On hydromagnetic critical layers

When a horizontal magnetic field B(z) is sheared vertically on a lengthscale L in a diffusionless fluid, critical layers occur at z_cwhere the local Alfven speed V(z_c) matches the phase speed c of the wave. However, when a vertical field B_zis introduced, all the critical layers disappear. The present study investigates the solution in the neighbourhood of z_cwhen BJB is very small, in order to clarify the manner in which the vertical field annihilates the critical layers. It is found that the solution across the critical layer is adjusted in a thin magnetic layer whose thickness is determined by the parameter ∊² (= U/αV, where U, V are measures of the vertical and horizontal components of the Alfven velocity and α/L is the horizontal wavenumber). The vertical field increases the order of the equation governing the vertical variations of the amplitude of the perturbations from two to four. Within the magnetic layer the two extra Alfven waves, one upgoing and the other downgoing, interact with those due to the horizontal field to make the solution regular everywhere. The mean vertical wave energy flux varies continuously from one constant value far on one side of the layer to another constant value far on the other side of the layer. The influence of the vertical field on the resistive instabilities present in its absence is also examined. It is found that the relative importance of resistivity and vertical field is measured by the ratio of the thicknesses of the resistive and magnetic layers. In general, the influence of the vertical field is to suppress resistive instabilities. The slow exchange resistive instabilities are suppressed by the presence of the vertical field if ∊≫a(Sα)^-⅓ while the localized gravitational modes are inhibited for ∊≫a(Sα)^-⅓, where a, b are constants whose values depend on the profile of the horizontal field and on the gravitational parameter G; and S is the Lundquist number.