Mixed-mode solutions in an air-filled differentially heated rotating annulus∗

We present an analysis of the primary bifurcations that occur in a mathematical model that uses the (three-dimensional) Navier-Stokes equations in the Boussinesq approximation to describe the flow of a near unity Prandtl number fluid (i.e. air) in the differentially heated rotating annulus. In particular, we investigate the double Hopf (Hopf-Hopf) bifurcations that occur along the axisymmetric to non-axisymmetric flow transition. Centre manifold reduction and normal forms are used to show that in certain regions in parameter space, stable quasiperiodic mixed-azimuthal mode solutions result as a nonlinear interaction of two bifurcating waves with different azimuthal wave numbers. These flows have been called wave dispersion and interference vacillation. The results differ from similar studies of the annulus with a higher Prandtl number fluid (i.e. water). In particular, we show that a decrease in Prandtl number can equilibrate these mixed-mode solutions. keywords: double Hopf (Hopf-Hopf) bifurcations, quasiperiodic solutions, wave dispersion, axisymmetric to non-axisymmetric transition, rotating waves, centre manifold reduction, numerical approximation of normal form coefficients, Navier-Stokes equations in the Boussinesq approximation PACS: 47.20.Ky, 05.45.a, 47.32.Ef, 47.15.Fe, 47.55.P-, 47.54.-r