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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

A three-dimensional anelastic model has been developed using the vorticity equation, in which the pressure gradient force is eliminated. The prognostic variables of the model dynamics are the horizontal components of vorticity at all heights and the vertical component of vorticity and the horizontally uniform part of the horizontal velocity at a selected height. To implement the anelastic approximation, vertical velocity is diagnostically determined from the predicted horizontal components of vorticity by solving an elliptic equation. This procedure replaces solving the elliptic equation for pressure in anelastic models based on the momentum equation. Discretization of the advection terms uses an upstream-weighted partially third-order scheme. When time is continuous, the solution of this scheme is quadratically bounded. As an application of the model, interactions between convection and its environment with vertical shear are studied without and with model physics from the viewpoint of vorticity dynamics, that is, the deceleration/acceleration process of the basic flow in particular. The authors point out that the process is purely three-dimensional, especially when the convection is relatively localized, involving the twisting terms and the horizontal as well as vertical transports of vorticity. Finally, it is emphasized that parameterization of cumulus friction is a resolution-dependent problem of vorticity dynamics associated with cumulus convection.

## Abstract

A three-dimensional anelastic model has been developed using the vorticity equation, in which the pressure gradient force is eliminated. The prognostic variables of the model dynamics are the horizontal components of vorticity at all heights and the vertical component of vorticity and the horizontally uniform part of the horizontal velocity at a selected height. To implement the anelastic approximation, vertical velocity is diagnostically determined from the predicted horizontal components of vorticity by solving an elliptic equation. This procedure replaces solving the elliptic equation for pressure in anelastic models based on the momentum equation. Discretization of the advection terms uses an upstream-weighted partially third-order scheme. When time is continuous, the solution of this scheme is quadratically bounded. As an application of the model, interactions between convection and its environment with vertical shear are studied without and with model physics from the viewpoint of vorticity dynamics, that is, the deceleration/acceleration process of the basic flow in particular. The authors point out that the process is purely three-dimensional, especially when the convection is relatively localized, involving the twisting terms and the horizontal as well as vertical transports of vorticity. Finally, it is emphasized that parameterization of cumulus friction is a resolution-dependent problem of vorticity dynamics associated with cumulus convection.

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## Abstract

A system of equations is presented that unifies the nonhydrostatic anelastic system and the quasi-hydrostatic compressible system for use in global cloud-resolving models. By using a properly defined quasi-hydrostatic density in the continuity equation, the system is fully compressible for quasi-hydrostatic motion and anelastic for purely nonhydrostatic motion. In this way, the system can cover a wide range of horizontal scales from turbulence to planetary waves while filtering vertically propagating sound waves of all scales. The continuity equation is primarily diagnostic because the time derivative of density is calculated from the thermodynamic (and surface pressure tendency) equations as a correction to the anelastic continuity equation. No reference state is used and no approximations are made in the momentum and thermodynamic equations. An equation that governs the time change of total energy is also derived. Normal-mode analysis on an *f* plane without the quasigeostrophic approximation and on a midlatitude *β* plane with the quasigeostrophic approximation is performed to compare the unified system with other systems. It is shown that the unified system reduces the westward retrogression speed of the ultra-long barotropic Rossby waves through the inclusion of horizontal divergence due to compressibility.

## Abstract

A system of equations is presented that unifies the nonhydrostatic anelastic system and the quasi-hydrostatic compressible system for use in global cloud-resolving models. By using a properly defined quasi-hydrostatic density in the continuity equation, the system is fully compressible for quasi-hydrostatic motion and anelastic for purely nonhydrostatic motion. In this way, the system can cover a wide range of horizontal scales from turbulence to planetary waves while filtering vertically propagating sound waves of all scales. The continuity equation is primarily diagnostic because the time derivative of density is calculated from the thermodynamic (and surface pressure tendency) equations as a correction to the anelastic continuity equation. No reference state is used and no approximations are made in the momentum and thermodynamic equations. An equation that governs the time change of total energy is also derived. Normal-mode analysis on an *f* plane without the quasigeostrophic approximation and on a midlatitude *β* plane with the quasigeostrophic approximation is performed to compare the unified system with other systems. It is shown that the unified system reduces the westward retrogression speed of the ultra-long barotropic Rossby waves through the inclusion of horizontal divergence due to compressibility.

