  We investigate exact nonlinear waves on surfaces locally approximating the
rotating sphere for two-dimensional inviscid incompressible flow. Our first
system corresponds to a beta-plane approximation at the equator and the second
to a gamma approximation, with the latter describing flow near the poles. We
find exact wave solutions in the Lagrangian reference frame that cannot be
written down in closed form in the Eulerian reference frame. The wave particle
trajectories, contours of potential vorticity and Lagrangian mean velocity take
relatively simple forms. The waves possess a non-trivial Lagrangian mean flow
that depends on the amplitude of the waves and on a particle label that
characterizes values of constant potential vorticity. The mean flow arises due
to potential vorticity conservation on fluid particles. Solutions over the
entire space are generated by assuming that the flow far from the origin is
zonal and there is a region of constant potential vorticity between this zonal
flow and the waves. In the gamma approximation a class of waves are found
which, based on analogous solutions on the plane, we call Ptolemaic vortex
waves. The mean flow of some of these waves resemble polar jet streams. Several
illustrative solutions are used as initial conditions in the fully spherical
rotating Navier-Stokes equations, where integration is performed via the
numerical scheme presented in Salmon and Pizzo (2023). The potential vorticity
contours found from these numerical experiments vary between stable permanent
progressive form and fully turbulent flows generated by wave breaking.
