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- Author or Editor: Akio Arakawa x

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## Abstract

The constraint on the coupled vertical profiles of cumulus heating and drying, which can be used as a partial closure in cumulus parameterization, is examined using observational data from convectively active regions in the summertime. The data used in this study include those derived from Global Atmospheric Research Programme (GARP) Atlantic Tropical Experiment Phase III, Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment over the intensive flux array region, and four subsets of the European Centre for Medium-Range Weather Forecasts Re-Analysis data that cover areas ranging from tropical to midlatitude continents. The profiles of **Q**
_{1} and **Q**
_{2} calculated from those data are analyzed using a statistical method. The proposed method is a revised version of the rotated principal component analysis based on the Promax rotation (RPCA_{Promax}), which is believed suitable for identifying basic structures embedded within a given dataset. It is designed in such a way that the distortion of identified structures due to the use of a linear model is minimized. The revised RPCA_{Promax}, together with some selected statistical tools, are evaluated using synthetic datasets before they are applied to observations.

The analysis of the observational data shows that, for all the convectively active regions examined, most of the variance of observed **Q**
_{1} and **Q**
_{2} can be explained by retaining only two modes. Moreover, while these two modes have different amplitudes in time and space, the shapes of the **Q**
_{1} and **Q**
_{2} profiles associated with each mode are similar from one region to another. In this sense, they are analogous to the cloud types in the spectral cumulus ensemble model of the Arakawa–Schubert cumulus parameterization, in which the spectral distribution of cloud-base mass flux varies with large-scale conditions while the vertical profile of normalized mass flux is fixed for each cloud type. It is suggested that, as far as deep convection is concerned, the cloud model in cumulus parameterization probably can be constructed based on the empirically determined **Q**
_{1} and **Q**
_{2} profiles.

## Abstract

The constraint on the coupled vertical profiles of cumulus heating and drying, which can be used as a partial closure in cumulus parameterization, is examined using observational data from convectively active regions in the summertime. The data used in this study include those derived from Global Atmospheric Research Programme (GARP) Atlantic Tropical Experiment Phase III, Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment over the intensive flux array region, and four subsets of the European Centre for Medium-Range Weather Forecasts Re-Analysis data that cover areas ranging from tropical to midlatitude continents. The profiles of **Q**
_{1} and **Q**
_{2} calculated from those data are analyzed using a statistical method. The proposed method is a revised version of the rotated principal component analysis based on the Promax rotation (RPCA_{Promax}), which is believed suitable for identifying basic structures embedded within a given dataset. It is designed in such a way that the distortion of identified structures due to the use of a linear model is minimized. The revised RPCA_{Promax}, together with some selected statistical tools, are evaluated using synthetic datasets before they are applied to observations.

The analysis of the observational data shows that, for all the convectively active regions examined, most of the variance of observed **Q**
_{1} and **Q**
_{2} can be explained by retaining only two modes. Moreover, while these two modes have different amplitudes in time and space, the shapes of the **Q**
_{1} and **Q**
_{2} profiles associated with each mode are similar from one region to another. In this sense, they are analogous to the cloud types in the spectral cumulus ensemble model of the Arakawa–Schubert cumulus parameterization, in which the spectral distribution of cloud-base mass flux varies with large-scale conditions while the vertical profile of normalized mass flux is fixed for each cloud type. It is suggested that, as far as deep convection is concerned, the cloud model in cumulus parameterization probably can be constructed based on the empirically determined **Q**
_{1} and **Q**
_{2} profiles.

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## Abstract

Two vertically discrete systems, one based on the “Charney-Phillips grid” and the other on the “Lorenz grid,” are compared in view of the quasi-geostrophic potential vorticity equation and baroclinic instability.

