Finite amplitude two-dimensional convection in a finite rotating system

Abstract

This paper considers the effect of rotation, measured by a Taylor number T , on two-dimensional Bénard convection between horizontal stress-free boundaries which are maintained at different constant temperatures. The fluid is confined laterally by rigid sidewalls which are assumed only approximately insulating, the possibility of small lateral heat losses, which are observed experimentally, being incorporated in the theory. The distance between the sidewalls is 2 L times the height of the layer. A weakly nonlinear theory based on the method of multiple scales is de­veloped to describe the motion for slightly supercritical Rayleigh numbers R , and large aspect ratios ( L ≫ 1), although the results are also valid for finite values of L if the speed of rotation is large ( T ≫ 1). In the exchange case a steady finite amplitude solution evolves if the Prandtl number of the fluid σ is greater than 0.577, but subcritical instability and bursting can occur for a certain range of Taylor numbers if σ < 0.577. In the over­stable case disturbances propagate between the sidewalls, and ultimately either decay or, for Rayleigh numbers greater than a critical value depending on both σ and T , attain an equilibrium state controlled by reflexion at the sidewalls.