Nonlinear evolution of hydrodynamical shear flows in two dimensions

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, voracity 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 |k_y,sw|k _x,axi| ≲ |ω/q|, i.e., that the ratio of wavenumbers (swinging azimuthal wavenumber to axisymmetric radial wavenumber) is less than the ratio of the perturbed 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. We explain how this shear (or KelvinHelmholtz) instability operates in real space as well. The instability occurs whenever the momentum transported by an energy-conserving perturbation opposes 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 outward while it operates. We verify all our conclusions in detail with pseudospectral numerical simulations. Simulations of the instability form vortices whose boundaries become highly convoluted.