Penetrative Convective Instabilities

Abstract

An approximate method for finding the critical conditions for stability with respect to penetrative cellular perturbation is proposed. The method is tested by finding the critical conditions for stability of flow between counter-rotating cylinders and by comparing them with those obtained by more rigorous methods and by experiments. The same method is later applied for Bénard convection arising in a horizontal layer of fluid with constant unstable density gradient and penetrating into a layer of fluid with constant stable density gradient situated above the unstable layer. It is found that the classical rigid-free boundary solution corresponds to the limiting case of infinite stability on top of the unstable layer. For small stability on top the solutions have some of the features of Chandra's columnar perturbations and the build up of cumulus towers in the tropics. The critical value of the Rayleigh number in this case is found to be much smaller than that obtained by not taking penetration into account.