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## Abstract

Available potential energy (APE) is defined as the difference between the total static energy of the atmosphere and that of a reference state that minimizes the total static energy after a sequence of reversible adiabatic transformations. Determining the rate at which APE is generated in the atmosphere allows one to estimate the amount of kinetic energy that can be generated by atmosphere flows. Previous expressions for the sources and sinks of APE rely on a dry framework and are limited by the fact that they require prior knowledge of the distribution of latent heat release by atmospheric motion. In contrast, this paper uses a moist APE framework to derive a general formula for the sources and sinks of APE that can be equally applied to dry and moist circulations.

Two key problems are addressed here. First, it is shown that any reorganization of the reference state due to diabatic heating or addition of water does not change its total static energy. This property makes it possible to determine the rate of change in APE even in the absence of an analytic formula for the reference state, as is the case in a moist atmosphere. Second, the effects of changing the total water content of an air parcel are also considered in order to evaluate the changes of APE due to precipitation, evaporation, and diffusion of water vapor. Based on these new findings, one can obtain the rate of change of APE from that of atmospheric entropy, water content, and pressure.

This result is used to determine the sources and sinks of APE due to different processes such as external energy sources, frictional dissipation, diffusion of sensible heat and water vapor, surface evaporation, precipitation, and reevaporation. These sources and sinks are then discussed in the context of an idealized atmosphere in radiative–convective equilibrium. For a moist atmosphere, the production of APE by the surface energy flux is much larger than any observational or theoretical estimates of frictional dissipation, and, as is argued here, must be balanced by a comparable sink of APE due to the diffusion of water vapor from unstable to stable air parcels.

## Abstract

Available potential energy (APE) is defined as the difference between the total static energy of the atmosphere and that of a reference state that minimizes the total static energy after a sequence of reversible adiabatic transformations. Determining the rate at which APE is generated in the atmosphere allows one to estimate the amount of kinetic energy that can be generated by atmosphere flows. Previous expressions for the sources and sinks of APE rely on a dry framework and are limited by the fact that they require prior knowledge of the distribution of latent heat release by atmospheric motion. In contrast, this paper uses a moist APE framework to derive a general formula for the sources and sinks of APE that can be equally applied to dry and moist circulations.

Two key problems are addressed here. First, it is shown that any reorganization of the reference state due to diabatic heating or addition of water does not change its total static energy. This property makes it possible to determine the rate of change in APE even in the absence of an analytic formula for the reference state, as is the case in a moist atmosphere. Second, the effects of changing the total water content of an air parcel are also considered in order to evaluate the changes of APE due to precipitation, evaporation, and diffusion of water vapor. Based on these new findings, one can obtain the rate of change of APE from that of atmospheric entropy, water content, and pressure.

This result is used to determine the sources and sinks of APE due to different processes such as external energy sources, frictional dissipation, diffusion of sensible heat and water vapor, surface evaporation, precipitation, and reevaporation. These sources and sinks are then discussed in the context of an idealized atmosphere in radiative–convective equilibrium. For a moist atmosphere, the production of APE by the surface energy flux is much larger than any observational or theoretical estimates of frictional dissipation, and, as is argued here, must be balanced by a comparable sink of APE due to the diffusion of water vapor from unstable to stable air parcels.

## Abstract

The primary goal of this paper is to validate the use of the anelastic approximation for fluids with a complex equation of state such as moist air or seawater. The anelastic approximation is based on a leading-order expansion of the equations of motion for a compressible fluid in terms of density. Its application to atmospheric flows has been based on a dry framework that treats phase transitions as an external energy source. However, cloudy air is more accurately described as a two-phase fluid in which condensed water and water vapor are in thermodynamic equilibrium. Thermodynamic equilibrium reduces to three the number of state variables necessary to describe the thermodynamic state of moist air, and leads to a discontinuity in the partial derivatives of the equation of state at the saturation point. A version of the anelastic approximation for a moist atmosphere is derived here by considering the atmospheric density as a small perturbation from a moist-adiabatic reference profile, and using moist entropy and total water content as prognostic variables, with buoyancy determined from the complete nonlinear equation of state.

