Wave-breaking probability - Breaking and Dissipation of Ocean Surface Waves - page 215

be U 10 / 2. Therefore, at some small scales (low phase speeds, high relative winds)

in the spectrum, the waves theoretically should be breaking very often.
The general pattern in real wave fields, however, is much more complicated, and this
is due to a number of reasons. If the frequencies f 1 and f 2 above represent respective
spectral peaks in the respective spectra (see Figure 5.29 for a typical view of the wave
power spectrum), then the U 10 /

c ( f 2 ) =

( )

wind forcing is applied to dominant waves in both
cases and the values of the forcing, as far as the flux of momentum/energy is concerned,
are unambiguous. If, however, the first frequency is the spectral peak f 1

c f

= f p and the

second frequency f 2 = into the wind forcing as such

is not that straightforward (the mean wind force per unit area is the mean wind stress, i.e.
equals the mean-over-wave-period momentum flux).
Indeed, as has been shown in a number of air-sea interaction studies (e.g. Donelan et al. ,
2006 ; Babanin et al. , 2007b ; Kudryavtsev & Makin , 2007 ), at strong wind forcing and in
the presence of steep dominant waves, particularly if those are breaking, air-flow separation
in the lee of these waves can occur. In this case, if the shorter waves riding the dominant
ones are under the separated air bubble they experience much lower wind stresses com-
pared to what would be expected from the respective values of U 10 /

2 f p , then the translation of U 10 / c ( f )

c ( f ) based on their

phase speed c .

At very strong wind forcing such a relative reduction of the wind stress/input is applica-
ble to the dominant waves themselves. The separated-over-the-crest-of-a-dominant-wave
air flow does not reattach to the surface until close to the crest of the next wave. As a result,
the wind effectively skips the wave troughs and 'does not know' how high/deep the waves
are. Consequently, the wave-induced pressure oscillations in the lower boundary layer are
weakened, and so is the wind input/stress ( Donelan et al. , 2006 ).
Absolute values of the stress always increase as the U 10 /

( f )

goes up, but relative val-
ues of the wind forcing can go down, depending on a combination of wind-speed and
wave-steepness magnitudes, or whether the waves with corresponding frequency f are
the spectral-peak or spectrum-tail waves, whether these waves experience the flow separa-
tion, are in fact under the separated bubble, or the flow remains attached along the wave
profile. The wave breaking occurrence/probability will respond accordingly. Figure 4.2 in
Section 4.1.1 illustrates this effect: the first wind-speed doubling led to the breaking hap-
pening four times faster, whereas subsequent doubling the wind only reduced the distance
to breaking three times.
Another complication of the general pattern in the spectral wave fields is dictated by the
physics depicted in radiative transfer equations (2.61) . There exists a competition between
source terms responsible for the wave evolution in RTE. As the wind grows, some of the
excessive energy/momentum flux provided to short waves by the wind will be transferred
to lower frequencies (faster waves in the spectrum) through nonlinear interactions and will
not contribute to the growth of the wave height/steepness of these short waves.
One way or another, however, but at some stage of the growing wind forcing, neither the
air-flow separation nor the nonlinear interactions seem to be able to digest the amount of
energy input by the wind, and wave breaking (dissipation) across the spectrum suddenly

c ( f )