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 the spectrum to some degree. For a stable equilibrium to exist, the sum of all three
sources must equal zero. In the equilibrium range, if the sum of the wind and wave
breaking source terms are not zero, Snl, which is equal to the divergence in the nonlinear
energy fluxes, must be non-zero to maintain a net zero balance. In other words, we
would have
ΓE
= Sin + Sds
(2)
ω
where ΓE is the flux of energy through the spectrum. If wind input and wave breaking
exactly cancel each other in this range, equation 2 would still hold. It would only mean
that the divergence would be identically zero over this range and the fluxes should be
exactly constant.
Nonlinear energy fluxes transfer energy from low to high frequencies and from
high to low frequencies simultaneously. In some previous papers, the term flux has been
used to describe the positive-directed fluxes (fluxes from low to high frequency), while
the negative-directed fluxes (fluxes from high to low frequency) are termed inverse
fluxes; however, this terminology will not be adopted here. We shall take the meaning of
ΓE to be the net difference of the fluxes in both directions, i.e.
+
-
ΓE = ΓE + ΓE
(3)
where the superscripts "+" and "-" denote fluxes toward higher and lower frequencies,
respectively. According to essentially all recent theoretical treatments of Snl, nonlinear
form in the absence of other significant source terms. However, as will be shown here,
even for typical wind inputs, the deviation from an ω -4 spectral form may be quite small.
Within the equilibrium range, the flux of energy from lower to higher frequencies is
given by (Resio et al., 2001) in a form equivalent to
Γ+ = Cnl g β 3
(4)
E
where Cnl is a dimensionless coefficient depending weakly on angular spreading and
proximity to the spectral peak and β is a wave steepness parameter equivalent to
α4u/(2g1/2) in equation 2. To a first approximation, the divergence of the flux can be
written as
Γ+
ξ3
E
~
(5)
ω
ω
where ξ is a compensated spectral density of the form
2F( k) g1/2 k5 / 2
ξ=
(6)
α 4u
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