It is shown that with the Charney-Phillips grid, the standard grid for the quasi-geostrophic system of equations, one can easily maintain important dynamical constraints on quasi-geostrophic flow, such as the conservation of quasi-geostrophic potential vorticity through horizontal advection and resulting integral constraints. With the Lorenz grid, however, in which horizontal velocity and (potential) temperature are carried at same levels, it is not straightforward even to define quasi-geostrophic potential vorticity. Moreover, due to an extra degree of freedom in potential temperature, the Lorenz grid can falsely satisfy the necessary condition for baroclinic instability near the lower and upper boundaries. In fact, eigenvalue solutions of the linear quasi-geostrophic equations show the existence of spuriously amplifying modes with short wavelengths, one trapped near the lower boundary and the other near the upper boundary. The former grows more rapidly then the latter when static stability increases with height. In a model discretized both in vertical and horizontal, the spurious amplification appears with *high* horizontal resolution unless vertical resolution is very high.

The existence of the spurious amplification of short waves in a nonlinear primitive equation model is also confirmed. Here the amplification also influences longer waves though nonlinearity and upper level presumably through vertical propagation of gravity waves.

It is shown that the spurious amplification can be removed at its origin by introducing additional terms in the thermodynamic equations for the bottom and top layers, which effectively eliminate the possibility of falsely satisfying the necessary condition for baroclinic instability.

## Abstract

Two vertically discrete systems, one based on the “Charney-Phillips grid” and the other on the “Lorenz grid,” are compared in view of the quasi-geostrophic potential vorticity equation and baroclinic instability.

It is shown that with the Charney-Phillips grid, the standard grid for the quasi-geostrophic system of equations, one can easily maintain important dynamical constraints on quasi-geostrophic flow, such as the conservation of quasi-geostrophic potential vorticity through horizontal advection and resulting integral constraints. With the Lorenz grid, however, in which horizontal velocity and (potential) temperature are carried at same levels, it is not straightforward even to define quasi-geostrophic potential vorticity. Moreover, due to an extra degree of freedom in potential temperature, the Lorenz grid can falsely satisfy the necessary condition for baroclinic instability near the lower and upper boundaries. In fact, eigenvalue solutions of the linear quasi-geostrophic equations show the existence of spuriously amplifying modes with short wavelengths, one trapped near the lower boundary and the other near the upper boundary. The former grows more rapidly then the latter when static stability increases with height. In a model discretized both in vertical and horizontal, the spurious amplification appears with *high* horizontal resolution unless vertical resolution is very high.

The existence of the spurious amplification of short waves in a nonlinear primitive equation model is also confirmed. Here the amplification also influences longer waves though nonlinearity and upper level presumably through vertical propagation of gravity waves.

It is shown that the spurious amplification can be removed at its origin by introducing additional terms in the thermodynamic equations for the bottom and top layers, which effectively eliminate the possibility of falsely satisfying the necessary condition for baroclinic instability.

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## Abstract

We have investigated baroclinic instability with cumulus heating using a vertically discrete, linearized, quasi-geostrophic model on a β-plane. Two formulations of cumulus heating were used. The first formulation (η-model) rests on the assumption that heating at all levels is proportional to the vertical *p*-velocity at the top of the lowest model layer. The second formulation (AS-model) follows the cumulus parameterization proposed by Arakawa and Schubert.

We present results for basic states with a constant temperature lapse rate and zonal flows linear in pressure. With both formulations, we found the Green modes for easterly shears destabilized by cumulus heating. We discuss the mechanism of this destabilization along with the vertical structure and energetics of the perturbations.

We extended the analyses for basic zonal flows similar to those observed during the Indian summer monsoon season, with the AS-model. The wavelength, phase speed, growth rate and vertical structure corresponding to a peak growth rate are very similar to some of the observed monsoon depressions. This similarity indicates that baroclinic instability with cumulus heating can be responsible for the development of monsoon depressions.

## Abstract

We have investigated baroclinic instability with cumulus heating using a vertically discrete, linearized, quasi-geostrophic model on a β-plane. Two formulations of cumulus heating were used. The first formulation (η-model) rests on the assumption that heating at all levels is proportional to the vertical *p*-velocity at the top of the lowest model layer. The second formulation (AS-model) follows the cumulus parameterization proposed by Arakawa and Schubert.

We present results for basic states with a constant temperature lapse rate and zonal flows linear in pressure. With both formulations, we found the Green modes for easterly shears destabilized by cumulus heating. We discuss the mechanism of this destabilization along with the vertical structure and energetics of the perturbations.

