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Akio Arakawa

Abstract

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

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

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Chichung Lin and Akio Arakawa

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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Chichung Lin and Akio Arakawa

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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Shrinivas Moorthi and Akio Arakawa

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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Chichung Lin and Akio Arakawa

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 (RPCAPromax), 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 RPCAPromax, 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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Akio Arakawa and Shrinivas Moorthi

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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Celal S. Konor and Akio Arakawa

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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Akio Arakawa and Wayne Howard Schubert

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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.

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Kuan-Man Xu and Akio Arakawa

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The Arakawa-Schubert (A-S) cumulus parameterization is evaluated by performing semiprognostic tests against data simulated by a cumulus ensemble model (CEM). The CEM is a two-dimensional cloud model for simulating the formation of an ensemble of cumulus clouds under prescribed large-scale conditions. Three simulations, two with vertical wind shear and one without, are performed with identical (time-varying) large-scale advective effects.

The semiprognostic tests follow a procedure similar to that used by Lord except that simulated data averaged over the entire domain or selected subdomains of the CEM provide the “observed” large-scale conditions. Detailed comparisons were made between the results of simulation and parameterization. The results include comparisons of surface precipitation rate, apparent heat source, apparent moisture sink, updraft mass flux, and downdraft mass flux. Two different sets of tests were performed. One is the standard A-S parameterization with the cloud work function (CWF) quasi equilibrium, and the other allows CWF nonequilibrium by taking into account the simulated time change of the CWF. The tests show that the A-S parameterization is basically valid in spite of the existence of mesoscale organization in cumulus convection. In particular, the assumption of CWF quasi equilibrium is more accurate for inputs averaged over smaller subdomain sizes that resolve some mesoscale processes. On the other hand, errors due to the nondiagnostic aspect of cumulus convection are more significant for inputs averaged over larger subdomain sizes. Errors due to the inherent nondeterministic aspect of cumulus convection appear to be more significant for inputs averaged over smaller subdomain sizes.

A modified A-S parameterization with a convective-scale downdraft formulation was also tested against the simulated data. The inclusion of downdrafts slightly improves the results of semiprognostic tests. The impact of downdrafts on the subcloud layer may depend significantly on the subdomain size.

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Chin-Hoh Moeng and Akio Arakawa

Abstract

A model for numerical simulation of stratus cloud layers is constructed by combining a second-order closure, turbulent transfer model with a thermal radiative transfer model. The turbulent transfer model allows water vapor saturation. The combined turbulence-radiation model is applied to both a horizontally uniform one-dimensional case and a horizontally nonuniform two-dimensional case. In the latter, the dynamics of mesoscale circulations are also incorporated.

Results of the two-dimensional simulation show that the layer cloud instability occurs where the sea surface temperature is high and the large-scale subsidence is weak. The simulated instability is analyzed in view of an instability criterion, the eddy kinetic energy budget, and evaporative cooling near the cloud top.

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