The generation and propagation of ship internal waves in a generally stratified ocean at high densimetric froude numbers, including nonlinear effects

A nonlinear theory for internal wave generation and propagation is derived here for slender ships traveling at high densimetric Froude number (F-h >> 1) in water of small density variation. It is based on an asymptotic equati on for the evolution of the internal wave vorticity generated under the shi p by a known inviscid ship flow and then self-propagating in the wake. In i ts numerical implementation, arbitrary pycnoclines and slender ship hulls m ay be used, and boundary conditions on the ship hull are satisfied; the fre e surface is treated here as rigid, although this may be relaxed. The theor y has been implemented by a suitable numerical method and numerous simulati ons have been carried out. The results have been compared with earlier OEL experiments. In the near field, emphasis is given to a triple-lobe pattern in the pycnocline, an upwelling along the centerline of motion with a troug h on either side, forming close behind the ship. Two distinct types of trip le lobes are identified: (a) dominant central lobe and very weak troughs, a nd; (b) weak central lobe and dominant troughs. The former (a) is shown to result in linear propagation Into the far field. The latter (b) results in far-field patterns preceded by a deep trough whose propagation is nonlinear . The comparisons of both simulated trends and actual amplitudes with measu rements are good, surprisingly so considering the small scale of the experi ment and the asymptotic nature of the theory. The effect of the turbulent w ake on the internal waves in the experiments is restricted to a very narrow region behind the ship; the bulk of the wave pattern including the leading waves seem unaffected. Simulations show that under certain conditions of s tratification, triple-lobe patterns with abnormally large troughs are gener ated and lead to strong nonlinear effects; these deep troughs propagate sid ewards to large distances aft (over 40 ship lengths) with slow decay, and r esult in much larger surface currents and strain rates than in the normal c ase. Correspondingly, fast waves of depression, which decay slowly, were di scovered through the simulation of two-dimensional initial value problems, where the initial area of depression was significantly less than required o f a true soliton; these "quasi-solitons" are briefly studied here.