Asymmetric internal solitary waves with a trapped core in deep fluids

We describe an asymptotic model for long large-amplitude internal solitary waves with a trapped core, propagating in a narrow layer of nearly uniformly stratified fluid embedded in an infinitely deep homogeneous fluid. We consider the case of a mode one asymmetric wave with an amplitude slightly greater than the critical amplitude, for which there is incipient over-turning, that is wave-breaking. We then incorporate a vortex core located near the point at which this incipient breaking occurs. 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. Thus the derived new equation for the wave amplitude incorporates both the nonlinearity arising due to the flow over the recirculation core, and the nonlinearity associated with the ambient stratification; the dispersion term however remains of the Benjamin-Ono type. 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 where a sharpening of the wave crest normally occurs. The limiting form of the broadening wave is “a deep fluid bore”. The wave speed is found to depend nonlinearly on the wave amplitude and the traditional linear dependence underestimates this speed.