Williams, G. P., 1972: Friction term formulation and convective instability in a shallow atmosphere. Journal of the Atmospheric Sciences, 29 (5), 870-876.

Abstract: The form of the friction terms for a shallow layer of fluid on a sphere is discussed for isotropic and transversely-isotropic fluids. We then examine the nature of convection in a transversely-isotropic fluid and find that long flat convection cells with a width to height ratio of 2 (v_H/v_V)^1/2 are produced, where v_H, v_V are the horizontal and vertical diffusion coefficients. The critical Rayleigh number is given by R ^c = 4 pi ⁴ v_H/v_V, but another Rayleigh number with a constant critical value is shown to be a more relevant parameter.