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## Abstract

This paper considers the problem of numerically integrating the primitive equations corresponding to B 2-level model of the atmosphere bounded by two zonal walls on a spherical earth. Inertio-gravitational motions of the external type are filtered *a priori;* for such a constraint it is possible to define a stream function corresponding to the vertically integrated motions. A system of integration is developed for initial conditions which specify the shear wind vector, the specific volume, and the vorticity of the vertically integrated flow. Methods for reducing truncation error and for increasing the rate of convergence of the elliptic part are discussed.

The question of boundary conditions is discussed at length. It is shown that the usual central difference methods yield independent solutions at alternate points, thus providing a source of computational instability to which the primitive equations are particularly sensitive. The solutions may be made compatible by suitable computational boundary conditions which can be deduced as sufficient conditions for insuring that the numerical solutions possess exact integrals. The application of these considerations to viscous flow is also discussed.

## Abstract

This paper considers the problem of numerically integrating the primitive equations corresponding to B 2-level model of the atmosphere bounded by two zonal walls on a spherical earth. Inertio-gravitational motions of the external type are filtered *a priori;* for such a constraint it is possible to define a stream function corresponding to the vertically integrated motions. A system of integration is developed for initial conditions which specify the shear wind vector, the specific volume, and the vorticity of the vertically integrated flow. Methods for reducing truncation error and for increasing the rate of convergence of the elliptic part are discussed.

The question of boundary conditions is discussed at length. It is shown that the usual central difference methods yield independent solutions at alternate points, thus providing a source of computational instability to which the primitive equations are particularly sensitive. The solutions may be made compatible by suitable computational boundary conditions which can be deduced as sufficient conditions for insuring that the numerical solutions possess exact integrals. The application of these considerations to viscous flow is also discussed.

# GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS

## I. THE BASIC EXPERIMENT

## Abstract

An extended period numerical integration of a baroclinic primitive equation model has been made for the simulation and the study of the dynamics of the atmosphere's general circulation. The solution corresponding to external gravitational propagation is filtered by requiring the vertically integrated divergence to vanish identically. The vertical structure permits as dependent variables the horizontal wind at two internal levels and a single temperature, with the static stability entering as a parameter.

The incoming radiation is a function of latitude only corresponding to the annual mean, and the outgoing radiation is taken to be a function of the local temperature. With the requirement for thermal equilibrium, the domain mean temperature is specified as a parameter. The role of condensation is taken into account only as it effectively reduces the static stability. All other external sources and sinks of heat are assumed to balance each other locally, and are thus omitted. The kinematics are that of a fluid on a sphere bounded by smooth zonal walls at the equator and at approximately 64° latitude. The dissipative sinks are provided by: (a) surface stresses proportional through a drag coefficient to the square of the surface wind which is suitably extrapolated from above, (b) internal convective stresses proportional to the vertical wind shear, and (c) lateral diffusion of momentum and heat through an exchange coefficient which depends on the local horizontal rate of strain—a horizontal length scale entering as the governing parameter.

For a given specification of the parameters, an integration for 60 days has been made from initial conditions where random temperature disturbances have been superimposed on a zonally symmetric regime which is baroclinically unstable according to linear theory. This experiment not only displays the scale selective character of baroclinic instability, yielding zonal wave number 5 to 6, but also predicts an index or energy cycle. The period of this cycle is 11 to 12 days for the first 40 days of the experiment, then lengthening to 17 days while diminishing in amplitude during the latter part.

The resulting mean zonal velocity profile is in good qualitative agreement with observation, but too intense, presumably because the effective static stability parameter is taken too large. Furthermore this profile is found to be no more than 5 percent super-geostrophic poleward of the angular momentum maximum and no more than 2 percent sub-geostrophic equatorward. The total zonal angular momentum remains constant to within 2 percent irrespective of the phase of the index cycle. This balance is controlled by the surface wind distribution which agrees quite well with observation. The poleward transport is mainly accomplished by the large-scale eddies, whereas the internal vertical flux is predominantly a transfer of the earth's angular momentum by the meridional circulation.

