Nonlinear Evolution of Hydrodynamical Shear Flows in Two Dimensions

Abstract

We examine how perturbed shear flows evolve in two-dimensional, incompressible, inviscid hydrodynamical fluids, with the ultimate goal of understanding the dynamics of accretion disks. To linear order, vorticity waves are swung around by the background shear, and their velocities are amplified transiently before decaying. It has been speculated that sufficiently amplified modes might couple nonlinearly, leading to turbulence. Here we show how nonlinear coupling occurs in two dimensions. This coupling is remarkably simple because it only lasts for a short time interval, when one of the coupled modes is in mid-swing. We focus on the interaction between a swinging and an axisymmetric mode. There is instability provided that k_{y,swing}/k_{x,axi} < omega/q, i.e., that the ratio of wavenumbers is less than the ratio of the axisymmetric mode's vorticity to the background vorticity. If this is the case, then when the swinging mode is in mid-swing it couples with the axisymmetric mode to produce a new leading swinging mode that has larger vorticity than itself; this new mode in turn produces an even larger leading mode, etc. Therefore all axisymmetric modes, regardless of how small in amplitude, are unstable to perturbations with sufficiently large azimuthal wavelength. We show that this shear instability occurs whenever the momentum transported by a perturbation has the sign required for it to diminish the background shear; only when this occurs can energy be extracted from the mean flow and hence added to the perturbation. For an accretion disk, this means that the instability transports angular momentum outwards while it operates.