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Hamiltonian Description of Idealized Binary Geophysical Fluids

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  • 1 Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania
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Abstract

A Hamiltonian formulation for the dynamics and thermodynamics of a compressible, rotating, binary fluid subject to gravity is developed. Here, binary refers to the presence of two components of the fluid, such as solids dissolved in a liquid or gaseous and liquid water existing along with dry air. These fluids are idealized in that the influences of diffusion processes are ignored and the binary flow is restricted to a single velocity.

The equations are presented in generic form applicable to an arbitrary binary geophysical flow. The relevant Poisson bracket satisfies Jacobi's identity. Three distinct Casimir invariants are described. The first reflects the conservation of entropy and concentration of the minor component. The second is a consequence of the conservation of the absolute circulation on curves formed by the intersection of surfaces of constant entropy with surfaces of constant concentration. The third is a generic potential vorticity of the form (ω  ·  ∇λ)/ρ. Here, ω is the absolute vorticity, ρ is the total density of the fluid, and λ is any thermodynamic variable. For example, λ can be the pressure, density, temperature, or mixing ratio as well as the more common choice of potential temperature.

Available energy of the system is defined locally in the finite-amplitude as well as in the small-amplitude limit. Both definitions are partitioned into available potential and available elastic energies.

A linear stability analysis indicates that the fluid is statically stable provided the square of the sound speed is positive, the total density decreases with height, and the square of a suitably defined buoyancy frequency is positive.

The formulation is applicable to a salty ocean and to a moist atmosphere. For the atmosphere, the full theory holds in the presence of either liquid water or ice in equilibrium with its vapor.

Corresponding author address: Peter R. Bannon, Dept. of Meteorology, The Pennsylvania State University, University Park, PA 16802. Email: bannon@ems.psu.edu

Abstract

A Hamiltonian formulation for the dynamics and thermodynamics of a compressible, rotating, binary fluid subject to gravity is developed. Here, binary refers to the presence of two components of the fluid, such as solids dissolved in a liquid or gaseous and liquid water existing along with dry air. These fluids are idealized in that the influences of diffusion processes are ignored and the binary flow is restricted to a single velocity.

The equations are presented in generic form applicable to an arbitrary binary geophysical flow. The relevant Poisson bracket satisfies Jacobi's identity. Three distinct Casimir invariants are described. The first reflects the conservation of entropy and concentration of the minor component. The second is a consequence of the conservation of the absolute circulation on curves formed by the intersection of surfaces of constant entropy with surfaces of constant concentration. The third is a generic potential vorticity of the form (ω  ·  ∇λ)/ρ. Here, ω is the absolute vorticity, ρ is the total density of the fluid, and λ is any thermodynamic variable. For example, λ can be the pressure, density, temperature, or mixing ratio as well as the more common choice of potential temperature.

Available energy of the system is defined locally in the finite-amplitude as well as in the small-amplitude limit. Both definitions are partitioned into available potential and available elastic energies.

A linear stability analysis indicates that the fluid is statically stable provided the square of the sound speed is positive, the total density decreases with height, and the square of a suitably defined buoyancy frequency is positive.

The formulation is applicable to a salty ocean and to a moist atmosphere. For the atmosphere, the full theory holds in the presence of either liquid water or ice in equilibrium with its vapor.

Corresponding author address: Peter R. Bannon, Dept. of Meteorology, The Pennsylvania State University, University Park, PA 16802. Email: bannon@ems.psu.edu

1. Introduction

Hamiltonian formulations of the equations of motion have been applied to a variety of idealized flows (see the reviews by Salmon 1988, 1998; Shepherd 1990; Morrison 1998). Particularly noteworthy is the ability of the Hamiltonian approach to systematize the derivation of the expression for the available energy (Shepherd 1993). Here available energy is the nonkinetic contribution to the pseudoenergy and has potential and elastic components.

The purpose of the present investigation is to extend the Hamiltonian description of idealized rotating compressible flows to the case where the fluid is a binary mixture of two flow components whose composition is not homogenous. The mixture is idealized to be in approximate local thermodynamic equilibrium. Diffusive processes are ignored. It is further assumed that the velocities of the two components are the same.

