The Generation and Propagation of Ship Internal Waves in a Generally Stratified Ocean at High Densimetric Froude Numbers, Including Nonlinear Effects

Abstract

A nonlinear theory for internal wave generation and propagation is derived here for slender ships traveling at high densimetric Froude number (Fh >> 1) in water of small density variation. It is based on an asymptotic equation for the evolution of the internal wave vorticity generated under the ship by a known inviscid ship flow and then self-propagating in the wake. In its numerical implementation, arbitrary pycnoclines and slender ship hulls may be used, and boundary conditions on the ship hull are satisfied; the free surface is treated here as rigid, although this may be relaxed. The theory has been implemented by a suitable numerical method and numerous simulations 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 trough on either side, forming close behind the ship. Two distinct types of triple lobes are identified:dominant central lobe and very weak troughs, and;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 measurements are good, surprisingly so considering the small scale of the experiment and the asymptotic nature of the theory. The effect of the turbulent wake 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 stratification, triple-lobe patterns with abnormally large troughs are generated and lead to strong nonlinear effects; these deep troughs propagate sidewards to large distances aft (over 40 ship lengths) with slow decay, and result in much larger surface currents and strain rates than in the normal case. Correspondingly, fast waves of depression, which decay slowly, were discovered through the simulation of two-dimensional initial value problems, where the initial area of depression was significantly less than required of a true soliton; these "quasi-solitons" are briefly studied here.