## 1. Introduction

Geophysical flows involve a large range of motions operating on multiple spatial and temporal scales, including propagating waves of various types, buoyancy-driven convective overturning, and geostrophic flow constrained by rotation. All these flows can be represented based on the compressible Navier–Stokes equations. However, because the compressible Navier–Stokes equations account for all types of fluid motions, they are often not needed to study specific aspects of the circulation. The details of sound wave propagation, in particular, are generally considered to be irrelevant for flows on scales of, say, 100 km and less, and this motivates the derivation and study of “soundproof” flow equations. These capture the net effect of sound waves (i.e., the rapid local equilibration of the thermodynamic pressure) while suppressing the need to numerically represent the propagation of sound waves.

Over the years, several mathematical approximations of the compressible Navier–Stokes equations have been developed with the specific goal of filtering out specific types of motions. The Boussinesq approximation (Boussinesq 1903) is probably the best known of these. It filters out sound waves by replacing the continuity equation by an incompressibility condition. The anelastic and pseudoincompressible approximations have further expanded the mathematical framework to account for large vertical variations of pressure and density and are the basis for a wide range of numerical models (Ogura and Phillips 1962; Dutton and Fichtl 1969; Lipps and Hemler 1982, 1985; Durran 1989; Bannon 1995, 1996; Prusa et al. 2008; Klein et al. 2010). See also Davies et al. (2003), Klein et al. (2010), and Achatz et al. (2010) for discussions of the regime of validity of soundproof approximations.

The Boussinesq and anelastic approximations rely on partial linearizations of the pressure and gravity terms in the momentum equation to filter out sound waves based on the assumptions of a low Mach number. The pseudoincompressible approximation does not invoke such linearizations in the case of an ideal gas equation of state and in the formulation using the Exner pressure as proposed originally in Durran (1989, 2008). Yet, our thermodynamically consistent extension to general equations of state will require an appropriate linearization [see (10) below]. In all the soundproof models, these approximations raise the issue of how they impact energy conservation and the underlying thermodynamics. Indeed, the fundamental laws of thermodynamics imply a nonlinear relationship among variations of enthalpy, pressure, and entropy. This problem is made worse when studying moist convection, as condensation itself implies a strong nonlinearity that results in a discontinuity in the partial derivatives of the equation of state.

This issue has been recognized for some time. In their original derivation of the anelastic approximation, Ogura and Phillips (1962) note that their derivation could not be extended to a moist atmosphere. The version obtained by Lipps and Hemler (1982) does not conserve energy. It is only recently that Pauluis (2008) showed that it is possible to obtain a thermodynamically consistent version of the anelastic approximation for a moist atmosphere if one accepts a restriction to nearly moist adiabatic background states. By thermodynamic consistency here we mean that the approximation

conserves energy, with an expression for the internal energy that is consistent with the nonlinear equation of state;

maintains the same conversion between internal and mechanical energy as found in a fully compressible fluid for reversible adiabatic flows; and

ensures that diabatic processes have the same impacts on the thermodynamic state variable as for the compressible Navier–Stokes equations.

*J. Fluid Mech.*) in this context.

The purpose of this article is to expand the results of Pauluis (2008) to obtain a thermodynamically consistent version of the pseudoincompressible approximation for a moist atmosphere. Whereas the anelastic approximation’s primary simplifying assumption is that of a time-independent spatial density distribution, the pseudoincompressible approximation relies instead on small pressure variations (Durran 1989, 2008). As shown recently by Klein et al. (2010) and Achatz et al. (2010) through formal asymptotic arguments, the anelastic and pseudoincompressible models share a broad regime of validity involving low Mach number atmospheric flows of an ideal gas with constant specific heat capacities. Almgren et al. (2006) developed a set of low–Mach number model equations for astrophysical applications that reduces to Durran’s pseudoincompressible model in the case of an ideal gas but is applicable to more general fluids. This work did not address the issue of thermodynamic consistency, however, and this motivates the present paper.

