Nonlinear, periodic, internal waves
Abstract
It is shown how to calculate largeamplitude, periodic, internal waves in a channel filled with continuously stratified fluid by using a Stokestype amplitude expansion along the channel and a modal expansion across the channel. We obtain explicit solutions for the case where the density increases exponentially with depth. It is found that periodic waves in exponentially stratified fluid are waves of depression: as the wave increases in amplitude the wave speed decreases and the mean density of the fluid at a given height decreases. The waves are limited in height by the formation of an eddy of fluid on the upper boundary above the trough of the wave. This is consistent with the description of waves of depression given by Amick and Toland (1984). There is no backflow before the eddy forms. By contrast, Amick and Toland (1984) have shown that solitary waves in the same system are waves of elevation, limited in height by a cusp in the streamlines at some interior point. A relationship is found between the mass flux and the mass displacement. The expressions necessary to calculate the potential and kinetic energies are given. A simple analytic solution for the internal wave is presented for a fluid with a weak exponential stratification, that is when the Boussinesq approximation is appropriate. For an Nth mode wave the limiting amplitude of the perturbation streamfunction is shown to be 1/ Nπ. This corresponds to a maximum streamline displacement of 2 z/ Nπ, where z (=0.7391) is the solution to z = cos z.
 Publication:

Fluid Dynamics Research
 Pub Date:
 March 1990
 DOI:
 10.1016/01695983(90)90025T
 Bibcode:
 1990FlDyR...5..301H
 Keywords:

 Computational Fluid Dynamics;
 Internal Waves;
 Nonlinearity;
 Boussinesq Approximation;
 Fourier Series;
 Stream Functions (Fluids);
 Two Dimensional Flow;
 Velocity Distribution;
 Vortices;
 Fluid Mechanics and Heat Transfer