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Most theoretical studies of solitary waves are for the weakly nonlinear regime, where models such as the Korteweg-de Vries equation are commonly obtained. However, observations of solitary waves often show that these waves can have large amplitudes, to the extent that they may contain vortex cores, that is, regions of recirculating flow. In this work, we report on theoretical asymptotic models, which describe explicitly the structure of solitary waves with recirculation zones, for certain special but important upstream flow configurations.
The key feature which enables this construction is that, both for stratified shear flows and for axisymmetric swirling flows, the steady state vorticity equation is almost linear when the upstream flow is almost uniform. That is, for stratified shear flows the upstream flow and the upstream stratification are almost constant, while for rotating flows the upstream axial flow and angular velocity are almost constant. This feature enables the asymptotic construction of solitary waves described by a steady-state generalised Korteweg-de Vries equation in an outer zone, matched to an inner zone containing a recirculation zone. These recirculation zones exist for wave amplitudes just greater than a certain critical amplitude for which there is incipient flow reversal. The recirculation zones have a universal structure such that their width increases without limit as the wave amplitude increases from the critical amplitude to a certain maximum amplitude, but their existence can be sensitive to the actual upstream flow configuration. Applications are made to observations and numerical simulations of large amplitude internal solitary waves, and to the phenomenon of vortexbreakdown.