The convective instability of a Maxwell-Cattaneo fluid in the presence of a vertical magnetic field

We study the instability of a Bénard layer subject to a vertical uniform magnetic field, in which the fluid obeys the Maxwell–Cattaneo (MC) heat flux–temperature relation. We extend the work of Bissell (Proc. R. Soc. A 472, 20160649 (doi:10.1098/rspa.2016.0649)) to non-zero values of the magnetic Prandtl number pm. With non-zero pm, the order of the dispersion relation is increased, leading to considerably richer behaviour. An asymptotic analysis at large values of the Chandrasekhar number Q confirms that the MC effect becomes important when C Q1/2 is O(1), where C is the MC number. In this regime, we derive a scaled system that is independent of Q. When CQ1/2 is large, the results are consistent with those derived from the governing equations in the limit of Prandtl number p → ∞ with pm finite; here we identify a new mode of instability, which is due neither to inertial nor induction effects. In the large pm regime, we show how a transition can occur between oscillatory modes of different horizontal scale. For Q ≫ 1 and small values of p, we show that the critical Rayleigh number is non-monotonic in p provided that C > 1/6. While the analysis of this paper is performed for stress-free boundaries, it can be shown that other types of mechanical boundary conditions give the same leading-order results.