  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.
