(Seminar on nonlinear mechanics)
Date : 21st April (Mon.), 1997 from 10:30am
Place : Conference Room ( Room No.217 of Eng. Building No.8 )
Speaker : Karsten Trulsen
(Department of Mathematics, University of Bergen, Norway)
Title : Evolution of gravity waves in two and three dimensions predicted by
a nonlinear Schrodinger equation for broader bandwidth
The classical third-order nonlinear Schrodinger equation, as well as
the modified fourth-order nonlinear Schrodinger equations of Dysthe (1979)
and Brinch--Nielsen & Jonsson (1986) all suffer from insufficient
resolution in bandwidth to fully capture the dispersive properties of
typical ocean waves. Rather than resorting to computationally intensive
integral equations or fully nonlinear integration, we extend the modified
nonlinear Schrodinger equation to broader bandwidth (Trulsen & Dysthe
1996). The new equation for broader bandwidth enables us to study
three-dimensional evolution of wave trains.
We have used the new model to study the frequency downshift of Stokes
waves, first reported by Lake et al. (1977).
Conservative models describing wave evolution in two dimensions
predict a temporary downshift in the strongly modulated stage of the wave
evolution following the initial modulational instability, but are not
capable of describing a permanent downshift. It has therefore been commonly
accepted that the downshift requires non-conservative effects like
dissipation or wave breaking.
Stokes waves are however unstable for perturbations transverse to
their direction of propagation. Several experiments showing downshift, have
been performed in tanks that are wide compared to the dominant wavelength.
This suggests that a three-dimensional theory should be used to investigate
the consequences of transverse modulation. We find that the evolution of
Stokes waves is qualitatively different in two and three dimensions. The
frequency downshift observed in wide tanks can be explained as a
consequence of conservative three-dimensional wave evolution, and does not
require dissipation or breaking.