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 ( 472, 20160649 (doi:10.1098/rspa.2016.0649)) to non-zero values of the magnetic Prandtl number . With non-zero , the order of the dispersion relation is increased, leading to considerably richer behaviour. An asymptotic analysis at large values of the Chandrasekhar number confirms that the MC effect becomes important when is (1), where is the MC number. In this regime, we derive a scaled system that is independent of . When is large, the results are consistent with those derived from the governing equations in the limit of Prandtl number  → ∞ with finite; here we identify a new mode of instability, which is due neither to inertial nor induction effects. In the large regime, we show how a transition can occur between oscillatory modes of different horizontal scale. For  ≫ 1 and small values of , we show that the critical Rayleigh number is non-monotonic in provided that  > 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.