In this paper, the instability of shallow water shear flow with a sheared parallel magnetic
field is studied. Waves propagating in such magnetic shear flows encounter critical levels
where the phase velocity relative to the basic flow c − U(y) matches the Alfven wave
velocities ±B(y)/√µρ, based on the local magnetic field B(y), the ...
In this paper, the instability of shallow water shear flow with a sheared parallel magnetic
field is studied. Waves propagating in such magnetic shear flows encounter critical levels
where the phase velocity relative to the basic flow c − U(y) matches the Alfven wave
velocities ±B(y)/√µρ, based on the local magnetic field B(y), the magnetic permeability
µ and the mass density of the fluid ρ. It is shown that when the two critical levels are
close to each other, the critical layer can generate an instability. The instability problem
is solved, combining asymptotic solutions at large wavenumbers and numerical solutions,
and the mechanism of instability explained using the conservation of momentum. For
the shallow water MHD system, the paper gives the general form of the local differential
equation governing such coalescing critical layers for any generic field and flow profiles,
and determines precisely how the magnetic field modifies the purely hydrodynamic
stability criterion based on the potential vorticity gradient in the critical layer. The
curvature of the magnetic field profile, or equivalently the electric current gradient,
J′ = −B′′/µ in the critical layer is found to play a complementary role in the instability.