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## Abstract

A spurious updraft pattern has been documented in some numerical simulations of squall lines. The pattern is notable because of a regular, repeating pattern of updrafts and downdrafts that are three–six grid lengths wide. This study examines the environmental and numerical conditions that lead to this problem. The spurious pattern is found only in simulations of upshear-tilted convective systems. Furthermore, the pattern coincides with deep (2–3 km) and wide (5–20 km) moist absolutely unstable layers (MAULs)—saturated layers of air that are statically unstable. In this physical environment, small-scale perturbations grow rapidly. The necessarily imperfect numerical schemes of the model introduce spurious small-scale perturbations into the MAULs, and these perturbations amplify owing to the unstable stratification. Some techniques are investigated that diffuse the perturbations or minimize their introduction in the statically unstable flow.

## Abstract

A spurious updraft pattern has been documented in some numerical simulations of squall lines. The pattern is notable because of a regular, repeating pattern of updrafts and downdrafts that are three–six grid lengths wide. This study examines the environmental and numerical conditions that lead to this problem. The spurious pattern is found only in simulations of upshear-tilted convective systems. Furthermore, the pattern coincides with deep (2–3 km) and wide (5–20 km) moist absolutely unstable layers (MAULs)—saturated layers of air that are statically unstable. In this physical environment, small-scale perturbations grow rapidly. The necessarily imperfect numerical schemes of the model introduce spurious small-scale perturbations into the MAULs, and these perturbations amplify owing to the unstable stratification. Some techniques are investigated that diffuse the perturbations or minimize their introduction in the statically unstable flow.

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## Abstract

A set of approximate equations for pseudoadiabatic thermodynamics is developed. The equations are derived by neglecting the entropy of water vapor and then compensating for this error by using a constant (but relatively large) value for the latent heat of vaporization. The subsequent formulations for entropy and equivalent potential temperature have errors that are comparable to those of previous formulations, but their simple form makes them attractive for use in theoretical studies. It is also shown that, if the latent heat of vaporization is replaced with a constant value, an optimal value should be chosen to minimize error; a value of 2.555 × 10^{6} J kg^{−1} is found in tests herein.

## Abstract

A set of approximate equations for pseudoadiabatic thermodynamics is developed. The equations are derived by neglecting the entropy of water vapor and then compensating for this error by using a constant (but relatively large) value for the latent heat of vaporization. The subsequent formulations for entropy and equivalent potential temperature have errors that are comparable to those of previous formulations, but their simple form makes them attractive for use in theoretical studies. It is also shown that, if the latent heat of vaporization is replaced with a constant value, an optimal value should be chosen to minimize error; a value of 2.555 × 10^{6} J kg^{−1} is found in tests herein.

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## Abstract

Using numerical simulations, this study examines the sensitivity of hurricane intensity and structure to changes in the surface exchange coefficients and to changes in the length scales of a turbulence parameterization. Compared to other recent articles on the topic, this study uses higher vertical resolution, more values for the turbulence length scales, a different initial environment (including higher sea surface temperature), a broader specification of surface exchange coefficients, a more realistic microphysics scheme, and a set of three-dimensional simulations. The primary conclusions from a recent study by Bryan and Rotunno are all upheld: maximum intensity is strongly affected by the horizontal turbulence length scale *l _{h}
* but not by the vertical turbulence length scale

*l*, and the ratio of surface exchange coefficients for enthalpy and momentum,

_{υ}*C*/

_{k}*C*, has less effect on maximum wind speed than suggested by an often-cited theoretical model. The model output is further evaluated against various metrics of hurricane intensity and structure from recent observational studies, including maximum wind speed, minimum pressure, surface wind–pressure relationships, height of maximum wind, and surface inflow angle. The model settings

_{d}*l*≈ 1000 m,

_{h}*l*≈ 50 m, and

_{υ}*C*/

_{k}*C*≈ 0.5 produce the most reasonable match to the observational studies. This article also reconciles a recent controversy about the likely value of

_{d}*C*/

_{k}*C*in high wind speeds by noting that simulations in a study by Emanuel used relatively large horizontal diffusion and low sea surface temperature. The model in this study can produce category 5 hurricanes with

_{d}*C*/

_{k}*C*as low as 0.25.

