Vacillatory convection in a rotating spherical fluid shell at infinite Prandtl number

The temporal and spatial behaviour of three-dimensional convection at infinite Prandtl number, in a rotating spherical fluid shell of radius ratio η = r i / r o = 0.4, has been investigated numerically for a range of Taylor number for which m c = 2, 3 and 4 are the critical values of azimuthal wavenumber at the onset of convection. When the Rayleigh number, R , exceeds a critical value, R 1c , primary nonlinear solutions with the predominant wavenumber m 0 = 4 in the form of azimuthally travelling waves give way to secondary solutions in the form of steadily drifting mixed-mode convection with two predominant wavenumbers, m 0 = 2 and 4. The secondary bifurcation solution becomes unstable at another critical value, R 2c , that leads to the tertiary solution in which the dominant wavelength of convection vacillates periodically between the two competing scales characterized by the azimuthal wavenumbers m 0 = 2 and 4. Instabilities and bifurcations associated with the evolution from a static state to wavenumber vacillation are discussed for a representative Taylor number of T = 10 4 . It is also shown that the interaction between the two spatially resonant wavenumbers m = 2 and 4 is much stronger than the interaction between the non-resonant wavenumbers m = 3 and 4 even though R c ( m = 3) is much closer to R c ( m = 4) than R c ( m = 2). For the convection of the dominant wavenumber m 0 = 2, the analysis is focused on the range of Taylor number T > T 4 , where T 4 is the Taylor number at which the critical wavenumber m c changes from 2 to 4 at the onset of convection. The m 0 = 2 steadily drifting nonlinear solutions, which are unstable at small amplitudes owing to the Eckhaus-type instability, gain their stability at large amplitudes at R 1c through nonlinear effects, and lose their stability at a higher Rayleigh number, R 2c , to the amplitudevacillating instability which leads to a periodic change in the amplitude of convection with little fluctuation in the pattern of flow.