## 1. Introduction

*ρ*is the density of the dry air,

**u**is its three-dimensional velocity field, and

*ρ*(

_{s}*z*) is a reference dry-air density field that is only a function of height

*z*. The summation convention holds over the subscript

*i*from 1 to 3. A second feature of the anelastic set is that the pressure field is determined using an elliptic diagnostic equation of the formHere, Φ

*=*

_{d}*p*/

_{d}*ρ*is the dynamic potential,

_{s}*p*is the dynamic pressure (i.e., the pressure perturbation from the reference state),

_{d}*θ*

_{vd}is the dynamic virtual potential temperature,

*θ*is the potential temperature of the base state, and

_{s}*r*is the liquid water mixing ratio. The notation follows that of Bannon (2002, appendix B) but without the subscript

_{l}*a*for the dry air. For simplicity, (1.3) ignores the effects of friction and rotation. The boundary conditions on (1.3) for rigid boundaries correspond to constraints on the derivative of the pressure and thus these conditions fail to specify the pressure uniquely. In particular, the horizontally uniform, homogeneous solution of (1.3) subject to rigid upper and lower boundary conditions iswhere Φ

_{0}is an arbitrary but constant dynamic potential. Ogura and Charney (1962) note that this nonuniqueness has no direct impact on the dynamics because it has no effect on the momentum equation that requires only the gradient of the dynamic potential. However, Schlesinger (1975) noted that the nonuniqueness does affect the temperature field and hence the microphysical calculations will be affected. Such an impact might be significant in deep moist convection.

*T*denotes temperature and

*θ*is the dry potential temperature perturbation. However, Bannon (1996) has noted that this approach does not obtain the correct temperature field in Lamb’s hydrostatic adjustment problem in a semi-infinite atmosphere.

_{d}*V*. Substituting from the anelastic version of Poisson’s relation,yields the required constraint on the pressure:where the vapor mixing ratio

*r*and virtual potential temperature are derived prognostically and

_{υ}*H*is the density scale height. We note that the constants in (1.6) and (1.8) would have to be generalized to be functions of time if open boundaries are employed in limited area models. Section 2 presents an analytic study of the problem of the injection of water vapor, heat, and momentum into compressible and anelastic atmospheres of finite vertical extent. [Bannon (1996) treated the case of a semi-infinite atmosphere for an injection of heat only.] The efficacy of the constraint of mass conservation (1.6) is demonstrated by the accurate prediction of the thermal field. Section 3a extends these analytic solutions for an isothermal atmosphere to numerical solutions in nonisothermal atmospheres with and without a stratosphere. Section 3b compares the mass conservation closure with two other closures. Section 4 incorporates the constraint in an anelastic numerical model and demonstrates its impact on deep moist convection. Nicholls and Pielke (1994a, b) also present some comparisons of anelastic and compressible responses to prescribed heating but they do not invoke the closure (1.6) to remove the degeneracy in the anelastic system. The mass conservation scheme was introduced independently by Lafore et al. (1998) but they did not do a comparison with other closures.

_{ρ}## 2. Analytic solutions of the hydrostatic adjustment problem

Because the degeneracy encountered when solving for the anelastic pressure using the elliptic Eq. (1.3) is isolated to the horizontal wavenumber zero contribution, this section considers a one-dimensional (horizontally invariant) moist atmosphere that is initially in hydrostatic balance but is subsequently perturbed by prescribed injections of water vapor, heat, and momentum. The atmosphere eventually adjusts to a new equilibrium in hydrostatic balance. The model extends the hydrostatic adjustment problem of Bannon (1995a, b) to include additional forcing mechanisms and considers atmospheres of finite vertical extent. These features make the analysis particularly relevant to numerical models. The compressible solution is presented first in order to establish a reference for the anelastic solution.

### a. Compressible adjustment

*s*) yields the equations governing the perturbations (denoted with a subscript

*d*):where

*w*is the vertical velocity,

_{d}*p*is the perturbation pressure,

_{d}*ρ*=

_{d}*ρ*

_{ad}+

*ρ*

_{s}r_{vd}is the moist air density perturbation,

*ρ*

_{ad}is the dry-air density perturbation,

*r*

_{vd}is the water vapor mixing ratio,

*γ*=

*c*/

_{p}*c*is the ratio of specific heats for dry air,

_{υ}*θ*

_{vd}is the virtual potential temperature perturbation,

*β*= 1/

*γε*− 1 is a parameter of the heating due to the introduction of mass through phase changes,

*ρ̇*

_{ph}, Θ̇ is a diabatic warming rate,

*ε*= 0.622 is the ratio of the molecular weights of water vapor and dry air, and

*ẇ*is an injection of vertical velocity. The form of (2.1) is very similar to the model used by Bannon (1995a) to study hydrostatic adjustment. The only significant differences are that the density and potential temperature are redefined to incorporate moisture, and injections of mass and momentum as well as heat are introduced. The injection of moisture can be explicitly coupled to an injection of heat representing condensational warming or evaporational cooling.

