Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 284

and

g f

T 2 ( f ) =− a 2 ρ w

f p (

F ( q ) − F threshold ( q )) dq . (7.52)

As discussed in Section 5.3.2 , normalisation by the directional spread A ( f ) (5.33) can be

omitted which is done here in (7.51) - (7.52) .
The expression uses dimensional wave spectrum F

( f ) , rather than its dimensionless

counterpart (5.31) like the saturation-based functions discussed in this section

above. The rationale behind the saturation use is that it allows nonlinear formulations for
the spectral dissipation, but the investigations of Tsagareli ( 2009 ) led to a conclusion that
linear whitecapping dissipation with the cumulative term is most suitable. If the (5.40) or
(7.50) - (7.52) term is to be made nonlinear, it can be done of course, but F

σ Phillips ( f )

has to be nor-
malised by a relevant spectral-density distribution first in order to satisfy the dimensional
argument ( Rogers et al. , 2011 ).
As discussed in Section 5.3.2 , since the saturation

( f )

(5.31) is the fifth moment
of the spectrum, experimental investigations of the breaking probabilities versus such a
parameter, for field-observed irregular spectra, led to formidable scatter (see Figure 5.23 ).
This is why it was preferred to keep the dissipation a function of the spectrum F

σ Phillips ( f )

)
itself; dependences of the breaking probabilities across the spectrum in terms of the spec-
tral density F

( f

are quite robust and therefore can be investigated experimentally (see
Babanin & Young , 2005 ; Babanin et al. , 2007c ).
The dimensionless threshold

( f )

(5.36) , on the contrary, is a steady value. In
the dimensional formulation of the dissipation function, this value can then be converted
into dimensional threshold at each frequency algebraically (5.37) .
The weighting coefficients a 1 and a 2 , of the inherent (7.51) and cumulative (7.52) dissi-
pation terms, were introduced by Babanin & Young ( 2005 ) and Young & Babanin ( 2006a )
on the basis of a single extreme-breaking record analysed. They obtained a 1

σ threshold ( f )

0065
and assumed the same value for a 2 , but obviously in a general case this issue needed to be
revisited in Babanin et al. ( 2010c ).
Computations of the spectral dissipation function (7.50) with such weighting coeffi-
cients, performed for a Combi spectrum (7.46) of moderately-forced waves with U 10 /

= 0 .

c p =

2 . 7 and wind speed U 10 = 10m/s, are presented in Figure 7.12 . The inherent break-

ing term T 1 ( (7.51) are also shown. The

shape of the wave-dissipation source function is the result of superposition of these two
terms in (7.50) .
As implied in (5.40) , the long-scale waves down to the size of dominant waves do not
experience induced dissipation. The contributions of the forced dissipation to the total start
at the spectral peak and then increase towards higher frequencies until it saturates. This
gradual transition is due to the integral of the forced dissipation term T2(f) (7.52) .
Obviously, the high-frequency waves, which have reached the saturated-dissipation
limit, mostly experience forced dissipation due to the influence of longer waves, and their
inherent-breaking dissipation can be neglected. The coefficients a 1 and a 2 are crucial in

f ) (7.51) and the forced-dissipation term T 2 ( f )