## 1. Introduction

Small-scale atmospheric phenomena are typically characterized by relatively slow dynamics—that is, low Mach number flows for which the fast acoustic modes are physically irrelevant. Thus, numerical modeling of these flows does not typically require explicitly resolving fast-propagating sound waves. Two different approaches have been widely used to remove the time step constraint that would result from resolving fast acoustic modes. The first and more common approach solves the fully compressible equations of motion but limits the impact of acoustic modes, for instance, by advancing the acoustic signal in time with an implicit time discretization or with multiple smaller time steps, as originally considered for cloud models in Tapp and White (1976) and Klemp and Wilhelmson (1978). A second alternative consists of analytically filtering acoustic modes from the original compressible equations, thus deriving a new set of equations, often called sound-proof equations. Within this category are anelastic (Batchelor 1953; Ogura and Phillips 1962; Dutton and Fichtl 1969; Gough 1969) and pseudoincompressible (Durran 1989) models.

Several anelastic formulations (see, e.g., Clark 1979; Lipps and Hemler 1982; Grabowski and Smolarkiewicz 2002), and recently a pseudoincompressible formalism (O’Neill and Klein 2014), have been developed for moist flows. In this paper we derive a low Mach number model for moist atmospheric flows with a general equation of state using the low Mach number formalism in Almgren et al. (2008) as a starting point. Here we use the term “low Mach number model” to refer to a model in which the equations are valid approximations to the fully compressible equations when the Mach number is small. In atmospheric modeling, the equations that follow from assuming the Mach number is small, thus the variation of the pressure from the background pressure is small, are also referred to as pseudoincompressible equations following Durran (1989). The anelastic equations require additional assumptions on the smallness of density and temperature variations to be valid. [For a complete discussion on sound-proof equations for atmospheric flows, and their differences, see, e.g., Klein (2009) and references therein.] We note that the low Mach number equations presented here do not guarantee that a flow that initially satisfies the low Mach number assumption will continue to satisfy it for all time. The buoyancy forcing from a large density perturbation in a domain with large vertical extent could accelerate the flow into a regime where the Mach number is no longer small. Once the flow reached this regime, the low Mach number equations would no longer be valid approximations to the fully compressible equations. However, until that point, the equations are valid.

Analogous to the moist pseudoincompressible model of O’Neill and Klein (2014) we consider only reversible processes given by water phase changes, using here an exact Clausius–Clapeyron formula for moist thermodynamics. In contrast to O’Neill and Klein (2014), however, we derive the equations of motion in terms of conserved variables [like Ooyama (1990) in a compressible framework]—that is, using appropriate invariant variables such that terms resulting from phase change are eliminated from the time evolution equations (Betts 1973; Tripoli and Cotton 1981; Hauf and Höller 1987). We also include the effects of the specific heats of both water vapor and liquid water and consider an isentropic expansion factor *γ* that accounts for variations in the water composition of moist air. Although the low Mach number formulation holds for any moist equation of state, for the purposes of numerical comparison with compressible solutions we consider the special case where both dry air and water vapor are assumed to be ideal gases.

While the larger time step allowed by the low Mach number formulation can lead to greater computational efficiency than a purely compressible formulation, it may also introduce larger errors in the dynamics of moist flows as investigated in Duarte et al. (2014). In addition to the latent heat release accompanying phase changes, thermodynamic properties such as the specific heat of moist air, as well as thermodynamic variables such as temperature, depend on the composition of the moist air and thus change over the time step. This motivates our use of invariant variables as prognostic variables—namely, total water content and a specific enthalpy of moist air that accounts for both sensible and latent heats. In models where source terms related to phase transition appear explicitly in the evolution equations, they are typically first neglected or lagged in time and then corrected or estimated during a given time step; the use of invariant variables removes the need of accounting for such terms. Nevertheless, the diabatic contribution of the latent heat release must appear in the source term for the low Mach divergence constraint on the velocity field. In practice, the latter involves computing the rate of evaporation (or condensation). Since no analytical expression exists for this rate, one of the most common ways to estimate it deduces it from the change in water vapor content necessary to ensure that there is no supersaturated water vapor at the end of the time step (cf. Soong and Ogura 1973). This variation is measured with respect to an initial estimate of water vapor that does not necessarily respect the saturation requirements either because it was initially advected without accounting for phase changes (see, e.g., Klemp and Wilhelmson 1978; Grabowski and Smolarkiewicz 1990; Bryan and Fritsch 2002; Satoh 2003; O’Neill and Klein 2014) or because it considers a lagged evaporation rate from the previous time step (see, e.g., Grabowski and Smolarkiewicz 2002). In our model, because water vapor is not used as a prognostic variable, we cannot compute this variation of water vapor. Instead we adopt a different approach, similar to Lipps and Hemler (1982), that estimates the evaporation rate based on the fact that whenever a parcel is saturated, a Clausius–Clapeyron formula relates the local values of water content to the thermodynamics within the parcel. The conservation equation for saturated water vapor then becomes a time-varying constraint that guarantees thermodynamic equilibrium from which the evaporation rate can be estimated. The latter is thus computed as required during a time step using the current values of water content and thermodynamic variables, diagnostically recovered from the invariant prognostic variables.