*δ*Φ = Φ

*− Φ*

^{f}*is the difference between the final and initial field. Here the scale heights are defined byand*

^{i}*N*

^{2}

_{s}=

*g*/

*θ*/

_{s}dθ_{s}*dz*is the buoyancy frequency squared. Note that the vertical momentum forcing does not appear in (2.2). Thus, it affects neither the potential vorticity field nor the steady state and is not discussed further. The net injections of heat, water vapor, and virtual potential temperature are given byFor prescribed injections, (2.2) is solved subject to the rigid lower (at

*z*=

*0*) and upper (at

*z*=

*D*) kinematic boundary conditions of zero vertical velocity. This condition implies a state of hydrostatic balance of the formwhere

*α*=

*H*

^{−1}

_{ρ}− (

*γH*)

_{s}^{−1}.

*T*= 255 K and surface pressure (at

_{s}*z*= 0 km)

*p*

_{*}= 1000 hPa. The pressure/density-scale height

*H*=

_{s}*RT*/

_{s}*g*= 7.46 km is constant. Here,

*R*is an ideal gas constant for dry air. The profiles of the basic-state fields are given by

*ρ*=

_{s}*ρ*

_{*}exp(−

*z*/

*H*),

_{s}*p*=

_{s}*p*

_{*}exp(−

*z*/

*H*), and

_{s}*θ*=

_{s}*θ*

_{*}exp(

*κ z*/

*H*), where

_{s}*ρ*

_{*}= 1.36 kg m

^{−3},

*κ*=

*R*/

*c*, and

_{p}*c*is the specific heat at constant pressure for dry air. In the isothermal case, the governing Eq. (2.2) and the boundary conditions (2.5) simplify towhere

_{p}*F*is the total forcing of the dynamic potential in the compressible case. Note that the operator on the left side of (2.2) is the limiting form of the operator on the dynamic anelastic pressure in (1.3) for an isothermal, horizontally invariant atmosphere. A unique solution to this isothermal problem is readily found. Integrating (2.6) once from the surface to the arbitrary height

_{c}*z*and invoking the lower boundary condition (2.7) yieldswhere the subscript zero denotes evaluation at the surface

*z*= 0. Dividing by the base-state density field and integrating a second time from the surface upward yields the pressure perturbation:The unknown constant

*δ*Φ

_{d}_{0}is determined by integrating the hydrostatic equation:In this manner the upper boundary condition (2.5) is incorporated into the solution. In the absence of a net mass injection, the perturbation pressures at the top and bottom are the same. A net mass injection will cause the pressure at the surface to be greater than that at the top.

*F*, the integral (2.9) is evaluated to find the steady-state pressure subject to the constraint (2.10). The density field is then calculated using the hydrostatic relation and the virtual potential temperature is calculated using the Poisson’s relation (2.10). The change in the vertical displacement of the air parcels is obtained from the temporally integrated form of (2.1c):The dry dynamic density field is found from the integrated dry continuity equation,and the moist continuity equation for the vapor dynamic density field yields

_{c}*δρ*

_{vd}= Δ

*ρ*

_{ph}. Finally, the temperature field is found from the gas law (2.1e) asAlternatively, the temperature field may be found from the first law of thermodynamics to satisfySolutions for specific injections are presented in section 2c alongside the anelastic solutions.

### b. Anelastic adjustment

*κ*set to zero. The arbitrary constant Φ

*is again determined from mass conservation but the hydrostatic relation may not be invoked. It is readily demonstrated that the hydrostatic relation is automatically satisfied in the interior by the homogeneous version of (1.3) and thus the anelastic boundary conditions (2.16) are redundant.*

_{d0}_{d0}. The density field is found using the anelastic Poisson relation (1.7) in the formand the temperature field is again determined from the ideal gas law (2.1e) in the formNoting that

*p*=

_{s}*ρ*in the isothermal case, the subtraction of (2.18) from (2.19) yields the solution for the anelastic temperature field directly from the forcing