There are two main difficulties when dealing with the thermodynamic relationship within the pseudoincompressible equation. First, the derivation of the pseudoincompressible model has been historically tied to the ideal gas law by introducing the Exner pressure and the dry potential temperature as primary prognostic variables. The associated particular transformations do not easily transfer to general equations of state, and this renders adaption of the pressure-based modeling approach to an arbitrary fluid rather difficult. Second, in contrast with the anelastic approximation, the pseudoincompressible approximation keeps a contribution from the pressure perturbation in the buoyancy, which must be accounted for in the conservation of total energy conservation. We manage to overcome these difficulties in this paper.

Our key result is a thermodynamically consistent pseudoincompressible model for fluids with general equation of state that allows for quite general background stratifications. Its derivation closely maintains the spirit of Durran’s original arguments in Durran (1989). An important difference between our model on the one hand and the pseudoincompressible equations of Durran (1989) and Almgren et al. (2006) is that the velocity divergence constraint cannot generally be cast in the usual form **∇** · (*β υ*) = 0 for a prescribed vertically stratified

*β*(

*z*) but must rather be deduced from mass continuity given the equation of state, background pressure distribution, and advection equations for additional scalars representing the internal degrees of freedom of the fluid under consideration.

## 2. Pseudoincompressible approximation for fluids with a general equation of state

*p*,

*S*,

*q*) are the pressure, entropy, and an additional scalar that represents any pertinent additional internal degree of freedom characterizing the fluid microstate. For example,

*q*could be the total water content in a model for moist atmospheric flows. These quantities will serve as our primary thermodynamic variables in what follows. Furthermore,

**is the velocity,**

*υ**g*the acceleration of gravity, and

**k**the vertical unit vector. Through an equation of state for the density

*p*.

On the right-hand sides of (1c) and (1d),

*υ*^{2}/2) +

*gz*. Importantly, changes in total energy are generally associated with sources either of entropy or of any additional internal degrees of freedom. The total energy changes are related to these sources through the thermodynamic potentials, temperature

*T*for entropy, and the Gibbs free energy

*μ*for any other internal degree of freedom. Equation (3) ensures that the total energy is conserved for reversible adiabatic flow (i.e., for vanishing

**F**—that is,

*H*on the primary thermodynamic variables [i.e.,

*H*=

*H*(

*p*,

*S*,

*q*)] defines the equation of state of the fluid. In fact, the inverse density, temperature, and chemical potential

*α*≡ 1/

*ρ*,

*T*, and

*μ*, respectively, satisfy the thermodynamic differential relations

*μ*=

*μ*−

_{υ}*μ*, where

_{d}*μ*and

_{υ}*μ*denote the chemical potentials for water vapor and dry air, respectively.

_{d}The central approximation in Durran’s pseudoincompressible model (Durran 1989) is that deviations of pressure from a given background hydrostatic distribution are small. Along with the original derivation of the model, Durran demonstrated that the equations satisfy a total energy conservation law, where the pseudoincompressible total energy was defined as the sum of internal, kinetic, and gravitational potential energy with the pressure constrained to equal its background hydrostatic value at the given vertical level. Durran’s original derivation of total energy conservation was based on an ideal gas equation of state with constant specific heat capacities and to the adiabatic case, and this allowed for a few simplifications. Here we generalize the model to account for general equations of state and we demonstrate thermodynamic consistency between entropy production, changes of internal degrees of freedom of the fluid, and the total energy balance given a suitable definition of “pseudo-incompressible thermodynamic potentials.” Our derivation borrows its key ideas from the analogous derivations for an anelastic model by Pauluis (2008).

