Numerical Solutions for the Shape-Preserving Two-Dimensional Thermal Convection Element

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  • 1 U.S. Weather Bureau, General Circulation Research Laboratory
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Abstract

The convective motions developed upon release of a line of fluid into a large body of miscible homogeneous fluid of a different density have been previously investigated experimentally by, J.M. Richards and found to be approximately shape-preserving. To investigate this phenomenon theoretically, the two-dimensional (xz) Boussinesq equations are here subjected to a space-time transformation such that shape-preserving buoyant elements (thermals) are steady state solutions in the transformed variables. Such solutions are approached asymptotically, for various values of spatially constant eddy viscosity and diffusion coefficients, by numerical time integration of a finite difference analog from given initial conditions. The solution for one set of coefficients is found to correspond rather closely with laboratory data obtained from Richards, while those for lower diffusion coefficients tend toward formation of a separated vortex pair. Solutions obtained by varying the eddy Prandtl number exhibit complex behavior. The effects of the finite boundary approximations used are found generally negligible except for the low diffusion case. The transient behavior helps relate the steady-state results to numerical solutions of transient behavior previously presented by Lilly and Ogura. The finite difference scheme used, adopted from work of A. Arakawa, conserves integrated momentum and quantities analogous to total energy and temperature variance.

Abstract

The convective motions developed upon release of a line of fluid into a large body of miscible homogeneous fluid of a different density have been previously investigated experimentally by, J.M. Richards and found to be approximately shape-preserving. To investigate this phenomenon theoretically, the two-dimensional (xz) Boussinesq equations are here subjected to a space-time transformation such that shape-preserving buoyant elements (thermals) are steady state solutions in the transformed variables. Such solutions are approached asymptotically, for various values of spatially constant eddy viscosity and diffusion coefficients, by numerical time integration of a finite difference analog from given initial conditions. The solution for one set of coefficients is found to correspond rather closely with laboratory data obtained from Richards, while those for lower diffusion coefficients tend toward formation of a separated vortex pair. Solutions obtained by varying the eddy Prandtl number exhibit complex behavior. The effects of the finite boundary approximations used are found generally negligible except for the low diffusion case. The transient behavior helps relate the steady-state results to numerical solutions of transient behavior previously presented by Lilly and Ogura. The finite difference scheme used, adopted from work of A. Arakawa, conserves integrated momentum and quantities analogous to total energy and temperature variance.

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