Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 235

main uncertainty in the probability-based dissipation theories is the difference between the
wave state before and after breaking.
The existing analytical approaches presume some constant lower level of wave height,
at which the breaking should stop. These levels are different in different models, some are
more and some are less realistic as described in Section 3.8 . Essentially, however, in real-
ity there is no constant lower wave-height level for the breaking to cease. As described in
Section 2.7 and Chapter 6 , the wave-height loss can range from 100% to 0%, i.e. the wave
can completely disappear as a result of breaking, or can just marginally lose a top of its
crest, with all possible variations in between. Therefore, even though conceptually attrac-
tive, the probability models, as they have been derived, are not quantitatively plausible at
this stage.
The second type of analytical dissipation model is what Donelan & Yuan ( 1994 ) called
quasi-saturated models ( Phillips , 1985 ; Donelan & Pierson , 1987 ). These models rely on
the equilibrium range of the wave spectrum, where some sort of saturation exists for the
wave spectral density. Phillips ( 1985 ) argued that in this region the wind input, the wave-
wave interactions and the dissipation should balance each other. Therefore, at each wave
scale (wavenumber), any excess energy contributed by combined wind input and nonlinear
interaction fluxes does not bring about spectral growth, but results in wave breaking and
can be interpreted as the spectral dissipation local in wavenumber space. Donelan & Yuan
( 1994 ) found additional support to Phillips' assumption in the experiments of Pierson et al.
( 1992 ) where the breaking energy-loss was found to associate with the peak of propagating
wave groups rather than being localised in the physical space. Thus, adjustments to the
spectrum are focused in the wavenumber and frequency domain.
Phillips ( 1985 ) found that such dissipation is cubic in terms of the spectral density, i.e.

n 3in (7.1) . Donelan & Pierson ( 1987 ) added consideration of wave directionality to

the energy balance of the equilibrium range, arguing that a simple balance between wind
input and dissipation is not observed at large angles to the wind. They also separated dis-
persive (gravity and capillary) waves and non-dispersive (gravity-capillary) waves, as the
nature of breaking differs for them because of different speeds of propagation relative to
wave groups. Donelan & Pierson ( 1987 ) obtained a local-in-wavenumber-space dissipa-
tion function, similar to that of Phillips ( 1985 ), but their exponent n depends on the wave
spectrum F

=

( k ) and on the wavenumber k as such. According to them, n can vary signifi-

cantly: n = 1-5. In most ranges of interest, however, n ∼ 5 - both for gravitational and

for capillary waves.
This model type has multiple shortcomings. The most important one is that, even if all
the physics implied is true and correct, the results are valid at the spectrum tail and not valid
at the spectral peak where a balance between the source functions cannot be expected. If
so, the dissipation function based on the quasi-saturated approach is only applicable to a
part of the spectrum, and is certainly not applicable to the most energetic waves which,
when they break, produce most of the dissipation in absolute values.
In terms of the physics, however, there are essential issues too. Donelan & Yuan ( 1994 )
based their argument in favour of the local-in-wavenumber dissipation on observations