Three Dimensional Compressible Hydrodynamic Simulations of Vortices in Disks

Abstract

We carry out three-dimensional, high resolution (up to $1024^2\times 256$) hydrodynamic simulations of the evolution of vortices in vertically unstratified Keplerian disks using the shearing sheet approximation. The transient amplification of incompressible, linear amplitude leading waves (which has been proposed as a possible route to nonlinear hydrodynamical turbulence in disks) is used as one test of our algorithms; our methods accurately capture the predicted amplification, converges at second-order, and is free from aliasing. Waves expected to reach nonlinear amplitude at peak amplification become unstable to Kelvin-Helmholtz modes when $\mid W_{\rm max}\mid\gtrsim Ω$ (where $W_{\rm max}$ is the local maximum of vorticity and $Ω$ the angular velocity). We study the evolution of a power-law distribution of vorticity consistent with Kolmogorov turbulence; in two-dimensions long-lived vortices emerge and decay slowly, similar to previous studies. In three-dimensions, however, vortices are unstable to bending modes, leading to rapid decay. Only vortices with a length to width ratio smaller than one survive; in three-dimensions the residual kinetic energy and shear stress is at least one order of magnitude smaller than in two-dimensions. No evidence for sustained hydrodynamical turbulence and transport is observed in three-dimensions. Instead, at late times the residual transport is determined by the amplitude of slowly decaying, large-scale vortices (with horizontal extent comparable to the scale height of the disk), with additional contributions from nearly incompressible inertial waves possible. Evaluating the role that large-scale vortices play in astrophysical accretion disks will require understanding the mechanisms that generate and destroy them.