  We investigate the weakly nonlinear dynamics of transient gravity waves at
infinite depth under the influence of a shear current varying linearly with
depth. An analytical solution is permitted via integration of the Euler
equations. Although similar problems were investigated in the 1960's and 70's
for special cases of resonance, this is to our knowledge the first general wave
interaction (mode coupling) solution derived to second order with a shear
current present. Wave interactions are integrable in a spectral convolution to
yield the second order dynamics of initial value problems.
  To second order, irrotational wave dynamics interacts with the background
vorticity field in a way that creates new vortex structures. A notable example
is the large parallel vortices which drive Langmuir circulation as oblique
plane waves interact with an ocean current. We also investigate the effect on
wave pairs which are misaligned with the shear current. In contrast to a
conjecture by Leibovich (1983) we find similar, but skewed, vortex structures
in every case except when the mean wave direction is perpendicular to the
direction of the current. Similar nonlinear wave-shear interactions are found
to also generate near-field vortex structures in the Cauchy-Poisson problem
with an initial surface elevation. These interactions create further groups of
dispersive ring waves in addition to those present in linear theory.
  The second order solution is derived in a general manner which accommodates
any initial condition through mode coupling over a continuous wave spectrum. It
is therefore applicable to a range of problems including special cases of
resonance. As a by--product of the general theory, a simple expression for the
Stokes drift due to a monochromatic wave propagating at oblique angle with a
current of uniform vorticity is derived, for the first time to our knowledge.
