Nonlinear Scale Invariance in Local Disk Flows

Abstract

An exact nonlinear scaling transformation is presented for the local three-dimensional dynamical equations of motion for differentially rotating disks. The result is relevant to arguments that have been put forth claiming that numerical simulations lack the necessary numerical resolution to resolve nonlinear instabilities that are supposedly present. We show here that any time dependent velocity field satisfying the local equations of motion and existing on small length scales, has an exact rescaled counterpart that exists on arbitrarily larger scales as well. Large scale flows serve as a microscope to view small scale behavior. The absence of any large scale instabilities in local numerical simulations of Keplerian disks suggests that the equations in this form have no instabilities at any scale, and that finite Reynolds number suppression is not the reason for the exhibited stable behavior. While this argument does not rule out the possibility of global hydrodynamical instability, it does imply that differential rotation {\em per se} is not unstable in a manner analogous to shear layers or high Reynolds number Poiseuille flow. Analogies between the stability behavior of accretion disks and these flows are specious.