Nonlinear convection in a rotating fluid

Abstract

The paper studies convection in a horizontal layer of fluid rotating about a vertical axis. The flows at large Rayleigh number R, with a single horizontal wave-number, are investigated using the mean-field approximation of Herring (1963). The flow that maximizes the heat flux is the same as that which gives an upper bound to the heat flux in the limit of infinite Prandtl number as calculated by the methods of Howard (1963) and Chan (1971, 1974).

Rotation is not significant until the Taylor number Ta exceeds O(R). For $O(R) \ll Ta \ll O[(R\log R)^{\frac{4}{3}}]$, it can increase the rate of heat transfer, a phenomenon noted experimentally by Rossby (1969). It does so because an Ekman layer is formed outside the thermal boundary layer, causing a thinning of the thermal layer. The maximum value of the Nusselt number N is approximately $0.177R^{\frac{1}{5}} Ta^{\frac{1}{10}}[\log Ta]^{\frac{1}{5}}$. As the Taylor number increases further into the region $O[(R\log R)^{\frac{4}{3}}] \ll Ta \ll O(R^{\frac{3}{2}})$, the maximum value of N drops sharply, and becomes approximately $0.029 R^{\frac{3}{2}} Ta^{-1} \log(R^{\frac{3}{2}}/Ta)$. Hence, N now decreases with a further increase of Ta and eventually becomes O(1) as $Ta\rightarrow O(R^{\frac{3}{2}})$ and the layer becomes stable.