_{s}gH_{ρ}### c. Comparison of anelastic and compressible solutions

*l*= 2.5 × 10

_{υ}^{6}J kg

^{−1}is the enthalpy of vaporization,

*R*= 461.5 J kg

_{υ}^{−1}K

^{−1}is the gas constant for water vapor, and

*H*is the Heaviside step function. The first term in parentheses in (2.21) has a value of 9.31 for

*T*= 255 K and dominates the second term

_{s}*β*= 0.15. The density Δ

*ρ*

_{cond}of condensed water is found from (2.20) by assuming the change in the virtual potential temperature at the surface would be 1 K. Then, Δ

*ρ*

_{0}= 5.76 × 10

^{−4}kg m

^{−3}. The heating is assumed to occur from the top of the boundary layer at

*z*= 2 km to the cloud top

_{b}*z*=

*z*= 13 km. The “top hat” structure of these injections makes the analytic solution for an isothermal atmosphere very convenient.

_{t}In contrast, the anelastic atmosphere cannot expand dynamically and must adjust instantaneously to the heating. Therefore, it may not generate any potential temperature anomaly due to advection. The anelastic potential temperature in Fig. 1c is identical to the net warming of Fig. 1a. However, the anelastic atmosphere instantly communicates a density anomaly throughout the column that is identical to the compressible density anomaly. Because the solution is hydrostatic, the pressure gradient corresponding to this density anomaly is also identical in the anelastic atmosphere. However, the amplitude of the pressure anomaly is underestimated by a constant amount. The anelastic temperature anomaly (Fig. 1e) given by (1.7) or (2.1d) is slightly less than the compressible anomaly in the finite atmosphere but is identical to that for a compressible, semi-infinite model. In this situation, the anelastic finite atmosphere is a better approximation to a compressible, semi-infinite atmosphere.

Figure 2 plots the vertical structure of the moistening given by (2.22) and the response of the thermodynamic state variables. In the compressible atmosphere, the loss of mass leads to an instantaneous drop in the pressure in the region of the forcing. The associated pressure gradient forces at the edge of this region will accelerate the flow into the region. In the semi-infinite atmosphere, as inferred from the positive potential temperature anomaly in Fig. 2c, the loss of mass from the column generates net downward displacement at all heights above and upward displacements below the middle of the forcing region. In a finite atmosphere, the displacements must vanish at the upper boundary as well. The loss of mass reduces the weight of the column and the adjusted surface pressure must fall. In contrast to (2.23), the pressure difference between the surface and the top of the atmosphere is given by this loss of mass. Concomitant with this pressure fall, there is a compression (rarefaction) of the air (Fig. 2f) and a warming (cooling) (Fig. 2e) inside (outside) the region of the forcing. Because of the discontinuous structure of the moistening (2.22), delta-function spikes in the temperature field appear at the boundaries of the forcing that arise from (2.13b).

*θ*= 0. The result (2.20) for just the moistening reduces toThus, the drying leads to a warming only in the region of the forcing. The anelastic temperature (Fig. 2e) is a good approximation to the semi-infinite compressible temperature. The anelastic pressure field (Fig. 2b) displays the same net pressure fall across the atmosphere as is required by hydrostatics but is always less than that of the compressible atmosphere.

_{d}## 3. Numerical solution of the hydrostatic adjustment problem

### a. Mass conservation closure

*z*= 100 m. The boundary condition (2.5) in the compressible case simplifies toThe anelastic solution is adjusted so that the mass conservation constraint (1.8) is satisfied. The one-dimensional heating is described byThe numerical procedure was benchmarked through comparison with the isothermal solutions of section 2.

Figure 3 shows the adjustments to this instantaneous heating in a 15-km-deep atmosphere with a standard tropospheric lapse rate (6.5 K km^{−1}) and a “tropical” surface temperature of 27°C. In Fig. 4, the domain extends to a depth of 30 km, with an isothermal stratosphere above 16 km. The results are qualitatively similar to those in the isothermal heating case (section 2c). Specifically, the anelastic solution slightly underestimates the pressure and temperature perturbations (Figs. 3b and 3e) in the 15-km-deep domain. The potential temperature is overestimated as a consequence of the absence of any vertical displacement in the anelastic case (Figs. 3c and 3f). The density perturbations are the same in the two cases (Fig. 3d). As in the isothermal atmosphere, the compressible expansion is larger in the deeper atmosphere (Figs. 3f and 4f). The anelastic solution for the temperature perturbation is again a better approximation to the deeper atmosphere (Fig. 4e) than to the shallower atmosphere (Fig. 3e).