The key finding of this paper is that this implementation of the anelastic approximation conserves energy. The total energy is equal to the sum of the kinetic energy and the thermodynamic energy. The latter is found to be equal to the sum of the enthalpy and geopotential energy of the parcel. Furthermore, the state relationships between this thermodynamic energy, entropy, and other state variables are the same as those for moist air after replacing the total pressure with the reference state pressure. This guarantees that, as long as the pressure perturbation remains small, the thermodynamic behavior of a fluid under the anelastic approximation is fully consistent with both the first and second laws of thermodynamics.

Two implications of this finding are also discussed. First, it is shown that the first and second laws of thermodynamics constrain the vertically integrated buoyancy flux. This is equivalent to deriving the total work performed in a compressible atmosphere from its entropy and energy budgets. Second, it is argued that an anelastic model can be built with temperature or enthalpy as a prognostic variable instead of entropy. The rate of change for this new state variable can be obtained from energy conservation, so that such a model explicitly obeys the first law of thermodynamics. The entropy in this model is equal to the entropy of the parcel evaluated at the reference pressure, and its evolution obeys the second law of thermodynamics.

## Abstract

The primary goal of this paper is to validate the use of the anelastic approximation for fluids with a complex equation of state such as moist air or seawater. The anelastic approximation is based on a leading-order expansion of the equations of motion for a compressible fluid in terms of density. Its application to atmospheric flows has been based on a dry framework that treats phase transitions as an external energy source. However, cloudy air is more accurately described as a two-phase fluid in which condensed water and water vapor are in thermodynamic equilibrium. Thermodynamic equilibrium reduces to three the number of state variables necessary to describe the thermodynamic state of moist air, and leads to a discontinuity in the partial derivatives of the equation of state at the saturation point. A version of the anelastic approximation for a moist atmosphere is derived here by considering the atmospheric density as a small perturbation from a moist-adiabatic reference profile, and using moist entropy and total water content as prognostic variables, with buoyancy determined from the complete nonlinear equation of state.

The key finding of this paper is that this implementation of the anelastic approximation conserves energy. The total energy is equal to the sum of the kinetic energy and the thermodynamic energy. The latter is found to be equal to the sum of the enthalpy and geopotential energy of the parcel. Furthermore, the state relationships between this thermodynamic energy, entropy, and other state variables are the same as those for moist air after replacing the total pressure with the reference state pressure. This guarantees that, as long as the pressure perturbation remains small, the thermodynamic behavior of a fluid under the anelastic approximation is fully consistent with both the first and second laws of thermodynamics.

Two implications of this finding are also discussed. First, it is shown that the first and second laws of thermodynamics constrain the vertically integrated buoyancy flux. This is equivalent to deriving the total work performed in a compressible atmosphere from its entropy and energy budgets. Second, it is argued that an anelastic model can be built with temperature or enthalpy as a prognostic variable instead of entropy. The rate of change for this new state variable can be obtained from energy conservation, so that such a model explicitly obeys the first law of thermodynamics. The entropy in this model is equal to the entropy of the parcel evaluated at the reference pressure, and its evolution obeys the second law of thermodynamics.

## Abstract

The impact of water vapor on the production of kinetic energy in the atmosphere is discussed here by comparing two idealized heat engines: the Carnot cycle and the steam cycle. A steam cycle transports water from a warm moist source to a colder dryer sink. It acts as a heat engine in which the energy source is the latent heat of evaporation. It is shown here that the amount of work produced by a steam cycle depends on relative humidity and is always less than that produced by the corresponding Carnot cycle.

The Carnot and steam cycles can be combined into a mixed cycle that is forced by both sensible and latent heating at the warm source. The work performed depends on four parameters: the total energy transport; the temperature difference between the energy source and sink; the Bowen ratio, which measures the partitioning between the sensible and latent heat transports; and the relative humidity of the atmosphere. The role of relative humidity on the work produced by a steam cycle is discussed in terms of the Gibbs free energy and in terms of the internal entropy production.

## Abstract

The impact of water vapor on the production of kinetic energy in the atmosphere is discussed here by comparing two idealized heat engines: the Carnot cycle and the steam cycle. A steam cycle transports water from a warm moist source to a colder dryer sink. It acts as a heat engine in which the energy source is the latent heat of evaporation. It is shown here that the amount of work produced by a steam cycle depends on relative humidity and is always less than that produced by the corresponding Carnot cycle.

