Chaos in models of double convection

Abstract

In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear.