Non-dissipative effects of breaking on the wave field - Breaking and Dissipation of Ocean Surface Waves - page 324

This question was first investigated by Trulsen & Dysthe ( 1990 ) by means of a modified
nonlinear Schrödinger equation (Dysthe equation). The effect of breaking was simulated
by adding a dissipation term to this equation which was solved numerically. As mentioned
in Section 4 , NLS and its modifications work in the physical space for the complex wave
envelope, and this dissipation was expressed as a function of wave steepness :

r

1 H

1
τ r thr

D NLS = − ( − thr ). (8.1)

This term has two very important features. These are the breaking limiting steepness thr

and the Heaviside unit-step function H . Here, r and τ r define relaxation of the wave

steepness back to its limiting value once the critical steepness is exceeded, that is the
breaking.
Nowadays, we do know that the limiting steepness exists (see Section 5.1.2 , Babanin
et al. ( 2007a ); Toffol i et al. ( 2010a )). Once the wave exceeds this steepness, it breaks, but
while breaking, its steepness is typically reduced to a level much lower than the critical
steepness. This issue, however, is of secondary importance here and can be easily modified
in the approach (see also Section 7.1.1 ). The significance of the Heaviside function will
be outlined with respect to the model of Hwung et al. ( 2009 ) and Hwung et al. ( 2010 )
discussed below.
Typical behaviour of the modulational instability in a uniform wave train is growth of
two resonant sidebands, both below and above the primary mode, that leads to strong
modulation of the wave train, which sidebands then subside and the initial near-uniform
train recurs. In the case of wave breaking happening in the course of such a cycle, how-
ever, as was shown by Trulsen & Dysthe ( 1990 ) theoretically and Tulin & Waseda ( 1999 )
experimentally, the grown lower-sideband becomes disconnected from the cycle and thus
a permanent downshift of the wave energy takes place.
The topic has been consistently developed further by Trulsen and his colleagues.
Trulsen & Dysthe ( 1992 ) demonstrated modification of this process subject not only to
the dissipation, but also to strong energy input (i.e. in a way simulating the wind forc-
ing, see also Sections 5.1 and 6.1 ). Trulsen & Dysthe ( 1997 ) extended the research into
three-dimensional wave fields. When using the Dysthe equation without dissipation, no
permanent downshifting was observed for two-dimensional waves, but if oblique perturba-
tions were allowed, the permanent downshift of the spectral peak did occur, with the energy
shifted to the oblique modes. Trulsen et al. ( 1999 ) confirmed such significant downshift to
the transverse modes experimentally, in a directional wave tank.
The most detailed and dedicated study of the downshifting due to breaking of waves
was conducted by Tulin & Waseda ( 1999 ) and Waseda & Tulin ( 1999 ). This was done in a
flume for two-dimensional waves, with and without the wind, and a theory for explaining
the results was also proposed (see discussion in Section 7.3.1 ).
The modulational instability in the experiments was, mostly, seeded, that is the fastest-
growing sidebands were imposed to accelerate development of the instability. While