_{d}## Abstract

Using numerical simulations, this study examines the sensitivity of hurricane intensity and structure to changes in the surface exchange coefficients and to changes in the length scales of a turbulence parameterization. Compared to other recent articles on the topic, this study uses higher vertical resolution, more values for the turbulence length scales, a different initial environment (including higher sea surface temperature), a broader specification of surface exchange coefficients, a more realistic microphysics scheme, and a set of three-dimensional simulations. The primary conclusions from a recent study by Bryan and Rotunno are all upheld: maximum intensity is strongly affected by the horizontal turbulence length scale *l _{h}
* but not by the vertical turbulence length scale

*l*, and the ratio of surface exchange coefficients for enthalpy and momentum,

_{υ}*C*/

_{k}*C*, has less effect on maximum wind speed than suggested by an often-cited theoretical model. The model output is further evaluated against various metrics of hurricane intensity and structure from recent observational studies, including maximum wind speed, minimum pressure, surface wind–pressure relationships, height of maximum wind, and surface inflow angle. The model settings

_{d}*l*≈ 1000 m,

_{h}*l*≈ 50 m, and

_{υ}*C*/

_{k}*C*≈ 0.5 produce the most reasonable match to the observational studies. This article also reconciles a recent controversy about the likely value of

_{d}*C*/

_{k}*C*in high wind speeds by noting that simulations in a study by Emanuel used relatively large horizontal diffusion and low sea surface temperature. The model in this study can produce category 5 hurricanes with

_{d}*C*/

_{k}*C*as low as 0.25.

_{d}^{ }

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## Abstract

This study uses large-eddy simulations to investigate processes of moist convection initiation (CI) over heterogeneous surface fluxes. Surface energy balance is imposed via a 180° phase lag of the surface moisture flux (relative to the sensible heat flux), such that the relatively warm surface is relatively dry (and the relatively cool surface is relatively wet). As shown in previous simulations, a mesoscale circulation forms in the presence of surface-flux heterogeneity, which coexists with turbulent fluctuations. The mesoscale convergence zone of this circulation develops over the relatively warm surface, and this is where clouds first form. Convection initiation occurs sooner as the amplitude of the heterogeneity increases, and as the surface moisture increases (i.e., Bowen ratio decreases). Shallow clouds initiate when boundary layer heights (*z _{i}
*) become greater than the lifting condensation level (LCL). Deep precipitating clouds initiate when the LCL and level of free convection (LFC) are roughly the same when averaged over the relatively warm surface, which is equivalent to the mean convective inhibition (CIN) becoming nearly zero. From the perspective of the entire (mesoscale) domain, cases with strongly heterogeneous surfaces have a wider distribution of both

*z*and LCL. Thus, a comparison of

_{i}*z*with LCL over a mesoscale area (i.e., within one mesoscale model grid box) may lead to misleading conclusions about CI and cloud-base height. It is also shown that as the amplitude of the surface-flux heterogeneity increases the mesoscale convergence zone becomes narrower and stronger. Furthermore, CI occurs earlier over relatively wet surfaces partly because turbulent eddies are more vigorous owing to slightly greater buoyancy.

_{i}## Abstract

This study uses large-eddy simulations to investigate processes of moist convection initiation (CI) over heterogeneous surface fluxes. Surface energy balance is imposed via a 180° phase lag of the surface moisture flux (relative to the sensible heat flux), such that the relatively warm surface is relatively dry (and the relatively cool surface is relatively wet). As shown in previous simulations, a mesoscale circulation forms in the presence of surface-flux heterogeneity, which coexists with turbulent fluctuations. The mesoscale convergence zone of this circulation develops over the relatively warm surface, and this is where clouds first form. Convection initiation occurs sooner as the amplitude of the heterogeneity increases, and as the surface moisture increases (i.e., Bowen ratio decreases). Shallow clouds initiate when boundary layer heights (*z _{i}
*) become greater than the lifting condensation level (LCL). Deep precipitating clouds initiate when the LCL and level of free convection (LFC) are roughly the same when averaged over the relatively warm surface, which is equivalent to the mean convective inhibition (CIN) becoming nearly zero. From the perspective of the entire (mesoscale) domain, cases with strongly heterogeneous surfaces have a wider distribution of both

*z*and LCL. Thus, a comparison of

_{i}*z*with LCL over a mesoscale area (i.e., within one mesoscale model grid box) may lead to misleading conclusions about CI and cloud-base height. It is also shown that as the amplitude of the surface-flux heterogeneity increases the mesoscale convergence zone becomes narrower and stronger. Furthermore, CI occurs earlier over relatively wet surfaces partly because turbulent eddies are more vigorous owing to slightly greater buoyancy.

