Convection in a rotating cylinder. Part 2. Linear theory for low Prandtl numbers

Abstract

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.