Figures 5 and 6 depict the results for a one-dimensional moisture sink, the magnitude of which is calculated from (2.21) with *β* = 0, given the heating profile of (3.4). The results are qualitatively similar to the isothermal case, except for the absence of the spikes in the compressible temperature perturbation. The pressure and density perturbations are negative at all levels in both the compressible and anelastic cases. The anelastic temperature perturbations are slightly more positive than the compressible case but the difference decreases as the domain increases. As in the isothermal case, upward displacements are found below the forcing region in the compressible adjustment, and downward displacements above the forcing (Figs. 5f and 6f); the vertical displacement and potential temperature perturbation are zero in the anelastic adjustment but the anelastic virtual potential temperature is proportional to the moistening (2.14).

### b. Alternative closures

An ad hoc alternative to the mass conservation closure of the previous section is to calculate the temperature perturbation using (1.5); this closure is referred to as thermal equivalency. Another ad hoc closure, following Schlesinger (1975), is to set the average value of the pressure perturbation over the lateral boundaries to zero; this closure is referred to as boundary pressure. Figures 7 and 8 compare the anelastic responses for the three closures for a warming and a moistening in a tropical atmosphere with a stratosphere corresponding to that in Figs. 4 and 6. The virtual potential temperature response in the three closures is the same and is given by (2.14). The thermal equivalency closure does not remove the ambiguity in the pressure field that may have an arbitrary function (1.4). To present the results, mass conservation has been applied to the thermal equivalency pressure field; then the pressure and density perturbations are the same as those for the mass conservation case. The pressure and density perturbations for the boundary pressure closure are less than those for the mass conservation closure for both a heating and a moistening. The thermal equivalency closure (1.5) underestimates the warming in the temperature field compared to the accurate mass conservation closure. In contrast the boundary pressure closure underestimates (overestimates) the warming compared to the mass conservation scheme in the warming (moistening) case.

## 4. Two-dimensional moist anelastic simulations

This section tests the impact of the mass conservation closure (1.8) in a new moist anelastic model described in section 4a and compares the closure (1.8) to the thermal equivalency and boundary pressure closures in warm bubble experiments of moist convection in section 4b. Unlike the one-dimensional solutions of sections 2 and 3, those presented here include buoyancy waves as part of the response.

### a. The anelastic model

The numerical model of Bryan and Fritsch (2002) is modified to allow the solution of the anelastic equation set presented in Bannon (2002). The primary changes are the addition of an anelastic pressure solver, and the alteration of the momentum and thermodynamic equations according to Bannon (2002). The time-split algorithm is not used here because the anelastic model has filtered the acoustic waves. The anelastic version of the model was tested by comparing one- and two-dimensional dry simulations to known linear, time-dependent, analytic solutions in an isothermal atmosphere (appendix D of Chagnon 2003).

The elliptic equation for the anelastic pressure field is solved at each time step using a Fourier transform technique. First, the elliptic equation is transformed in the horizontal direction, and the vertical derivatives are written in finite-difference form. The tridiagonal matrix in the vertical direction is then solved at each grid point, to obtain the pressure field in wavenumber space. An inverse Fourier transform produces the pressure perturbation field to within a multiple of the base-state density. The boundary conditions at the top and bottom boundaries are given by (2.16), and periodic boundary conditions are used at the lateral boundaries. The horizontal and vertical grid intervals are both either 200 or 100 m. The domain size is 100 km in the horizontal (*x*) direction and 20 km in the vertical (*z*) direction.

The model uses a saturation adjustment technique to account for phase changes, as described in Bryan and Fritsch (2002). This approach involves first integrating the model equations in a dynamical step, ignoring microphysical tendencies, then separately applying the microphysical step. In the dynamical step, third-order Runge–Kutta time differencing is used, with a 3-s time step. The order of the calculations is as follows. The advection of potential temperature and momentum is calculated first using fifth-order spatial derivatives. The anelastic pressure perturbation is then found, followed by the momentum tendency attributable to the pressure-gradient force. After the completion of the dynamical step, the warm-rain microphysical calculations are performed, following Kessler (1969).

### b. Warm bubble experiments

^{−1}of convective available potential energy (CAPE). A warm bubble with horizontal (vertical) radius of 10 km (2.5 km) is placed in the center of the domain at 3.5 km above ground level, with a maximum virtual potential temperature perturbation of 4 K. The virtual potential temperature perturbation iswithwhere

*x*and

_{o}*z*are the horizontal and vertical locations of the center of the bubble with half widths

_{o}*x*= 10 km and

_{r}*z*= 2.5 km. Specifying the warm bubble as a virtual potential temperature perturbation ensures that the initial buoyancy distribution is the same for all simulations, even though the initial temperature fields differ as a consequence of the different pressure closures.

