{'fraction_non_alphanumeric': 0.03745743473325766, 'fraction_numerical': 0.006242905788876277, 'mean_word_length': 5.017064846416382, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
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 δΣ, δvy ~ t-Ri (for |Ri| ≪ 1), where Ri ≡ N/(Ω)2 is the Richardson number, N is the square of the radial Brunt-Väisälä frequency, and Ω is the effective shear rate; and (4) the energy of an ensemble of incompressive shwaves in the stratified shearing sheet behaves asymptotically as Rit1-4Ri for |Ri| ≪ 1. For Keplerian disks with modest radial gradients, |Ri| 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.