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.

中文翻译：

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存在垂直磁场时麦克斯韦-卡塔内奥流体的对流不稳定性

我们研究了受垂直均匀磁场影响的贝纳德层的不稳定性，其中流体遵循麦克斯韦-卡塔内奥 (MC) 热通量-温度关系。我们将 Bissell 的工作 (Proc. R. Soc. A 472, 20160649 (doi:10.1098/rspa.2016.0649)) 扩展到磁普朗特数 pm 的非零值。当 pm 非零时，色散关系的阶数增加，从而导致更加丰富的行为。对钱德拉塞卡数 Q 大值的渐近分析证实，当 C Q1/2 为 O(1)（其中 C 是 MC 数）时，MC 效应变得很重要。在这种情况下，我们推导了一个独立于Q的标度系统。当CQ1/2很大时，结果与在普朗特数p→∞且pm有限的极限下由控制方程推导的结果一致；在这里，我们确定了一种新的不稳定模式，它既不是由于惯性效应也不是感应效应造成的。在大 pm 状态下，我们展示了不同水平尺度的振荡模式之间如何发生转变。对于 Q ≫ 1 和较小的 p 值，我们表明只要 C > 1/6，临界瑞利数在 p 中是非单调的。虽然本文的分析是针对无应力边界进行的，但可以表明其他类型的机械边界条件给出了相同的主阶结果。