GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS

I. THE BASIC EXPERIMENT

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  • 1 General Circulation Research Laboratory, U.S. Weather Bureau, Washington, D.C.
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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.

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