Global numerical simulations of differentially rotating disks with free eccentricity

We study the nonlinear evolution of global $m=1$ modes with low pattern speeds in a differentially rotating non magnetic disk, by means of two and three dimensional numerical simulations. The modes make disk streamlines eccentric with maximum eccentricity in the range $0.13 {-} 0.24.$ We found that long lived patterns corresponding to eccentricities ~0.1 lasted for the duration of the simulations of ~64 orbits at the disk outer boundary. They had slow retrograde precession with period ~10 orbits at the outer boundary, in good agreement with that found from linear normal mode analysis. As expected from linear stability analysis, which leads one to expect a parametric instability associated with the non circular streamlines, we also found that three dimensional simulations showed the growth of a local instability to producing vertical motions on a local scale that eventually results in small amplitude turbulence with root mean square vertical velocity typically ∼$ 0.03 c_{\rm s},$ cs being the sound speed. This turbulence together with much more effective shocks produced by the interaction of the eccentric disk with the inner boundary were responsible for damping the disk eccentricity. Our first estimate of the damping time associated with turbulence for eccentricities ~0.1, corresponded to that associated with a turbulent viscosity acting to circularize the eccentric streamlines with “α” parameter ~ 10-3. For parameters appropriate to protostellar disks at 5 AU this corresponds to a decay time ∼$10^{5{-}6} ~\rm y.$ Thus although the indication is that disks with eccentric streamlines are long lived, there is associated turbulence even in the non magnetic case leading to estimated decay times that may be significant when the possibility of the growth of orbital eccentricity for extrasolar planets is considered through disk planet interaction on sufficiently long timescales.