The Carnot and steam cycles can be combined into a mixed cycle that is forced by both sensible and latent heating at the warm source. The work performed depends on four parameters: the total energy transport; the temperature difference between the energy source and sink; the Bowen ratio, which measures the partitioning between the sensible and latent heat transports; and the relative humidity of the atmosphere. The role of relative humidity on the work produced by a steam cycle is discussed in terms of the Gibbs free energy and in terms of the internal entropy production.

## Abstract

The behavior of the Hadley circulation is analyzed in the context of an idealized axisymmetric atmosphere. It is argued that the cross-equatorial Hadley circulation exhibits two different regimes depending on the depth of the planetary boundary layer and the sea surface temperature gradient in the equatorial regions. The first regime corresponds to a classic direct circulation from the summer to winter hemisphere. The second regime differs in that the return flow rises above the boundary layer in the winter hemisphere and crosses the equator within the free troposphere. This equatorial jump is associated with a secondary maximum in precipitation on the winter side of the equator.

The transition between these two regimes can be understood through the dynamical constraints on the low- level flow. Strong virtual temperature gradients are necessary for the return flow to cross the equator within the planetary boundary layer. However, the mass transport driven by such a temperature gradient is highly sensitive to the thickness of the boundary layer. For a weak temperature gradient or a shallow boundary layer, the return flow is prevented from crossing the equator within the the boundary layer and, instead, must do so in the free troposphere. These dynamical constraints act equally in a dry and a moist atmosphere. However, a comparison between dry and moist simulations shows that the equatorial jump is much deeper in a moist atmosphere. This is interpreted as resulting from the feedbacks between the large-scale flow and moist convection, which results in establishing a very weak gross moist stability for the equatorial jump.

## Abstract

The behavior of the Hadley circulation is analyzed in the context of an idealized axisymmetric atmosphere. It is argued that the cross-equatorial Hadley circulation exhibits two different regimes depending on the depth of the planetary boundary layer and the sea surface temperature gradient in the equatorial regions. The first regime corresponds to a classic direct circulation from the summer to winter hemisphere. The second regime differs in that the return flow rises above the boundary layer in the winter hemisphere and crosses the equator within the free troposphere. This equatorial jump is associated with a secondary maximum in precipitation on the winter side of the equator.

The transition between these two regimes can be understood through the dynamical constraints on the low- level flow. Strong virtual temperature gradients are necessary for the return flow to cross the equator within the planetary boundary layer. However, the mass transport driven by such a temperature gradient is highly sensitive to the thickness of the boundary layer. For a weak temperature gradient or a shallow boundary layer, the return flow is prevented from crossing the equator within the the boundary layer and, instead, must do so in the free troposphere. These dynamical constraints act equally in a dry and a moist atmosphere. However, a comparison between dry and moist simulations shows that the equatorial jump is much deeper in a moist atmosphere. This is interpreted as resulting from the feedbacks between the large-scale flow and moist convection, which results in establishing a very weak gross moist stability for the equatorial jump.

## Abstract

This paper introduces the Mean Airflow as Lagrangian Dynamics Approximation (MAFALDA), a new method designed to extract thermodynamic cycles from numerical simulations of turbulent atmospheric flows. This approach relies on two key steps. First, mean trajectories are obtained by computing the mean circulation using height and equivalent potential temperature as coordinates. Second, thermodynamic properties along these trajectories are approximated by using their conditionally averaged values at the same height and *θ*
_{
e
}. This yields a complete description of the properties of air parcels that undergo a set of idealized thermodynamic cycles.

MAFALDA is applied to analyze the behavior of an atmosphere in radiative–convective equilibrium. The convective overturning is decomposed into 20 thermodynamic cycles, each accounting for 5% of the total mass transport. The work done by each cycle can be expressed as the difference between the maximum work that would have been done by an equivalent Carnot cycle and a penalty that arises from the injection and removal of water at different values of its Gibbs free energy. The analysis indicates that the Gibbs penalty reduces the work done by all thermodynamic cycles by about 55%. The cycles are also compared with those obtained for doubling the atmospheric carbon dioxide, which in the model used here leads to an increase in surface temperature of about 3.4 K. It is shown that warming greatly increases both the energy transport and work done per unit mass of air circulated. As a result, the ratio of the kinetic energy generation to the convective mass flux increases by about 20% in the simulations.