The poleward heat transport is primarily accomplished by a Hadley circulation at low latitudes but by the large-scale horizontal eddies in mid-latitudes, where a Ferrel circulation tends to compensate through an equatorward flux. This compensation at mid-latitudes by an indirect meridional circulation is also quite evident, in the potential-kinetic energy transformations. Comparison of the momentum and heat transfer with observed data when available shows reasonably good quantitative agreement.

The lateral transfer of momentum and heat by the non-linear diffusion, which parametrically is supposed to simulate the action of motions of sub-grid scale, accounts for a significant portion of the total eddy transfer. Although no direct comparison with the corresponding transfer in the real atmosphere is available, intuitively our small-scale diffusion appears to play too large a role.

A diagnosis is made of the transformations among the baratropic and baroclinic parts of the kinetic energy as well as the zonal mean and zonal perturbation parts of the available potential and kinetic energy. This reveals the dominant paths that the energy passes through from source to ultimate sinks and the processes responsible for these transformations. It is found that the partitioning of dissipation by the energy components may differ considerably from estimates made from observation.

## Abstract

An extended period numerical integration of a baroclinic primitive equation model has been made for the simulation and the study of the dynamics of the atmosphere's general circulation. The solution corresponding to external gravitational propagation is filtered by requiring the vertically integrated divergence to vanish identically. The vertical structure permits as dependent variables the horizontal wind at two internal levels and a single temperature, with the static stability entering as a parameter.

The incoming radiation is a function of latitude only corresponding to the annual mean, and the outgoing radiation is taken to be a function of the local temperature. With the requirement for thermal equilibrium, the domain mean temperature is specified as a parameter. The role of condensation is taken into account only as it effectively reduces the static stability. All other external sources and sinks of heat are assumed to balance each other locally, and are thus omitted. The kinematics are that of a fluid on a sphere bounded by smooth zonal walls at the equator and at approximately 64° latitude. The dissipative sinks are provided by: (a) surface stresses proportional through a drag coefficient to the square of the surface wind which is suitably extrapolated from above, (b) internal convective stresses proportional to the vertical wind shear, and (c) lateral diffusion of momentum and heat through an exchange coefficient which depends on the local horizontal rate of strain—a horizontal length scale entering as the governing parameter.

For a given specification of the parameters, an integration for 60 days has been made from initial conditions where random temperature disturbances have been superimposed on a zonally symmetric regime which is baroclinically unstable according to linear theory. This experiment not only displays the scale selective character of baroclinic instability, yielding zonal wave number 5 to 6, but also predicts an index or energy cycle. The period of this cycle is 11 to 12 days for the first 40 days of the experiment, then lengthening to 17 days while diminishing in amplitude during the latter part.

The resulting mean zonal velocity profile is in good qualitative agreement with observation, but too intense, presumably because the effective static stability parameter is taken too large. Furthermore this profile is found to be no more than 5 percent super-geostrophic poleward of the angular momentum maximum and no more than 2 percent sub-geostrophic equatorward. The total zonal angular momentum remains constant to within 2 percent irrespective of the phase of the index cycle. This balance is controlled by the surface wind distribution which agrees quite well with observation. The poleward transport is mainly accomplished by the large-scale eddies, whereas the internal vertical flux is predominantly a transfer of the earth's angular momentum by the meridional circulation.

The poleward heat transport is primarily accomplished by a Hadley circulation at low latitudes but by the large-scale horizontal eddies in mid-latitudes, where a Ferrel circulation tends to compensate through an equatorward flux. This compensation at mid-latitudes by an indirect meridional circulation is also quite evident, in the potential-kinetic energy transformations. Comparison of the momentum and heat transfer with observed data when available shows reasonably good quantitative agreement.

The lateral transfer of momentum and heat by the non-linear diffusion, which parametrically is supposed to simulate the action of motions of sub-grid scale, accounts for a significant portion of the total eddy transfer. Although no direct comparison with the corresponding transfer in the real atmosphere is available, intuitively our small-scale diffusion appears to play too large a role.