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## Abstract

Idealized simulations of the 15 May 2009 squall line from the Second Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX2) are evaluated in this study. Four different microphysical setups are used, with either single-moment (1M) or double-moment (2M) microphysics, and either hail or graupel as the dense (rimed) ice species. Three different horizontal grid spacings are used: Δ*x* = 4, 1, or 0.25 km (with identical vertical grids). Overall, results show that simulated squall lines are sensitive to both microphysical setup *and* horizontal resolution, although some quantities (i.e., surface rainfall) are more sensitive to Δ*x* in this study. Simulations with larger Δ*x* are slower to develop, produce more precipitation, and have higher cloud tops, all of which are attributable to larger convective cells that do not entrain midlevel air. The highest-resolution simulations have substantially more cloud water evaporation, which is partly attributable to the development of *resolved* turbulence. For a given Δ*x*, the 1M simulations produce less rain, more intense cold pools, and do not have trailing stratiform precipitation at the surface, owing to excessive rainwater evaporation. The simulations with graupel as the dense ice species have unrealistically wide convective regions. Comparison against analyses from VORTEX2 data shows that the 2M setup with hail and Δ*x* = 0.25 km produces the most realistic simulation because (i) this simulation produces realistic distributions of reflectivity associated with convective, transition, and trailing stratiform regions, (ii) the cold pool properties are reasonably close to analyses from VORTEX2, and (iii) relative humidity in the cold pool is closest to observations.

## Abstract

Idealized simulations of the 15 May 2009 squall line from the Second Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX2) are evaluated in this study. Four different microphysical setups are used, with either single-moment (1M) or double-moment (2M) microphysics, and either hail or graupel as the dense (rimed) ice species. Three different horizontal grid spacings are used: Δ*x* = 4, 1, or 0.25 km (with identical vertical grids). Overall, results show that simulated squall lines are sensitive to both microphysical setup *and* horizontal resolution, although some quantities (i.e., surface rainfall) are more sensitive to Δ*x* in this study. Simulations with larger Δ*x* are slower to develop, produce more precipitation, and have higher cloud tops, all of which are attributable to larger convective cells that do not entrain midlevel air. The highest-resolution simulations have substantially more cloud water evaporation, which is partly attributable to the development of *resolved* turbulence. For a given Δ*x*, the 1M simulations produce less rain, more intense cold pools, and do not have trailing stratiform precipitation at the surface, owing to excessive rainwater evaporation. The simulations with graupel as the dense ice species have unrealistically wide convective regions. Comparison against analyses from VORTEX2 data shows that the 2M setup with hail and Δ*x* = 0.25 km produces the most realistic simulation because (i) this simulation produces realistic distributions of reflectivity associated with convective, transition, and trailing stratiform regions, (ii) the cold pool properties are reasonably close to analyses from VORTEX2, and (iii) relative humidity in the cold pool is closest to observations.

