A Model for Investigating Eddy Viscosity Effects on Mesoscale Cellular Convection

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  • 1 Dept. of Geosciences, Purdue University, West Lafayette, Ind. 47907
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Abstract

A mathematical model consisting of a complete set of the Boussinesq equations governing Rayleigh convection in the atmosphere has been solved for the marginal stability case in order to study the geometry and circulation patterns of mesoscale cellular convection. The significant new feature of the model is the inclusion of eddy viscosity variation with height through the convecting layer. Results obtained show that the direction of convection circulations are controlled by the sign of the vertical gradient of eddy viscosity. It is also concluded that variable convective depth has a significant but small effect on the geometry of the convection, with the degree of cell flatness being principally controlled by the degree of anisotropy.

Using an assumed periodic form of the perturbation solutions, the problem reduced to solving a sixth-order differential stability equation with variable coefficients and accompanying boundary conditions. By applying the two-variable, small-perturbation technique, the stability equation was solved yielding a marginal stability diagram relating an “atmospheric” Rayleigh number to the wavenumber of a given disturbance. Further, a family of neutral stability curves was generated for different convective depths. Marginal stability solutions obtained for different eddy viscosity changes with height show the dependency of the circulation reversal on changing the sign of the eddy viscosity gradient. In order to obtain the direction of the vertical motion it was necessary to impose Stommel's argument of minimum frictional loss. Therefore, convection cells with descending motions in the center, occurring in an environment of decreasing eddy viscosity with height, are seen to change their circulation direction from an “open” cell to a “closed” cell by simply changing the eddy viscosity to increase with height (and conversely).

Abstract

A mathematical model consisting of a complete set of the Boussinesq equations governing Rayleigh convection in the atmosphere has been solved for the marginal stability case in order to study the geometry and circulation patterns of mesoscale cellular convection. The significant new feature of the model is the inclusion of eddy viscosity variation with height through the convecting layer. Results obtained show that the direction of convection circulations are controlled by the sign of the vertical gradient of eddy viscosity. It is also concluded that variable convective depth has a significant but small effect on the geometry of the convection, with the degree of cell flatness being principally controlled by the degree of anisotropy.

Using an assumed periodic form of the perturbation solutions, the problem reduced to solving a sixth-order differential stability equation with variable coefficients and accompanying boundary conditions. By applying the two-variable, small-perturbation technique, the stability equation was solved yielding a marginal stability diagram relating an “atmospheric” Rayleigh number to the wavenumber of a given disturbance. Further, a family of neutral stability curves was generated for different convective depths. Marginal stability solutions obtained for different eddy viscosity changes with height show the dependency of the circulation reversal on changing the sign of the eddy viscosity gradient. In order to obtain the direction of the vertical motion it was necessary to impose Stommel's argument of minimum frictional loss. Therefore, convection cells with descending motions in the center, occurring in an environment of decreasing eddy viscosity with height, are seen to change their circulation direction from an “open” cell to a “closed” cell by simply changing the eddy viscosity to increase with height (and conversely).

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