Linear theory of thin, radially stratified disks

We consider the nonaxisymmetric linear theory of radially stratified disks. We work in a shearing-sheet-like approximation, in which the vertical structure of the disk is neglected, and develop equations for the evolution of a plane-wave perturbation comoving with the shear flow ( a shearing wave, or "shwave"). We calculate a complete solution set for compressive and incompressive short-wavelength perturbations in both the stratified and unstratified shearing-sheet models. We develop expressions for the late-time asymptotic evolution of an individual shwave, as well as for the expectation value of the energy for an ensemble of shwaves that are initially distributed isotropically in k-space. We find that (1) incompressive, short-wavelength perturbations in the unstratified shearing sheet exhibit transient growth and asymptotic decay, but the energy of an ensemble of such shwaves is constant with time; (2) short-wavelength compressive shwaves grow asymptotically in the unstratified shearing sheet, as does the energy of an ensemble of such shwaves; (3) incompressive shwaves in the stratified shearing sheet have density and azimuthal velocity perturbations delta Sigma, delta v(y) similar to t(-Ri) ( for vertical bar Ri vertical bar << 1), where Ri equivalent to N-x(2)/(q Omega)(2) is the Richardson number, N-x(2) is the square of the radial Brunt-Vaisala frequency, and q Omega is the effective shear rate; and (4) the energy of an ensemble of incompressive shwaves in the stratified shearing sheet behaves asymptotically as Rit(1-4Ri) for vertical bar Ri vertical bar << 1. For Keplerian disks with modest radial gradients, vertical bar Ri vertical bar is expected to be << 1, and there is therefore weak growth in a single shwave for Ri < 0 and near-linear growth in the energy of an ensemble of shwaves, independent of the sign of Ri.