A Linear Stability of Convective Motion in a Thermally Unstable Layer Below a Stable Region

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Abstract
A linearized stability analysis is made for convection arising in a horizontal layer of fluid with constant unstable temperature gradient and penetrating into a layer of fluid with constant stable temperature gradient lying above the unstable layer. With the Boussinesq approximation, solutions for the entire fluid are obtained by matching solutions for the unstable and stable layers at the interface. The critical Rayleigh number is calculated as a function of the depth and the static stability of the stable layer. Eigenfunctions for the vertical distributions of vertical velocity and temperature are also calculated. It is found that, for a fixed depth of the stable layer, the horizontal scale of convective cells at the marginal state is increased as the stability in the stable layer is decreased. However, with the infinitely deep stable layer, the vertical scale of convective cells is also increased and cells look more slender as the stability becomes smaller. For a fixed stability, as the epth of the stable layer is increased, both the critical Rayleigh number and the corresponding horizontal wavenumber are decreased first, attain minimums and then show oscillatory variations about asymptotic values. This oscillatory behavior of the critical Rayleigh number appears to be associated with the transition of the number of secondary cells occurring in the stable layer.

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