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  • View in gallery

    Comparison with the compressible solution for the moist thermal simulations at 1000 s. Perturbational potential temperature is shown with contours every 1 K: (left) first and (right) second approach. (bottom) The simulations include the δΓ1 correction.

  • View in gallery

    Solution to the isentropic background problem neglecting the specific heats of liquid water and vapor water. (left) Perturbational potential temperature shown with contours every 0.5 K; negative contours are dashed. Contrast to Fig. 6a in Bryan and Fritsch (2002). (right) Comparison with the low Mach number solution shown in Fig. 1 (bottom left); perturbational potential temperature is shown with contours every 1.5 K.

  • View in gallery

    Initially saturated, nonisentropic background state. (top) Liquid water mass fraction and (bottom) vertical velocity are shown for (left) the reference compressible and low Mach number solution at 300 s using the (center) first and (right) second approach. Contours are every (top) 10−4 and (bottom) 1 m s−1; negative contours are dashed.

  • View in gallery

    Nonisentropic background state with a saturated perturbation. Moist specific entropy is shown with contours every 50 J kg−1 K−1. The low Mach number solution at 300 s (red) using the (left) first and (right) second approach overlays the reference compressible solution (black). (bottom) Simulations using the δΓ1 correction.

  • View in gallery

    Nonisentropic background state with a saturated perturbation. Moist specific entropy is shown with contours every 50 J kg−1 K−1. The low Mach solution at 300 s (red) using the δΓ1 correction given by (38) overlays the reference compressible solution (black).

  • View in gallery

    Two interacting thermals in a three-dimensional nonisentropic background state. Isosurfaces of liquid water for (left) the reference compressible solutions and (right) the low Mach number ones at times (top) 500 s (ql = 3.073 × 10−5) and (bottom) 1000 s (ql = 3.214 × 10−5). The low Mach number solver uses the modified divergence constraint [(31)] (second approach) with the δΓ1 correction given by (38).

  • View in gallery

    Horizontal budgets of liquid water, , computed with (33) at t = 200, 500, and 1000 s. Reference compressible and low Mach number solutions are represented with solid and dashed lines, respectively. (right) Zoomed region to highlight details.

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A Low Mach Number Model for Moist Atmospheric Flows

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  • 1 Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California
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Abstract

A low Mach number model for moist atmospheric flows is introduced that accurately incorporates reversible moist processes in flows whose features of interest occur on advective rather than acoustic time scales. Total water is used as a prognostic variable, so that water vapor and liquid water are diagnostically recovered as needed from an exact Clausius–Clapeyron formula for moist thermodynamics. Low Mach number models can be computationally more efficient than a fully compressible model, but the low Mach number formulation introduces additional mathematical and computational complexity because of the divergence constraint imposed on the velocity field. Here, latent heat release is accounted for in the source term of the constraint by estimating the rate of phase change based on the time variation of saturated water vapor subject to the thermodynamic equilibrium constraint. The authors numerically assess the validity of the low Mach number approximation for moist atmospheric flows by contrasting the low Mach number solution to reference solutions computed with a fully compressible formulation for a variety of test problems.

Corresponding author address: Max Duarte, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd. MS 50A-1148, Berkeley, CA 94720. E-mail: mdgonzalez@lbl.gov

Abstract

A low Mach number model for moist atmospheric flows is introduced that accurately incorporates reversible moist processes in flows whose features of interest occur on advective rather than acoustic time scales. Total water is used as a prognostic variable, so that water vapor and liquid water are diagnostically recovered as needed from an exact Clausius–Clapeyron formula for moist thermodynamics. Low Mach number models can be computationally more efficient than a fully compressible model, but the low Mach number formulation introduces additional mathematical and computational complexity because of the divergence constraint imposed on the velocity field. Here, latent heat release is accounted for in the source term of the constraint by estimating the rate of phase change based on the time variation of saturated water vapor subject to the thermodynamic equilibrium constraint. The authors numerically assess the validity of the low Mach number approximation for moist atmospheric flows by contrasting the low Mach number solution to reference solutions computed with a fully compressible formulation for a variety of test problems.

