  Oceanic internal waves often have curvilinear fronts and propagate over
various currents. We present the first study of long weakly-nonlinear internal
ring waves in a three-layer fluid in the presence of a background linear shear
current. The leading order of this theory leads to the angular adjustment
equation - a nonlinear first-order differential equation describing the
dependence of the linear long wave speed on the angle to the direction of the
current. Ring waves correspond to singular solution (envelope of the general
solution) of this equation, and they can exist only under certain conditions.
The constructed solutions reveal qualitative differences in the shapes of the
wavefronts of the two baroclinic modes: the wavefront of the faster mode is
elongated in the direction of the current, while the wavefront of the slower
mode is squeezed. Moreover, different regimes are identified according to the
vorticity strength. When the vorticity is weak, part of the wavefront is able
to propagate upstream. However, when the vorticity is strong enough, the whole
wavefront propagates downstream. A richer behaviour can be observed for the
slower mode. As the vorticity increases, singularities of the swallowtail-type
may arise and, eventually, solutions with compact wavefronts crossing the
downstream axis cease to exist. We show that the latter is related to the
long-wave instability of the base flow. We obtain analytical expressions for
the coefficients of the cKdV-type amplitude equation, and numerically model the
evolution of the waves for both modes. The initial evolution is in agreement
with the leading-order predictions for the deformations of the wavefronts.
Then, as the wavefronts expand, strong dispersive effects in the upstream
direction are revealed. Moreover, when nonlinearity is enhanced, fission of
waves can occur in the upstream part of the ring waves.
