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)^-1/2B ₀ 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.