Non-dissipative effects of breaking on the wave field - Breaking and Dissipation of Ocean Surface Waves - page 330

level has only changed by less than two times (7.30) . That is, the wave-breaking control
still maintains approximately the same tail shape and balances the much grown wind input
(7.31) - (7.32) by adjusting its severity up: that is the extra wind input is lost locally to the
dissipation, but the spectrum tail more or less holds its level.
While talking about the saturation level, it must be mentioned of course that using the
same dimensional argument as Phillips ( 1958 ), but assuming the role of the wind still to be
essential, Toba ( 1973 ) obtained a different parameterisation for the spectrum tail:

ω − 4 F (ω) = β ug , (8.17)

i.e. he argued in favour of a quasi-equilibrium interval, whose level also depends on some
characteristic wind speed u . Well before that, Zakharov & Filonenko ( 1966 ) achieved
the exact analytical solution of the Hasselmann equation for the spectrum defined by
interactions in the system of nonlinear waves, which also produced an

ω − 4 tail:

c K p 1 / 3 g ω − 4 F (ω) = . (8.18)

Here, the spectral level varies as a function of energy flux p across the spectrum, and c K is
the Kolmogorov constant (see Zakharov, 2010).
The observations were divided and produced both f − 5 and f − 4 parameterisations,
e.g. JONSWAP spectrum ( (2.7) , Hasselmann et al. , 1973 ) and the Donelan et al. ( 1985 )
spectrum, respectively. The f − 4 interval required more attention as the initial formulations
(8.17) and (8.18) seemed contradictory: the first one associated the equilibrium with the
external wind forcing and the second one with the internal nonlinear flux. The contradiction
was basically resolved by Resio et al. ( 2004 ) who concluded, based on a comprehensive
collection of field data sets, that

U 10 c p )

1 / 3

u = ( − u 0 (8.19)

where the first term defines the external energy flux. This flux can now be related to the
internal flux p , and the contradiction is answered. Here, u 0 is a dimensional offset of the
experimental dependences, which complicates the theoretical argument, but is apparently
the robust feature of the experimental analysis by Resio et al. ( 2004 ).
In the meantime, based on experimental evidence and theoretical argument, there came
understanding that both the f − 4 and f − 5 subintervals are present at the tail, the first one
being closer to the peak ( Forristall , 1981 ; Evans & Kibblewhite , 1990 ; Kahma & Calkoen ,
1992 ; Babanin & Soloviev , 1998b ; Resio et al. , 2004 ). Moreover, it became clear that
the total wind input, integrated over the entire spectrum F

, cannot converge to the
total wind stress known independently, unless the f − 5 subinterval does exist ( Tsagareli
et al. , 2010 ; Babanin et al. , 2010c ). For the oceanic waves, the transition was observed
at frequency

( f )

)ω p ( Forristall , 1981 ; Evans & Kibblewhite , 1990 ). Kahma
& Calkoen ( 1992 ), based on a 'grand average' of many measurements of the saturation
spectra from different data sets, indicated a dimensional value for such a frequency:

ω t ≈ ( 2 . 5 − 3

g
U 10 .

ω t ≈ 5 (8.20)