The next section presents the generic equations for such a binary geophysical fluid. Section 3 casts the system into a Hamiltonian formulation. This formulation extends that for the single-component system to include a prognostic equation for the relative concentration of the two components. Section 4 describes the range of Casimir invariants. These include a generalized potential vorticity that is implicit in the kinematic discussions of Truesdell (1951, 1954) and Haynes and McIntyre (1987). Section 5 describes the available energy for the system, its partitioning into available potential and available elastic energies, and its small amplitude limit. The approach follows that of Shepherd (1993) to yield a local expression like that of Andrews (1981) relative to an arbitrary reference state. Section 6 presents a linear stability analysis of the flow. Sections 7 and 8 apply these findings to a moist atmosphere and a salty ocean. The formulation for the atmosphere allows the water vapor to exist in equilibrium with either liquid or solid water and to undergo a change of phase. It generalizes the moist available potential energy of Lorenz (1979) to a local expression relative to an arbitrary reference state. It is also shown that, in principle, the nonlinear theory is applicable to the oceans.

2. The model geophysical fluid

The equations for three-dimensional, rotating, compressible, inviscid flow of a single-temperature, single-velocity, binary system are
i1520-0469-60-22-2809-e21a
where v is the three-dimensional velocity, Ω is the rotation rate, p is the pressure, ρ is the total density, Φ is the geopotential, η is the total entropy per unit total mass, and χ is the generic concentration describing the relative amount of mass of the sparse component to that of the more abundant component. It is assumed that both components move with the same velocity v and have the same temperature T.
We write the thermodynamic identity for the internal energy u per unit total mass as
i1520-0469-60-22-2809-e22
where μ is the chemical potential of the mixture. Because u is a function of ρ, η, and χ, we may write
i1520-0469-60-22-2809-e23
Comparison of these last two expressions implies
i1520-0469-60-22-2809-e24
and the commutability of the second derivatives yields Maxwell's relations in the form
i1520-0469-60-22-2809-e25
It is also convenient to introduce the enthalpy, h = u + p/ρ. Then the thermodynamic identity becomes
i1520-0469-60-22-2809-e26
For the present we assume the definitions for u, h, η, χ, and μ are known. Their specification is postponed until sections 7 and 8 in which applications to the atmosphere and ocean are made.

3. Hamiltonian description

The prognostic system of Eqs. (2.1) is Hamiltonian in the variables u = (u1, u2, u3, u4, u5, u6)T = (v, ρ, η, χ)T with the Hamiltonian function
i1520-0469-60-22-2809-e31
describing the total energy of the system. The prognostic equations may be written in Cartesian tensor notation:
i1520-0469-60-22-2809-e32
where δH/δuj denotes a functional derivative, and Jij is the tensor:
i1520-0469-60-22-2809-e33
where the velocity and absolute vorticity are
i1520-0469-60-22-2809-e34
respectively. We note that the functional derivatives of the Hamiltonian are
i1520-0469-60-22-2809-e35a
Then the Hamiltonian description (3.2) becomes
i1520-0469-60-22-2809-e36
The first three equations represent the momentum equation in the form
i1520-0469-60-22-2809-e37
where the pressure gradient is related to the other flow variables by the spatial derivative form of the thermodynamic identity (2.6). The fourth, fifth, and sixth equations in this set represent the equation of continuity for total density, and the equations of conservation of entropy and concentration, respectively.
It remains to be shown that J satisfies the Jacobi condition. Its Poisson bracket is
i1520-0469-60-22-2809-e38a
where the integration is over the total volume V of fluid. Using the divergence theorem this expression becomes
i1520-0469-60-22-2809-e38b
where we have assumed that
i1520-0469-60-22-2809-e39
on the boundaries of the domain. Here, is the unit normal on the boundaries. It is traditional (e.g., Morrison 1998) to neglect these boundary contributions. The bracket (3.8b) may be put into Lie–Poisson form by a transformation from the variables v, η, and χ to the variables M (=ρv), σ (=ρη), and X (=ρχ) in a manner similar to Morrison (1998, p. 490). This transformation amounts to writing (2.1) in flux form. The result,
i1520-0469-60-22-2809-e310
implies that the Jacobi identity is satisfied because the contribution of X is isomorphic with that of σ and ρ and, apart from the contributions from F and G, (3.10) is linear in [M, ρ, σ, X] (Morrison 1982).

