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- Author or Editor: Akio Arakawa x

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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

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

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

## Abstract

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

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

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## Abstract

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

## Abstract

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

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## Abstract

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

## Abstract

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

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## Abstract

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

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

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

## Abstract

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

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

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