A diagnosis is made of the transformations among the baratropic and baroclinic parts of the kinetic energy as well as the zonal mean and zonal perturbation parts of the available potential and kinetic energy. This reveals the dominant paths that the energy passes through from source to ultimate sinks and the processes responsible for these transformations. It is found that the partitioning of dissipation by the energy components may differ considerably from estimates made from observation.

## Abstract

With the three-dimensional field of velocity predicted by numerical methods it is possible to predict the moisture distribution and hence the occurrence of large-scale saturation. A three-parameter model was used to predict the 12-hour precipitation for the early stages of the storms of November 24, 1950 and November 5, 1953, neglecting cloud storage, supersaturation, a possible lack of condensation nuclei, evaporation from falling droplets, and moisture sources. Large-scale orographic influences were taken into account.

A quantitative comparison of the predicted rainfall with the correspondingly large-scale smoothed observed precipitation indicates a skill comparable to that of the predicted flow. An examination of the small-scale observed rainfall indicates that in these cases convective instability resulted in large standard deviations from the large-scale average. Numerical prediction of regions of convective instability, which is also shown, could for the time being be utilized for subjective interpretation.

## Abstract

With the three-dimensional field of velocity predicted by numerical methods it is possible to predict the moisture distribution and hence the occurrence of large-scale saturation. A three-parameter model was used to predict the 12-hour precipitation for the early stages of the storms of November 24, 1950 and November 5, 1953, neglecting cloud storage, supersaturation, a possible lack of condensation nuclei, evaporation from falling droplets, and moisture sources. Large-scale orographic influences were taken into account.

A quantitative comparison of the predicted rainfall with the correspondingly large-scale smoothed observed precipitation indicates a skill comparable to that of the predicted flow. An examination of the small-scale observed rainfall indicates that in these cases convective instability resulted in large standard deviations from the large-scale average. Numerical prediction of regions of convective instability, which is also shown, could for the time being be utilized for subjective interpretation.

## Abstract

Two-week predictions were made for two winter cases by applying the Geophysical Fluid Dynamics Laboratory high-resolution, nine-level, hemispheric, moist general circulation model. Three versions of the model are discussed: Experiment 1 includes the orography but not the radiative transfer or the turbulent exchange of heat and moisture with the lower boundary; Experiment 2 accounts for all of these effects as well as land-sea contrast; Experiment 3 allows, in addition, the difference in thermal properties between the land-ice and sea-ice surfaces, as well as an 80% relative humidity condensation criterion reduced from the 100% criterion in Experiments 1 and 2.

The computed results are compared with observed data in terms of the evolution of individual cyclonic and anticyclonic patterns, the zonal mean structure of temperature, wind, and humidity, the precipitation over the United States, and the hemispheric energetics.

The forecast near sea level was considerably improved in Experiments 2 and 3 over Experiment 1. The experiment succeeded in forecasting the birth of second and third generation extratropical cyclones and their behavior thereafter. The hemispheric sum of precipitation was increased five times in Experiment 2 over that in Experiment 1, and even more in Experiment 3, the greatest contribution occurring in the Tropics. Two winter cases were considered. The correlation coefficients between the observed and the forecast patterns for the change of 500-mb geopotential height from the initial time remained above 0.5 for 13 days in one case and for 9 days in the other.

There are, however, several defects in the model. The forecast temperature was too low. In the flow pattern the intensities of the Highs and Lows weakened appreciably after 6 or 8 days, reflecting the fact that the forecast of eddy kinetic energy was less than the observed. On the other hand, the intensity of the tropospheric westerlies was too great.

## Abstract

Two-week predictions were made for two winter cases by applying the Geophysical Fluid Dynamics Laboratory high-resolution, nine-level, hemispheric, moist general circulation model. Three versions of the model are discussed: Experiment 1 includes the orography but not the radiative transfer or the turbulent exchange of heat and moisture with the lower boundary; Experiment 2 accounts for all of these effects as well as land-sea contrast; Experiment 3 allows, in addition, the difference in thermal properties between the land-ice and sea-ice surfaces, as well as an 80% relative humidity condensation criterion reduced from the 100% criterion in Experiments 1 and 2.

