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

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Abstract
The temporal and spatial behaviour of three-dimensional convection at infinite Prandtl number, in a rotating spherical fluid shell of radius ratio η = ri/ro = 0.4, has been investigated numerically for a range of Taylor number for which mc = 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, R1c, primary nonlinear solutions with the predominant wavenumber m0 = 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, m0 = 2 and 4. The secondary bifurcation solution becomes unstable at another critical value, R2c, 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 m0 = 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 = 104. 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 Rc(m = 3) is much closer to Rc(m = 4) than Rc(m = 2). For the convection of the dominant wavenumber m0 = 2, the analysis is focused on the range of Taylor number T > T4, where T4 is the Taylor number at which the critical wavenumber mc changes from 2 to 4 at the onset of convection. The m0 = 2 steadily drifting nonlinear solutions, which are unstable at small amplitudes owing to the Eckhaus-type instability, gain their stability at large amplitudes at R1c through nonlinear effects, and lose their stability at a higher Rayleigh number, R2c, to the amplitudevacillating instability which leads to a periodic change in the amplitude of convection with little fluctuation in the pattern of flow.