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## Abstract

For time integrations of the wave equation, it is desirable to use a scheme that is stable over a wide range of the Courant number. Implicit schemes are examples of such schemes, but they do that job at the expense of global calculation, which becomes an increasingly serious burden as the horizontal resolution becomes higher while covering a large horizontal domain. If what an implicit scheme does from the point of view of explicit differencing is looked at, it is a multipoint scheme that requires information at all grid points in space. Physically this is an overly demanding requirement because wave propagation in the real atmosphere has a finite speed. The purpose of this study is to seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme with a finite number of grid points in space. In this paper, a space-centered trapezoidal implicit scheme is used as the target scheme as an example. It is shown that an explicit space-centered scheme with forward time differencing using an infinite number of grid points in space can be made equivalent to the trapezoidal implicit scheme. To avoid global calculation, a truncated version of the scheme is then introduced that only uses a finite number of grid points while maintaining stability. This approach of constructing a stable explicit scheme is called multipoint explicit differencing (MED). It is shown that the coefficients in an MED scheme can be numerically determined by single-time-step integrations of the target scheme. With this procedure, it is rather straightforward to construct an MED scheme for an arbitrarily shaped grid and/or boundaries. In an MED scheme, the number of grid points necessary to maintain stability and, therefore, the CPU time needed for each time step increase as the Courant number increases. Because of this overhead, the MED scheme with a large time step can be more efficient than a usual explicit scheme with a smaller time step only for complex multilevel models with detailed physics. The efficiency of an MED scheme also depends on how the advantage of parallel computing is taken.

## Abstract

For time integrations of the wave equation, it is desirable to use a scheme that is stable over a wide range of the Courant number. Implicit schemes are examples of such schemes, but they do that job at the expense of global calculation, which becomes an increasingly serious burden as the horizontal resolution becomes higher while covering a large horizontal domain. If what an implicit scheme does from the point of view of explicit differencing is looked at, it is a multipoint scheme that requires information at all grid points in space. Physically this is an overly demanding requirement because wave propagation in the real atmosphere has a finite speed. The purpose of this study is to seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme with a finite number of grid points in space. In this paper, a space-centered trapezoidal implicit scheme is used as the target scheme as an example. It is shown that an explicit space-centered scheme with forward time differencing using an infinite number of grid points in space can be made equivalent to the trapezoidal implicit scheme. To avoid global calculation, a truncated version of the scheme is then introduced that only uses a finite number of grid points while maintaining stability. This approach of constructing a stable explicit scheme is called multipoint explicit differencing (MED). It is shown that the coefficients in an MED scheme can be numerically determined by single-time-step integrations of the target scheme. With this procedure, it is rather straightforward to construct an MED scheme for an arbitrarily shaped grid and/or boundaries. In an MED scheme, the number of grid points necessary to maintain stability and, therefore, the CPU time needed for each time step increase as the Courant number increases. Because of this overhead, the MED scheme with a large time step can be more efficient than a usual explicit scheme with a smaller time step only for complex multilevel models with detailed physics. The efficiency of an MED scheme also depends on how the advantage of parallel computing is taken.

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## Abstract

One of the important roles of the PBL is to transport moisture from the surface to the cloud layer. However, how this transport process can be accounted for in cloud-resolving models (CRMs) is not sufficiently clear and has rarely been examined. A typical CRM can resolve the bulk feature of large convection systems but not the small-scale convection and turbulence motions that carry a large portion of the moisture fluxes. This study uses a large-eddy simulation of a tropical deep-convection system as a benchmark to examine the subgrid-scale (SGS) moisture transport into a cloud system.

It is shown that most of the PBL moisture transport to the cloud layer occurs in the areas under low-level updrafts, with rain, or under cloudy skies, although these PBL regimes may cover only a small fraction of the entire cloud-system domain. To develop SGS parameterizations to represent the spatial distribution of this moisture transport in CRMs, three models are proposed and tested. An updraft–downdraft model performs exceptionally well, while a statistical-closure model and a local-gradient model are less satisfactory but still perform adequately. Each of these models, however, has its own closure issues to be addressed. The updraft–downdraft model requires a scheme to estimate the mean SGS updraft–downdraft properties, the statistical-closure model needs a scheme to predict both SGS vertical-velocity and moisture variances, while the local-gradient model requires estimation of the SGS vertical-velocity variance.

