Bending of vertical Hopf bifurcation branches in rotating thermal convection with an imperfection

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Abstract
The cascading bifurcation of two-dimensional, steady and periodic, thermal convection states in a rotating box with a uniform temperature distribution on the walls was studied by Magnan and Reiss, SIAM J. Appl. Math. (to be published). We extend that perturbation analysis by considering the effect of a slightly nonuniform temperature distribution on a wall of the box. This imperfection can qualitatively alter the response of the convection system. In particular, our two-parameter study finds that (a) the ‘‘vertical’’ Hopf bifurcation branch of periodic solutions, previously obtained at third order for a uniform temperature distribution, is now ‘‘bent’’ by the imperfection; (b) the number and position of the Hopf bifurcation points on the steady solution branches can vary with the strength of the imperfection and thus the stabilities of these branches can accordingly change; (c) the periodic solution branches, which now bifurcate either to the right or left, terminate in either a Hopf bifurcation point or in an infinite period bifurcation point; and (d) the imperfection can dominate the effects of higher-order terms in the amplitude equations. Furthermore, we establish for a semibounded rotating convection layer that when an imperfection is present the nonlinear interaction of two primary bifurcation steady states can generate stable, periodic solution branches, which has not been shown to occur for ordinary (nonrotating) Rayleigh-Benard convection, or for any other convection system.

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