Low-Prandtl-number convection in a rotating cylindrical annulus

Abstract

Motivated by recent experimental results obtained in a low-Prandtl-number fluid (Jaletzky 1999), we study theoretically the rotating cylindrical annulus model with rigid boundary conditions. A boundary layer theory is presented which allows a systematic study of the linear properties of the system in the asymptotic regime of very fast rotation rates. It shows that the Stewartson layers have a (de)stabilizing influence at (high) low Prandtl numbers. In the weakly nonlinear regime and for low Prandtl numbers, a strong retrograde mean flow develops at quadratic order. The Poiseuille part of this mean flow is determined by an equation obtained by averaging of the Navier–Stokes equation. It thus gives rise to a new global-coupling term in the envelope equation describing modulated waves, which can be used for other systems. The influence of this global-coupling term on the sideband instabilities of the waves is studied. In the strongly nonlinear regime, the waves restabilize against these instabilities at small rotation rates, but they are destabilized by a short-wavelength mode at larger rotation rates. We also find an inversion in the dependence of the amplitude on the Rayleigh number at low Prandtl numbers and intermediate rotation rates.