Ring Spreading Tests

Among the tests of the viscosity treatment for the planet in a disc calculations, runs of the viscous spreading of a ring of matter in a given gravitational field were performed. The initial matter distribution consists of a Delta-function initially located at a specific radius r. The ring spreads under the influence of a constant viscosity, with no pressure forces included. This problem can be solved analytically (see Pringle, 1981, Ann. Rev. Astron. Astrophys., 19, p137).

The runs are performed by solving the Navier-Stokes equations without pressure and no energy equation but with all viscous terms in the radial and azimuthal equation included. We start the evolution imposing the analytic solution

at a viscous time of t=0.016. The ring is located initially at r=1 and the inner and outer radii are located at r=0.2 and r=1.8. The interval is typically covered by 128 radial gridcells.

  • One dimensional runs

    • Before we go to the two-dimensional runs we present cases where only the radial dependence has been taken into account (only one gridcell in the azimuthal direction). For a low (dimensionless) viscosity the results for different viscous times are shown in Fig. 1. The results agree well with the analytic solution. For larger viscosity however, one finds instabilities in the flow. Three different test cases solving the following dimensionless system of equations

    were performed. Note that we use two different viscosities in the radial and angular momentum equation. The initial conditions are the same as above and the dimensionless viscosity is 1.0e-5, i.e. ten times larger as above. The only parameter varied in the three test cases is the magnitude of the radial viscosity. In the first (model 1) it is switched off, in the second (Model 2) it is equal to the angular viscosity, and in the third (model 3) it is 1.5 times larger.

    The solid lines represent the analytic solution (at about 47 orbital periods) which refers to a viscous time of about 0.036, and crosses/dashed-lines to the numerical solution. Note, that the initial time of 0.016 viscous timescales refers to about 21 orbital periods for this viscosity. For each of the runs about 4000 timesteps were used.

    It can be seen that without radial viscosity an instability exists which can be damped by increasing the radial viscosity coefficient. This is exactly the Viscous Overstability which was described first by Kato (1978, MN, 185, 629), for a one-dimensional radial disk by Papaloizou & Stanley (1986, MN, 220, 593), and for a two-dimensional (r-z) disk by Kley, Papaloizou & Lin (1993, ApJ, 409, 739). We remark that decreasing the resolution also damps the instability because the smallest wavelengths show the largest growth rates.

  • Two dimensional runs

    • In the two-dimensional runs one also finds an instability which develops eventually into a non-axisymmetric pattern. The behaviour is similar to the one-dimensional case, a higher viscosity and higher resolution increase the growth rates. The non-axisymmetry appears to be triggered by the initial onset the radial instability, and can as such also be attributed to the viscous overstability. An example of such a calculation for a viscosity of 4.48e-5 is shown in the following figure.

    The resolution is 128x128 gridcells (only half are displayed), and the time refers to 0.184 viscous timescales corresponding to 50 orbital periods. Here, radial and azimuthal viscosity are the same. A two-dimensional color plot at the same time shows also more clearly the tilted ridges created by the instability.

    A one-dimensional (radial) plot of the surface density is given in the next plot.

    where the blue crosses (dotted line) refers to a radial cut in the middle (Phi=Pi), and the yellow triangles refers to an azimuthally averaged cut. The averaged curve follows the analytical one closely, while the non-averaged one displays the positions of the ridges.

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