Pattern selection in oscillatory rotating convection

Three-dimensional pattern selection in a low Prandtl number Boussinesq fluid with stress-free boundaries, where the onset of convection is oscillatory, is explored. Restricting the problem to a square lattice, the normal form coefficients are calculated as functions of τ (the square root of the Taylor number) and the Prandtl number σ. There is a large region of the (σ,τ) plane where a heteroclinic cycle connecting four Travelling Roll states is stable. As σ is decreased the cycle undergoes a transverse loss of stability, creating quasiperiodic orbits which may themselves become chaotic. All these stable dynamics occur at onset. Although conjectured on the basis of general results from symmetric bifurcation theory (and well-known for steady convection as the Küppers–Lortz instability [J. Fluid Mech. 35 (1969) 609]), cycling behaviour has not previously been demonstrated directly from the hydrodynamic equations in the oscillatory case. A second region of the (σ,τ) plane contains stable Travelling Roll solutions: we examine their stability to perturbations at varying angles and demonstrate the existence of small-angle instabilities of travelling rolls.