4. Casimir invariants

The Hamiltonian description presented in section 3 is an Eulerian one for the six variables u. In principle, a Lagrangian description may be constructed for nine variables (u plus the three spatial coordinates). This reduction in the number of variables from the Lagrangian to the Eulerian description implies that the Eulerian description is noncanonical. Mathematically this reduction is reflected in the fact that the matrix 𝗝 is singular. As a consequence, Eulerian Hamiltonian descriptions possess global invariants. Because of the reduction in the number of invariants (Kuroda 1990), the Hamiltonian system should possess three distinct Casimir invariants. These Casimirs are functionals C that have a vanishing Poisson bracket:
i1520-0469-60-22-2809-e41
If the arbitrary functional F is a function of ρ, η, χ, or v separately, then C must satisfy the four constraints
i1520-0469-60-22-2809-e42a
Three distinct solutions to (4.2) are
i1520-0469-60-22-2809-e43a
where f and g are arbitrary functions of η and χ, and va is the absolute velocity. Here, the generalized potential vorticity is
i1520-0469-60-22-2809-e44
where λ = λ(ρ, η, χ) is an arbitrary thermodynamic scalar function.
The first Casimir, C1, is a consequence of the material conservation of entropy and concentration and expresses the global invariance of these two quantities (and combinations thereof). This thermodynamic Casimir C1 has the functional derivatives
i1520-0469-60-22-2809-e45a
The first three constraints of (4.2) are trivially satisfied while the last is satisfied by the chain rule. Thus, C1 is a Casimir.
The second Casimir, C2, is a consequence (Salmon 2003, personal communication) of the conservation of the absolute circulation
i1520-0469-60-22-2809-e46
where L is a curve formed by the intersection of a surface of constant entropy η with a surface of constant concentration χ. If the curve L is closed, then the circulation is conserved. If the curve L does not close, then the circulation is also conserved by the assumption to neglect any boundary contributions. Letting η, χ, and a be the mass-labeling coordinates such that the density is the Jacobian of the transformation between the mass-labeling and Cartesian coordinates,
i1520-0469-60-22-2809-e47
Because only a varies along the curve L, we can rewrite the circulation as
i1520-0469-60-22-2809-e48
Then
i1520-0469-60-22-2809-e49
is also conserved because Γ is conserved on each curve L of constant η and χ. Transforming the integral to physical space yields
i1520-0469-60-22-2809-e410
which is equivalent to (4.3b). The functional derivatives of C2 are
i1520-0469-60-22-2809-e411a
and it is straightforward to show that these satisfy (4.2). Thus, C2 is a Casimir.
The third Casimir, C3, is a consequence of the kinematic structure of potential vorticity (see Truesdell 1951, 1954; Haynes and McIntyre 1987). This kinematic Casimir arises directly from the fact that the divergence of a curl vanishes. The functional derivatives of C3 are
i1520-0469-60-22-2809-e412a
which all vanish. This result implies that the functional derivatives of the Casimir are identically zero and C3 satisfies the four conditions trivially. This is a very special case, but it holds for an arbitrary thermodynamic scalar function λ because of the vector identities for the vanishing of the divergence of a curl and the curl of a gradient. Thus, Eqs. (4.3) are the Casimir invariants.

