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, 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 latetime 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 δΣ, δvy ∼ t−Ri (for |Ri| ≪ 1), where Ri ≡ N2 x/(q̃Ω) is the Richardson number, N2 x is the square of the radial Brunt-Väisälä frequency and q̃Ω is the effective shear rate; (iv) the energy of an ensemble of incompressive shwaves in the stratified shearing sheet behaves asymptotically as Ri t1−4Ri 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. Subject headings: accretion, accretion disks, solar system: formation, galaxies: nuclei