The computed results are compared with observed data in terms of the evolution of individual cyclonic and anticyclonic patterns, the zonal mean structure of temperature, wind, and humidity, the precipitation over the United States, and the hemispheric energetics.

The forecast near sea level was considerably improved in Experiments 2 and 3 over Experiment 1. The experiment succeeded in forecasting the birth of second and third generation extratropical cyclones and their behavior thereafter. The hemispheric sum of precipitation was increased five times in Experiment 2 over that in Experiment 1, and even more in Experiment 3, the greatest contribution occurring in the Tropics. Two winter cases were considered. The correlation coefficients between the observed and the forecast patterns for the change of 500-mb geopotential height from the initial time remained above 0.5 for 13 days in one case and for 9 days in the other.

There are, however, several defects in the model. The forecast temperature was too low. In the flow pattern the intensities of the Highs and Lows weakened appreciably after 6 or 8 days, reflecting the fact that the forecast of eddy kinetic energy was less than the observed. On the other hand, the intensity of the tropospheric westerlies was too great.

## Abstract

The “primitive equations of motion” are adopted for this study. The nine levels of the model are distributed so as to resolve surface boundary layer fluxes as well as radiative transfer by ozone, carbon dioxide, and water vapor. The lower boundary is a kinematically uniform land surface without any heat capacity. The stabilizing effect of moist convection is implicitly incorporated into the model by requiring an adjustment of the lapse rate whenever it exceeds the moist adiabatic value. The numerical integrations are performed for the mean annual conditions over a hemisphere starting with an isothermal atmosphere at rest. The spatial distribution of gaseous absorbers is assumed to have the annual mean value of the actual atmosphere and to be constant with time.

A quasi-equilibrium is attained about which a cyclic energy variation occurs with an irregular period of about 2 weeks. The dominant wave number of the meridional component of the wind is 5 to 6 in the troposphere but is reduced to about 3 in the stratosphere. The gross structure and behavior of the tropopause and stratosphere below 30 km. agree reasonably well with observation. The meridional circulation obtained from the computation has a 3-cell structure in the troposphere and tends toward a 2-cell structure with increasing altitude in the stratosphere. Although the level of the jet stream as well as that of the maximum northward transport of momentum coincides with observation, the intensity of the jet stream turns out to be much stronger than the observed annual mean. In the stratosphere the temperature increases with increasing latitude because of the effect of large-scale motion. The magnitude of the increase, however, is smaller than that observed.

A detailed study of the vertical distribution of the budget of kinetic energy, of available potential energy, of heat, and of angular momentum is made. The mechanism for maintaining the kinetic energy of the jet stream and of the stratosphere is discussed. It is concluded that in the model the kinetic energy in the stratosphere is maintained against its conversion into potential energy and dissipation through interaction with the troposphere, which is in qualitative agreement with the results derived from an analysis of the actual atmosphere. In the troposphere, the conversion of potential energy reaches a maximum at about the 500-mb. level. This energy is then transferred to the level of the jet stream and to the surface boundary layer by the so-called pressure interaction term, thus providing the source of kinetic energy for these two levels at which dissipation is predominant. As with the results of Phillips [27] and Smagorinsky [37], the ratio of eddy kinetic energy to zonal kinetic energy and that of eddy to zonal available potential energy are computed to be much smaller than those of the actual atmosphere.

## Abstract

The “primitive equations of motion” are adopted for this study. The nine levels of the model are distributed so as to resolve surface boundary layer fluxes as well as radiative transfer by ozone, carbon dioxide, and water vapor. The lower boundary is a kinematically uniform land surface without any heat capacity. The stabilizing effect of moist convection is implicitly incorporated into the model by requiring an adjustment of the lapse rate whenever it exceeds the moist adiabatic value. The numerical integrations are performed for the mean annual conditions over a hemisphere starting with an isothermal atmosphere at rest. The spatial distribution of gaseous absorbers is assumed to have the annual mean value of the actual atmosphere and to be constant with time.

