The Steady Circulation of a Nonrotating, Viscous, Heat-Conducting Atmosphere

Robert E. Dickinson National Center for Atmospheric Research, Boulder, Colo.

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Abstract

The problem of small-amplitude, steady motions and temperature perturbations in a stratified, ideal gas atmosphere with viscosity and heat conduction is investigated. Horizontally varying sources of heat, mass and momentum force these motions. The model assumes two-dimensionality and no rotation. The solutions illustrate the coupling between motions and the temperature field in a stratified fluid with exponential increase in the vertical of the coefficients of kinematic viscosity and thermometric conductivity. Horizontal variation of temperature sets up pressure forces which drive horizontal winds. By continuity the divergence of these winds gives rise to vertical motions which alter the temperature fields. The solutions oscillate with height. Sources at relatively low levels can produce at high levels temperatures and circulations inverse to the directly driven temperature and wind oscillations.

The mathematical analysis reduces to the solution of a sixth-order ordinary differential equation of the hypergeometric type. Power series solutions are obtained in the pressure variable. These are used to satisfy boundary conditions at the zero pressure boundary. Another fundamental set of solutions is constructed in the form of integral representations in order to satisfy a boundedness condition at large pressure. The two solution bases are matched by an expansion of the integrals in power series. The solutions are used to construct a Green's function for the inhomogeneous problem. A numerical example assuming a heat source at zero pressure is discussed. Tabulated values and asymptotic approximations are used to provide a detailed numerical description of the solutions, their derivatives, and the Green's function coefficients.

Abstract

The problem of small-amplitude, steady motions and temperature perturbations in a stratified, ideal gas atmosphere with viscosity and heat conduction is investigated. Horizontally varying sources of heat, mass and momentum force these motions. The model assumes two-dimensionality and no rotation. The solutions illustrate the coupling between motions and the temperature field in a stratified fluid with exponential increase in the vertical of the coefficients of kinematic viscosity and thermometric conductivity. Horizontal variation of temperature sets up pressure forces which drive horizontal winds. By continuity the divergence of these winds gives rise to vertical motions which alter the temperature fields. The solutions oscillate with height. Sources at relatively low levels can produce at high levels temperatures and circulations inverse to the directly driven temperature and wind oscillations.

The mathematical analysis reduces to the solution of a sixth-order ordinary differential equation of the hypergeometric type. Power series solutions are obtained in the pressure variable. These are used to satisfy boundary conditions at the zero pressure boundary. Another fundamental set of solutions is constructed in the form of integral representations in order to satisfy a boundedness condition at large pressure. The two solution bases are matched by an expansion of the integrals in power series. The solutions are used to construct a Green's function for the inhomogeneous problem. A numerical example assuming a heat source at zero pressure is discussed. Tabulated values and asymptotic approximations are used to provide a detailed numerical description of the solutions, their derivatives, and the Green's function coefficients.

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