Rapidly rotating Maxwell-Cattaneo convection

Motivated by astrophysical and geophysical applications, the classical problem of ro-tating Rayleigh-Benard convection has been widely studied. Assuming a classical Fourier heat law, in which the heat flux is directly proportional to the temperature gradient, the evolution of temperature is governed by a parabolic advection-diffusion equation; this, in turn, implies an infinite speed of propagation of information. In reality, the system is rendered hyperbolic by extending the Fourier law to include an advective derivative of the flux-the Maxwell-Cattaneo (M-C) effect. Although the correction (measured by the parameter I', a nondimensional representation of the relaxation time) is nominally small, it represents a singular perturbation and hence can lead to significant effects when the rotation rate (measured by the Taylor number T ) is sufficiently high. In this paper, we investigate the linear stability of rotating convection, incorporating the M-C effect, concentrating on the regime of T >> 1, I' << 1. On increasing I' for a fixed T >> 1, the M-C effect first comes into play when I' = O(T -1/3). Here, as in the classical problem, the preferred mode can be either steady or oscillatory, depending on the value of the Prandtl number a. For I' > O(T -1/3), the influence of the M-C effect is sufficiently strong that the onset of instability is always oscillatory, regardless of the value of a. Within this regime, the dependence on a of the critical Rayleigh number and of the scale of the preferred mode are explored through the analysis of specific distinguished limits.