This paper is concerned with a theoretical model for large amplitude solitary waves with a vortex core in a shallow layer of stratified fluid with a nearly uniform stratification. Previous work has shown that solitary waves can be calculated up to a critical amplitude for which the horizontal velocity, in a frame for which the wave is at rest, approaches zero at the boundary point beneath the wave crest for a wave of elevation (or above the wave crest for a wave of depression). Here we calculate waves with amplitudes slightly greater than the critical amplitude by incorporating a vortex core located near the aforementioned boundary point. The effect of the vortex core is to introduce into the governing equation for the wave amplitude an extra nonlinear term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude. We find that as the wave amplitude increases above the critical amplitude, the wave broadens, which is in marked contrast to the case of small amplitude waves. Further, the wave speed is found to depend nonlinearly on the wave amplitude, again in marked contrast to the linear dependence for small amplitude waves. The specific case of linearly stratified fluid is examined in detail.

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 traveling 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. Furthermore, we show that the vortex core family has a limiting amplitude, with the limiting amplitude corresponding to a semi-infinite bore.

Nonlinear dipolar vortices/gyres on a ␥-plane are investigated both experimentally and theoretically. The solutions describe a fundamental dipolar mode of large scale barotropic motion of the polar ocean or atmosphere on the rotating planet. The entire dipolar gyre is predicted to rotate anticyclonically with a specific angular velocity. The existence and stability of the theoretically predicted flow are confirmed in a laboratory experiment on a rotating platform. The laboratory flows are induced by an electromagnetic method and are observed using the nonintrusive optical method of altimetric imaging velocimetry. The rotation rate of the experimental flow is in good agreement with that predicted theoretically. Detailed measurements of the velocity field and surface elevation demonstrate that an assumption of linearity of the relation between the relative vorticity and the stream function is valid.