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- Author or Editor: Akio Arakawa x

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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.

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## Abstract

Preliminary tests of the multiscale modeling approach, also known as the cloud-resolving convective parameterization, or superparameterization, are performed using an idealized framework. In this approach, a two-dimensional cloud-system resolving model (CSRM) is embedded within each vertical column of a general circulation model (GCM) replacing conventional cloud parameterization. The purpose of this study is to investigate the coupling between the GCM and CSRMs and suggest a revised method of coupling that abandons the cyclic lateral boundary condition for each CSRM used in the original cloud-resolving convective parameterization. In this way, the CSRM extends into neighboring GCM grid boxes while sharing approximately the same mass fluxes with the GCM at the borders of the grid boxes.

With the original and revised methods of coupling, numerical simulations of the evolution of cloud systems are conducted using a two-dimensional model that couples CSRMs with a lower-resolution version of the CSRM with no physics [large-scale dynamics model (LSDM)]. The results with the revised method show that cloud systems can propagate from one LSDM grid column to the next as expected. Comparisons with a straightforward application of a single CSRM to the entire domain (CONTROL) show that the biases of the large-scale thermodynamic fields simulated by the coupled model are significantly smaller with the revised method. The results also show that the biases are near the smallest when the velocity fields of the LSDM and CSRM are nudged to each other with the time scale of a few hours and the thermodynamic field of the LSDM is instantaneously updated at each time step with the domain-averaged CSRM field.

## Abstract

Preliminary tests of the multiscale modeling approach, also known as the cloud-resolving convective parameterization, or superparameterization, are performed using an idealized framework. In this approach, a two-dimensional cloud-system resolving model (CSRM) is embedded within each vertical column of a general circulation model (GCM) replacing conventional cloud parameterization. The purpose of this study is to investigate the coupling between the GCM and CSRMs and suggest a revised method of coupling that abandons the cyclic lateral boundary condition for each CSRM used in the original cloud-resolving convective parameterization. In this way, the CSRM extends into neighboring GCM grid boxes while sharing approximately the same mass fluxes with the GCM at the borders of the grid boxes.

With the original and revised methods of coupling, numerical simulations of the evolution of cloud systems are conducted using a two-dimensional model that couples CSRMs with a lower-resolution version of the CSRM with no physics [large-scale dynamics model (LSDM)]. The results with the revised method show that cloud systems can propagate from one LSDM grid column to the next as expected. Comparisons with a straightforward application of a single CSRM to the entire domain (CONTROL) show that the biases of the large-scale thermodynamic fields simulated by the coupled model are significantly smaller with the revised method. The results also show that the biases are near the smallest when the velocity fields of the LSDM and CSRM are nudged to each other with the time scale of a few hours and the thermodynamic field of the LSDM is instantaneously updated at each time step with the domain-averaged CSRM field.

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## Abstract

Advantages of using an isentropic vertical coordinate in atmospheric models are well recognized. In particular, the use of an isentropic coordinate virtually eliminates discretization errors for vertical advection since isentropic surfaces are material surfaces under dry-adiabatic processes. This is also advantageous for predicting moist-adiabatic condensation processes because their occurrence and maintenance largely depend on the converging moisture transport through the surrounding unsaturated regions.

In this paper, a basic problem in incorporating condensation heating into an isentropic coordinate model is discussed: that is, the problem of choosing a proper vertical grid for predicting moisture and computing condensation amount and condensation heating. Two different vertical grids are described, one of which predicts moisture for each model layer (M grid) and the other predicts it at each interface separating the model layers (N grid). The models based on these two vertical grids become identical without condensation. To illustrate the different impacts of these grids on dynamics, simulations of horizontally standing oscillations with two models based on these grids are presented. Results indicate that the model based on the M grid has difficulty in correctly recognizing the reduction of effective static stability due to condensation heating, while the model based on the N grid does not. The difficulty with the M grid is due to decoupling of condensation and heating for vertically small scales.

In view of these results, it is desirable to use the N grid in a model based on an isentropic vertical coordinate. The vertically discrete moisture continuity equation and a method to calculate condensation amount and heating on the N grid are presented.

## Abstract

Advantages of using an isentropic vertical coordinate in atmospheric models are well recognized. In particular, the use of an isentropic coordinate virtually eliminates discretization errors for vertical advection since isentropic surfaces are material surfaces under dry-adiabatic processes. This is also advantageous for predicting moist-adiabatic condensation processes because their occurrence and maintenance largely depend on the converging moisture transport through the surrounding unsaturated regions.

In this paper, a basic problem in incorporating condensation heating into an isentropic coordinate model is discussed: that is, the problem of choosing a proper vertical grid for predicting moisture and computing condensation amount and condensation heating. Two different vertical grids are described, one of which predicts moisture for each model layer (M grid) and the other predicts it at each interface separating the model layers (N grid). The models based on these two vertical grids become identical without condensation. To illustrate the different impacts of these grids on dynamics, simulations of horizontally standing oscillations with two models based on these grids are presented. Results indicate that the model based on the M grid has difficulty in correctly recognizing the reduction of effective static stability due to condensation heating, while the model based on the N grid does not. The difficulty with the M grid is due to decoupling of condensation and heating for vertically small scales.

In view of these results, it is desirable to use the N grid in a model based on an isentropic vertical coordinate. The vertically discrete moisture continuity equation and a method to calculate condensation amount and heating on the N grid are presented.

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## 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.

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## 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.