  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 Alfven 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
Alfven 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. The implications of this work for astrophysical and fusion
applications and the potential for future research extending the results to
include compressibility are discussed.