We extended the analyses for basic zonal flows similar to those observed during the Indian summer monsoon season, with the AS-model. The wavelength, phase speed, growth rate and vertical structure corresponding to a peak growth rate are very similar to some of the observed monsoon depressions. This similarity indicates that baroclinic instability with cumulus heating can be responsible for the development of monsoon depressions.

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## Abstract

Parameterization of cumulus convection requires a model that describes the statistical properties of a cumulus ensemble under given large-scale conditions. Such a model is called a cloud model for cumulus parameterization (CMCP). It would be best if the development of a CMCP were guided by synchronous observations covering a population of clouds. Unfortunately, observations for cumulus clouds are usually confined to individual clouds, leaving many uncertainties in designing a CMCP.

In an attempt to improve the formulation of entrainment effects in a CMCP, the data simulated by a two-dimensional cloud-resolving model are used to investigate sources of entrainment into cumulus clouds. The authors first plot the Paluch diagram using the data from a nonprecipitating experiment. It is found that typical patterns on the Paluch diagram obtained by observational studies can be reproduced using the simulated data and can be interpreted in ways other than two-point mixing. The authors further examine entrainment sources through extensive trajectory analysis using the data from a precipitating experiment. We find that cloud air parcels at one level usually originate from locations of various heights, indicating a continuous series of entrainment events occurring throughout the cloud depth. However, the authors do not find a cloud air parcel descending more than several hundred meters. Penetrative downdrafts produced by mixing between cloud air and entrained air are not observed in the cases simulated. It seems that, as far as tropical deep convection is concerned, ignoring the contribution from descendent cloud air in a CMCP is an acceptable simplification.

## Abstract

Parameterization of cumulus convection requires a model that describes the statistical properties of a cumulus ensemble under given large-scale conditions. Such a model is called a cloud model for cumulus parameterization (CMCP). It would be best if the development of a CMCP were guided by synchronous observations covering a population of clouds. Unfortunately, observations for cumulus clouds are usually confined to individual clouds, leaving many uncertainties in designing a CMCP.

In an attempt to improve the formulation of entrainment effects in a CMCP, the data simulated by a two-dimensional cloud-resolving model are used to investigate sources of entrainment into cumulus clouds. The authors first plot the Paluch diagram using the data from a nonprecipitating experiment. It is found that typical patterns on the Paluch diagram obtained by observational studies can be reproduced using the simulated data and can be interpreted in ways other than two-point mixing. The authors further examine entrainment sources through extensive trajectory analysis using the data from a precipitating experiment. We find that cloud air parcels at one level usually originate from locations of various heights, indicating a continuous series of entrainment events occurring throughout the cloud depth. However, the authors do not find a cloud air parcel descending more than several hundred meters. Penetrative downdrafts produced by mixing between cloud air and entrained air are not observed in the cases simulated. It seems that, as far as tropical deep convection is concerned, ignoring the contribution from descendent cloud air in a CMCP is an acceptable simplification.

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## Abstract

According to Part I of this paper, it seems that ignoring the contribution from descendent cloud air in a cloud model for cumulus parameterization (CMCP), such as the spectral cumulus ensemble model in the Arakawa–Schubert parameterization, is an acceptable simplification for tropical deep convection. Since each subensemble in the spectral cumulus ensemble model is formally analogous to an entraining plume, the latter is examined using the simulated data from a cloud-resolving model (CRM). The authors first follow the analysis procedure of Warner. With the data from a nonprecipitating experiment, the authors show that the entraining-plume model cannot simultaneously predict the mean liquid water profile and cloud top height of the clouds simulated by the CRM. However, the mean properties of active elements of clouds, which are characterized by strong updrafts, can be described by an entraining plume of similar top height.

