Rapidly rotating thermal convection at low prandtl number

Rotating Boussinesq convection in a plane layer is governed by two dimensionless groups in addition to the Rayleigh number R: the Prandtl number σ and the Taylor number Ta. Scaled equations for fully nonlinear rotating convection in the limit of rapid rotation and small Prandtl number, where the onset of convection is oscillatory, are derived by considering distinguished limits where σⁿ Ta¹/2 ∼ 1 but σ → 0 and Ta → ∞, for different n > 1. In the resulting asymptotic expansion in powers of Ta ^-1/2 the leading-order equations, which are independent of n, are solved to provide an analytic description of fully nonlinear convection. Three distinct asymptotic regimes are identified, distinguished by the relative importance of the subdominant buoyancy and inertial terms. For the most interesting case, n = 4, the stability of different planforms near onset is investigated using a double expansion in powers of Ta ^-1/8 and the amplitude of convection ε. The lack of a buoyancy term at leading order demands that the perturbation expansion be continued through six orders to derive amplitude equations determining the dynamics. The case n = 1 is also analysed. The relevance of this theory to experimental results is briefly discussed.