## Abstract

This paper introduces the Mean Airflow as Lagrangian Dynamics Approximation (MAFALDA), a new method designed to extract thermodynamic cycles from numerical simulations of turbulent atmospheric flows. This approach relies on two key steps. First, mean trajectories are obtained by computing the mean circulation using height and equivalent potential temperature as coordinates. Second, thermodynamic properties along these trajectories are approximated by using their conditionally averaged values at the same height and *θ*
_{
e
}. This yields a complete description of the properties of air parcels that undergo a set of idealized thermodynamic cycles.

MAFALDA is applied to analyze the behavior of an atmosphere in radiative–convective equilibrium. The convective overturning is decomposed into 20 thermodynamic cycles, each accounting for 5% of the total mass transport. The work done by each cycle can be expressed as the difference between the maximum work that would have been done by an equivalent Carnot cycle and a penalty that arises from the injection and removal of water at different values of its Gibbs free energy. The analysis indicates that the Gibbs penalty reduces the work done by all thermodynamic cycles by about 55%. The cycles are also compared with those obtained for doubling the atmospheric carbon dioxide, which in the model used here leads to an increase in surface temperature of about 3.4 K. It is shown that warming greatly increases both the energy transport and work done per unit mass of air circulated. As a result, the ratio of the kinetic energy generation to the convective mass flux increases by about 20% in the simulations.

## Abstract

The dynamics of convectively coupled gravity waves traveling over a precipitating region are analyzed in an idealized model for the large-scale atmospheric circulation. The model is composed of a shallow water system coupled to an advection equation for moisture through the convection term, utilizing a quasi-equilibrium relaxation to moisture closure. Here the authors investigate the model in the strict quasi-equilibrium (SQE) of infinitely short relaxation time. This framework is applied to study the behavior of a disturbance propagating along a narrow precipitation band, similar to the intertropical convergence zone (ITCZ). For an ITCZ width on the order of the equatorial Rossby radius, Kelvin waves propagate at the moist gravity wave speed (about 15 m s^{−1}), whereas for a narrow ITCZ, the propagation speed is comparable to the dry gravity wave (about 50 m s^{−1}). It is also shown that a Kelvin wave propagating along a narrow precipitation region exhibits a meridional circulation that modulates the precipitation rate and affects the propagation speed of the wave.

## Abstract

The dynamics of convectively coupled gravity waves traveling over a precipitating region are analyzed in an idealized model for the large-scale atmospheric circulation. The model is composed of a shallow water system coupled to an advection equation for moisture through the convection term, utilizing a quasi-equilibrium relaxation to moisture closure. Here the authors investigate the model in the strict quasi-equilibrium (SQE) of infinitely short relaxation time. This framework is applied to study the behavior of a disturbance propagating along a narrow precipitation band, similar to the intertropical convergence zone (ITCZ). For an ITCZ width on the order of the equatorial Rossby radius, Kelvin waves propagate at the moist gravity wave speed (about 15 m s^{−1}), whereas for a narrow ITCZ, the propagation speed is comparable to the dry gravity wave (about 50 m s^{−1}). It is also shown that a Kelvin wave propagating along a narrow precipitation region exhibits a meridional circulation that modulates the precipitation rate and affects the propagation speed of the wave.

## Abstract

A new relationship between the surface distribution of equivalent potential temperature and the potential temperature at the tropopause is proposed. Using a Gaussian approximation for the distribution of equivalent potential temperature, the authors argue that the tropopause potential temperature is approximately given by the mean equivalent potential temperature at the surface plus twice its standard derivation. This relationship is motivated by the comparison of the meridional circulation on dry and moist isentropes. It is further tested using four reanalysis datasets: the Interim ECMWF Re-Analysis (ERA-Interim); the NCEP–Department of Energy (DOE) Reanalysis II; the NCEP Climate Forecast System Reanalysis; and the Twentieth-Century Reanalysis (20CR), version 2. The proposed relationship successfully captures the annual cycle of the tropopause for both hemispheres. The results are robust among different reanalysis datasets, albeit the 20CR tends to overestimate the tropopause potential temperature. Furthermore, the proposed mechanism also works well in obtaining the interannual variability (with climatological annual cycle removed) for Northern Hemisphere summer with an above-0.6 correlation across different reanalyses. On the contrary, this mechanism is rather weak in explaining the interannual variability in the Southern Hemisphere and no longer works for Northern Hemisphere wintertime. This work suggests the important role of the moist dynamics in determining the midlatitude tropopause.