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## Abstract

Rawinsonde data were collected before and during passage of a squall line in Oklahoma on 15 May 2009 during the Second Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX2). Nine soundings were released within 3 h, allowing for unprecedented analysis of the squall line’s internal structure and nearby environment. Four soundings were released in the prestorm environment and they document the following features: low-level cooling associated with the reduction of solar isolation by a cirrus anvil; abrupt warming (1.5 K in 30 min) above the boundary layer, which is probably attributable to a gravity wave; increases in both low-level and deep-layer vertical wind shear within 100 km of the squall line; and evidence of ascent extending at least 75 km ahead of the squall line. The next sounding was released ∼5 km ahead of the squall line’s gust front; it documented a moist absolutely unstable layer within a 2-km-deep layer of ascent, with vertical air velocity of approximately 6 m s^{−1}. Another sounding was released after the gust front passed but before precipitation began; this sounding showed the cold pool to be ∼4 km deep, with a cold pool intensity *C* ≈ 35 m s^{−1}, even though this sounding was located only 8 km behind the surface gust front. The final three soundings were released in the trailing stratiform region of the squall line, and they showed typical features such as: “onion”-shaped soundings, nearly uniform equivalent potential temperature over a deep layer, and an elevated rear inflow jet. The cold pool was 4.7 km deep in the trailing stratiform region, and extended ∼1 km above the melting level, suggesting that sublimation was a contributor to cold pool development. A mesoscale analysis of the sounding data shows an upshear tilt to the squall line, which is consistent with the cold pool intensity *C* being much larger than a measure of environmental vertical wind shear Δ*U*. This dataset should be useful for evaluating cloud-scale numerical model simulations and analytic theory, but the authors argue that additional observations of this type should be collected in future field projects.

## Abstract

Rawinsonde data were collected before and during passage of a squall line in Oklahoma on 15 May 2009 during the Second Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX2). Nine soundings were released within 3 h, allowing for unprecedented analysis of the squall line’s internal structure and nearby environment. Four soundings were released in the prestorm environment and they document the following features: low-level cooling associated with the reduction of solar isolation by a cirrus anvil; abrupt warming (1.5 K in 30 min) above the boundary layer, which is probably attributable to a gravity wave; increases in both low-level and deep-layer vertical wind shear within 100 km of the squall line; and evidence of ascent extending at least 75 km ahead of the squall line. The next sounding was released ∼5 km ahead of the squall line’s gust front; it documented a moist absolutely unstable layer within a 2-km-deep layer of ascent, with vertical air velocity of approximately 6 m s^{−1}. Another sounding was released after the gust front passed but before precipitation began; this sounding showed the cold pool to be ∼4 km deep, with a cold pool intensity *C* ≈ 35 m s^{−1}, even though this sounding was located only 8 km behind the surface gust front. The final three soundings were released in the trailing stratiform region of the squall line, and they showed typical features such as: “onion”-shaped soundings, nearly uniform equivalent potential temperature over a deep layer, and an elevated rear inflow jet. The cold pool was 4.7 km deep in the trailing stratiform region, and extended ∼1 km above the melting level, suggesting that sublimation was a contributor to cold pool development. A mesoscale analysis of the sounding data shows an upshear tilt to the squall line, which is consistent with the cold pool intensity *C* being much larger than a measure of environmental vertical wind shear Δ*U*. This dataset should be useful for evaluating cloud-scale numerical model simulations and analytic theory, but the authors argue that additional observations of this type should be collected in future field projects.

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## Abstract

An axisymmetric numerical model is used to evaluate the maximum possible intensity of tropical cyclones. As compared with traditionally formulated nonhydrostatic models, this new model has improved mass and energy conservation in saturated conditions. In comparison with the axisymmetric model developed by Rotunno and Emanuel, the new model produces weaker cyclones (by ∼10%, in terms of maximum azimuthal velocity); the difference is attributable to several approximations in the Rotunno–Emanuel model. Then, using a single specification for initial conditions (with a sea surface temperature of 26°C), the authors conduct model sensitivity tests to determine the sensitivity of maximum azimuthal velocity (*υ*
_{max}) to uncertain aspects of the modeling system. For fixed mixing lengths in the turbulence parameterization, a converged value of *υ*
_{max} is achieved for radial grid spacing of order 1 km and vertical grid spacing of order 250 m. The fall velocity of condensate (*V _{t}
*) changes

*υ*

_{max}by up to 60%, and the largest

*υ*

_{max}occurs for pseudoadiabatic thermodynamics (i.e., for

*V*> 10 m s

_{t}^{−1}). The sensitivity of

*υ*

_{max}to the ratio of surface exchange coefficients for entropy and momentum (

*C*/

_{E}*C*) matches the theoretical result,

_{D}*υ*

_{max}∼ (

*C*/

_{E}*C*)

_{D}^{1/2}, for nearly inviscid flow, but simulations with increasing turbulence intensity show less dependence on

