Local Gravitational Instability of Magnetized Shear Flows

The effect of magnetic shear and shear flow on local gravitationally induced instabilities is investigated. A simple model is constructed allowing for an arbitrary entropy gradient and a shear plasma flow in the Boussinesq approximation. A transformation to shearing magnetic coordinates achieves a model with plasma flow along the magnetic field lines where the coordinate lines are coincident with the field lines. The solution for the normal modes of the system depends on two parameters: the Alfvén Mach number of the plasma flow and the entropy gradient. The behavior of the unstable normal modes of this system is summarized by a stability diagram. Important characteristics of this stability diagram are the following: magnetic shear is stabilizing and the entropy gradient must exceed a threshold value for unstable mode growth to occur; flow acts to suppress mode growth in a substantially unstable regime as expected, yet near marginal stability it can lessen the stabilizing effect of magnetic shear and enhance the growth rates of the instability; and, as the Alfvén Mach number approaches one, the instability is completely stabilized. Analytical work is presented supporting the characteristics of the stability diagram and illuminating the physical mechanisms controlling the behavior of the model. A derivation of the stability criterion for the case without shear flow, asymptotic solutions in the limit that the Alfvén Mach number approaches one and in the limit of zero growth rate, a complete WKB solution for large growth rates, an exactly soluble bounded straight field case, and energy conservation relations are all presented. The implications of this work for astrophysical and fusion applications and the potential for future research extending the results to include compressibility are discussed. Subject headings: MHD—gravitational instability—magnetic shear—velocity shear