Nonlinear thermal inertial waves in rotating fluid spheres

Abstract

ABSTRACTWhen the Prandtl number is sufficiently small, the rotational constraint on convection in rotating systems is largely broken by inertial effects and the corresponding convective instability in leading order is described by non-dissipative thermal inertial waves propagating in either prograde or retrograde direction while buoyancy forces maintain the waves against viscous damping at next order. We investigate, via direct numerical simulation using a finite element method, the nonlinear properties of thermal inertial waves in rapidly rotating, self-gravitating, internally heated Boussinesq fluid spheres, concentrating on the liquid metal gallium with the Prandtl number . We analyze the nonlinear numerical solution by decomposing it into the spectrum of mathematically complete spherical inertial modes. Different forms of nonlinear convection, representing a sequence of bifurcations, are identified with increasing Rayleigh number. Near the threshold of convective instability, we show that the primary bifurcation is supercritical, dominated by a single thermal-inertial-wave mode, and consistent with the prediction of the existing asymptotic theory. At moderately supercritical Rayleigh numbers, we reveal that various thermal-inertial-wave modes with different radial/azimuthal/equatorial symmetries – which are convectively excited and maintained – become nonlinearly interactive, progressively breaking the temporal, radial, azimuthal and equatorial symmetries of the primary solution and, then, leading to a weakly turbulent state.