Transition from subcritical to supercritical bifurcation in penetrative convection in a vertical cylinder

The stability of a vertical cylindrical column of water with the horizontal bottom boundary maintained at 0°C and the temperature of the top boundary varying between 4 and 8°C is considered. A weakly non-linear analysis shows that the bifurcation is either supercritical or subcritical depending upon the values of two parameters: the aspect ratio of the cylinder (height/radius) which is varied between 2 and 4, and the penetration parameter μ defined as the ratio of the whole height over the height of the 4°C isotherm. For a given value of the aspect ratio, the bifurcation is supercritical when the whole height is unstable (μ = 1) and becomes subcritical above a particular value of μ when penetration occurs (μ > 1).