*C*/

_{E}*C*; this result suggests that the effect of

_{D}*C*/

_{E}*C*is less important than has been argued previously. The authors find that

_{D}*υ*

_{max}is most sensitive to the intensity of turbulence in the radial direction. However, some settings, such as inviscid flow, yield clearly unnatural structures; for example,

*υ*

_{max}exceeds 110 m s

^{−1}, despite a maximum observed intensity of ∼70 m s

^{−1}for this environment. The authors show that turbulence in the radial direction limits maximum axisymmetric intensity by weakening the radial gradients of angular momentum (which prevents environmental air from being drawn to small radius) and of entropy (which is consistent with weaker intensity by consideration of thermal wind balance). It is also argued that future studies should consider parameterized turbulence as an important factor in simulated tropical cyclone intensity.

## Abstract

An axisymmetric numerical model is used to evaluate the maximum possible intensity of tropical cyclones. As compared with traditionally formulated nonhydrostatic models, this new model has improved mass and energy conservation in saturated conditions. In comparison with the axisymmetric model developed by Rotunno and Emanuel, the new model produces weaker cyclones (by ∼10%, in terms of maximum azimuthal velocity); the difference is attributable to several approximations in the Rotunno–Emanuel model. Then, using a single specification for initial conditions (with a sea surface temperature of 26°C), the authors conduct model sensitivity tests to determine the sensitivity of maximum azimuthal velocity (*υ*
_{max}) to uncertain aspects of the modeling system. For fixed mixing lengths in the turbulence parameterization, a converged value of *υ*
_{max} is achieved for radial grid spacing of order 1 km and vertical grid spacing of order 250 m. The fall velocity of condensate (*V _{t}
*) changes

*υ*

_{max}by up to 60%, and the largest

*υ*

_{max}occurs for pseudoadiabatic thermodynamics (i.e., for

*V*> 10 m s

_{t}^{−1}). The sensitivity of

*υ*

_{max}to the ratio of surface exchange coefficients for entropy and momentum (

*C*/

_{E}*C*) matches the theoretical result,

_{D}*υ*

_{max}∼ (

*C*/

_{E}*C*)

_{D}^{1/2}, for nearly inviscid flow, but simulations with increasing turbulence intensity show less dependence on

*C*/

_{E}*C*; this result suggests that the effect of

_{D}*C*/

_{E}*C*is less important than has been argued previously. The authors find that

_{D}*υ*

_{max}is most sensitive to the intensity of turbulence in the radial direction. However, some settings, such as inviscid flow, yield clearly unnatural structures; for example,

*υ*

_{max}exceeds 110 m s

^{−1}, despite a maximum observed intensity of ∼70 m s

^{−1}for this environment. The authors show that turbulence in the radial direction limits maximum axisymmetric intensity by weakening the radial gradients of angular momentum (which prevents environmental air from being drawn to small radius) and of entropy (which is consistent with weaker intensity by consideration of thermal wind balance). It is also argued that future studies should consider parameterized turbulence as an important factor in simulated tropical cyclone intensity.

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## Abstract

In idealized simulations of convective storms, which are almost always run as large-eddy simulations (LES), the planetary boundary layers (PBLs) are typically laminar (i.e., they lack turbulent eddies). When compared with turbulent simulations, theory, or simulations with PBL schemes, the typically laminar LES used in the severe-storms community produce unrealistic near-surface vertical wind profiles containing excessive vertical wind shear when the lower boundary condition is nonfree slip. Such simulations are potentially problematic given the recent interest within the severe storms community in the influence of friction on vorticity generation within tornadic storms. Simulations run as LES that include surface friction but lack well-resolved turbulent eddies thus probably overestimate friction’s effects on storms.

## Abstract

In idealized simulations of convective storms, which are almost always run as large-eddy simulations (LES), the planetary boundary layers (PBLs) are typically laminar (i.e., they lack turbulent eddies). When compared with turbulent simulations, theory, or simulations with PBL schemes, the typically laminar LES used in the severe-storms community produce unrealistic near-surface vertical wind profiles containing excessive vertical wind shear when the lower boundary condition is nonfree slip. Such simulations are potentially problematic given the recent interest within the severe storms community in the influence of friction on vorticity generation within tornadic storms. Simulations run as LES that include surface friction but lack well-resolved turbulent eddies thus probably overestimate friction’s effects on storms.

