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## Abstract

A review of the cumulus parameterization problem is presented with an emphasis on its conceptual aspects covering the history of the underlying ideas, major problems existing at present, and possible directions and approaches for future climate models. Since its introduction in the early 1960s, there have been decades of controversies in posing the cumulus parameterization problem. In this paper, it is suggested that confusion between budget and advection considerations is primarily responsible for the controversies. It is also pointed out that the performance of parameterization schemes can be better understood if one is not bound by their authors' justifications. The current trend in posing cumulus parameterization is away from deterministic diagnostic closures, including instantaneous adjustments, toward prognostic or nondeterministic closures, including relaxed and/or triggered adjustments. A number of questions need to be answered, however, for the merit of this trend to be fully utilized.

Major practical and conceptual problems in the conventional approach of cumulus parameterization, which include artificial separations of processes and scales, are then discussed. It is rather obvious that for future climate models the scope of the problem must be drastically expanded from “cumulus parameterization” to “unified cloud parameterization,” or even to “unified model physics.” This is an extremely challenging task, both intellectually and computationally, and the use of multiple approaches is crucial even for a moderate success. “Cloud-resolving convective parameterization” or “superparameterization” is a promising new approach that can develop into a multiscale modeling framework (MMF). It is emphasized that the use of such a framework can unify our currently diversified modeling efforts and make verification of climate models against observations much more constructive than it is now.

## Abstract

A review of the cumulus parameterization problem is presented with an emphasis on its conceptual aspects covering the history of the underlying ideas, major problems existing at present, and possible directions and approaches for future climate models. Since its introduction in the early 1960s, there have been decades of controversies in posing the cumulus parameterization problem. In this paper, it is suggested that confusion between budget and advection considerations is primarily responsible for the controversies. It is also pointed out that the performance of parameterization schemes can be better understood if one is not bound by their authors' justifications. The current trend in posing cumulus parameterization is away from deterministic diagnostic closures, including instantaneous adjustments, toward prognostic or nondeterministic closures, including relaxed and/or triggered adjustments. A number of questions need to be answered, however, for the merit of this trend to be fully utilized.

Major practical and conceptual problems in the conventional approach of cumulus parameterization, which include artificial separations of processes and scales, are then discussed. It is rather obvious that for future climate models the scope of the problem must be drastically expanded from “cumulus parameterization” to “unified cloud parameterization,” or even to “unified model physics.” This is an extremely challenging task, both intellectually and computationally, and the use of multiple approaches is crucial even for a moderate success. “Cloud-resolving convective parameterization” or “superparameterization” is a promising new approach that can develop into a multiscale modeling framework (MMF). It is emphasized that the use of such a framework can unify our currently diversified modeling efforts and make verification of climate models against observations much more constructive than it is now.

## 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.

## 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.

## 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.

## 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.

## 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.

## Abstract

Two types of vertical grids are used for atmospheric models: the Lorenz grid (L grid) and the Charney–Phillips grid (CP grid). Although the CP grid is the standard grid for quasigenstrophic models, it is not widely used in the primitive equation models because it is easier with the L grid to maintain some of the integral properties of the continuous system.

In this paper, problems with the L grid are pointed out that are due to the existence of an extra degree of freedom in the vertical distribution of the temperature (and the potential temperature). Then a vertical differencing of the primitive equations based on the CP grid is presented, while most of the advantages of the L grid in a hybrid σ–*p* vertical coordinate are maintained. The discrete hydrostatic equation is constructed in such a way that it is free from the vertical computational mode in the thermal field. Also, the vertical advection of the potential temperature in the discrete thermodynamic equation is constructed in such a way that it reduces to the standard (and most straightforward) vertical differencing of the quasigeostrophic equations based on the CP grid.

Simulations of standing oscillations superposed on a resting atmosphere are presented using two vertically discrete models, one based on the L grid and the other on the CP grid. The comparison of the simulations shows that with the L grid a stationary vertically zigzag pattern dominates in the thermal field, while with the CP grid no such pattern is evident. Simulations of the growth of an extratropical cyclone in a cyclic channel on a β plane are also presented using two different σ-coordinate models, again one with the L grid and the other with the CP grid, starting from random disturbances. The L grid simulation is dominated by short waves, while there is no evidence of short-wave growth in the CP grid simulation.

## Abstract

Two types of vertical grids are used for atmospheric models: the Lorenz grid (L grid) and the Charney–Phillips grid (CP grid). Although the CP grid is the standard grid for quasigenstrophic models, it is not widely used in the primitive equation models because it is easier with the L grid to maintain some of the integral properties of the continuous system.

In this paper, problems with the L grid are pointed out that are due to the existence of an extra degree of freedom in the vertical distribution of the temperature (and the potential temperature). Then a vertical differencing of the primitive equations based on the CP grid is presented, while most of the advantages of the L grid in a hybrid σ–*p* vertical coordinate are maintained. The discrete hydrostatic equation is constructed in such a way that it is free from the vertical computational mode in the thermal field. Also, the vertical advection of the potential temperature in the discrete thermodynamic equation is constructed in such a way that it reduces to the standard (and most straightforward) vertical differencing of the quasigeostrophic equations based on the CP grid.

