Stability of sheared magnetic field in dependence on its critical level position

A horizontal layer rotating about a vertical axis z at angular velocity Omega = Omega(0)(z) over cap and heated from below by an adverse temperature gradient beta = (dT/dz)(z) over cap, T being the temperature, is investigated. The layer filled by inviscid Boussinesq fluid of finite conductivity between the boundaries at z = +/-d/2 is under a gravitational field oriented vertically downwards, g = -g (z) over cap and permeated by a horizontal right-lined sheared magnetic field B(0). The finite electrical conductivity is a destabilizing factor in such a system and resistive instabilities may rise under certain conditions. They are usually associated with the presence of a critical level defined by the condition k.B(0) = 0, where k is the wave vector of perturbation and B(0) is the ambient magnetic field. This condition is automatically satisfied by the fields featuring zero point(s). In this work the imposed magnetic field of the form B(0) = B(0) tanh gamma(z-z(0))(y) over cap, is considered, where B(0) is the magnitude of the field and z(0) measures its asymmetry with respect to the central plane z = 0. We examine the effect of varying z(0), the position of the critical level within the layer, on the stability of the system. The model is interesting from the geophysical and astrophysical point of view because of presence of an internal shear magnetic layer near the critical level associated with current sheet. The shear layer is controlled by the parameter gamma y which enables to change the sharpness of the magnetic field gradient across the shear layer and thereby it influences its thickness. For sufficiently large gradient of the field, the shear layer is found to be a source of hydromagnetic resistive instabilities and it is called the critical layer. A shift in the critical level location from the mid-plane towards a boundary causes the system to change in terms of the basic magnetic field (specifically its overall gradient). Boundaries were chosen to be either both perfectly conducting, insulating or a combination thereof. Linear stability analysis was performed. In general, the most preferred instabilities were found to take the form of stationary rolls. Sufficiently large gradient of the ambient magnetic field may provide conditions for exciting two distinct modes of instabilities, the whole-layer mode depending on global current distribution with its convective rolls extending throughout the whole layer, and the critical-layer mode confined to the region of the critical layer. It was found that for the layer with different electrical properties of the boundaries, the crucial importance for the onset of a particular mode (and its convective pattern) had not only the shift itself but also the fact to which boundary it was carried. Unlike the previous studies of the purely antisymmetric basic-state fields of linear and tanh profile (Tucker and Jones, 1997), and of the asymmetric field of linear profile (Marsenic and Sevcik, 2008) where only the stationary motions developed throughout the layer were possible, the presented model allows for qualitatively different modes localized mainly around the critical level. The confinement of the convection to the region of the critical layer was possible just for a sufficiently steep shear of the magnetic field and for the thin shear layer being shifted to the perfectly conducting boundary. (C) 2010 Elsevier B.V. All rights reserved.