A stability criterion for steady finite amplitude convection with an external magnetic field

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Abstract
In a horizontal layer of fluid, thermal expansion or the presence of dissolved salt may cause a density gradient opposite to the direction of gravity. In such cases, when the buoyancy forces are sufficient to overcome the dissipative effects, the static state becomes unstable and convective motions arise. If the layer is infinitely large in horizontal extent, the non-linear convection problem is highly degenerate, admitting many different steady-state solutions. A general necessary criterion for stability of such non-linear steady solutions is developed here for the case in which a homogeneous vertical magnetic field acts on the fluid. The criterion is demonstrated for two rigid bounding surfaces which are perfect thermal and electrical conductors, but it is applicable to more general kinds of boundary conditions.

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