A quasi-equilibrium is attained about which a cyclic energy variation occurs with an irregular period of about 2 weeks. The dominant wave number of the meridional component of the wind is 5 to 6 in the troposphere but is reduced to about 3 in the stratosphere. The gross structure and behavior of the tropopause and stratosphere below 30 km. agree reasonably well with observation. The meridional circulation obtained from the computation has a 3-cell structure in the troposphere and tends toward a 2-cell structure with increasing altitude in the stratosphere. Although the level of the jet stream as well as that of the maximum northward transport of momentum coincides with observation, the intensity of the jet stream turns out to be much stronger than the observed annual mean. In the stratosphere the temperature increases with increasing latitude because of the effect of large-scale motion. The magnitude of the increase, however, is smaller than that observed.

A detailed study of the vertical distribution of the budget of kinetic energy, of available potential energy, of heat, and of angular momentum is made. The mechanism for maintaining the kinetic energy of the jet stream and of the stratosphere is discussed. It is concluded that in the model the kinetic energy in the stratosphere is maintained against its conversion into potential energy and dissipation through interaction with the troposphere, which is in qualitative agreement with the results derived from an analysis of the actual atmosphere. In the troposphere, the conversion of potential energy reaches a maximum at about the 500-mb. level. This energy is then transferred to the level of the jet stream and to the surface boundary layer by the so-called pressure interaction term, thus providing the source of kinetic energy for these two levels at which dissipation is predominant. As with the results of Phillips [27] and Smagorinsky [37], the ratio of eddy kinetic energy to zonal kinetic energy and that of eddy to zonal available potential energy are computed to be much smaller than those of the actual atmosphere.

# SIMULATED CLIMATOLOGY OF A GENERAL CIRCULATION MODEL WITH A HYDROLOGIC CYCLE

## III. Effects of Increased Horizontal Computational Resolution

## Abstract

The results of a numerical time integration of a hemispheric general circulation model of the atmosphere with moist processes and a uniform earth's surface has already been published by Manabe, Smagorinsky, and Strickler. In this study, the integration is repeated after halving the midlatitude grid size from approximately 500 to 250 km.

This increase in the resolution of the horizontal finite differences markedly improves the features of the model atmosphere. For example, the system of fronts and the associated cyclone families in the high resolution atmosphere is much more realistic than that in the low resolution atmosphere. Furthermore, the general magnitude and the spectral distribution of eddy kinetic energy are in better agreement with the actual atmosphere as a result of the improvement in resolution.

In order to explain these improvements, an extensive analysis of the energetics of both the low and high resolution atmospheric models is carried out. It is shown that these improvements are due not only to the increase of the accuracy of the finite differences but also to the shift in the scale of dissipation by the nonlinear lateral viscosity toward a smaller scale resulting from the decrease in grid size. In the low resolution atmospheric model, the transfer of energy from eddy to zonal kinetic energy is missing because of excessive subgrid scale dissipation at medium wave numbers, whereas it has significant magnitude in the high resolution atmospheric model. It is speculated that further increase of resolution should improve the results because it tends to separate the characteristic scale of dissipation from that of the source of eddy kinetic energy.

The analysis of the energetics in wave number space clearly demonstrates the differences between the energetics of the different parts of the atmosphere. In middle latitudes there are essential differences between the energetics of the model troposphere and that of the model stratosphere. In the model troposphere, the eddy kinetic energy is produced by the conversion of eddy potential energy in the range of wave numbers from 2 to 8. Part of the energy thus produced is dissipated by the subgrid scale dissipation, and most of the remainder is decascaded to zonal kinetic energy. In the model stratosphere, where very long waves predominate, the eddy kinetic energy is generated in the range of wave numbers from 2 to 3 by the energy supplied from the troposphere. Most of this energy is then decascaded barotropically to zonal kinetic energy.

In the Tropics, eddy kinetic energy is mainly produced by the release of eddy available potential energy generated by the heat of condensation. Although the rate of conversion is maximum at very low wave numbers, the conversion spectrum extends to very high wave numbers.

