Long-Wavelength Rotating Convection Between Poorly Conducting Boundaries

The onset of thermal convection in a horizontal layer of fluid rotating about a vertical axis is examined by means of a nonlocal model partial differential equation (PDE). This PDE is obtained asymptotically from the Navier--Stokes and heat equations in the limit of small conductivity of the horizontal boundaries. The model describes the onset of convection near a steady bifurcation from the conduction state and is valid 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 restricted version of our model PDE for convection in a nonrotating fluid layer predicts a preference for convection in a square planform rather than 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 Küppers--Lortz instability when rotation is introduced. We analyze the stability of weakly nonlinear rolls and square planforms and supplement our analysis with numerical simulations of the model PDE. The most notable feature of the numerical simulations in square periodic domains of moderate size is the strong preference for convection in a square planform.