere as an isolated system, i.e., with no exchange of
work,heatormatterwithitssurroundings(space,oceanorearth’ssurface). Therighthand
side of the entropy equation in (2.17) thus only consists of a net production of entropy due
to the irreversible processes.
Heatexchanges between the atmosphere andits surroundings,such as radiativeheating
andcooling,surfacesensibleheatfluxandsurfacelatentheatflux,canbeeasilyincorporated
in our variational formulation, as long as these are considered as external processes.
We incorporate the heating into our variational formalism (2.4)–(2.6) by modifying the
phenomenological constraint (2.5) to
∂L
Σ˙ =−Pfr :∇ϕ˙ +J ·∇Γ˙ −R, (2.22)
S
∂S
whereRdenotestheheatingratedensityinLagrangianrepresentation. Thevariationalcon-
straint (2.6) is however kept unchanged. This follows a general principle on the variational
2.5 Potential vorticity and circulation theorem 9
formulation of thermodynamics stated in Gay-Balmaz and Yoshimura [2017a,b], namely,
that external effects only affect the phenomenological constraint and not the variational
constraint. Schematically, one passes from the phenomenological constraint to the varia-
tional constraint as J Λ˙ +P J δΛ , where P denotes the power density
α α α ext α α α ext
associated to heat transfer between the system and the exterior.
P P
In Eulerian representation, the variational formalism (2.11)–(2.14) is thus modified by
adding the contribution of the heating in (2.12) as
∂L
D¯ σ =−σfr :∇v+j ·∇D γ−r, (2.23)
t s t
∂s
whereRandrarerelatedasR(t,X)=r(t,ϕ(t,X))|∇ϕ(t,X)|. Thisresultsinamodification
of the entropy equation in (2.15) into
T(D¯ s+divj )=σfr :∇v−j ·∇T +r (2.24)
t s s
and corresponding modifications in the temperature and potential temperature equations.
2.5 Potential vorticity and circulation theorem
Potentialvorticityis a scalarfield constructedby projecting the absolute vorticityonto the
gradientofthepotentialtemperature(or,moregenerally,thegradientofanadvectedscalar
field) anddividing the resultbythe totalmassdensity. Potentialvorticityis a fundamental
quantity since it allows one to understand many processes taking place in the atmosphere
and the ocean (e.g., Hoskins, McIntyre, and Robertson [1985]). In absence of irreversible
processes, it turns out to be a conserved quantity.
We first recall here how the irreversible processes affect the evolution of the potential
vorticity. Then we extend the definition of potential vorticity to the general setting of the
Lagrangiansystem (2.15) and write its evolution equation. Having such a general equation
is useful for a unified treatment of potential vorticity for various approximations of the
equations of atmospheric dynamics, with the irreversible processes included.
TheRossby-Ertelpotentialvorticityq isdefinedbyρq =ζ ·∇θ,whereζ =curlv+2Ω
a a
is the absolute vorticity and θ is the potential temperature defined in (2.19). Its evolution
equation
∂θ
ρD q =div ρ−1divσfr×∇θ+ Qζ ,
t ∂T a
(cid:18) (cid:19)
follows from the evolution of the absolute vorticity ∂ ζ +curl(ζ ×v) = ρ−2∇ρ×∇p+
t a a
curl(ρ−1divσfr), the conservation of mass, and the potential temperature equation (2.21)
(see e.g., Haynes and McIntyre [1986]). In absence of viscosity and heat conduction, q is
conserved.
In the more general situation of the system (2.15) associated to a general expression of
the Lagrangian density L, we can consider a potential vorticity q˜defined as ρq˜= ζ ·∇ψ,
a
where ζ :=curl ρ−1∂L/∂v and ψ is a scalar field satisfying an evolution equation of the
a
type D ψ =ψ˙. The first equation in (2.15) implies
t (cid:0) (cid:1)
∂L
∂ ζ +curl(ζ ×v)=∇η×∇ +curl(ρ−1divσfr). (2.25)
t a a ∂s
From this equation and D ψ =ψ˙, one obtains the evolution of potential vorticity as
t
∂L
ρD q˜=div η∇ ×∇ψ+ρ−1divσfr×∇ψ+ψ˙ζ . (2.26)
t ∂s a
(cid:18) (cid:19)
2.6 Phenomenological relations and entropy production 10
If, for example, ψ is chosen as the specific entropy η, then (2.26) simplifies to
ρD q˜=div ρ−1divσfr×∇η+η˙ζ ,
t a
where, from the second equation in (2.(cid:0)15), η˙ = 1 (σfr :∇v−(cid:1)div(Tj )) with T =−∂L.
Tρ s ∂s
LetusnowconsiderKelvin’scirculationtheorem. Forthesystem(2.17)itdirectlyfollows
from the balance of momentum as
d
(v+R)·dx= ρ−1divσ·dx, σ =−pδ+σfr,
dt
Ict Ict
where c is a loop advected by the wind flow. Its more general version for system (2.15) is
t
found to be
d 1∂L ∂L
·dx= η∇ +ρ−1divσfr ·dx.
dt ρ∂v ∂s
Ict Ict(cid:18) (cid:19)
In absence of viscosity it recovers the general form of Kelvin’s circulation theorem in the
Euler-Poincar´esetting derived in Holm, Marsden and Ratiu [2002].
2.6 Phenomenological relations and entropy production
Thephenomenologicalconstraints(2.5)and(2.12)tobeusedinthevariationalformulations
are determined by the expressions of the thermodynamic fluxes σfr and j in terms of the
s
thermodynamic forces ∇v and ∇T. The phenomenological expression of these fluxes must
be in agreement with the second law of thermodynamics, which requires that the internal
entropy production, given in equation (2.16), is positive. When only viscosity and heat
conduction are considered, we have the well-known relations
2
σfr =2µ(Defv)+ ζ− µ (divv)δ and Tj =−k∇T (Fourier law), (2.27)
s
3
(cid:18) (cid:19)
with Defv= 1(∇v+∇vT) the deformation tensor, and where µ≥0 is the first coefficient
2
of viscosity (shear viscosity), ζ ≥ 0 is the second coefficient of viscosity (bulk viscosity),
and k ≥ 0 is the thermal conductivity. In general, these coefficients may all depend on
thermodynamic variables. For the atmosphere below 100 km, µ is so small (µ = 1.7×
10−5kg m−1s−1 for standard atmospheric conditions) that molecular viscosity is negligible
except in a thin layer within a few centimeters of the earth’s surface where the vertical
shear is very large. The second coefficient of viscosity is irrelevant in incompressible flows
and notoriouslydifficult to measure in compressible flows. A common assumptionis Stokes
hypothesis ζ =0.
3 Variational derivation of the thermodynamic of a moist
atmosphere
Moist air consists of dry air and water substance in its different phases, namely, water
vapor, liquid water, and solid water. The gas component of moist air is thus a mixture of
dry air and water vapor. The liquid and solid phases of water both consist of an airborne
component (cloud) that follows the motion of the gas component, and of a precipitating
component (rain or snow) that has relative motion with respect to the gas component.
Weshallnotconsiderthe precipitatingcomponentinthis paper. Also,forsimplicity,we
do not consider the ice phase, but it can be easily added. We thus assume that moist air