A box diagram of the energetics of the high resolution moist model shows that the eddy available potential energy is generated by the heat of condensation as well as by energy transfer from the zonal available potential energy. Furthermore, it is noteworthy that the zonal kinetic energy is maintained not only by the barotropic exchange from the eddy kinetic energy but also from the conversions of zonal potential energy. The intensification of the direct tropical cell and the weakening of the indirect Ferrel cell in middle latitudes caused by the moist processes are responsible for this positive zonal conversion.

One of the highlights of the results from the integration of the high resolution moist model is the successful simulation of the evolution of fronts and the associated cyclone families. The influence of moist processes upon frontal structure as well as other synoptic features is investigated by comparing the moist model atmosphere with the dry model atmosphere without the effect of the selective heating of condensation. It is found that the heat of condensation significantly reduces the width of fronts and the characteristic scale of cyclones in the lower troposphere.

## Abstract

The results of a numerical time integration of a hemispheric general circulation model of the atmosphere with moist processes and a uniform earth's surface has already been published by Manabe, Smagorinsky, and Strickler. In this study, the integration is repeated after halving the midlatitude grid size from approximately 500 to 250 km.

This increase in the resolution of the horizontal finite differences markedly improves the features of the model atmosphere. For example, the system of fronts and the associated cyclone families in the high resolution atmosphere is much more realistic than that in the low resolution atmosphere. Furthermore, the general magnitude and the spectral distribution of eddy kinetic energy are in better agreement with the actual atmosphere as a result of the improvement in resolution.

In order to explain these improvements, an extensive analysis of the energetics of both the low and high resolution atmospheric models is carried out. It is shown that these improvements are due not only to the increase of the accuracy of the finite differences but also to the shift in the scale of dissipation by the nonlinear lateral viscosity toward a smaller scale resulting from the decrease in grid size. In the low resolution atmospheric model, the transfer of energy from eddy to zonal kinetic energy is missing because of excessive subgrid scale dissipation at medium wave numbers, whereas it has significant magnitude in the high resolution atmospheric model. It is speculated that further increase of resolution should improve the results because it tends to separate the characteristic scale of dissipation from that of the source of eddy kinetic energy.

The analysis of the energetics in wave number space clearly demonstrates the differences between the energetics of the different parts of the atmosphere. In middle latitudes there are essential differences between the energetics of the model troposphere and that of the model stratosphere. In the model troposphere, the eddy kinetic energy is produced by the conversion of eddy potential energy in the range of wave numbers from 2 to 8. Part of the energy thus produced is dissipated by the subgrid scale dissipation, and most of the remainder is decascaded to zonal kinetic energy. In the model stratosphere, where very long waves predominate, the eddy kinetic energy is generated in the range of wave numbers from 2 to 3 by the energy supplied from the troposphere. Most of this energy is then decascaded barotropically to zonal kinetic energy.

In the Tropics, eddy kinetic energy is mainly produced by the release of eddy available potential energy generated by the heat of condensation. Although the rate of conversion is maximum at very low wave numbers, the conversion spectrum extends to very high wave numbers.

A box diagram of the energetics of the high resolution moist model shows that the eddy available potential energy is generated by the heat of condensation as well as by energy transfer from the zonal available potential energy. Furthermore, it is noteworthy that the zonal kinetic energy is maintained not only by the barotropic exchange from the eddy kinetic energy but also from the conversions of zonal potential energy. The intensification of the direct tropical cell and the weakening of the indirect Ferrel cell in middle latitudes caused by the moist processes are responsible for this positive zonal conversion.

One of the highlights of the results from the integration of the high resolution moist model is the successful simulation of the evolution of fronts and the associated cyclone families. The influence of moist processes upon frontal structure as well as other synoptic features is investigated by comparing the moist model atmosphere with the dry model atmosphere without the effect of the selective heating of condensation. It is found that the heat of condensation significantly reduces the width of fronts and the characteristic scale of cyclones in the lower troposphere.