Square patterns in rotating Rayleigh-Bénard convection

The Küppers-Lortz instability occurs in rotating Rayleigh-Bénard convection and is a paradigmatic example of spatiotemporal chaos. Since the steady state of convection rolls is unstable to disturbance rolls oriented with an angle of about 60 degrees with respect to the given rolls in the prograde direction [G. Küppers and D. Lortz, J. Fluid Mech. 35, 609 (1969)], a spatiotemporally chaotic pattern is realized with patches of rolls continuously replaced by other patches in which the roll axis is switched by about 60 degrees. Surprisingly and contrary to this established scenario, Bajaj [Phys. Rev. Lett. 81 (1998)] observed experimentally square patterns in a cylindrical layer in the range of parameters where Küppers-Lortz instability was expected. In this paper we present square patterns which we have obtained in a numerical study by taking into account realistic boundary conditions. The Navier-Stokes and heat transport equations have been solved in the Oberbeck-Boussinesq approximation. The numerical method is pseudospectral and second order accurate in time. The rotation velocity of the square pattern increases linearly with the control parameter epsilon=Ra/R a(c) -1 , as in the experiment of Bajaj Furthermore, it was observed that this velocity decreases when the aspect ratio of the cylinder increases. These results indicate that the square pattern appears when the flow is laterally confined. The range of epsilon for which this pattern is stable tends to vanish for more extended layers.