*g*is the acceleration of gravity. In the following no further assumptions are made regarding

*δp*. In fact, the leading-order terms in the momentum equation from (1b) are those constituting the hydrostatic balance in (7). The actual acceleration, (∂

*+*

_{t}**·**

*υ***∇**)

**, therefore balances with the first- and higher-order perturbations of the pressure gradient term. Keeping all terms up to first order in the pressure perturbation**

*υ**δp*, we find the pseudo-incompressible model

*δp*are not determined by the mass balance through the equation of state. Rather,

*δp*is a perturbation pressure that adjusts to guarantee compliance of the velocity field with a divergence constraint. To reveal this constraint, we carry out the differentiations in (11a), insert (8), (11c), and (11d), and find

*δp*satisfies a second-order elliptic pressure equation for externally prescribed diabatic source terms,

*p**, using the pseudoincompressible mass balance from (11a), and considering that

**and collecting terms:**

*υ**+*

_{t}**·**

*υ***∇**)

*z*≡

*w*, employing again the pseudoincompressible mass balance from (11a) to recast terms in conservation form, and letting

**·**

*υ***∇**

*δp*=

**∇**· (

*δp*) −

**υ***δp*

**∇**·

**, we have**

*υ**δp*. From a physical point of view, the additional corrections to the temperature and Gibbs free energy associated with the dynamic pressure perturbation account for the small changes in these thermodynamics potentials that result from adiabatic fluctuations of the pressure field. For example, if the dynamical pressure perturbation is positive, it implies that the parcel is slightly more compressed than the background reference state, and thus has a slightly higher temperature.

**F**

*and*

_{T}*h*

_{υ}**F**

*:*

_{q}*h*is the specific enthalpy of the water vapor. The diabatic source terms

_{υ}*s*being the specific entropy of water vapor. For consistency, we consider here that the specific enthalpy of water vapor can be related to its entropy and free energy by

_{υ}*h*=

_{υ}*μ*** +

*T**

*s*. This allows us to write the irreversible entropy production as

_{υ}*T*)

**F**

*· (*

_{q}*s*

_{υ}**∇**

*T*+

**∇**

*μ*) = −

*R*

_{υ}**F**

*·*

_{q}**∇**ln

*e*, with

*R*being the specific gas constant and

_{υ}*e*the partial pressure of water vapor. According to the second law of thermodynamics, the internal entropy production must always be positive,

**F**

*and*

_{q}**F**

*. Other processes, such as radiation or frictional dissipation, can be treated similarly by following the classic approach for a compressible fluid but using the value of*

_{T}*T*** and

*μ*** for the corresponding thermodynamic potentials.

It is well known that deviations of the pressure from a hydrostatic background distribution *δp* ~ *M ^{α}* with 0 <

*α*≤ 2; see Klein et al. (2010) for a detailed recent discussion]. Thus, for realistic values

*M*< 0.1, first, the linearization in (27) will be a quite accurate approximation to the real temperature; and second, with

*T*** will generally remain positive and thus represent a meaningful thermodynamic potential. In turn, the pseudoincompressible approximation loses its validity when the perturbation term in (27) is no longer small compared with the leading approximation [i.e., when

Interestingly, the thermodynamically consistent temperature approximation *T*** is obtained by combining the result of a leading-order evolution equation for *T**, which can be derived straightforwardly from (11), with the result of the determining, in many cases elliptic, equation for *δp* (see the appendix).

## 3. Conclusions

A thermodynamically consistent pseudoincompressible model according to the three criteria listed in the introduction has been achieved: As regards the first criterion, the internal energy *p* with *α υ* ·

**∇**

*p*. In the pseudoincompressible case, the pressure field is the sum of the reference pressure and the dynamical pressure perturbation, and the specific volume

*α*includes the leading- and first-order expansion of the density to the pressure perturbation (9). As regards the third criterion, a diabatic process associated with a net heating is related to the change of entropy and water content by

Pauluis (2008) derives a total energy budget analogous to our (20) for the anelastic system. He argues that this equation may be discretized directly and in conservation form instead of the entropy transport equation so as to guarantee total energy conservation. The situation is somewhat more subtle here as the effective thermodynamic potentials *T*** and *μ*** involve the (elliptic) pressure perturbation *δp*. If one follows the line of development in Klein (2009), however, one may consider the total energy equation as providing the divergence constraint while the mass conservation equation effectively describes entropy transport. Then, in a semi-implicit discretization, the terms involving *δp* on the right-hand side of (20) will naturally appear as additional diagonal entries in the elliptic pressure equation. We leave the detailed development of a related numerical method for future work.