5. Available energy

A pseudoenergy may be constructed following Shepherd (1993) in terms of the Hamiltonian and a Casimir C relative to a base state u0 as
i1520-0469-60-22-2809-e51
Then we define the available energy as the nonkinetic energy contribution to the pseudoenergy. To construct the pseudoenergy for a generic geophysical flow we consider a type-1 Casimir invariant (4.3a) with functional derivatives (4.5) and choose a resting base-state u0 such that
i1520-0469-60-22-2809-e52
that is, in hydrostatic balance
i1520-0469-60-22-2809-e53
Because the density is a positive quantity, this relation implies that the base-state pressure is a monotonic function of height. Thermodynamically, we have that
p0p0ρ0η0χ0
But with the hydrostatic relation, this implies
i1520-0469-60-22-2809-e55
Similarly, ρ0 = ρ0(η0, χ0). Here, the Casimir C must satisfy
i1520-0469-60-22-2809-e56
when evaluated in the base state. The last condition is satisfied trivially. The remaining three conditions require that f satisfy
i1520-0469-60-22-2809-e57a
where the subscript 0 indicates the quantity in parentheses has been evaluated in the base state (5.2). For example, u0 is the base-state internal energy. In order to satisfy the first condition we choose f such that
i1520-0469-60-22-2809-e58
where
Zη,χ0z.
This relation is implicit in the hydrostatic relation. Clearly, (5.8) satisfies (5.7a) when evaluated in the base state. We next consider the second condition (5.7b) and evaluate
i1520-0469-60-22-2809-e510
Using (2.4a,b) and evaluating in the base state yields
i1520-0469-60-22-2809-e511
The last equality holds because p0 = p0(η, χ) and p0 = p0(z), so
i1520-0469-60-22-2809-e512
and the second condition is satisfied. Similarly, using (2.4b,c),
i1520-0469-60-22-2809-e513
and the third condition (5.7c) is satisfied.
The available energy per unit volume is the nonkinetic contribution to (5.1) or
i1520-0469-60-22-2809-e514
Then using (5.8) we have
i1520-0469-60-22-2809-e515
which is the generalization of the result (9.12) of Shepherd (1993).
The available energy may be partitioned into available potential energy (APE) and available elastic energy (AEE), A = AEE + APE. Following Andrews (1981) the available elastic energy per unit volume is
i1520-0469-60-22-2809-e516
where h is the enthalpy. Then APE = A − AEE. If the departures of u from the base state u0 are small, then (5.15) may be expanded in a Taylor series about u0. To leading order, one finds
i1520-0469-60-22-2809-e517
where the subscripts ρ, η, and χ denote differentiation with respect to those variables but the subscript 0 denotes evaluation in the base state. Evaluation of the second derivatives is relatively straightforward but tedious. Some details are summarized in the appendix. In addition, use is made of the relation
i1520-0469-60-22-2809-e518
With some effort, one finds that
i1520-0469-60-22-2809-e519
where the first term on the right-hand side is the linear APE and the second the linear AEE. Here, c0 is the speed of sound
i1520-0469-60-22-2809-e520
and the square of the buoyancy frequency is
i1520-0469-60-22-2809-e521
It may also be verified that the linearized form of (5.16) is the second term in (5.19). These results are the generalization of the one-component expressions to a binary fluid.

6. Linear stability analysis

We next examine the stability of the hydrostatic base state (5.2). The pseudoenergy (5.1) vanishes when evaluated in the base state. In addition, its first variation vanishes in the base state by the construct (5.6) of the Casimir C. Thus, the state u0 is an extremum. For linear stability, it is sufficient that the second variation of the pseudoenergy be positive. This convexity condition is
i1520-0469-60-22-2809-e61
where the subscripts denote differentiation. For linear stability it is sufficient that the integrand be positive definite. The kinetic energy contribution is positive. The available energy contribution is a third-order quadratic form. Stability is ensured if this contribution is nonnegative. Based on the theory of quadratic forms (e.g., Hildebrand 1965), this contribution is positive semidefinite if
i1520-0469-60-22-2809-e62a
when evaluated in the base state. The vanishing of the third condition (6.2c) means that the 3 × 3 symmetric matrix associated with the quadratic form is singular and that a 2 × 2 submatrix contains the necessary information. Then it is sufficient for stability that
i1520-0469-60-22-2809-e63a
Using the results in the appendix, we find
i1520-0469-60-22-2809-e64a
where c0 is the speed of sound (5.20) and the square of the buoyancy frequency is given by (5.21). Then stability is ensured if the sound speed is real, the density decreases with height, and the square of the buoyancy frequency is positive. While these stability conditions are well known, their derivation here for a compressible binary fluid using a Hamiltonian description is new. Identical stability requirements result from conditions analogous to (6.3) but for the derivatives with respect to ρ and χ. Those conditions with respect to η and χ fail to capture all of the stability requirements. This result reflects the singularity of the 3 × 3 matrix.

