The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios τ ≡ d/h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow. The precession velocity of the slow modes vanishes as τ → ∞. Thus it is these modes which provide the connection between the results for a finite-aspect-ratio System and the unbounded layer in which the instability is a steady-state one, except in low Prandtl number fluids.The linear stability problem is solved for several different sets of boundary conditions, and the results compared with recent experiments. Results are presented for Prandtl numbers σ in the range 6.7 ≤ σ ≤ 7.0 as a function of both the rotation rate and the aspect ratio. The results for rigid walls, thermally conducting top and bottom and an insulating sidewall agree well with the measured critical Rayleigh numbers and precession frequencies for water in a τ = 1 cylinder. A conducting sidewall raises the critical Rayleigh number, while free-slip boundary conditions lower it. The difference between the critical Rayleigh numbers with no-slip and free-slip boundaries becomes small for dimensionless rotation rates Ωh2/v ≥ 200, where v is the kinematic viscosity.

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A numerical study of several time integration methods for solving the threedimensional Boussinesq thermal convection equations in rotating spherical shells is presented. Implicit and semi-implicit time integration techniques based on backward differentiation and extrapolation formulae are considered. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The codes are validated with a known benchmark, and their efficiency is studied. The results show that the use of high order methods, especially those with time step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions.

Direct numerical simulations of chevrons, blinking states, and repeated transients in binary fluid mixtures with a negative separation ratio heated from below are described. The calculations are performed in two-dimensional containers using realistic boundary conditions and the parameter values used in the experiments of Kolodner [Phys. Rev. E 47, 1038 (1993)]. Particular attention is paid to the multiplicity of states, and their dependence on the applied Rayleigh number and the aspect ratio of the container. Quantitative agreement with the experiments is obtained, and a mechanism explaining the origin and properties of the repeated transients observed in the experiments is proposed.

The onset of convection in a low-Prandtl-number fluid confined in a uniformly rotating vertical cylinder heated from below is studied. The linear stability problem is solved for perfectly conducting stress-free or rigid boundary conditions at the top and bottom; the sidewalls are taken to be insulating and rigid. For these Prandtl numbers axisymmetric overstability leads to an oscillating concentric pattern of rolls. When the instability breaks axisymmetry the resulting pattern must in addition precess. The relationship between these two types of oscillatory behaviour is explored in detail. The complex interaction between different types of neutrally stable modes is traced out as a function of the Prandtl and Taylor numbers, as well as the aspect ratio. A qualitative explanation is provided for the multiplicity of modes of a given azimuthal wavenumber and its dependence on the parameters. Specific predictions are made for the Prandtl numbers 0.025, 0.49 and 0.78, corresponding to mercury, liquid helium 4 and compressed carbon dioxide gas.