This paper is organized as follows. We introduce the new low Mach number model for moist atmospheric flows in section 2 and describe the moist thermodynamics in section 3. In section 4 we discuss the numerical implementation. Last, in section 5, we present several numerical comparisons and discuss our findings.

## 2. Low Mach number formulation

*ρ*is the total density,

*p*, is defined by an equation of state (EOS). The term

*ρ*is the total density. Positive values of

*M*stands for the Mach number. In contrast to anelastic models, the perturbations in density and temperature themselves do not need to be small for the equations to remain valid; as long as those perturbations do not result in the flow violating the assumption of a low Mach number, hence small

*ρ*with respect to

*s*, and is based on the low Mach number form of the equation of state; that is,

*p*by

*α*and

*S*in the divergence constraint [(12)] depend on the water composition of moist air,

*p*, is now decoupled from the density and the other state variables in the enthalpy and momentum equations [(16) and (17); to be compared with (3) and (2)]. Instead of having

An underlying assumption in the low Mach number approximation is that the pressure remains close to the background pressure. Heat release and large-scale convective motions in a convectively unstable background can both cause the background state to evolve in time. As discussed in Almgren (2000) and demonstrated numerically in Almgren et al. (2006a) for an externally specified heating profile, if the base state does not evolve in response to heating, the low Mach number method quickly becomes invalid. For the small-scale motions of interest here, the base state can effectively be viewed as independent of time; however, for the sake of completeness we retain the full time dependence of the base state in the development of the methodology.

## 3. Moist thermodynamics

Phase changes and thus variations in water composition of moist air are introduced in the flow dynamics through the divergence constraint [(18)], specifically through *α* and *S*. These parameters are evaluated at a given time accounting for the current water composition in terms of liquid and vapor, and thus accounting for phase transitions and the current saturation requirements, given the prognostic state variables. To define *α* and *S* according to (13) in (18), we here consider dry air and water vapor to be ideal gases (see, e.g., Ooyama 1990; Satoh 2003; Klemp et al. 2007) and note that while the low Mach number formulation allows a more general equation of state, this is a standard assumption in atmospheric modeling.

*R*the universal gas constant for ideal gases, and

*α*and

*S*depend on the composition of the moist mixture even when there is no phase transition—that is, when

*T*, which are not explicitly computed in (14)–(18), are obtained by solving the following nonlinear system of equations:which satisfies the Clausius–Clapeyron relation and the saturation requirements, given

*S*[(21)] involves having a low Mach divergence constraint where the source term depends on the pressure and the velocity field; that is,A simpler expression can be obtained by rearranging the terms in (30), as shown in appendix C, which leads to a modified divergence constraint:similar to the general low Mach divergence constraint [(12)], withboth depending on

## 4. Numerical methodology

To solve the low Mach number equation set of (14)–(18) we begin with the MAESTRO code, which was originally designed to simulate low Mach number stratified, reacting flows in astrophysical settings (Almgren et al. 2006a,b; Almgren et al. 2008; Nonaka et al. 2010).

*S*and

*α*if the modified divergence of (31) is considered]. Moreover, the momentum equation [(17)], including the correction term introduced by Klein and Pauluis (2012), can be equivalently recast asas derived in Vasil et al. (2013), and discussed in Almgren et al. (2014).

MAESTRO thus solves the low Mach number equation set now given by (14)–(16) with the momentum equation [(35)] and the divergence constraint [(34)]. A predictor–corrector formalism is implemented to solve the flow dynamics, as detailed in Nonaka et al. (2010). In the predictor step an estimate of the expansion of the base state is first computed, and then an estimate of the flow variables at the new time level. In the corrector step results of the previous state update are used to compute a new base-state update as well as the full-state update.