_{r}The temporal evolution of the domain-wide vertical velocity maxima from the three simulations is plotted in Fig. 9; the maximum rainwater mixing ratios are shown in Fig. 10. Until 25–30 min, the vertical velocity and rainwater maxima are similar in the three closures, but differences emerge thereafter. The vertical velocity maxima are rather noisy after 30 min. However, Fig. 10 shows that there is a tendency for the thermal equivalency closure to produce lower peak rain mixing ratios than the mass conservation closure; the boundary pressure closure tends to produce higher values.

Figures 11 and 12 show the evolution of the domain-averaged root-mean-square (rms) error for the pressure, temperature, vertical velocity, and liquid water mixing ratio every 3 min for the thermal equivalency and boundary pressure closures, respectively. Here, the error is defined relative to the runs with the mass conservation closure that was shown in sections 2 and 3b to capture the hydrostatic adjustment process accurately relative to a compressible atmosphere. The rms errors are comparable for the two closures except for the pressure field for which the boundary pressure is greater by a factor of 10. In addition the error growth is larger at the higher resolution. This behavior is consistent with the idea of the smallest-scale errors growing faster (e.g., Leith and Kraichnan 1972). It is important to note that the initial error occurs in the temperature field for the horizontal wavenumber zero mode and it then enters the other fields through the microphysical calculations. Runs without any condensation would show no differences between the three closures.

Two other domain-integrated quantities are the mass of dry air and the total rainfall. By construction, the mass conservation closure conserves mass to machine precision. A plot of the total mass as a function of time for the boundary pressure run (not shown) reveals a monotonic decease in mass from 9.580 × 10^{8} kg to 9.530 × 10^{8} kg over the first 50 min, corresponding to a 0.5% loss. The subsequent 10 min indicates a slight increase to 9.533 × 10^{8} kg. Because the closure (1.5) for the thermal equivalency does not provide a closure for the pressure, mass conservation may also be applied in that closure. The evolution of the total rainfall with time (not shown) shows a monotonic increase. After 60 min, the mass conservation case has a total rainfall of 4.87 × 10^{7} kg. In contrast, the thermal equivalency has a smaller rainfall of 4.78 × 10^{7} kg and the boundary pressure case a larger rainfall of 4.99 × 10^{7} kg.

Figure 13 exhibits contour plots of the horizontally averaged temperature difference as a function of time and height for the thermal equivalency and boundary pressure closures relative to the mass conservation closure for the 100-m-resolution runs. Negative values indicate that the predicted temperature field is smaller than that with the mass conservation closure. The figure indicates a systematic cool bias for the two closures in the upper troposphere of over 0.5 K.

The growth of differences between the three simulations may be attributed to the nonlinearity of the microphysical calculations, which are highly sensitive to the temperature field. Clearly, the simulations diverge as time advances; the contrasting evolution of the convective overturning demonstrates the sensitivity of simulations of moist convection to the pressure closure and the nature of the temperature calculation.

## 5. Conclusions

The constraint of mass conservation has been successfully incorporated into an anelastic formulation of the equations of motion for deep convection. It is demonstrated that this constraint provides a closure to the solution of the diagnostic equation for the pressure field that removes the ambiguity in the solution of the horizontal mean. Analytic and numerical solutions for the one-dimensional problem of the hydrostatic adjustment to idealized heat sources and moisture sinks indicate the anelastic model compares favorably to the solutions in a compressible model. The comparison improves as the depth of the model atmosphere is increased.

The effect of the mass conservation closure has also been tested in idealized two-dimensional numerical simulations of deep convection generated by the ascent of a warm bubble in a conditionally unstable atmosphere. The results are compared to those with a closure that sets the average value of the dynamic pressure to zero on the lateral boundaries and with the thermal equivalency closure of (1.5). The differences among the three closures are shown to increase with time and with resolution as the closures impact the nonlinear microphysical calculations during the simulated precipitation event. Both the boundary pressure and thermal equivalency closures exhibit a systematic cool bias in the upper troposphere.

## Acknowledgments

The National Science Foundation (NSF), under NSF Grants ATM-0215358 and ATM0539969, provided partial financial support. We thank the anonymous reviewers for their constructive comments.

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