Simulations of standing oscillations superposed on a resting atmosphere are presented using two vertically discrete models, one based on the L grid and the other on the CP grid. The comparison of the simulations shows that with the L grid a stationary vertically zigzag pattern dominates in the thermal field, while with the CP grid no such pattern is evident. Simulations of the growth of an extratropical cyclone in a cyclic channel on a β plane are also presented using two different σ-coordinate models, again one with the L grid and the other with the CP grid, starting from random disturbances. The L grid simulation is dominated by short waves, while there is no evidence of short-wave growth in the CP grid simulation.

## Abstract

A vertical finite-difference scheme for the primitive equations in sigma coordinates is obtained by requiring that the discrete equations retain some important properties of the continuous equations. A family of schemes is derived whose members conserve total energy, maintain an integral constraint on the vertically integrated pressure gradient force, have a local differencing of the hydrostatic equation, and give exact forms of the hydrostatic equation and the pressure gradient force for particular atmospheres. The proposed scheme is a member of this family that in addition conserves the global mass integral of the potential temperature under abiabatic processes.

## Abstract

A vertical finite-difference scheme for the primitive equations in sigma coordinates is obtained by requiring that the discrete equations retain some important properties of the continuous equations. A family of schemes is derived whose members conserve total energy, maintain an integral constraint on the vertically integrated pressure gradient force, have a local differencing of the hydrostatic equation, and give exact forms of the hydrostatic equation and the pressure gradient force for particular atmospheres. The proposed scheme is a member of this family that in addition conserves the global mass integral of the potential temperature under abiabatic processes.

## Abstract

To improve the simulation of nonlinear aspects of the flow over steep topography, a potential enstrophy and energy conserving scheme for the shallow water equations is derived. It is pointed out that a family of schemes can conserve total energy for general flow and potential enstrophy for flow with no mass flux divergence. The newly derived scheme is a unique member of this family, that conserves both potential enstrophy and energy for general flow. Comparison by means of numerical experiment with a scheme that conserves (potential) enstrophy for purely horizontal nondivergent flow demonstrated the considerable superiority of the newly derived potential enstrophy and energy conserving scheme, not only in suppressing a spurious energy cascade but also in determining the overall flow regime. The potential enstrophy and energy conserving scheme for a spherical grid is also presented.

## Abstract

To improve the simulation of nonlinear aspects of the flow over steep topography, a potential enstrophy and energy conserving scheme for the shallow water equations is derived. It is pointed out that a family of schemes can conserve total energy for general flow and potential enstrophy for flow with no mass flux divergence. The newly derived scheme is a unique member of this family, that conserves both potential enstrophy and energy for general flow. Comparison by means of numerical experiment with a scheme that conserves (potential) enstrophy for purely horizontal nondivergent flow demonstrated the considerable superiority of the newly derived potential enstrophy and energy conserving scheme, not only in suppressing a spurious energy cascade but also in determining the overall flow regime. The potential enstrophy and energy conserving scheme for a spherical grid is also presented.

## Abstract

In constructing a numerical model of the atmosphere, we must choose an appropriate vertical coordinate. Among the various possibilities, isentropic vertical coordinates such as the θ-coordinate seem to have the greatest potential, in spite of the technical difficulties in treating the intersections of coordinate surfaces with the lower boundary. The purpose of this paper is to describe the θ-coordinate model we have developed and to demonstrate its potential through simulating the nonlinear evolution of a baroclinic wave.

In the model we have developed, vertical discretization maintains important integral constraints, such as conservation of the angular momentum and total energy. In treating the intersections of coordinate surfaces with the lower boundary, we have followed the massless-layer approach in which the intersecting coordinate surfaces are extended along the boundary by introducing massless layers. Although this approach formally eliminates the intersection problem, it raises other computational problems. Horizontal discretization of the continuity and momentum equations in the model has been carefully designed to overcome these problems.

Selected results from a 10-day integration with the 25-layer, β-plane version of the model are presented. It seems that the model can simulate the nonlinear evolution of a baroclinic wave and associated dynamical processes without major computational difficulties.

## Abstract

In constructing a numerical model of the atmosphere, we must choose an appropriate vertical coordinate. Among the various possibilities, isentropic vertical coordinates such as the θ-coordinate seem to have the greatest potential, in spite of the technical difficulties in treating the intersections of coordinate surfaces with the lower boundary. The purpose of this paper is to describe the θ-coordinate model we have developed and to demonstrate its potential through simulating the nonlinear evolution of a baroclinic wave.

In the model we have developed, vertical discretization maintains important integral constraints, such as conservation of the angular momentum and total energy. In treating the intersections of coordinate surfaces with the lower boundary, we have followed the massless-layer approach in which the intersecting coordinate surfaces are extended along the boundary by introducing massless layers. Although this approach formally eliminates the intersection problem, it raises other computational problems. Horizontal discretization of the continuity and momentum equations in the model has been carefully designed to overcome these problems.

Selected results from a 10-day integration with the 25-layer, β-plane version of the model are presented. It seems that the model can simulate the nonlinear evolution of a baroclinic wave and associated dynamical processes without major computational difficulties.