Three‐dimensional Waves Generated at Lindblad Resonances in Thermally Stratified Disks

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Abstract
We analyze the linear, three-dimensional response to tidal forcing of a disk that is thin and thermally stratified in the direction normal to the disk plane. We model the vertical disk structure locally as a polytrope that represents a disk of high optical depth. We solve the three-dimensional gasdynamics equations semianalytically in the neighborhood of a Lindblad resonance. These solutions match asymptotically onto those valid away from resonances (previously obtained by Korycansky & Pringle) and provide solutions valid at all radii r. We obtain the following results: (1) A variety of waves are launched at the resonance, including r-modes and g-modes. However, the f-mode carries more than 95% of the torque exerted at the resonance. (2) These three-dimensional waves collectively transport exactly the amount of angular momentum predicted by the standard two-dimensional resonant torque formula. (3) Near resonance, the f-mode behaves compressibly and occupies the full vertical extent of the disk. Away from resonance, the f-mode behaves incompressibly, becomes confined near the surface of the disk, and, in the absence of other dissipation mechanisms, damps via shocks. In general, the radial length scale for this process is roughly rL/m (for resonant radius rL and azimuthal tidal forcing wavenumber m), independent of the disk thickness H. This wave-channeling process is due to the variations of physical quantities in r and is not due to wave refraction. (4) However, the inwardly propagating f-mode launched from an m = 2 inner Lindblad resonance experiences relatively minor channeling (accompanied by about a factor of 5 increase in nonlinearity), all the way to the radial center of the disk. We conclude that for binary stars, tidally generated waves at Lindblad resonances in highly optically thick circumbinary disks are subject to strong nonlinear damping by the channeling mechanism, while those in circumstellar accretion disks are subject to weaker nonlinear effects. We also apply our results to waves excited by young planets for which m ≈ r/H and conclude that the waves are likely damped on the scale of a few H.
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