With the data from a precipitating experiment, the authors examine the spectral cumulus ensemble model using the Paluch diagram. It is found that the spectral cumulus ensemble model appears adequate if different types of clouds in the spectrum are interpreted as subcloud elements with different entrainment characteristics. The resolved internal structure of clouds can thus be viewed as a manifestation of a cloud spectrum. To further investigate whether the fractional rate of entrainment is an appropriate parameter for characterizing cloud types in the spectral cumulus ensemble model, the authors stratify the simulated saturated updrafts (subcloud elements) into different types according to their eventual heights and calculate the cloud mass flux and mean moist static energy for each type. Entrainment characteristics are then inferred through the cloud mass flux and in-cloud moist static energy. It is found that different types of subcloud elements have distinguishable thermodynamic properties and entrainment characteristics. However, for each cloud type, the fractional rate of entrainment is not a constant in height but tends to be larger at lower levels and near cloud top. In addition, the in-cloud moist static energy at cloud base considerably deviates from the mean in the subcloud layer, indicating that the effects due to inhomogeneity of the planetary boundary layer should be taken into account in a CMCP as well.

## Abstract

According to Part I of this paper, it seems that ignoring the contribution from descendent cloud air in a cloud model for cumulus parameterization (CMCP), such as the spectral cumulus ensemble model in the Arakawa–Schubert parameterization, is an acceptable simplification for tropical deep convection. Since each subensemble in the spectral cumulus ensemble model is formally analogous to an entraining plume, the latter is examined using the simulated data from a cloud-resolving model (CRM). The authors first follow the analysis procedure of Warner. With the data from a nonprecipitating experiment, the authors show that the entraining-plume model cannot simultaneously predict the mean liquid water profile and cloud top height of the clouds simulated by the CRM. However, the mean properties of active elements of clouds, which are characterized by strong updrafts, can be described by an entraining plume of similar top height.

With the data from a precipitating experiment, the authors examine the spectral cumulus ensemble model using the Paluch diagram. It is found that the spectral cumulus ensemble model appears adequate if different types of clouds in the spectrum are interpreted as subcloud elements with different entrainment characteristics. The resolved internal structure of clouds can thus be viewed as a manifestation of a cloud spectrum. To further investigate whether the fractional rate of entrainment is an appropriate parameter for characterizing cloud types in the spectral cumulus ensemble model, the authors stratify the simulated saturated updrafts (subcloud elements) into different types according to their eventual heights and calculate the cloud mass flux and mean moist static energy for each type. Entrainment characteristics are then inferred through the cloud mass flux and in-cloud moist static energy. It is found that different types of subcloud elements have distinguishable thermodynamic properties and entrainment characteristics. However, for each cloud type, the fractional rate of entrainment is not a constant in height but tends to be larger at lower levels and near cloud top. In addition, the in-cloud moist static energy at cloud base considerably deviates from the mean in the subcloud layer, indicating that the effects due to inhomogeneity of the planetary boundary layer should be taken into account in a CMCP as well.

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## Abstract

The goal of this paper is to gain insight into the resolution dependence of model physics, the parameterization of moist convection in particular, which is required for accurately predicting large-scale features of the atmosphere. To achieve this goal, experiments using a two-dimensional nonhydrostatic model with different resolutions are conducted under various idealized tropical conditions. For control experiments (CONTROL), the model is run as a cloud-system-resolving model (CSRM). Next, a “large-scale dynamics model” (LSDM) is introduced as a diagnostic tool, which is a coarser-resolution version of the same model but with only partial or no physics. Then, the LSDM is applied to an ensemble of realizations selected from CONTROL and a “required parameterized source” (RPS) is identified for the results of the LSDM to become consistent with CONTROL as far as the resolvable scales are concerned.

The analysis of RPS diagnosed in this way confirms that RPS is highly resolution dependent in the range of typical resolutions of mesoscale models even in ensemble/space averages, while “real source” (RS) is not. The time interval of implementing model physics also matters for RPS. It is emphasized that model physics in future prediction models should automatically produce these resolution dependencies so that the need for retuning parameterizations as resolution changes can be minimized.

## Abstract

The goal of this paper is to gain insight into the resolution dependence of model physics, the parameterization of moist convection in particular, which is required for accurately predicting large-scale features of the atmosphere. To achieve this goal, experiments using a two-dimensional nonhydrostatic model with different resolutions are conducted under various idealized tropical conditions. For control experiments (CONTROL), the model is run as a cloud-system-resolving model (CSRM). Next, a “large-scale dynamics model” (LSDM) is introduced as a diagnostic tool, which is a coarser-resolution version of the same model but with only partial or no physics. Then, the LSDM is applied to an ensemble of realizations selected from CONTROL and a “required parameterized source” (RPS) is identified for the results of the LSDM to become consistent with CONTROL as far as the resolvable scales are concerned.

