Non-linear dynamics of barotropic Rossby waves in a meridional shear flow

The barotropic vorticity equation is solved numerically and analytically in order to study the non-linear evolution of a barotropic forced Rossby wave in a meridional shear flow. The flow configuration is an idealised model for a western boundary current in an oceanic basin. There is a critical layer at the longitude where the shear flow speed is equal to the wave phase speed, and the wave is incident on the critical layer from the east or west. The numerical solutions show that a non-linear eastward-propagating Rossby wave is absorbed by the mean flow at the critical layer at early times and reflected at later times, but a westward-propagating wave is transmitted through the critical layer. The asymptotic analyses show that the critical layer thickness is O(epsilon(1/2)), whereis the relative magnitude of the perturbation to the basic flow, and that if the solution includes an eastward-propagating component the wave-induced mean flow increases with time in the critical layer. The westward-propagating part of the solution is non-singular and does not on its own affect the mean flow in the critical layer.