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, as well as other astrophysical shear flows. 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, i.e., when its phasefronts are aligned with the radial direction. We focus on the interaction between a swinging and an axisymmetric mode. There is instability provided that |ky,sw/kx,axi| . |ωaxi/q|, i.e., that the ratio of wavenumbers (swinging azimuthal wavenumber to axisymmetric radial wavenumber) 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. (Of course in a disk the azimuthal wavelength cannot exceed the circumferential distance.) We show that this shear (or Kelvin-Helmholtz) 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. We verify all our conclusions in detail with full hydrodynamical simulations, done with a pseudospectral method in a shearing box. Simulations of the instability form vortices whose boundaries become highly convoluted. Whether this nonlinear instability plays a role in a transition to turbulence is an interesting possibility that awaits three dimensional investigation. Subject headings: accretion, accretion disks — instabilities — solar system: formation —turbulence