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## Abstract

A synthesis of previous studies suggests the need for new, more accurate approximations for ice–liquid water potential temperature (*θ*
_{il}), a thermodynamic variable utilized in some numerical models. Starting from equations presented in a previous study, two new approximate formulations of *θ*
_{il} are derived, along with their governing equations. The new formulations are significant improvements over previous ones because no terms are dropped during their derivation and no Taylor series approximations are utilized. The governing equations for the new formulations reveal conditions under which *θ*
_{il} can be considered a conserved variable.

Potential temperature lapse rates determined from a reference thermodynamic equation are compared numerically against lapse rates determined from several approximations of *θ*
_{il}. Many of the findings agree with previous studies. However, the results show that a commonly used formulation does not account for the specific heats of water, and thus has an inherent cold bias. When the tendency to *θ*
_{il} is assumed to be zero (by approximation), one of the new formulations is superior to all other formulations that have been presented previously. When the full governing equation is used, the new *θ*
_{il} formulations produce results identical to those from the reference equation.

## Abstract

A synthesis of previous studies suggests the need for new, more accurate approximations for ice–liquid water potential temperature (*θ*
_{il}), a thermodynamic variable utilized in some numerical models. Starting from equations presented in a previous study, two new approximate formulations of *θ*
_{il} are derived, along with their governing equations. The new formulations are significant improvements over previous ones because no terms are dropped during their derivation and no Taylor series approximations are utilized. The governing equations for the new formulations reveal conditions under which *θ*
_{il} can be considered a conserved variable.

Potential temperature lapse rates determined from a reference thermodynamic equation are compared numerically against lapse rates determined from several approximations of *θ*
_{il}. Many of the findings agree with previous studies. However, the results show that a commonly used formulation does not account for the specific heats of water, and thus has an inherent cold bias. When the tendency to *θ*
_{il} is assumed to be zero (by approximation), one of the new formulations is superior to all other formulations that have been presented previously. When the full governing equation is used, the new *θ*
_{il} formulations produce results identical to those from the reference equation.

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## Abstract

A benchmark solution that facilitates testing the accuracy, efficiency, and efficacy of moist nonhydrostatic numerical model formulations and assumptions is presented. The solution is created from a special configuration of moist model processes and a specific set of initial conditions. The configuration and initial conditions include: reversible phase changes, no hydrometeor fallout, a neutrally stable base-state environment, and an initial buoyancy perturbation that is identical to the one used to test nonlinearly evolving dry thermals. The results of the moist simulation exhibit many of the properties found in its dry counterpart. Given the similar results, and acceptably small total mass and total energy errors, it is argued that this new moist simulation design can be used as a benchmark to evaluate moist numerical model formulations.

The utility of the benchmark simulation is highlighted by running the case with approximate forms of the governing equations found in the literature. Results of these tests have implications for the formulation of numerical models. For example, it is shown that an equation set that conserves both mass *and* energy is crucial for obtaining the benchmark solution. Results also suggest that the extra effort required to conserve mass in a numerical model may not lead to significant improvements in results unless energy is also conserved.

## Abstract

A benchmark solution that facilitates testing the accuracy, efficiency, and efficacy of moist nonhydrostatic numerical model formulations and assumptions is presented. The solution is created from a special configuration of moist model processes and a specific set of initial conditions. The configuration and initial conditions include: reversible phase changes, no hydrometeor fallout, a neutrally stable base-state environment, and an initial buoyancy perturbation that is identical to the one used to test nonlinearly evolving dry thermals. The results of the moist simulation exhibit many of the properties found in its dry counterpart. Given the similar results, and acceptably small total mass and total energy errors, it is argued that this new moist simulation design can be used as a benchmark to evaluate moist numerical model formulations.

The utility of the benchmark simulation is highlighted by running the case with approximate forms of the governing equations found in the literature. Results of these tests have implications for the formulation of numerical models. For example, it is shown that an equation set that conserves both mass *and* energy is crucial for obtaining the benchmark solution. Results also suggest that the extra effort required to conserve mass in a numerical model may not lead to significant improvements in results unless energy is also conserved.