Corresponding author address: Max Duarte, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd. MS 50A-1148, Berkeley, CA 94720. E-mail: mdgonzalez@lbl.gov

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

We begin by writing the fully compressible equations of motion expressing conservation of mass, momentum, and enthalpy in a constant gravitational field in which we neglect Coriolis forces and viscous terms:
e1
e2
e3
where the material derivative is defined as . Here ρ is the total density, is the velocity, and is the specific enthalpy of moist air. The pressure, p, is defined by an equation of state (EOS). The term represents a source of heat (per unit mass and time) to the system, such as thermal conduction or radiation. We include gravitational acceleration given by , where is the unit vector in the vertical direction.
We formulate the moist atmospheric processes as in Romps (2008), with the additional simplification that at any grid point all phases have the same velocity and temperature. We treat moist air as an ideal mixture of dry air, water vapor, and liquid water, with the water phases in thermodynamic equilibrium, so that only reversible processes are taken into account; ice-phase microphysics, precipitation fallout, and subgrid-scale turbulence are ignored. Denoting by , , and the mass fraction of dry air, water vapor, and liquid water, respectively, we note that and write
e4
e5
e6
where ρ is the total density. Positive values of correspond to evaporation; negative values correspond to condensation. If we define the mass fraction of total water, , then we can replace (5)(6) by
e7
[We note that a system including both (1) and (4)(6) or (1), (4), and (7) is overspecified; in practice (1) need not be solved separately.]
We define the internal energy of moist air, as in Romps (2008),
e8
where the constant-volume specific heat of moist air is given by with the specific heats at constant volume: , , and for dry air, water vapor, and liquid water, respectively. Here is the triple-point temperature and is the specific internal energy of water vapor at the triple point, while the enthalpy of moist air is given by
e9
Note that an evolution equation for the internal energy,
e10
can be used instead of (3). In either case, there are no source terms related to phase change in (3) or (10), as observed in Ooyama (1990) and Duarte et al. (2014). Finally, to close the system (1)(4) with (7), we consider a general equation of state for moist air written as . (Because we could remove one of these three arguments from the equation of state; for clarity of exposition, however, we leave all three.)
In the low Mach number model, we write the pressure, , with , as the sum of a base-state pressure, , and a perturbational, or dynamic, pressure, , such that , where 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 , the equations remain valid. The base state is assumed to be in hydrostatic equilibrium; that is, , where is the base-state density. The fully compressible momentum equation [(2)] could be rewritten as
eq1
with no approximation. The low Mach number momentum equation has an additional contribution to the buoyancy term,
e11
as introduced by Klein and Pauluis (2012). Here the derivative of ρ with respect to is taken at constant entropy, s, and is based on the low Mach number form of the equation of state; that is, . As pointed out in Vasil et al. (2013), this form of the momentum equation is analytically equivalent to the momentum equation in the pseudoincompressible equation set; that is,
eq2
with Exner pressure, ( is a reference pressure, while and stand for the specific gas constant and constant-pressure specific heat of dry air, respectively), potential temperature, and the base-state potential temperature, , if we define a linearized relationship between the perturbational pressures, and .
To complete the low Mach number model for moist atmospheric flows we first replace p by in the equation of state and we then differentiate along particle paths. Following the derivation in Almgren et al. (2008), with details as given in appendix A, we derive the following divergence constraint for the low Mach number model:
e12
with
e13
where , , , , , , and , where is the specific heat of moist air at constant pressure. Here, as in (11), refers to the derivative at constant entropy. Notice that both α and S in the divergence constraint [(12)] depend on the water composition of moist air, and . Most importantly the constraint on the divergence of the velocity field retains compressibility effects from stratification as well as latent heat release and compositional changes.
Summarizing the low Mach number equation set for moist atmospheric flows in the form we will use, we have
e14
e15
e16
e17
e18
where the total mass density is defined as
e19
Contrary to the original compressible equation set (1)(3) together with (4) and (7) and a given equation of state, the total pressure field, 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 in the momentum equation [(17)], we now have , where is a perturbational pressure on the background pressure , which is no longer related to the other state variables through the equation of state. That is, pressure is no longer advanced in time through system (14)(18), and hence acoustic modes are filtered out from the original compressible governing equations. In practice, is computed in such a way that the divergence constraint [(18)] is satisfied at a given time, analogous to what is done for incompressible flows. While the equation of state might at first glance appear to be missing from the low Mach number equation set, it is in fact encapsulated in the constraint, (18), which was derived by differentiation of the equation of state along particle paths. This differs substantially from the anelastic approximation, which results from replacing the density by the background density in the continuity equation, which thus contains no information about the equation of state.