## Abstract

A new relationship between the surface distribution of equivalent potential temperature and the potential temperature at the tropopause is proposed. Using a Gaussian approximation for the distribution of equivalent potential temperature, the authors argue that the tropopause potential temperature is approximately given by the mean equivalent potential temperature at the surface plus twice its standard derivation. This relationship is motivated by the comparison of the meridional circulation on dry and moist isentropes. It is further tested using four reanalysis datasets: the Interim ECMWF Re-Analysis (ERA-Interim); the NCEP–Department of Energy (DOE) Reanalysis II; the NCEP Climate Forecast System Reanalysis; and the Twentieth-Century Reanalysis (20CR), version 2. The proposed relationship successfully captures the annual cycle of the tropopause for both hemispheres. The results are robust among different reanalysis datasets, albeit the 20CR tends to overestimate the tropopause potential temperature. Furthermore, the proposed mechanism also works well in obtaining the interannual variability (with climatological annual cycle removed) for Northern Hemisphere summer with an above-0.6 correlation across different reanalyses. On the contrary, this mechanism is rather weak in explaining the interannual variability in the Southern Hemisphere and no longer works for Northern Hemisphere wintertime. This work suggests the important role of the moist dynamics in determining the midlatitude tropopause.

## Abstract

Responses of the atmospheric circulation to a doubling of CO_{2} are examined in a global climate model, focusing on the circulation on both dry and moist isentropes. The isentropic circulations are reconstructed using the statistical transformed Eulerian mean (STEM), which approximates the isentropic flow from the Eulerian-mean and second-order moments. This approach also makes it possible to decompose the changes in the circulation into changes in zonal mean and eddy statistics.

It is found that, as a consequence of CO_{2} doubling, the dry isentropic circulation weakens across all latitudes. The weaker circulation in the tropics is a result of the reduction in mean meridional circulation while the reduction in eddy sensible heat flux largely contributes to the slowdown of the circulation in the midlatitudes. The heat transport on dry isentropes, however, increases in the tropics because of the increase in dry effective stratification whereas it decreases in the extratropics following the reduction in eddy sensible heat transport. Distinct features are found on moist isentropes. In the tropics, the circulation weakens, but without much change in heat transport. The extratropical circulation shifts poleward with an intensification (weakening) on the poleward (equatorward) flank, primarily because of the change in eddy latent heat transport. The total heat transport in the midlatitudes also shows a poleward shift but is of smaller magnitude. The differences between the dry and moist circulations reveal that in a warming world the increase in midlatitude eddy moisture transport is associated with an increase in warm moist air exported from the subtropics into the midlatitude storm tracks.

## Abstract

Responses of the atmospheric circulation to a doubling of CO_{2} are examined in a global climate model, focusing on the circulation on both dry and moist isentropes. The isentropic circulations are reconstructed using the statistical transformed Eulerian mean (STEM), which approximates the isentropic flow from the Eulerian-mean and second-order moments. This approach also makes it possible to decompose the changes in the circulation into changes in zonal mean and eddy statistics.

It is found that, as a consequence of CO_{2} doubling, the dry isentropic circulation weakens across all latitudes. The weaker circulation in the tropics is a result of the reduction in mean meridional circulation while the reduction in eddy sensible heat flux largely contributes to the slowdown of the circulation in the midlatitudes. The heat transport on dry isentropes, however, increases in the tropics because of the increase in dry effective stratification whereas it decreases in the extratropics following the reduction in eddy sensible heat transport. Distinct features are found on moist isentropes. In the tropics, the circulation weakens, but without much change in heat transport. The extratropical circulation shifts poleward with an intensification (weakening) on the poleward (equatorward) flank, primarily because of the change in eddy latent heat transport. The total heat transport in the midlatitudes also shows a poleward shift but is of smaller magnitude. The differences between the dry and moist circulations reveal that in a warming world the increase in midlatitude eddy moisture transport is associated with an increase in warm moist air exported from the subtropics into the midlatitude storm tracks.

