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J Schmidt (University of Potsdam, Germany), H Salo (University of Oulu, Finland), F Spahn (University of Potsdam, Germany)
Viscous overstability was suggested to cause radial structure in an opaque planetary ring (Schmit and Tscharnuter, Icarus, 1995, 115, p304). We extended that model by the hydrodynamic heat flow equation (Spahn et al., Icarus, 2000, 145, p657) and used expressions for the transport coefficients determined in direct N-particle simulations of a dense ring (see the accompanying poster by Salo et al.). The overstable modes of the extended model are in good quantitative agreement with the overstability observed in simulations where the disk's self-gravity is included via an enhancement of the frequency of vertical oscillations. In the model ring (meter sized smooth spherical particles, Bridges' velocity dependent inelasticity law for ice spheres) overstability sets in for optical depths larger than about one. In particular, the growth rates in the linear regime are predicted correctly by the hydrodynamic model, as well as the critical wavelength (wavelengths larger than about 100m are unstable), and the phase--shifts between the perturbations of density and radial and tangential velocities. A weakly nonlinear stability analysis of the isothermal hydrodynamic model yields a nonlinear saturation of the growth of the overstable modes and predicts standing waves to be unstable with respect to traveling waves. This is also observed in our simulations.
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