Rapidly rotating Maxwell-Cattaneo convection

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.

Figure 1

${\sigma }_t$, determined by expression (28), as a function of $T$. Steady (oscillatory) convection is favored for $\sigma >{\sigma }_t$ ($\sigma <{\sigma }_t$). The lines $T=T_c=27(\sqrt{2}-1)/2$ and $\sigma ={\sigma }_c$ (the positive root of $8{\sigma }^4=\sigma +1$) are marked.

Figure 2

The number of T-B points, governed by the real positive roots for ${\tilde{k}}^2$ of (33) in $(\tilde{T},\sigma )$ space.

Figure 3

Examples of marginal stability curves, for $\tilde{R}$ as a function of ${\tilde{k}}^2$, corresponding to the four different regions in Fig. 2; the red lines denote where the steady mode is marginal $(s=0)$, the blue lines where the oscillatory mode is marginal ($s=\pm i\omega$). (a) (Two T-B points), $\tilde{T}=1, \sigma =0.3$; (b) (no T-B points), $\tilde{T}=10, \sigma =0.3$; (c) (three T-B points), $\tilde{T}=0.5, \sigma =1.05$; (d) (one T-B point), $\tilde{T}=0.5, \sigma =10$.

Figure 4

Marginal stability curves, for $\tilde{R}$ as a function of ${\tilde{k}}^2$ with $\sigma ={10}^2$; the red and blue lines denote, respectively, where the steady and oscillatory modes are marginal. (a) $\tilde{T}=0.5$; (b) $\tilde{T}=1$; (c) $\tilde{T}=1.5$.

Figure 5

(a) $R_c$, (b) $k_c^2$, and (c) ${\omega }_c^2$ as functions of $\sigma$ for the case of $\mathrm{\Gamma }={10}^{-4}, T={10}^{16}$ (i.e. $\lambda =1/4$); the onset of instability is oscillatory for all $\sigma$. The magenta lines denote the asymptotic expressions (46); the black dashed lines are the expressions (50) (a), (51) (b), and (52) (c), with the respective limiting values (57), (54), and (55) shown by the blue dotted lines with circle markers; the green lines are given by (60) (a), (61) (b), and (62) (c), with the respective limiting expressions (66) shown as red dashed lines. Numerical results of the full system are shown as squares with red edges. For comparison, the red dotted lines denote $R_c, k_c^2$, and ${\omega }_c^2$ for the classical problem; here the onset of instability is oscillatory for $\sigma <{\sigma }_c$ and steady for $\sigma >{\sigma }_c$.