Nonlinear disintegration of the internal tide
preprintposted on 11.05.2007, 13:57 authored by K.R. Helfrich, Roger Grimshaw
The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Because observations frequently show both tides and waves with amplitudes beyond the restrictions of weakly nonlinear theory, the evolution is studied using a fully-nonlinear, weakly nonhydrostatic two-layer theory that included the effects of rotation. In the hydrostatic limit, the governing equations have periodic, nonlinear inertia-gravity solutions that are explored as models of the nonlinear internal tide. These are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initially sinusoidal, linear internal tide is closely linked to the presence of these periodic waves. The initial tide steepens due to nonlinearity and sheds energy into short solitary waves. The disintegration is halted as the longwave part of the solution settles onto a state close to one of the nonlinear, hydrostatic solutions, with the short solitary waves superimposed. The degree of disintegration depends upon the initial amplitude of the tide and the properties of the underlying nonlinear solutions, which, depending on stratification and tidal frequency, exist only for a finite range of amplitudes (or energies). There is a lower threshold below which no short solitary waves are produced. However, for initial amplitudes above another threshold, given approximately by the energy of the limiting nonlinear inertia-gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations of large amplitude solitary waves in the South China Sea are discussed in the context of these model results.
- Mathematical Sciences