Thermal inertial waves in a rotating fluid layer: Exact and asymptotic solutions

The problem of the onset of convective instability in a rapidly rotating fluid layer heated from below with various velocity boundary conditions is investigated by constructing exact solutions and by asymptotic analysis. It is shown that convective motions at sufficiently small Prandtl numbers are described in leading order by a thermal inertial wave. It is at the next order that buoyancy forces drive the wave against the weak effect of viscous dissipation. On the basis of the perturbation of the thermal inertial wave, asymptotic convection solutions for rigid boundaries can be expressed in simple analytic form. A new asymptotic power law between the critical Rayleigh number and the Ekman number is derived. In addition, solutions for both stationary and time-dependent convection are calculated numerically. Comparison between the numerical and asymptotic solutions is then made to show that a satisfactory quantitative agreement has been achieved.