7. Application to a moist atmosphere

In order to apply the preceding results to a moist atmosphere it is necessary to provide the equation of state and the definitions for the thermodynamic-state variables for entropy, internal energy, and enthalpy such that the fundamental thermodynamic relation (2.2) is satisfied. Assuming an ideal gas behavior, the equation of state for the total pressure p is
i1520-0469-60-22-2809-e71
where ε = Ra/Rυ = 0.622 is the ratio of the molecular weight of water vapor to that of dry air, and Ra and Rυ are the gas constants for dry air and water vapor. The mixing ratios of water vapor rυ and total water r are
i1520-0469-60-22-2809-e72a
where ρa, ρυ, and ρw are the densities of the dry air, water vapor, and total water. The total density is related to the density of dry air by ρ = ρa(1 + r). The vapor pressure e is
i1520-0469-60-22-2809-e73
where the partial pressure due to dry air is pa = pe. Then the mixing ratio is rυ = εe/pa. In keeping with atmospheric conventions we replace the symbol η with s for atmospheric entropy and replace the symbol χ with r for the mixing ratio of total water. Because all the flow components move with the same velocity, there is no fallout and the total mixing ratio r satisfies a conservation statement of the form (2.1d). Following Lorenz (1979) and Emanuel (1994), the thermodynamic fields may be expressed in a general manner so that they are valid for unsaturated flow (r = rυ) without liquid water (rlrrυ = 0) and for saturated flow (rυ = rrl) with liquid water (rl > 0).
The entropy s per unit mass of cloudy air is
i1520-0469-60-22-2809-e74
The specific entropies for the dry air, water vapor, and liquid water are
i1520-0469-60-22-2809-e75a
where sa0, sυ0, and sl0 are reference values of the entropy of dry air, water vapor, and liquid water. Here, cpa and cpv are the specific heats at constant pressure for the dry air and water vapor, while cl is the specific heat of the liquid water. We assume that these specific heats are constants independent of temperature but allow the enthalpy of vaporization lυ to satisfy Kirchhoff's relation dlυ = −(clcpv)dT. In saturated conditions, the saturation vapor pressure esat is defined as a function of temperature by the Clausius–Clapeyron equation
i1520-0469-60-22-2809-e76
such that esat = 6.11 mb at T = 273.16 K.
Similarly the internal energy is
i1520-0469-60-22-2809-e77
where ua0 and uυ0 are reference values of the internal energy:
i1520-0469-60-22-2809-e78a
and T00 is a reference temperature. Then the enthalpy h per unit mass of cloudy air is
i1520-0469-60-22-2809-e79
Note that this relation defines the enthalpies of the dry air and water vapor as
i1520-0469-60-22-2809-e710a
The thermodynamic identity (2.2) for the present system can be shown to have the form
i1520-0469-60-22-2809-e711
where the chemical potential for this system is (1 + r)2μ = μυμa and the chemical potentials of the dry air and water vapor are
i1520-0469-60-22-2809-e712a
This result follows in a clear and easily interpreted fashion because the definitions for the entropy and internal energy retained the reference values of the fields (cf. Lorenz 1979). Having served their purpose, these reference values do not alter (2.1) or the Hamiltonian formulation and may be dropped in the subsequent analysis. [We note that Eq. (17) of Lorenz is valid for his problem where the differentials are material ones following a cloudy air parcel that conserves its total water, dr = 0.]
In order to utilize the expression for the available energy (5.15), it is necessary to express the internal energy in terms of the state variables ρ, s, and r. Formally this requirement is met by the preceding set of relations. The ideal gas laws may be used to eliminate the partial pressures from the definition of the entropy (7.4) to yield an equation for the temperature field as a function of ρ, s, and r. Then the internal energy (7.7) and total pressure (7.1) are also functions of ρ, s, and r. This relation for the pressure holds in the unsaturated case because rυ = r and in the saturated case because rυ = rsat, where
i1520-0469-60-22-2809-e713
and the saturation vapor pressure is only a function of temperature by the Clausius–Clapeyron Eq. (7.6). In order to partition the available energy into available elastic and potential energies, the enthalpy (7.9) must be expressed in terms of the pressure, entropy, and total mixing ratio. Again this is formally possible. Because (5.15) and (5.16) involve differences in the internal energy and enthalpy for the same values of χ = r, the reference values ua0 and uυ0 need not be specified and may be dropped from the subsequent analysis.
The small-amplitude expressions for the available energies (5.19) are more readily accessible as they only require expressions for the speed of sound and buoyancy frequency. It is a tedious exercise to demonstrate from the above relations that these are given by
i1520-0469-60-22-2809-e714a
for the unsaturated case. Here,
i1520-0469-60-22-2809-e715a
In the saturated case we find results that agree with Lalas and Einaudi (1974) and Durran and Klemp (1982).

We note that the formulation presented here for a moist atmosphere with water vapor and liquid water is readily modified for one with water vapor and solid water.

