Large amplitude Rossby waves riding on a background flow with a weak shear can be calculated up to a critical amplitude for which the meridional velocity, in a frame travelling with the wave, approaches zero at some point. Here we consider waves with an amplitude slightly greater than the critical amplitude by incorporating a region of recirculating fluid ( vortex core ) near this critical point. The effect of the vortex core is to introduce an extra nonlinear term into the equation for the wave amplitude proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude. The main effect due to the vortex core is a broadening of the wave profile. Further we show that the vortex core family has a limiting amplitude, with the limiting amplitude corresponding to a semi-infinite bore.