Three-dimensional waves generated at Lindblad resonances in thermally stratified disks

We analyze the linear, three-dimensional response to tidal forcing of a dis k 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 re presents a disk of high optical depth. We solve the three-dimensional gasdy namics equations semianalytically in the neighborhood of a Lindblad resonan ce. These solutions match asymptotically onto those valid away from resonan ces (previously obtained by Korycansky & Pringle) and provide solutions val id 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) Th ese three-dimensional waves collectively transport exactly the amount of an gular momentum predicted by the standard two-dimensional resonant torque fo rmula. (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 r(L)/m (for resonant radius r(L) and azimuthal tidal forcing wavenumber m), independent of the disk th ickness H. This wave-channeling process is due to the variations of physica l quantities in r and is not due to wave refraction. (4) However, the inwar dly propagating f-mode launched from an m = 2 inner Lindblad resonance expe riences relatively minor channeling (accompanied by about a factor of 5 inc rease in nonlinearity), all the way to the radial center of the disk. We co nclude that for binary stars, tidally generated waves at Lindblad resonance s in highly optically thick circumbinary disks are subject to strong nonlin ear damping by the channeling mechanism, while those in circumstellar accre tion disks are subject to weaker nonlinear effects. We also apply our resul ts to waves excited by young planets for which m approximate to r/H and con clude that the waves are likely damped on the scale of a few H.