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.

The partial pressures of dry air and water vapor are then given by and where and are the specific gas constants for dry air and water vapor, respectively, with R the universal gas constant for ideal gases, and and the molar masses of dry air and water, respectively. The sum of the partial pressures defines the total pressure of a parcel,
e20
where the specific gas constant of moist air is defined as
eq3
Additionally, the specific heat capacities at constant pressure can be defined as
eq4
for dry air, water vapor, and moist air, respectively. A common approximation in cloud models is to neglect the specific heats of water vapor and liquid water [see, e.g., Bryan and Fritsch (2002) for a study and discussion on this topic]. This was also assumed in the moist pseudoincompressible model of O’Neill and Klein (2014). Here we consider specific heats for all three phases.
With this choice for the equation of state for moist air [(20)], we have
e21
where is the isentropic expansion factor of moist air, , and the latent heat of vaporization, , is defined as
e22
Note that both α and S depend on the composition of the moist mixture even when there is no phase transition—that is, when . We can now replace as used in (12) and (13) by for moist atmospheric flows.
The saturation vapor pressure with respect to liquid water, , is defined by the following Clausius–Clapeyron relation:
e23
with constants and given, for instance, by
e24
as in Romps (2008). The saturated mass fraction of water vapor, , can be then computed from the EOS, given in this case by
e25
or equivalently by
e26
Following Ooyama (1990) and Satoh (2003), we assume that air parcels cannot be supersaturated, and thus cannot exceed its saturated value, . The water mass fractions, and , as well as the temperature T, which are not explicitly computed in (14)(18), are obtained by solving the following nonlinear system of equations:
e27
which satisfies the Clausius–Clapeyron relation and the saturation requirements, given .
No approximation has been introduced so far to evaluate the water composition and therefore the thermodynamic properties of the moist fluid. We now need to estimate the evaporation rate that is required to quantify the latent heat release in the divergence constraint. Considering that there is no analytical expression for such a rate and that we cannot derive it from an approximate value of vapor water as previously discussed in the introduction, we introduce the following approach. Taking into account that is different from zero only when a change of phase is taking place—that is, when —we can rewrite the conservation equation for [(5)] as
e28
We show in appendix B that if , then
e29
where parameters , , and are in general functions of . Otherwise, whenever . In Lipps and Hemler (1982), the term in (28) is replaced with , where stands for the water vapor content advected without considering the evaporation rate in (5): .
Finally, using (28) to estimate in S [(21)] involves having a low Mach divergence constraint where the source term depends on the pressure and the velocity field; that is,
e30
A simpler expression can be obtained by rearranging the terms in (30), as shown in appendix C, which leads to a modified divergence constraint:
e31
similar to the general low Mach divergence constraint [(12)], with
e32
both depending on . In particular if no heat source is considered, we have that and all the information related to phase change is included in . Now in the general low Mach constraint (12) and (13) is given by a modified isentropic expansion factor of moist air: [(C2)]. Both divergence constraints, (30) and (31), are analytically equivalent.

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).