The analysis of RPS diagnosed in this way confirms that RPS is highly resolution dependent in the range of typical resolutions of mesoscale models even in ensemble/space averages, while “real source” (RS) is not. The time interval of implementing model physics also matters for RPS. It is emphasized that model physics in future prediction models should automatically produce these resolution dependencies so that the need for retuning parameterizations as resolution changes can be minimized.

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## Abstract

A generalized framework for cumulus parameterization applicable to any horizontal resolution between those typically used in general circulation and cloud-resolving models is presented. It is pointed out that the key parameter in the generalization is *σ*, which is the fractional area covered by convective updrafts in the grid cell. Practically all conventional cumulus parameterizations assume *σ* ≪ 1, at least implicitly, using the gridpoint values of the thermodynamic variables to define the thermal structure of the cloud environment. The proposed framework, called “unified parameterization,” eliminates this assumption from the beginning, allowing a smooth transition to an explicit simulation of cloud-scale processes as the resolution increases. If clouds and the environment are horizontally homogeneous with a top-hat profile, as is widely assumed in the conventional parameterizations, it is shown that the *σ* dependence of the eddy transport is through a simple quadratic function. Together with a properly chosen closure, the unified parameterization determines *σ* for each realization of grid-scale processes. The parameterization can also provide a framework for including stochastic parameterization. The remaining issues include parameterization of the in-cloud eddy transport because of the inhomogeneous structure of clouds.

## Abstract

A generalized framework for cumulus parameterization applicable to any horizontal resolution between those typically used in general circulation and cloud-resolving models is presented. It is pointed out that the key parameter in the generalization is *σ*, which is the fractional area covered by convective updrafts in the grid cell. Practically all conventional cumulus parameterizations assume *σ* ≪ 1, at least implicitly, using the gridpoint values of the thermodynamic variables to define the thermal structure of the cloud environment. The proposed framework, called “unified parameterization,” eliminates this assumption from the beginning, allowing a smooth transition to an explicit simulation of cloud-scale processes as the resolution increases. If clouds and the environment are horizontally homogeneous with a top-hat profile, as is widely assumed in the conventional parameterizations, it is shown that the *σ* dependence of the eddy transport is through a simple quadratic function. Together with a properly chosen closure, the unified parameterization determines *σ* for each realization of grid-scale processes. The parameterization can also provide a framework for including stochastic parameterization. The remaining issues include parameterization of the in-cloud eddy transport because of the inhomogeneous structure of clouds.

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## Abstract

In Part I of this paper, a generalized modeling framework for representing deep moist convection was presented. The framework, called unified parameterization, effectively unifies the parameterizations in general circulation models (GCMs) and cloud-resolving models (CRMs) and thus is applicable to any horizontal resolution between those typically used in those models. The key parameter in the unification is the fractional convective cloudiness *σ*, which is the fractional area covered by convective updrafts in the grid cell. The central issue of Part I is to formulate the *σ* dependence of vertical eddy transports of thermodynamic variables and to determine *σ* for each realization of grid-scale processes. The present paper completes the formulation through further analysis of the simulated data. The analyzed fields include the vertical structure of the *σ* dependence of vertical and horizontal eddy transports of moist static energy and horizontal momentum and that of cloud microphysical sources. For the momentum transport, the analysis results clearly show the limits of the traditional approach of parameterization based on an effectively one-dimensional model. For cloud microphysical conversions, it is shown that those taking place primarily inside and outside the updrafts are roughly proportional to *σ* and 1 − *σ*, respectively.

