# Keplerian Disk

## February 6, 2017

So we’ve seen how Cholla does with rotating flows in general, but how about the classic problem of a disk in Keplerian rotation? Generally, there are two challenges in simulating such disks with a static grid code: first, the orbital speeds near the center of the disk can get extremely large as the resolution of the simulation is increased, and second, advection errors (like those seen in the Gresho test) lead to artificial accretion through the disk.

To set up a rotating gas disk, I’ve followed the initial conditions from Krumholtz et al. (2004). There, the potential is set by assuming a single solar mass at the center, and the velocities are Keplerian, $v^2 = GM/r$. The disk is assumed to be 10K and isothermal, with a radius $r_{0} = 2\times10^{15}$ cm and a power-law surface density profile $\Sigma = \Sigma_{0}(r_{0}/r)$, with $\Sigma_{0} = 0.1$ $\mathrm{g}$ $\mathrm{cm}^{-2}$. Because the disk is in Keplerian rotation, the surface density should not change with time.

Results

The simulation is run for approximately 25 orbits of the cells at the smallest radius, or a single orbit at $r = 1\times10^{15}$ cm. I used a 128x128 grid with custom boundaries set to reproduce the correct velocites. I’m again using PPMC, an exact Riemann solver, and CTU. The movies below show the evolution in the surface density profile and velocities as a function of radius, as well as a log-scale image of the surface density. For the radial plots, the exact solution is plotted as a black line, while red points show the simulation results. The point plotted on the surface density image represents the inferred motion of a particle at that radius, and is merely there to guide the eye.

Clearly, Cholla has the most trouble at the center of the grid, as expected. The average L1 surface density and velocity errors by the end of the simulation are about 6% and 0.1%, respectively, where I’ve normalized by the average surface density and velocity. However, the errors for points near the center of the disk are much larger, as can be seen in the movies. When I increase the resolution by a factor of two, the simulation breaks. That said, as a test of the 2D potential, I’d say this is pretty good (and since I’m not going to be simulating Keplerian disks, I don’t think I need to investigate sink particles or other fixes to these issues at this time).