Numerical studies of penetrative convective instabilities

The stability of thermal convection for a statically unstable layer surmounted by a statically stable layer has been attacked by direct numerical integration of the linearized equations of motion and heat conduction. Two types of vertical temperature variation are considered, namely a piecewise-linear distribution and a piecewise-parabolic distribution. In both cases the ratio of the vertical gradient of temperature in the upper layer to the corresponding gradient in the lower layer is given by a specified parameter χ. Numerical values of a critical Rayleigh number, the most unstable wavelengths, and the depths of penetration of the convective cells into the stable region are given for several values of χ. The calculated forced cells that occur in the stable region are in good agreement with an appropriate linear theory. Comparison of the results for χp (parabolic) = −1 with the case of counter-rotating cylinders (D. L. Harris and W. H. Reid, 1964) shows a significant difference in the penetration of the cells despite equality of the parameters at instability.