We recall that for the moist equation of state considered here we replace in the original MAESTRO notation by . Since in general varies in space and time, the solution procedure in the original MAESTRO algorithm replaces by ; that is, in the divergence constraint [(30)], where is the lateral average of ,
e33
where is a region at constant height for the plane-parallel atmosphere, and represents an area measure. The same follows for in the modified divergence constraint [(31)]: . The introduction of an averaged (or ) allows us to rewrite the constraint of (30) [or (31)] as
e34
as shown in appendix B in Almgren et al. (2006a), with
eq5
[with and instead of 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 as
e35
as 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 and , as well as the local temperature T, must be computed pointwise in order to define α, S, and (or , , and ). Pointwise values of are thus diagnostically recovered by solving the nonlinear system [(27)], given the values of at a given time; we use the Newton solver described in Duarte et al. (2014).

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 and are used to estimate during the prediction step, which is later recomputed with the updated values of and during the correction step. Notice, however, that during both steps the moist composition and hence phase transitions are diagnostically recovered based on the current values of . The second approach consists of considering the modified divergence constraint [(31)], in which case there is no need to estimate the evaporation rate [(29)]. Parameters , , and in (34) are computed with given the current values of throughout the predictor–corrector scheme for the flow dynamics.

Notice that considering the lateral average of through (33) is the only approximation introduced in the present model in terms of thermodynamic properties of the moist fluid. The replacement of by its lateral average was demonstrated in Almgren et al. (2008) to have little effect on the astrophysical flows studied there. In the case of moist atmospheric flows, is given by , which varies according to the local moist air composition at a given time and position. If the modified divergence constraint (31) is considered, is given by , which varies not only according to the moist air composition but also the local values of . We can thus consider the effects of localized variations in following Almgren et al. (2008), by writing , and hence,
eq6
instead of (18). Assuming that , we then have
e36
which leads to
e37
We will refer to the correction whenever (37) is considered instead of (34). Notice that , and therefore, to solve (37) a lagged is used in evaluating the right-hand side, as described in Almgren et al. (2008).

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 θe0 = 320 K and is saturated; that is, and everywhere in the domain. A warm perturbation in potential temperature is introduced in the domain, which leads to a rising thermal. Here we use the configuration and parameters as given in Duarte et al. (2014). The normal velocity is set equal to zero on the side and bottom boundaries; at the top boundary the normal velocity is set equal to the velocity corresponding to the base-state evolution. [See Almgren et al. (2006b) for further detail about the base-state evolution.] Homogeneous Neumann boundary conditions are used in solving for the perturbational pressure. For the thermodynamic variables, we impose homogeneous Neumann boundary conditions on the horizontal sides; the background state is reconstructed by extrapolation at vertical boundaries and the full state is set equal to the base one there in order to determine the corresponding fluxes. The time step for the low Mach number computations is dictated by the advective CFL number, which is based on the velocity but not the sound speed; we set this to 0.9.

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, in S, and hence it is not necessary to compute in (29). For a 256 × 128 grid of similar spatial resolution to that considered in Bryan and Fritsch (2002), the maximum and minimum values for the perturbational wet equivalent potential temperature are 4.055 402 and −0.289 31 K, respectively, compared with the original 4.095 21 and −0.305 695 K in Bryan and Fritsch (2002). Additionally, our computation yields 15.8199 and −9.455 86 m s−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 but rather considers a modified divergence constraint [(31)] with and defined by (32). Since for this particular problem, , we have . For a 256 × 128 grid, the maximum and minimum values for the perturbational wet equivalent potential temperature are now given by 4.129 11 and −0.301 90 K, respectively. The time steps taken with the second approach are slightly larger than with the first approach, and the overall computational time is roughly 10% to 15% less with this approach than with the first approach.