Since we are considering the time evolution of total water [(15)] and the definition of enthalpy of moist air [(9)] involves a conservation equation [(16)] without source terms related to phase change, all the information related to variations in the moist composition and latent heat release is contained in the divergence constraint [(34)]. Here *T*, must be computed pointwise in order to define *α*, *S*, and

We consider two approaches for handling phase transitions depending on which divergence constraint is used to define (34) in the numerical implementation: (30) or (31). In the first case, which uses (30), the evaporation rate is evaluated according to (29) and introduced in the source term *S* of the constraint. As previously remarked, there is a dependence of *S* on the velocity field and the base-state pressure. Consequently, approximate or lagged values of

## 5. Numerical simulations

To validate the low Mach number method we compare simulations using the low Mach number method to simulations using a fully compressible approach. The first problem we consider is a benchmark problem presented in Bryan and Fritsch (2002) for moist flows in an isentropically stratified background. We investigate both the first and second approach to implementing phase transitions and find very good agreement between the low Mach number and compressible simulations. We then consider a problem described in Grabowski and Clark (1991) for nonisentropic background states and both saturated and only partially saturated media, which was also studied in Duarte et al. (2014). Finally, we show a comparison of three-dimensional simulations.

### a. Isentropic background state

Two-dimensional simulations of a benchmark test case are investigated in Bryan and Fritsch (2002). The computational domain is 10 km high and 20 km wide; the background atmosphere is isentropically stratified at a uniform wet equivalent potential temperature *θ*_{e0} = 320 K and is saturated; that is,

In the first approach to account for phase transitions the divergence constraint is given by (30) with *α* and *S* defined by (21) and the evaporation rate [(29)]. For this particular problem, *S*, and hence it is not necessary to compute ^{−1} for the maximum and minimum vertical velocities, respectively, to be compared with 16.7130 and −9.926 98 m s^{−1} in Bryan and Fritsch (2002). The low Mach number code takes roughly a factor of 6 less computational time than the compressible simulation at the same resolution run with the code described in Duarte et al. (2014) at an acoustic CFL number of 0.9.

The second approach does not explicitly estimate

Figure 1 shows results from computations of the problem described above; in this figure ^{−5} to 6.5 × 10^{−4}, while replacing ^{−3} to 1.5 × 10^{−3} This behavior follows naturally from the fact that in the modified divergence constraint [(31)],

The formalism adopted for the present low Mach model considers straightforwardly the effects of the specific heats of liquid water and water vapor in the evaluation of the thermodynamic properties of the moist fluid and, in particular, in the definition of the internal energy and specific enthalpy of moist air [(8) and (9)]. The latter is not possible within the moist pseudoincompressible model introduced in O’Neill and Klein (2014), which relies on a

### b. Nonisentropic background state

We next consider a nonisentropic background state given by the hydrostatically balanced profiles in (2) of Clark and Farley (1984). For the following computations we define a computational domain 4 km high and wide, with periodic horizontal boundary conditions. The boundary conditions at the top and bottom boundaries are as described for the isentropic case, except for the thermodynamic variables, which are extrapolated to determine the corresponding fluxes. Again we use the configuration and parameters as given in Duarte et al. (2014). All simulations with the low Mach formulation were performed on a uniform grid of 256 × 256. As before, the reference compressible solutions were computed on a finer grid—here 1024 × 1024.

The initial distributions of water vapor and liquid water in the atmosphere are set by the relative humidity in the atmosphere, RH, measured in percentage and defined as _{0} < 100%, then no liquid water should be initially present in the atmosphere in order to guarantee the thermodynamic equilibrium of the initial state. Following Duarte et al. (2014), we consider in this study two cases: first, a saturated medium with RH_{0} = 100% and _{0} = 20%, and hence, no liquid water in the initial background state.

Let us consider the first configuration with an initially saturated environment. As described in Duarte et al. (2014), we initially introduce a warm perturbation of temperature. Figure 3 shows solutions obtained with the low Mach formulation using the first and second approaches for the divergence constraint, as well as the compressible reference solution. Solutions are very similar in all three cases even though the low Mach approximations yield thermals rising slightly faster. In contrast to the previous example, introducing the

For the second configuration with RH_{0} = 20%, we consider the same previous temperature perturbation and an additional circular perturbation in the relative humidity, which is set to 100%, with a transition layer, as detailed in Duarte et al. (2014). Initially there is no liquid water in the domain. Like Fig. 1, Fig. 4 compares the two low Mach number solutions to the reference compressible solution; the top row shows the results using laterally averaged ^{−3} to 6.9 × 10^{−4} which is considerably smaller than the deviation of ^{−2}.

