On the reflection of a train of finite-amplitude internal waves from a uniform slope

Abstract

The reflection of a train of two-dimensional finite-amplitude internal waves propagating at an angle β to the horizontal in an inviscid fluid of constant buoyancy frequency and incident on a uniform slope of inclination α is examined, specifically when β > α. Expressions for the stream function and density perturbation are derived to third order by a standard iterative process. Exact solutions of the equations of motion are chosen for the incident and reflected first-order waves. Whilst these individually generate no harmonics, their interaction does force additional components. In addition to the singularity at α = β when the reflected wave propagates in a direction parallel to the slope, singularities occur for values of α and β at which the incident-wave and reflected-wave components are in resonance; strong nonlinearity is found at adjacent values of α and β. When the waves are travelling in a vertical plane normal to the slope, resonance is possible at second order only for α < 8.4° and β < 30°. At third order the incident wave is itself modified by interaction with reflected components. Third-order resonances are only possible for α < 11.8° and, at a given α, the width of the β-domain in which nonlinearities connected to these resonances is important is much less than at second order. The effect of nonlinearity is to reduce the steepness of the incident wave at which the vertical density gradient in the wave field first becomes zero, and to promote local regions of low static stability remote from the slope. The importance of nonlinearity in the boundary reflection of oceanic internal waves is discussed.In an Appendix some results of an experimental study of internal waves are described. The boundary layer on the slope is found to have a three-dimensional structure.