Spreading ring calculations -- Marvin

To measure the magnitude of numerical viscosity in disk simulations with AstroBEAR I have performed calculations of a spreading ring.

A short reminder on the spreading ring problem:

If we assume an accretion disk that is rotationally symmetric and geometrically thin and that its angular frequency does not change with time we can derive the following equation that describes the time dependent evolution of such an accretion disk:

Here is the accretion disk's surface density, its angular frequency and the viscosity of the gas. For a Keplerian gravitational potential, constant viscosity and an initial condition in the form of a delta peak at position

this equation has the following analytical solution (eq. 1):

with

(eq. 2)

and the modified Bessel function of the first kind .

As a delta peak is numerically difficult to handle I use this analytical solution as initial condition, with and initial values for of and , respectively.

The following movie shows the general behavior of such a ring. The numerical and physical parameters are the same than those of my simulations of the CND unless stated otherwise, e.g., the resolution at the location of the ring is about 0.04 pc. However, I switched of cooling and magnetic fields for the spreading ring calculations.

First animation: surface density for $\tau_{\text{i}} = 0.001$​

The following two figures show the radial surface density profiles, the first one for and the second one for . For each snapshot of the simulation I fitted a curve according to eq. 1 to the surface density profiles, which are also shown in these figures.

This fit allows to determine the corresponding value of . Thus we have as a function of time, determining the slope of this function gives the viscosity (see eq. 2).

I define the parameter

where is the maximal alpha viscosity, with and I use typical values of the CND.

For the simulation I get a value of

and for the simulation a value of

Considering that in the literature a value of is often used to account for viscous disk evolution, this result is not extraordinarily good, but I think it shows that our simulations are not dominated by numerical viscosity.

I furthermore did a spreading ring calculation with central outflow, the following movie shows that the material of the ring is slowly blown away by the wind, contrary to the simulations of the CND where the material of the disk was moving towards the central black hole due to its angular momentum loss.

Second animation: surface density for $\tau = 0.001$, with central outflow​

The following figure shows the corresponding radial surface density profiles. Because the development of the ring is completely different from the analytical solution of the spreading ring, I do not show any fits to eq. 1 here.

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