Instabilities of the Ekman‐Hartmann Boundary Layer

Gilman and Benton have demonstrated the existence of composite Ekman‐Hartmann layer flow in rotating, electrically conducting fluids permeated by a magnetic field normal to the boundary. This flow was shown to evolve smoothly from a pure Ekman layer to a pure Hartmann layer as the parameter $α ≡ (2μρλΩ0)−1/2 B0$ increases (μ is magnetic permeability, $ρ$ is density, $λ$ is magnetic diffusivity, $Ω0$ is the rotation rate, and $B0$ is the imposed magnetic field). Here, it is shown that in the Cartesian, low magnetic Prandtl number limit, this flow exhibits the two instabilities to two‐dimensional rolls characteristic of the pure Ekman layer, but at Reynolds numbers that increase rapidly as $α$ increases. Both rolls decrease in horizontal scale, orient more nearly parallel to the flow far from the boundary, and acquire smaller phase velocities. Coriolis forces are seen to give a parallel roll instability of the Hartmann layer, at much lower Reynolds numbers than given by Roberts for Hartmann layer instability in the absence of rotation.