Figure 1 shows results from computations of the problem described above; in this figure from the low Mach number simulations is overlaid on the reference compressible solution, which was computed with the fully coupled compressible solver described in Duarte et al. (2014) and validated against results from Bryan and Fritsch (2002). The low Mach number simulations were carried out on a uniform grid of 512 × 256; the compressible solution was performed on a finer grid of 1024 × 512 in order to get a more accurate reference solution. In the top row of Fig. 1 we see that the low Mach number computation using the first approach agrees well with the compressible solution. The computation with the second approach, however, shows a significant height difference of the top of the thermal. In the bottom row of Fig. 1 we see that accounting for the local variation of and through the correction described in section 4 improves the fidelity of both approaches. With the first approach we see the improvement mostly in the tips; with the second approach the height of the thermal is noticeably improved. The fact that including the correction impacts the second approach more dramatically than the first is consistent with the observation that replacing by in the first approach meant neglecting local variations in from −8 × 10−5 to 6.5 × 10−4, while replacing by in the second approach meant neglecting local variations in at least an order of magnitude larger—that is, from −6.7 × 10−3 to 1.5 × 10−3 This behavior follows naturally from the fact that in the modified divergence constraint [(31)], accounts not only for the varying water composition but also for all the latent heat released during phase transitions (see appendix C); therefore, local variations in are expected to be more important.

Fig. 1.
Fig. 1.

Comparison with the compressible solution for the moist thermal simulations at 1000 s. Perturbational potential temperature is shown with contours every 1 K: (left) first and (right) second approach. (bottom) The simulations include the δΓ1 correction.

Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0248.1

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 (potential temperature minus Exner pressure) formalism defined with the specific heat of dry air. Neglecting the specific heats of liquid water and water vapor in our model amounts to consider the equation set A investigated in Bryan and Fritsch (2002) for the same isentropic background configuration. This is also the benchmark problem considered in O’Neill and Klein (2014) to validate their model. Figure 2 shows the low Mach number solution when the specific heats of water are neglected and contrasts it to the general case where all specific heats are taken into account, like in Fig. 1. In Bryan and Fritsch (2002), the maximum and minimum values for are 2.136 47 and −1.396 27 K, respectively. As discussed in Bryan and Fritsch (2002), the impact of approximating thermodynamic properties of moist air can lead to important variations for certain configurations, as illustrated in this example.

Fig. 2.
Fig. 2.

Solution to the isentropic background problem neglecting the specific heats of liquid water and vapor water. (left) Perturbational potential temperature shown with contours every 0.5 K; negative contours are dashed. Contrast to Fig. 6a in Bryan and Fritsch (2002). (right) Comparison with the low Mach number solution shown in Fig. 1 (bottom left); perturbational potential temperature is shown with contours every 1.5 K.

Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0248.1

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 . In particular if RH0 < 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 RH0 = 100% and everywhere in the domain as in the isentropic background problem; and a second configuration with RH0 = 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 correction does not visibly change the results. For a 256 × 256 grid and a CFL factor of 0.9, the time steps for the low Mach approximations are about 1.4 s compared to 0.04 s with the compressible formulation. The low Mach number simulation takes roughly a factor of 13 less computational time than the compressible simulation.

Fig. 3.
Fig. 3.

Initially saturated, nonisentropic background state. (top) Liquid water mass fraction and (bottom) vertical velocity are shown for (left) the reference compressible and low Mach number solution at 300 s using the (center) first and (right) second approach. Contours are every (top) 10−4 and (bottom) 1 m s−1; negative contours are dashed.

Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0248.1

For the second configuration with RH0 = 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 and while the bottom row shows the results using the correction. Here the deviation of from in the first approach ranges from −4.2 × 10−3 to 6.9 × 10−4 which is considerably smaller than the deviation of from in the second approach, which ranges from −0.17 to 3.3 × 10−2.

Fig. 4.
Fig. 4.

Nonisentropic background state with a saturated perturbation. Moist specific entropy is shown with contours every 50 J kg−1 K−1. The low Mach number solution at 300 s (red) using the (left) first and (right) second approach overlays the reference compressible solution (black). (bottom) Simulations using the δΓ1 correction.

Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0248.1

Better agreement can be seen in Fig. 5, where a term of order is also considered in (36) for the correction:
e38
The time steps used in the low Mach computations are about 2.1 s, compared to 0.04 s with the compressible formulation. The low Mach number simulation takes roughly a factor of 15 less computational time than the compressible simulation.
Fig. 5.
Fig. 5.

Nonisentropic background state with a saturated perturbation. Moist specific entropy is shown with contours every 50 J kg−1 K−1. The low Mach solution at 300 s (red) using the δΓ1 correction given by (38) overlays the reference compressible solution (black).

Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0248.1

c. Three-dimensional simulation

Finally, we consider two interacting thermals rising in a three-dimensional, nonisentropically stratified background in a domain 10 km on a side and 15 km high. The background state is defined as in section 5b, with a relative humidity of RH0 = 20% and no liquid water in the initial configuration. The following temperature perturbation is then introduced:
eq7
where
eq8
eq9
with x1 = y1 = 5 km, z1 = 7.5 km and x2 = y2 = 7 km, and z2 = 7.5 km. Within the regions where the temperature is perturbed, we also perturb the relative humidity by setting RH equal to 50% for r1 < 3 km and r2 < 2 km, for each initial thermal, with corresponding transition layers:
eq10
eq11

We consider the low Mach number formalism using the second approach [modified divergence constraint (31)] to numerically implement phase transitions, with the δΓ1 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 ql 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, , computed using Eq. (33), at 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 ql. Recalling that ql 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 ql 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.

Fig. 6.
Fig. 6.

Two interacting thermals in a three-dimensional nonisentropic background state. Isosurfaces of liquid water for (left) the reference compressible solutions and (right) the low Mach number ones at times (top) 500 s (ql = 3.073 × 10−5) and (bottom) 1000 s (ql = 3.214 × 10−5). The low Mach number solver uses the modified divergence constraint [(31)] (second approach) with the δΓ1 correction given by (38).

Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0248.1

Fig. 7.
Fig. 7.

Horizontal budgets of liquid water, , computed with (33) at t = 200, 500, and 1000 s. Reference compressible and low Mach number solutions are represented with solid and dashed lines, respectively. (right) Zoomed region to highlight details.

Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0248.1

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 code1 (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 δΓ1 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.

Acknowledgments

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

We rewrite the conservation of mass [(1)] as an expression for the divergence of velocity:
ea1
Differentiating the equation of state, written as , along particle paths, we obtain
eq12
with , , and . An expression for can be obtained by differentiating the definition of moist enthalpy [(9)] and comparing terms with the enthalpy equation [(3)]:
eq13
or, gathering terms,
ea2
where is the specific heat of moist air at constant pressure, , and . Coming back to (A1) and replacing p by , we can write the divergence constraint on the velocity field as (12) with α and S given by (13).

APPENDIX B

Derivation of the Evaporation Rate

Considering that from (28) and differentiating [(25)] along particle paths, we obtain
eq14
with
eq15
according to (25) and (23). With the equation of state [(20)], (A2) becomes
eb1
and hence,
eb2
that is,
eb3
into (29), replacing also p by .
Notice that another expression for can be derived considering [(26)] in . Moreover, two more expressions for can be found using
eq16
instead of (B1), deduced from the conservation equation for internal energy instead of enthalpy . Numerical computations using different formulations for yield practically identical results.

APPENDIX C

Modified Divergence Constraint

Rearranging terms in (30), after having introduced the estimate of [(29)] in S [(21)] yields the modified divergence constraint [(31)] with
eq17
eq18
and thus
ec1
After some manipulation, we can write that with
ec2
using (B3). In particular, if there is no phase transition , then and ; similarly, in (C1) and .

As pointed out in appendix B, three more expressions for , other than (B2), can be derived yielding four different formulations for and in the modified divergence constraint [(31)]. Nevertheless, all of them yield practically identical numerical results.

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