Inertial convection in a rotating narrow annulus: Asymptotic theory and numerical simulation

Abstract

An important way of breaking the rotational constraint in rotating convection is to invoke fast oscillation through strong inertial effects which, referring to as inertial convection, is physically realizable when the Prandtl number Pr of rotating fluids is sufficiently small. We investigate, via both analytical and numerical methods, inertial convection in a Boussinesq fluid contained in a narrow annulus rotating rapidly about a vertical symmetry axis and uniformly heated from below, which can be approximately realizable in laboratory experiments [R. P. Davies-Jones and P. A. Gilman, “Convection in a rotating annulus uniformly heated from below,” J. Fluid Mech. 46, 65-81 (1971)]. On the basis of an assumption that inertial convection at leading order is represented by a thermal inertial wave propagating in either prograde or retrograde direction and that buoyancy forces appear at the next order to maintain the wave against the effect of viscous damping, we derive an analytical solution that describes the onset of inertial convection with the non-slip velocity boundary condition. It is found that there always exist two oppositely traveling thermal inertial waves, sustained by convection, that have the same azimuthal wavenumber, the same size of the frequency, and the same critical Rayleigh number but different spatial structure. Linear numerical analysis using a Galerkin spectral method is also carried out, showing a quantitative agreement between the analytical and numerical solutions when the Ekman number is sufficiently small. Nonlinear properties of inertial convection are investigated through direct three-dimensional numerical simulation using a finite-difference method with the Chorin-type projection scheme, concentrating on the liquid metal gallium with the Prandtl number Pr = 0.023. It is found that the interaction of the two counter-traveling thermal inertial waves leads to a time-dependent, spatially complicated, oscillatory convection even in the vicinity of the onset of inertial convection. The nonlinear properties are analyzed via making use of the mathematical completeness of inertial wave modes in a rotating narrow annulus, suggesting that the laminar to weakly turbulent transition is mainly caused by the nonlinear interaction of several inertial wave modes that are excited and maintained by thermal convection at moderately supercritical Rayleigh numbers.