The trapping and vertical focusing of internal waves in a pycnocline due to the horizontal inhomogeneities of density and currents

Abstract

This paper studies the propagation of a wave packet in regions where the central packet frequency ω is close to the local maximum of the effective Väisälä frequency Nf(z) = N(z)/[1 − k·U(z)/ω], where k is the central wavevector of the packet and U is the mean current with a vertical velocity shear. The wave approaches the layer ω = Nfm asymptotically, i.e. trapping of the wave takes place. The trapping of guided internal waves is investigated within the framework of the linearized equations of motion of an incompressible stratified fluid in the WKB approximation, with viscosity, spectral bandwidth of the packet, vertical shear of the mean current and non-stationarity of the environment taken into account. As the packet approaches the layer of trapping, the growth of the wavenumber k is restricted only by possible wave-breaking and viscous dissipation. The growth of k is accompanied by the transformation of the vertical structure of internal-wave modes. The wave motion focuses at a certain depth determined by the maximum effective Väisälä frequency Nfm. The trapping of the wave packet results in power growth of the wave amplitude and steepness. At larger times the viscous dissipation becomes a dominating factor of evolution as a result of strong slowing down of the packet motion.The role of trapping in the energy exchange of internal waves, currents and small-scale turbulence is discussed.