LONG-WAVELENGTH ROTATING CONVECTION BETWEEN POORLY CONDUCTING BOUNDARIES

The onset of thermal convection in a horizontal layer of fluid rotatin g about a vertical axis is examined by means of a nonlocal model parti al differential equation (PDE). This PDE is obtained asymptotically fr om the Navier-Stokes and heat equations in the limit of small conducti vity of the horizontal boundaries. The model describes the onset of co nvection near a steady bifurcation from the conduction state and is va lid provided the Prandtl number of the fluid is not too small and the rotation rate of the layer is not too great. It is known that a restri cted version of our model PDE for convection in a nonrotating fluid la yer predicts a preference for convection in a square planform rather t han two-dimensional roll motions. We find that this preference carries over to the rotating layer. The instability of rolls in a nonrotating layer is compounded by the Kuppers-Lortz instability when rotation is introduced. We analyze the stability of weakly nonlinear rolls and sq uare planforms and supplement our analysis with numerical simulations of the model PDE. The most notable feature of the numerical simulation s in square periodic domains of moderate size is the strong preference for convection in a square planform.