Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 234

Donelan & Yuan ( 1994 ) broadly classified theoretical models of spectral dissipation into
three types: probability models, quasi-saturated models and whitecap models. In Babanin
et al. ( 2007e ), we also added a turbulent model class to this classification, based on work
by Polnikov ( 1993 ).
Polnikov ( 1993 ) proposed a dissipation model where residual features of after-breaking
turbulence could supposedly be interpreted. He argued that, no matter what the cause of
the breaking, the result is turbulence in the water. In this approach, the rate of wave-
spectrum dissipation would be governed by the effective turbulent viscosity. Therefore,
to describe the wave energy dissipation in a wave spectrum form, it was necessary to find
a link between the wave spectrum and the water-turbulence spectrum.
This link was postulated, but was never further elaborated. Polnikov ( 1993 ) assumed a
simplified representation of wave dynamics equations, and applied that to monochromatic
waves. The assumptions in general may or may not be suitable, but even then, treating wave
motion in terms of eddy viscosity is questionable. Most importantly, however, spectral
waves of different scales interact, and the turbulent vortices of particular scales are not only
generated as a result of dissipation of counterpart waves of the same scales, but also as a
result of collapse of larger vortices (Kolmogorov cascade). Besides, the ocean is always
turbulent, and not only is the spectrum of this background turbulence unrelated to the wave
spectrum, but also a large proportion of such turbulence is produced by sources other than
the waves. Therefore, the basic postulate appears unfeasible in principle, and this kind of
model will be omitted here.
7.1.1 Probability, quasi-saturated and whitecap models
Historically, theories of breaking dissipation started with the work of Longuet-Higgins
( 1969a ). This was an analytical probability model that considered the waves prior to break-
ing. It was further developed by Yuan et al. ( 1986 , 2009 ) and Hua & Yuan ( 1992 ). The
approach was described in detail in Section 3.8 , and here we will only provide the most
relevant summary.
The approach employs the probability distribution of surface elevations, assuming waves
to be linear, in order to predict the likelihood of wave heights, or surface accelerations, or
surface velocities exceeding those characteristic of the Stokes limiting wave. Such waves
are unstable and will break, and as a result their height, as well as surface accelerations and
velocities will reduce. The difference can be interpreted in terms of energy loss, and the
resulting dissipation function appears to be a linear function of the wave spectrum.
The model is sensible in a physical sense, and in our view in this regard is the most
robust out of the three approaches described in this subsection. There can be qualitative
and quantitative concerns brought forward, but they are not of principal concern and can
potentially be addressed in the future.
Indeed, as has been discussed in Sections 2.9 , 5.1 , and throughout the topic, the Stokes
limiting criteria mostly hold, even if with small variations and even though the wave at
the breach of breaking onset does not actually appear to be the classical Stokes wave. The