Nonlinear Bending Waves in Vertically Stratified Planetary Rings

The theory of bending waves in planetary rings generally neglects the finite vertical thickness of the rings and simply describes the vertical warping of a razor-thin ring plane under the influence of self-gravity and resonant forcing by satellites (Shu 1984). The effects of viscosity in the bending wave was then added as a diffusion of the vertical velocity in the bending wave, without any consideration of radial motions in the ring. This rather extreme simplification omits the strong coupling between vertical and horizontal motions that can occur in warped disks with a finite vertical scale height, as has been extensively discussed in the context of warped circumstellar gas disks. In a more realistic disk with vertical stratification, the pressure will be maximum at the disk mid-plane and somewhat less near the upper and lower surfaces of the disk. When a vertically stratified disk is warped out of the reference plane, horizontal pressure gradients are created. Since these horizontal pressure gradients are antisymmetric about the reference mid-plane of the unperturbed disk, they force a vertically sheared horizontal motion at the same frequency as the vertical oscillation. In nearly Keplerian disks, such as Saturn’s rings, the vertical oscillation frequency is close to the horizontal epicycle frequency, so one can expect significant excitation of horizontal motions in a bending wave. The theoretical framework of Ogilvie (2018) for warped circumstellar disks has been adapted to bending waves in planetary rings by adding self-gravity. Preliminary results show that (1) viscous damping of the vertically sheared horizonal motions can be more important than the traditional viscous damping used by Shu (1984) and (2) that a viscous instability can develop in nonlinear bending waves, which could explain the vertical rift observed in the Titan -1:0 nodal bending wave in Saturn’s C ring.