The steepening of long, internal waves

Abstract

An investigation is made of the deformation of long waves in a stratified fluid as influenced by the curvature of the vertical profile of density, by the terms neglected in the Boussinesq approximation and by the vertical shear. In the continuous case these are taken to be small effects and the results for waves with an even number of nodal surfaces are as follows: (1) If the density profile is concave upward, a wave of elevation tends to break in the direction of the propagation, i.e. the forward part of the wave steepens; if the profile is convex upward, it tends to break backward. (2) The terms neglected in the Boussinesq approximation cause a wave of elevation to tend to break backward. (3) If the wave is propagating toward negative x and the fluid speed along the x-axis increases with height, the wave of elevation tends to break forward. To the first order there is no tendency for breaking if the waves have an odd number of nodal surfaces. An analysis is also made of a two-fluid system. The same tendencies exist in (1) above provided the concave-upward profile corresponds to a shallower lower fluid. In (2) and (3) above the tendencies for breaking in the continuous system and in the two-fluid system are opposite.