8. Application to the ocean

To apply the Hamiltonian description to the ocean we choose the fundamental thermodynamic state variables to be the density ρ, the entropy η, and the concentration is taken to be the salinity χS. The salinity S is the mass of sea salt per unit mass of seawater. In contrast to the atmospheric thermodynamics that may be expressed completely and analytically, the oceanic case is complicated by the reliance on tabular expressions for the state variables (e.g., Fofonoff 1962, 1985). The approach of Feistel (1993; see also Feistel and Hagen 1994, 1995) is adopted here because it presents a synthesis of earlier results that is internally mathematically consistent. In this approach, the equation of state for seawater and the Gibbs free energy G are given as functions of the measurable state variables of salinity, temperature, and pressure:
i1520-0469-60-22-2809-e81a
In addition the temperature is expressed as a function of the salinity, entropy, and pressure:
TTS,η,p
These three relations are sufficient in principle to express the internal energy and enthalpy in forms suitable for use in the preceding results. Use of the temperature relation (8.2) in (8.1a) implies that the density is a function of salinity, entropy, and pressure. Formally, this may be inverted to yield an expression for the pressure as a function of density, entropy, and salinity. Similarly, (8.2) may be used in the Gibbs function (8.1b) to eliminate temperature. Thus, we find
i1520-0469-60-22-2809-e83a
Using (8.3) we find that the Gibbs free energy is also a function of density, entropy, and salinity:
GGρ,η,S
Again, using (8.3) in (8.2) we have
TTρ,η,S
Because the internal energy u = G + p/ρ, the results (8.3)–(8.5) imply
uuρ,η,S
and the thermodynamic identity (2.2) becomes
i1520-0469-60-22-2809-e87
where μsw is the chemical potential of seawater (i.e., the difference between the specific chemical potential of salt is seawater and that of water in seawater; Fofonoff 1962; Feistel 1993). Formally then, this idealized ocean system is Hamiltonian.

Lastly because the enthalpy h = G + Tη, it may be expressed using (8.2) and (8.3) as h = h(p, η, S). Thus, the available energy relations of section 5 are applicable to the ocean in both their finite- and small-amplitude forms.

9. Conclusions

The Hamiltonian description of a compressible, rotating flow has been extended to the case of a binary fluid. This generic description in section 3 is applicable to any fluid system satisfying the thermodynamic identity (2.2). This constraint is satisfied by an ocean idealized to contain a single solute, the salinity, and by a moist atmosphere idealized to contain water in gaseous and either liquid or solid form. [A moist atmosphere containing water vapor and liquid and solid water is not Hamiltonian in the present description because the thermodynamic identity is more general than that of (2.2).] The Casimir invariants of the generic binary fluid are given by (4.3). They reflect the conservation of entropy and concentration, the conservation of circulation on lines formed on the intersection of surfaces of constant entropy and surfaces of constant concentration, and the conservation of a generalized potential vorticity. In (4.3c) the thermodynamic scalar function λ may be the pressure, density, temperature, salinity, or water mixing ratio.

The result (5.15) for the available energy extends the result of Andrews (1981) to a binary system. Like that of Andrews, the available energy is a local, positive-definite definition of the nonkinetic contribution to the pseudoenergy. Following Andrews (1981, section 5) the present moist result is similarly related to the moist available potential energy of Lorenz (1979) as his dry result is to that of Lorenz (1955). The utility of the present result in an Eulerian framework is severely restricted because it is required that every value of entropy and concentration be present in the reference state. In a Lagrangian framework, one could use (5.15) to track the evolution and partitioning of the available and kinetic energies for individual fluid parcels. Such an application to moist parcel dynamics will be presented in another forum.

Acknowledgments

The National Science Foundation under Grants ATM-9820233 and ATM-0215358 provided partial support for this research. I thank Dennis Lamb for fruitful discussions on moist thermodynamics and Raymond G. Najjar for bringing the work of Rainer Feistel to my attention. Reviewer Rick Salmon graciously provided Casimir C2 after noting a deficiency in my original formulation.

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APPENDIX

Derivatives of the Available Energy

This appendix presents some details of the evaluation of the second derivatives of the available energy per unit volume A. It employs the thermodynamic relations (2.4), as well as Maxwell's relations (2.5), and the fact that
i1520-0469-60-22-2809-ea1
implies
i1520-0469-60-22-2809-ea2
Similar relations hold for differentiation with respect to χ. The second derivative with respect to η is one of the most involved calculations. An intermediate result is
i1520-0469-60-22-2809-ea3
We note that, in general,
i1520-0469-60-22-2809-ea4
but for the base state p0 = p0(η, χ), so the composite derivatives (Callen 1985) are
i1520-0469-60-22-2809-ea5a
Thus, (A.3) may be written as
i1520-0469-60-22-2809-ea6a
In summary, we find
i1520-0469-60-22-2809-ea7a
Then we have
i1520-0469-60-22-2809-ea9a
Lastly, one can show that (6.2c) holds using these results and (A.5).
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