Steady plane fronts in a rotating fluid

Abstract

A derivation is given of the equstions describing a steady plane front in a rotating differentially heated fluid bounded by a horizontal non-conducting wall. The kinematic viscosity v and the thermal conductivity x are assumed small and of the same order. The thickness of the front is of order (v/Ω)1/2 where Ω is the angular velocity, and the characteristic length along the front is of order (ga/Ω2) tgΔT, where g is the accelersation of gravity, a the coefficient of thermal expansion, ε the slope of the front, and ΔT the temperature variation across the front. There is a circulation in a vertical plane perpendicular to the front surface driven by the friction boundary layer. In this plane Coriolis acceleration, non-linear acceleration and friction forces are all of the same order. Considering the mass and momentum transport equations and requiring the solution to be gravitationally stable, one can show that there must be a frictional inflow towards the front surface from both the cold and the warm side. A similarity form satisfies the front equations and leads to a system of ordinary differential equstions. Some qualitative features of the solution can be deduced, but the complete solution, requiring an analysis of the transition region where the front joins the friction boundary layer, is still lacking. The result obtained is in qualitative agreement with observations of fronts in some laboratory experiments, and with observations of natural fronts in the atmosphere and the oceans.

In the latter cases the parameters v and x in the theory should stand for the turbulent coefficients of viscosity and conductivity.