Multiple instabilities in rotating convection

Abstract

Abstract The linearised Boussinesq equations, which describe two-dimensional convection in a rotating layer of fluid, can support multiple instabilities associated with the same as well as different wavenumbers. These instabilities involve both steady and oscillatory convection rolls, and the most complicated situations involve triple interactions between two steady and one oscillatory or two oscillatory and one steady mode. In this paper we consider the weakly nonlinear dynamics of a triple instability involving a steady and an oscillatory roll of wavenumber k 2 and an oscillatory mode of wavenumber k 1. The methods of centre manifolds and normal forms are used to derive equations for the evolving wave amplitudes and bifurcation theory is employed to find and analyse the stability of the equilibrium states. Periodic, quasi-periodic and period-doubling bifurcations are all possible close to the codimension three point. This paper forms the first part of a two-part presentation of the results of our investigation and only the periodic and quasi-periodic solutions are discussed here. The period-doubling behaviour together with a more complete scenario, placing the results in a physical context, will be discussed elsewhere.