THE PARKER INSTABILITY IN DIFFERENTIALLY-ROTATING DISKS

[[abstract]]We investigate Parker’s instability for a differentially rotating system comprised of thermal gas, magnetic field, and cosmic-ray particles. The rotation axis coincides with the direction of the vertical gravity, and the rotation is modeled to occur with linear shear. The general initial-value problem is formulated, and the condition for normal modes is obtained from this formulation. A dispersion relation is obtained for the limiting case when the growth rate (or frequency) of the wave is large in comparison with the kinematic shear rate. This dispersion relation suffices to show, in the absence of dissipation, that no finite amount of shear and rotation can ever completely stabilize Parker’s mode although the growth rate of certain Fourier components can be materially reduced. Eigenvalues and eigenfunctions are computed analytically for a number of limiting cases of interest, and a few numerical examples are given. The effect of rotation is to suppress waves which are very long in the horizontal directions; the full effects of shear are more difficult to access –numerical methods are suggested for future work, Subsidiary issues examined in this study are (i) the derivation of an alternative equation, valid in the non-linear regime and for arbitrary geometries, for the usual fluid equation adopted to describe the behavior of the cosmic-ray pressure, (ii) the distinction between environments under which Parker’s mode of instability is likely to lead to convection, to cosmic-ray inflation, or to gas drainage downward to form dense clumps of matter, (iii) an explanation of the physical reasonableness of normal mode solutions with finite energy densities at infinity and the relation of such solutions to the initial-value problem.[[fileno]]2010118010126[[department]]物理