On radiating solitons in a model of the internal wave–shear flow resonance

Abstract

The work concerns the nonlinear dynamics of oceanic internal waves in resonance with a surface shear current. The resonance occurs when the celerity of the wave matches the mean flow speed at the surface. The evolution of weakly nonlinear waves long compared to the thickness of the upper mixed layer is found to be described by two linearly coupled equations (a linearized intermediate long wave equation and the Riemann wave equation). The presence of a pseudodifferential operator leads to qualitatively new features of the wave dynamics compared to the previously studied case of shallow water. The system is investigated primarily by means of numerical analysis. It possesses a variety of both periodic and solitary wave stationary solutions, including ‘delocalized solitons’ with a localized core and very small non-decaying oscillatory tails (throughout the paper we use the term ‘soliton’ as synonymous with ‘solitary wave’ and do not imply any integrability of the system). These ‘solitons’ are in linear resonance with infinitesimal waves, which in the evolutionary problem normally results in radiative damping. However, the rate of the energy losses proves to be so small, that these delocalized radiating solitons can be treated as quasi-stationary, that is, effectively, as true solitons at the characteristic time scales of the system. Moreover, they represent a very important class of intermediate asymptotics in the evolution of initial localized pulses. A typical pulse evolves into a sequence of solitary waves of all kinds, including the ‘delocalized’ ones, plus a decaying train of periodic waves. The remarkable feature of this evolution is that of all the products of the pulse fission (in a wide range of parameters of the initial pulse) the radiating solitons have by far the largest amplitudes. We argue that the radiating solitons acting as intermediate asymptotics of initial-value problems are a generic phenomenon not confined to the particular model under consideration.