Finite prandtl number convection in spherical shells

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Abstract
Finite amplitude convection in spherical shells with spherically symmetric gravity and heat source distribution is considered. The nonlinear problem of three-dimensional convection in shells with stress-free and isothermal boundaries is solved by expanding the dependent variables in terms of powers of the amplitude of convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. The shell is assumed to be thick and only shells for which the ratio ζ of outer radius to inner radius is 2 or 3 are considered. Three cases, two of which lead to a self adjoint problem, are treated in this paper. The stable solutions are found to be l=2 modes for ζ=3 where l is the degree of the spherical harmonics and an l=3 non-axisymmetric mode which exhibits the symmetry of a tetrahedron for ζ=2. These stable solutions transport the maximum amount of heat. The Prandtl number dependence of the heat transport is computed for the various solutions analyzed in the paper.

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