## Abstract

One of the important roles of the PBL is to transport moisture from the surface to the cloud layer. However, how this transport process can be accounted for in cloud-resolving models (CRMs) is not sufficiently clear and has rarely been examined. A typical CRM can resolve the bulk feature of large convection systems but not the small-scale convection and turbulence motions that carry a large portion of the moisture fluxes. This study uses a large-eddy simulation of a tropical deep-convection system as a benchmark to examine the subgrid-scale (SGS) moisture transport into a cloud system.

It is shown that most of the PBL moisture transport to the cloud layer occurs in the areas under low-level updrafts, with rain, or under cloudy skies, although these PBL regimes may cover only a small fraction of the entire cloud-system domain. To develop SGS parameterizations to represent the spatial distribution of this moisture transport in CRMs, three models are proposed and tested. An updraft–downdraft model performs exceptionally well, while a statistical-closure model and a local-gradient model are less satisfactory but still perform adequately. Each of these models, however, has its own closure issues to be addressed. The updraft–downdraft model requires a scheme to estimate the mean SGS updraft–downdraft properties, the statistical-closure model needs a scheme to predict both SGS vertical-velocity and moisture variances, while the local-gradient model requires estimation of the SGS vertical-velocity variance.

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## Abstract

Although there are important advantages in the use of an isentropic vertical coordinate in atmospheric models, it requires overcoming computational difficulties associated with intersections of coordinate surfaces with the earth’s surface. In this paper, the authors present a model based on the generalized vertical coordinate, *ζ* = *F*(*θ, p, p*
_{
S
}), in which an isentropic coordinate can be combined with a terrain-following *σ* coordinate near the surface with a smooth transition between the two. One of the key issues in developing such a model is to satisfy consistency between the predictions of the pressure and the potential temperature. In the model presented in this paper, consistency is maintained by the use of an equation that determines the vertical mass flux. A procedure to properly choose *ζ* = *F*(*θ, p, p*
_{
S
}) is also presented, which guarantees that *ζ* is a monotonic function of height even when unstable stratification occurs.

In the vertical discretization, the Charney–Phillips grid is used since, with this grid, it is straightforward to satisfy the thermodynamic equation when *ζ* = *θ.* In the generalized vertical coordinate, determining the pressure gradient force requires both the Montgomery potential and the geopotential at the same levels. The discrete hydrostatic equation is designed to maintain consistency between the two. The vertically discrete equations also satisfy two important integral constraints. With these features, the model becomes identical to the isentropic coordinate model developed by when *ζ* = *θ.*

To demonstrate the performance of the model, the simulated nonlinear evolution of a midlatitude disturbance starting from random disturbances is presented. In the simulation, physical processes are represented by simple thermal forcing in the form of Newtonian heating and friction in the form of Rayleigh damping. During the evolution of the disturbance, the model generates sharp fronts both at the surface and in the upper and middle troposphere. No serious computational difficulties are found in this simulation.

## Abstract

Although there are important advantages in the use of an isentropic vertical coordinate in atmospheric models, it requires overcoming computational difficulties associated with intersections of coordinate surfaces with the earth’s surface. In this paper, the authors present a model based on the generalized vertical coordinate, *ζ* = *F*(*θ, p, p*
_{
S
}), in which an isentropic coordinate can be combined with a terrain-following *σ* coordinate near the surface with a smooth transition between the two. One of the key issues in developing such a model is to satisfy consistency between the predictions of the pressure and the potential temperature. In the model presented in this paper, consistency is maintained by the use of an equation that determines the vertical mass flux. A procedure to properly choose *ζ* = *F*(*θ, p, p*
_{
S
}) is also presented, which guarantees that *ζ* is a monotonic function of height even when unstable stratification occurs.

In the vertical discretization, the Charney–Phillips grid is used since, with this grid, it is straightforward to satisfy the thermodynamic equation when *ζ* = *θ.* In the generalized vertical coordinate, determining the pressure gradient force requires both the Montgomery potential and the geopotential at the same levels. The discrete hydrostatic equation is designed to maintain consistency between the two. The vertically discrete equations also satisfy two important integral constraints. With these features, the model becomes identical to the isentropic coordinate model developed by when *ζ* = *θ.*

To demonstrate the performance of the model, the simulated nonlinear evolution of a midlatitude disturbance starting from random disturbances is presented. In the simulation, physical processes are represented by simple thermal forcing in the form of Newtonian heating and friction in the form of Rayleigh damping. During the evolution of the disturbance, the model generates sharp fronts both at the surface and in the upper and middle troposphere. No serious computational difficulties are found in this simulation.