## Acknowledgments

R.K. thanks Deutsche Forschungsgemeinschaft for their support through Grants KL 611/14 and SPP 1276 “MetStröm” and the Leibniz Gemeinschaft (WGL) for support through their PAKT program, project “Hochauflösende Modellierung von Wolken und Schwerewellen.” O.P. is supported by the U.S. National Science Foundation under Grant ATM-0545047.

## APPENDIX

### The Elliptic Pressure Equation

In section 2, in the context of (11), we have claimed the pressure perturbation field *δp* to satisfy a second-order elliptic equation as a result of the divergence constraint in (13) for externally prescribed source terms

*δp*on the left-hand side we obtain the Helmholtz-type elliptic equation:

*ρ**,

**; the background state pressure and density**

*υ**p**,

**, or**

*υ*Note that the perturbation pressure equation may become more complex when molecular or turbulent diffusion are included and when

## REFERENCES

Achatz, U., R. Klein, and F. Senf, 2010: Gravity waves, scale asymptotics and the pseudo-incompressible equations.

,*J. Fluid Mech.***663**, 120–147.Almgren, A. S., J. B. Bell, C. A. Rendleman, and M. Zingale, 2006: Low Mach number modeling of type Ia supernovae. I. Hydrodynamics.

,*Astrophys. J.***637**, 922–936.Bannon, P. R., 1995: Potential vorticity conservation, hydrostatic adjustment, and the anelastic approximation.

,*J. Atmos. Sci.***52**, 2302–2312.Bannon, P. R., 1996: On the anelastic approximation for a compressible atmosphere.

,*J. Atmos. Sci.***53**, 3618–3628.Boussinesq, J., 1903:

*Théorie des Analytique de la Chaleur*. Vol. 2. Gauthier-Villars, 645 pp.Davies, T., A. Staniforth, N. Wood, and J. Thuburn, 2003: Validity of anelastic and other equation sets as inferred from normal-mode analysis.

,*Quart. J. Roy. Meteor. Soc.***129**, 2761–2775.Durran, D. R., 1989: Improving the anelastic approximation.

,*J. Atmos. Sci.***46**, 1453–1461.Durran, D. R., 2008: A physically motivated approach for filtering acoustic waves from the equations governing compressible stratified flow.

,*J. Fluid Mech.***601**, 365–379.Dutton, J. A., and G. H. Fichtl, 1969: Approximate equations of motion for gases and liquids.

,*J. Atmos. Sci.***26**, 241–254.Klein, R., 2009: Asymptotics, structure, and integration of sound-proof atmospheric flow equations.

,*Theor. Comput. Fluid Dyn.***23**, 161–195.Klein, R., U. Achatz, D. Bresch, O. M. Knio, and P. K. Smolarkiewicz, 2010: Regime of validity of sound-proof atmospheric flow models.

,*J. Atmos. Sci.***67**, 3226–3237.Lipps, F., and R. Hemler, 1982: A scale analysis of deep moist convection and some related numerical calculations.

,*J. Atmos. Sci.***29**, 2192–2210.Lipps, F., and R. Hemler, 1985: Another look at the scale analysis of deep moist convection.

,*J. Atmos. Sci.***42**, 1960–1964.Ogura, Y., and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere.

,*J. Atmos. Sci.***19**, 173–179.Pauluis, O., 2008: Thermodynamic consistency of the anelastic approximation for a moist atmosphere.

,*J. Atmos. Sci.***65**, 2719–2729.Prusa, J., P. Smolarkiewicz, and A. Wyszogrodzki, 2008: EULAG, a computational model for multiscale flows.

,*Comput. Fluids***37**, 1193–1207.Waters, L. K., G. J. Fix, and C. L. Cox, 2004: The method of Glowinski and Pironneau for the unsteady Stokes problem.

,*Comput. Math. Appl.***48**, 1191–1211.