This paper presents a linear-stability analysis for the transition
from a steady, two-dimensional thermocapillary convection in a liquid-metal layer to a periodic,
three-dimensional flow involving hydrothermal waves which propagate in the directionzw normal to the plane of the base flow. There is a uniform magnetic field
applied
parallel to the free surface in the plane of the base flow, and there is
a linear
temperature gradient along the free surface in the base flow. The ratio
of the layer's
length to its depth, 2L, is large. The magnetic Reynolds number
is small.
A key parameter is λ, the ratio of the large Hartmann number
based on depth to
L. The value of λ increases as either the magnetic field
strength is increased or L is
decreased. The results for very small values of λ agree with the
results of a previous
treatment of this instability without a magnetic field. As λ is
increased, the critical
Marangoni number and the wavenumber for the hydrothermal rolls both increase.
For large values of λ, the base flow and the hydrothermal waves
are confined to a
free-surface layer with O(λ−2) dimensionless
thickness.