Motivated by astrophysical and geophysical applications, the classical problem of rotating Rayleigh-Bénard 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 $\mathrm{\Gamma }$, 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 $T\gg 1, \mathrm{\Gamma }\ll 1$. On increasing $\mathrm{\Gamma }$ for a fixed $T\gg 1$, the M-C effect first comes into play when $\mathrm{\Gamma }=O\left(T^{-1/3}\right)$. Here, as in the classical problem, the preferred mode can be either steady or oscillatory, depending on the value of the Prandtl number $\sigma$. For $\mathrm{\Gamma }>O\left(T^{-1/3}\right)$, the influence of the M-C effect is sufficiently strong that the onset of instability is always oscillatory, regardless of the value of $\sigma$. Within this regime, the dependence on $\sigma$ of the critical Rayleigh number and of the scale of the preferred mode are explored through the analysis of specific distinguished limits.

Close

PrevNext

View figure in article

• 
• 
• 
• 
• 

• Flow instability
• Rayleigh-Bénard convection