Linear Theory of Thin, Radially-Stratified Disks

Abstract

We consider the nonaxisymmetric linear theory of radially-stratified disks. We work in a shearing-sheet-like approximation, where 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: (i) 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 (consistent with Afshordi, Mukhopadhyay & Narayan 2004); (ii) short-wavelength compressive shwaves grow asymptotically in the unstratified shearing sheet, as does the energy of an ensemble of such shwaves; (iii) incompressive shwaves in the stratified shearing sheet have density and azimuthal velocity perturbations \delta \Sigma,\delta v_y ~ t^{-Ri} (for |Ri| << 1), where Ri = 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; (iv) the energy of an ensemble of incompressive shwaves in the stratified shearing sheet behaves asymptotically as Ri t^{1-4Ri} for |Ri| << 1. For Keplerian disks with modest radial gradients, |Ri| is expected to be << 1, and there will therefore be 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.