## Abstract

In Part I of this paper, a generalized modeling framework for representing deep moist convection was presented. The framework, called unified parameterization, effectively unifies the parameterizations in general circulation models (GCMs) and cloud-resolving models (CRMs) and thus is applicable to any horizontal resolution between those typically used in those models. The key parameter in the unification is the fractional convective cloudiness *σ*, which is the fractional area covered by convective updrafts in the grid cell. The central issue of Part I is to formulate the *σ* dependence of vertical eddy transports of thermodynamic variables and to determine *σ* for each realization of grid-scale processes. The present paper completes the formulation through further analysis of the simulated data. The analyzed fields include the vertical structure of the *σ* dependence of vertical and horizontal eddy transports of moist static energy and horizontal momentum and that of cloud microphysical sources. For the momentum transport, the analysis results clearly show the limits of the traditional approach of parameterization based on an effectively one-dimensional model. For cloud microphysical conversions, it is shown that those taking place primarily inside and outside the updrafts are roughly proportional to *σ* and 1 − *σ*, respectively.

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## Abstract

A theory of the interaction of a cumulus cloud ensemble with the large-scale environment is developed. In this theory, the large-scale environment is divided into the subcloud mixed layer and the region above. The time changes of the environment are governed by the heat and moisture budget equations for the subcloud mixed layer and for the region above, and by a prognostic equation for the depth of the mixed layer. In the environment above the mixed layer, the cumulus convection affects the temperature and moisture fields through cumulus-induced subsidence and detrainment of saturated air containing liquid water which evaporates in the environment. In the subcloud mixed layer, the cumulus convection does not act directly on the temperature and moisture fields, but it affects the depth of the mixed layer through cumulus-induced subsidence. Under these conditions the problem of parameterization of cumulus convection reduces to the determination of the vertical distributions of the total vertical mass flux by the ensemble, the total detrainment of mass from the ensemble, and the thermodynamical properties of the detraining air.

The cumulus ensemble is spectrally divided into sub-ensembles according to the fractional entrainment rate, given by the ratio of the entrainment per unit height to the vertical mass flux in the cloud. For these sub-ensembles, the budget equations for mass, moist static energy, and total water content are obtained. The solutions of these equations give the temperature excess, the water vapor excess, and the liquid water content of each sub-ensemble, and further reduce the problem of parameterization to the determination of the mass flux distribution function, which is the sub-ensemble vertical mass flux at the top of the mixed layer.

The cloud work function, which is an integral measure of the buoyancy force in the clouds, is defined for each sub-ensemble; and, under the assumption that it is in quasi-equilibrium, an integral equation for the mass flux distribution function is derived. This equation describes how a cumulus ensemble is forced by large-scale advection, radiation, and surface turbulent fluxes, and it provides a closed parameterization of cumulus convection for use in prognostic models of large-scale atmospheric motion.

## Abstract

A theory of the interaction of a cumulus cloud ensemble with the large-scale environment is developed. In this theory, the large-scale environment is divided into the subcloud mixed layer and the region above. The time changes of the environment are governed by the heat and moisture budget equations for the subcloud mixed layer and for the region above, and by a prognostic equation for the depth of the mixed layer. In the environment above the mixed layer, the cumulus convection affects the temperature and moisture fields through cumulus-induced subsidence and detrainment of saturated air containing liquid water which evaporates in the environment. In the subcloud mixed layer, the cumulus convection does not act directly on the temperature and moisture fields, but it affects the depth of the mixed layer through cumulus-induced subsidence. Under these conditions the problem of parameterization of cumulus convection reduces to the determination of the vertical distributions of the total vertical mass flux by the ensemble, the total detrainment of mass from the ensemble, and the thermodynamical properties of the detraining air.

The cumulus ensemble is spectrally divided into sub-ensembles according to the fractional entrainment rate, given by the ratio of the entrainment per unit height to the vertical mass flux in the cloud. For these sub-ensembles, the budget equations for mass, moist static energy, and total water content are obtained. The solutions of these equations give the temperature excess, the water vapor excess, and the liquid water content of each sub-ensemble, and further reduce the problem of parameterization to the determination of the mass flux distribution function, which is the sub-ensemble vertical mass flux at the top of the mixed layer.

The cloud work function, which is an integral measure of the buoyancy force in the clouds, is defined for each sub-ensemble; and, under the assumption that it is in quasi-equilibrium, an integral equation for the mass flux distribution function is derived. This equation describes how a cumulus ensemble is forced by large-scale advection, radiation, and surface turbulent fluxes, and it provides a closed parameterization of cumulus convection for use in prognostic models of large-scale atmospheric motion.