Linear corner-mode instability of magnetohydrodynamic Rayleigh-Bénard convection in a rectangular domain

Rayleigh-Bénard convection in a horizontal fluid layer is sustained only above a finite critical Rayleigh number. When the fluid is electrically conducting, a vertical magnetic field increases the critical Rayleigh number for the instability of the purely heat-conducting state in the bulk of the layer. When electrically insulating side walls are taken into account, the least stable mode becomes localized near these walls. Previous works studied the linear stability of such a wall-attached convection mode in circular containers or along a straight periodic wall. The present work considers a rectangular container, where induced eddy currents are presumably even more obstructed at the corners between insulating side walls. The linear stability problem is formulated using the quasistatic approximation of the induction equation. It is discretized and solved numerically for free-slip container walls by means of a two-dimensional spectral collocation method. The localization of the least stable mode at the corner is confirmed, and its properties are characterized by various geometric quantities. The effect of the magnetic field strength quantified by the Hartmann number is also examined, and similarities and differences between the wall-attached convection mode on a straight wall and in a corner are pointed out.