Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 237

spreads laterally and longitudinally (e.g. Phillips et al. , 2001 ) and may or may not satisfy
the similarity assumption even in the mean. Therefore, both assumptions need experi-
mental verification which has not been done. Most essentially, however, even the very
concept of correlation of the whitecap-exerted pressure with the forward wave slope has
never been verified experimentally. We should also point out that, before the distribu-
tion of established whitecaps is formed and they commence the negative work on the
wave, some energy is already lost from the wave, which cannot be accounted for by such
a model.

7.1.2 Kinetic-dynamic model

Zakharov et al. ( 2007 ) proposed a new analytical approach to the dissipation-function
problem which, here, we will conventionally call the kinetic-dynamic model. Unlike the
whitecap model in Section 7.1.1 , which assumes the dissipation to be weak in the mean,
this approach principally relies on co-existence of the so-called 'weak turbulence' and
strong and rapid isolated dissipation events.
The weak turbulence comprises weakly-nonlinear background wave fields whose reso-
nant four-wave interactions can be described by the kinetic equation for waves ( Hassel-
mann , 1962 ) and result in energy flux across the spectrum from large to small scales where
they eventually dissipate to viscosity ( Zakharov & Zaslavskii , 1982 ). The strong turbulence
is associated with wave breaking:

“Even if the weak turbulent resonant interaction effects dominate in the greater part of space, strongly
nonlinear effects could appear as rare localized coherent events they could be catastrophic, in

which case they are wave collapses ... Even rare sporadic collapse events can essentially affect the
physical picture of wave turbulence”

...

( Zakharov et al. , 2007 ).

The coexistence of wave collapses and weak turbulence was verified and confirmed
by means of numerical solutions of the primitive dynamic equations for the wave fields
( Dyachenko et al. , 1992 ; Dias et al. , 2004 ). In Zakharov et al. ( 2007 ), the authors further
simulated the wave evolution numerically be means of both the Hasselmann kinetic equa-
tion and basic Euler equations for the three-dimensional potential flow with a free surface.
The first evolution is for a statistical ensemble and, as said above, is due to weakly nonlin-
ear interactions only, and the second evolution is for dynamic variables whose statistics is
then compared with the outcomes of the first method.
Qualitatively, agreement between the statistical and dynamic models was good and
such key features of the wave evolution as spectral-peak downshifting, directional-spread
broadening and ω − 4 tail were observed in both simulations. Quantitatively, however, the

dissipation due to breaking had to be introduced in the statistical model for the outcome to
conform with the dynamic numerical experiment.
Thus, the quantitative estimate of whitecapping dissipation was obtained based on first
principles. The analytical model, however, does not provide an explicit expression for this