### c. Three-dimensional simulation

_{0}= 20% and no liquid water in the initial configuration. The following temperature perturbation is then introduced:where

*x*

_{1}=

*y*

_{1}= 5 km,

*z*

_{1}= 7.5 km and

*x*

_{2}=

*y*

_{2}= 7 km, and

*z*

_{2}= 7.5 km. Within the regions where the temperature is perturbed, we also perturb the relative humidity by setting RH equal to 50% for

*r*

_{1}< 3 km and

*r*

_{2}< 2 km, for each initial thermal, with corresponding transition layers:

We consider the low Mach number formalism using the second approach [modified divergence constraint (31)] to numerically implement phase transitions, with the *δ*Γ_{1} correction given by (38). As before, the low Mach number solutions are contrasted to reference compressible solutions. For a uniform grid of size 256 × 256 × 384, the time step in the low Mach number simulation is approximately 3 s, compared to 0.1 s in the compressible simulation. For this particular problem, the total run time of the low Mach number simulation is roughly a factor of 5 less than that of the compressible simulation. Figure 6 illustrates the formation of liquid water as computed with both formulations. The relative difference between the maximum values of *q*_{l} in the compressible and the low Mach number solutions is roughly 8% at *t* = 500 s and 4% at *t* = 1000 s—the two times shown in Fig. 6. Figure 7 shows the horizontal budgets of liquid water, *t* = 200 s and at the two simulation times shown in Fig. 6. For *t* = 200 s practically the same solution is recovered with both formulations, with a relative difference of 0.2% between the maximum values of *q*_{l}. Recalling that *q*_{l} is diagnostically recovered in both the compressible and the low Mach number approach, we can track the formation of liquid water by computing it after each time step for comparison purposes. Local values of *q*_{l} larger than 10^{−10} appear after 111 and 112.5 s for the compressible and low Mach number simulations, respectively. The Mach number *M* associated with this particular problem remains lower than 0.05 during the entire numerical simulation.

## 6. Summary

We have presented a new low Mach number model for moist atmospheric flows with a general equation of state, based on the low Mach number model for stratified reacting flows introduced in Almgren et al. (2008). In our model we consider only reversible processes—namely, water phase changes as in O’Neill and Klein (2014), using an exact Clausius–Clapeyron formula for moist thermodynamics and considering the effects of the specific heats of water and the temperature dependency of the latent heat. A set of invariant variables was used as prognostic variables in the equations of motion, including in particular the total water content and a specific enthalpy of moist air that accounts for the contribution of both sensible and latent heats. The evolution equations can thus be solved without needing to estimate or neglect source terms related to phase change during the time integration. The mass fractions of water vapor and liquid water are diagnostically recovered as required during a time step by imposing the saturation requirements of an atmosphere at thermodynamic equilibrium. The latter is an important property since updating the solution while ignoring the varying water composition may negatively impact the accuracy of the moist flow dynamics, as investigated in Duarte et al. (2014).

We then considered a moist thermodynamic model that treats dry air and water vapor as ideal gases to define the equation of state for moist air. To account for the latent heat release in the low Mach divergence constraint for the velocity field, the evaporation rate is estimated from the time variation of saturated water vapor within a parcel. The amount of saturated water vapor within a parcel is determined by the Clausius–Clapeyron formula as a function of the local thermodynamical state; the evolution of the state depends on the local advected motions. An analytical expression for the evaporation rate was thus derived that depends on local parameters given by the temperature and pressure, the water composition, and the velocity field. Two approaches were then considered. In the first, the rate of phase change can be computed to evaluate the latent heat release; in the second, a modified divergence constraint can be analytically deduced by introducing the derived expression for the evaporation rate in the original divergence constraint. Both approaches are analytically equivalent and together with the low Mach number equation set allow us to characterize moist atmospheric flows.

The MAESTRO code^{1} (Nonaka et al. 2010), originally designed to simulate stratified reacting flows arising in astrophysical settings, was adapted to model moist atmospheric flows. A series of test problems was investigated with both isentropic and nonisentropic background states, as well as saturated and partially saturated regions in the atmosphere. Results were contrasted to reference solutions obtained with a fully compressible formulation. Very good agreement with the reference moist dynamics was shown using both the first and second approaches (with the *δ*Γ_{1} correction), thus demonstrating that low Mach number models can serve as a reasonably accurate and computationally efficient alternative to compressible codes for small-scale moist atmospheric applications.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contract DE-AC02005CH11231.

# APPENDIX A

## Derivation of the Low Mach Divergence Constraint

*p*by

*α*and

*S*given by (13).

# APPENDIX B

## Derivation of the Evaporation Rate

*p*by

# APPENDIX C

## Modified Divergence Constraint

*S*[(21)] yields the modified divergence constraint [(31)] withand thusAfter some manipulation, we can write that

As pointed out in appendix B, three more expressions for

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^{1}

Available online at http://bender.astro.sunysb.edu/Maestro/download/.