# Nonlinear disintegration of the internal tide

preprint

posted on 11.05.2007, 13:57 by K.R. Helfrich, Roger GrimshawThe 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.

## History

## School

- Science

## Department

- Mathematical Sciences