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, focusing on the interaction between an axisymmetric mode and a swinging mode. We find that all axisymmetric modes, regardless of how small in amplitude, are unstable when they interact with swinging modes that have sufficiently large azimuthal wavelength. Quantitatively, the criterion for instability is that |ky,sw/kx,axi|