Instability of swirl in low-Prandtl-number thermal convection

Abstract

In the present paper we examine low-Prandtl-number thermal convection using a highly truncated modal approach. For the horizontal structure we assume a hexagonal planform as in Toomre Gough & Spiegel (1977) but including a vertical vorticity mode. The system develops a non-zero vertical vorticity component through a finite-amplitude instability. Following this, the system displays a Hopf bifurcation giving rise to periodic oscillations. The mechanism for this instability is associated with the growth of swirl in the azimuthal direction. We have found three different types of periodic solutions, possibly associated with subharmonic bifurcations, and their structure has been examined.A large part of the present work is devoted to exploring the cases of mercury and liquid helium - or air - as the best-known examples of low and intermediate-Prandtl-number fluids. Results for mercury are quite satisfactory as far as frequencies and fluxes are concerned and they show reasonable agreement with experimental measurements at mildly supercritical Rayleigh values. On the other hand, for liquid helium or air agreement is poor.