Abstract

The onset of thermal convection in a horizontal layer of ∞uid rotating about a vertical axis is examined by means of a nonlocal model partial dierential 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 ∞uid 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 ∞uid layer predicts a preference for convection in a square planform rather than two-dimensional roll motions. We nd 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 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.

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