Hydrothermal wave instability of thermocapillary-driven convection in a transverse magnetic field

We study the linear stability of thermocapillary-driven convection in a planar unbounded layer of an electrically conducting low-Prandtl-number liquid heated from the side and subjected to a transverse magnetic field. The thresholds of convective instability for both longitudinal and oblique disturbances are calculated numerically and also asymptotically by considering the Hartmann and Prandtl numbers as large and small parameters, respectively. The magnetic field has a stabilizing effect on the flow with the critical temperature gradient for the transition from steady to oscillatory convection increasing as square of the field strength, as also does the critical frequency, while the critical wavelength reduces inversely with field strength. These asymptotics develop in a strong enough magnetic field when the instability is entirely due to the jet of the base flow confined in the Hartmann layer at the free surface. In contrast to the base flow, the critical disturbances, having a long wavelength at small Prandtl numbers, extend from the free surface into the bulk of the liquid layer over a distance exceeding the thickness of the Hartmann layer by a factor O(Pr(- 1/2 )). For Ha ≤ Pr(- 1/2 ) the instability is influenced by the actual depth of the layer. For such moderate magnetic fields the instability threshold is sensitive to the thermal properties of the bottom of the layer and the dependences of the critical parameters on the field strength are more complicated. In the latter case, various instability modes are possible depending on the thermal boundary conditions and the relative magnitudes of Prandtl and Hartmann numbers.