On the Vertical Stability of a Rotating Fluid Subject to a Horizontal Temperature Gradient

Abstract

The vertical potential temperature gradient in the atmosphere is the result of heat transport by radiation and by small- and large-scale motions, with complications due to latent heat release and absorption in condensation and cloud. An idealized model, characterized by the absence of latent heat effects and of a vertical component to the gradient of the impressed temperature field, is considered in this paper. As the Péclét number is very large, the governing mathematical equations are highly non-linear and generally intractable. Nevertheless, simple physical reasoning suggests that when the flow is axisymmetric the average vertical gradient of potential temperature set up by the flow itself should be close to 0.67|ΔT|/d (and that the corresponding expression for a non-rotating fluid is 0.50|ΔT|/d), where ΔT is the impressed horizontal temperature contrast and d is the depth of the fluid. This result and an expression based on it for convective heat transfer agree satisfactorily with laboratory and numerical studies discussed in detail elsewhere. It is also argued that when, owing to baroclinic instability, non-axisymmetric flow occurs, the average vertical temperature gradient is maintained by the baroclinic waves at that value for which the ratio of the Brunt-Väisälä frequency to the Coriolis parameter is approximately equal to the ratio of the horizontal dimension of the system to d. This conclusion is also consistent with observations.