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

Γ

, 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

,

Γ

≪

1

. On increasing

Γ

for a fixed

T

≫

1

, the M-C effect first comes into play when

Γ

=

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

σ

. For

Γ

>

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

σ

. Within this regime, the dependence on

σ

of the critical Rayleigh number and of the scale of the preferred mode are explored through the analysis of specific distinguished limits.