Non-dissipative effects of breaking on the wave field - Breaking and Dissipation of Ocean Surface Waves - page 329

We must notice that this is exactly the view of the breaking onset that we have these days.
As discussed in detail in Chapters 4 and 5 , the breaking onset is defined by the limiting
steepness (2.47) , (2.51) and (2.52) , and it is only in extreme conditions that the wind can
reduce this steepness, and even then only marginally.
If so, the original argument of Phillips ( 1958 ) is right and the only relevant dimensional
parameter for this part of the spectrum is g . This led to the frequency spectrum of the
saturated interval

g 2 ω − 5 F (ω) = α (8.15)

and wavenumber spectrum

k − 4 ( k ) = α k (θ) . (8.16)

10 − 3 . Expression (8.16)
defines spectral densities in different directions of wavenumber vector k , and the dimen-
sionless coefficient

Following (8.13) , Phillips ( 1958 ) defined his constant as α = 7 . 4 ·

α k depends on this direction θ

.
Parameterisations (8.15) - (8.16) must have been the first physical and even quantitative
account for wave-breaking effects on the behaviour of the wave system. By limiting the
growth of waves, they define the shape of the spectrum tail and even set its level.
Many more measurements of the equilibrium interval have been conducted since 1958.
The level

α

proved to be not that constant. The early observations of Burling ( 1955 )were
conducted at what we would now classify as relatively light winds, and for stronger winds
even Phillips ( 1977 ) himself reconsidered this value. As shown in Babanin & Soloviev
( 1998a ), indeed remains approximately constant for the best part of the wave-spectrum

development, but for mature waves (which would usually accommodate the light-wind
conditions too) it starts dropping towards the value of Phillips ( 1958 ). This is expression
(5.75) in this topic.
As discussed in Section 5.3.4 , the tail level

α

α responds to the wind forcing if U 10 / c p <

f p ≤

1 23). At such forcing, apparently, the average Phillips' 'geometry of the

limiting shape' of the wave crests can change due to, perhaps, changing rates of the break-
ing occurrence in the high-frequency spectral bands.
Once it reaches, however,

. 45 (i.e. 0 .

45, the level of the f − 5 tail
does not grow up further on average even if the wind forcing increases very significantly
( Figure 5.42 ). As shown in Figure 5.27 , such a situation does not actually mean that the
breaking rates of short waves reach 100%, they stay essentially below this ultimate per-
centage. Therefore, this condition can only signify a change of the breaking severity. Thus,
for the younger waves, the breaking reacts to the changing wind forcing by altering the
dissipation rates rather than the rate of breaking occurrence, in its role of maintaining the
spectrum-tail level ( Figure 6.6 ).
At hurricane-like winds, that is for the very young waves, with the 100% breaking rate of
the short waves occurring, the tail is still roughly at the same level range (see Section 7.3.5
and Figure 7.6 ). This fact further highlights the significant role of breaking in the formation
of the wave spectrum as we know it. Now, the forcing is most extreme, but the spectrum

10 − 3 at U 10 /

α ≈ 13 . 2 · c p ≥ 1 .