## Abstract

This paper presents a theoretical study of the effects of moist convection on geostrophic adjustment in an infinite channel. The governing equations correspond to a linearized shallow water system of equations for the atmosphere first vertical baroclinic mode, which is coupled to a vertically averaged moisture equation. The coupling is through a parameterization that represents precipitation. The transient behavior and final state of the flow initially at rest with active precipitation limited to half of the channel is investigated, both numerically and analytically. It is shown that an initial imbalance resulting from precipitation induces a circulation that dries out the nonprecipitating region and further enhances precipitation. This interaction between precipitation and dynamics leads to a sharper temperature gradient and stronger jet in the final state, when compared to the dry adjustment. Unlike in the dry case, the moist geostrophic adjustment cannot be entirely determined from the initial unbalanced flow, since it depends on the time scale for convection. Analytic approximations are derived in limits of both fast and slow convective adjustment time.

## Abstract

This paper presents a theoretical study of the effects of moist convection on geostrophic adjustment in an infinite channel. The governing equations correspond to a linearized shallow water system of equations for the atmosphere first vertical baroclinic mode, which is coupled to a vertically averaged moisture equation. The coupling is through a parameterization that represents precipitation. The transient behavior and final state of the flow initially at rest with active precipitation limited to half of the channel is investigated, both numerically and analytically. It is shown that an initial imbalance resulting from precipitation induces a circulation that dries out the nonprecipitating region and further enhances precipitation. This interaction between precipitation and dynamics leads to a sharper temperature gradient and stronger jet in the final state, when compared to the dry adjustment. Unlike in the dry case, the moist geostrophic adjustment cannot be entirely determined from the initial unbalanced flow, since it depends on the time scale for convection. Analytic approximations are derived in limits of both fast and slow convective adjustment time.

## Abstract

The present work analyzes the impacts of radiative cooling in three-dimensional high-resolution direct numerical simulations of moist Rayleigh–Bénard convection. An atmospheric slab is destabilized by imposing a warm, moist lower boundary and a colder, dryer upper boundary. These boundary conditions are chosen such that the atmosphere is relaxed toward a conditionally unstable state in which unsaturated air parcels experience a stable stratification and unsaturated parcels experience an unstable one. Conditionally unstable moist Rayleigh–Bénard convection in the absence of radiative transfer produces self-aggregated convectively active cloudy regions separated by a quiescent unsaturated environment. Such convection is strongly limited by diffusion and is unable to transport much energy. As radiative cooling partially compensates for the adiabatic warming in the unsaturated environment and destabilizes the lower unsaturated boundary, its inclusion results in a significant enhancement of convective activity and cloud cover. A dry convectively unstable region develops at the lower boundary in a way that is reminiscent of the planetary boundary layer. Convective transport increases through the entire layer, leading to a significant enhancement of the upward transport of energy and water.

## Abstract

The present work analyzes the impacts of radiative cooling in three-dimensional high-resolution direct numerical simulations of moist Rayleigh–Bénard convection. An atmospheric slab is destabilized by imposing a warm, moist lower boundary and a colder, dryer upper boundary. These boundary conditions are chosen such that the atmosphere is relaxed toward a conditionally unstable state in which unsaturated air parcels experience a stable stratification and unsaturated parcels experience an unstable one. Conditionally unstable moist Rayleigh–Bénard convection in the absence of radiative transfer produces self-aggregated convectively active cloudy regions separated by a quiescent unsaturated environment. Such convection is strongly limited by diffusion and is unable to transport much energy. As radiative cooling partially compensates for the adiabatic warming in the unsaturated environment and destabilizes the lower unsaturated boundary, its inclusion results in a significant enhancement of convective activity and cloud cover. A dry convectively unstable region develops at the lower boundary in a way that is reminiscent of the planetary boundary layer. Convective transport increases through the entire layer, leading to a significant enhancement of the upward transport of energy and water.