4.4 Influence of Mesoscale Effects on Wind Wave Evolution
145
According to the experimental data (Andreev, 1988; Zaslavskii & Krasitskii, 1993) it is assumed that the spectrum maximum frequency Wmax is
changed significantly less at a given time interval than the spectrum value
itself.
According to the experimental data the spectrum peakness parameter
"( shows a decreasing tendency as waves are developing (Mitsuyasu et al.,
1980; Donelan et al., 1985; Babanin & Soloviev, 1998). The JONSWAP spectrum transforms to the Pierson~Moskowitz spectrum with"(= 1, in case the
fetch X is sufficiently large. High variability of the spectrum peakness parameter "( is well known according to measurement data. It is so high that
a systematic dependence "f(x) cannot be determined for a wide range of dimensionless fetch range x = Xg/U 2 • That is why the dependence "((x) was
not found in the JONSWAP experiment (Hasselmann et al., 1973).
Now the non-linear spectrum evolution depending on quasi-oscillations in
the spatially homogenous case has to be calculated numerically. Consider the
following equation:
as at = Gni(S) - S h(w) cos(2ntjT),
(4.71)
where h(w) = J-lf(w) ln("f) 2rr/T, and Gni(S) is the non-linear interaction
integral in the wind wave spectrum relative to the spectral energy density
S(w,cp).
It should be noted that in case Gn,(S) is equal to zero, the expression
(4.70) is the solution to equation (4.71). On the other hand if oscillations are
excluded (h(w) = 0) the traditional non-linear spectrum evolution can be
described by (4.71).
Starting with fundamental papers by Hasselman (1962) and Zakharov
(1969), where the right-hand term of (4.71) contains only the non-linear interaction term Gn1(S), (4.71) is traditionally applied in studying the nonlinear wave spectrum evolution. But, unlike the aforementioned papers, the
right-hand term of (4.71) includes an additional component, periodic in time,
not used in the wind wave models. The second component of the right-hand
term ( 4. 71) may be assumed to be an approximation of the wind wave energy input and energy dissipation. But the problems of wave development
under the wind effect and dissipation are not discussed here. An attempt
to describe the spectrum evolution due to non-linear energy transfer and
spectrum quasi-oscilation is undertaken below.
In order to solve the problem correctly the quasi-oscillation period should
be sufficiently large for the waves "to have time" to interact, the latter being
determined by the characteristic time of phase intermixing (Yuen & Lake,
1987):
_!____ ~ w"(k) (~k) 2 ,
Tph
( 4. 72)
where t1k is the wave spectrum width.
145
According to the experimental data (Andreev, 1988; Zaslavskii & Krasitskii, 1993) it is assumed that the spectrum maximum frequency Wmax is
changed significantly less at a given time interval than the spectrum value
itself.
According to the experimental data the spectrum peakness parameter
"( shows a decreasing tendency as waves are developing (Mitsuyasu et al.,
1980; Donelan et al., 1985; Babanin & Soloviev, 1998). The JONSWAP spectrum transforms to the Pierson~Moskowitz spectrum with"(= 1, in case the
fetch X is sufficiently large. High variability of the spectrum peakness parameter "( is well known according to measurement data. It is so high that
a systematic dependence "f(x) cannot be determined for a wide range of dimensionless fetch range x = Xg/U 2 • That is why the dependence "((x) was
not found in the JONSWAP experiment (Hasselmann et al., 1973).
Now the non-linear spectrum evolution depending on quasi-oscillations in
the spatially homogenous case has to be calculated numerically. Consider the
following equation:
as at = Gni(S) - S h(w) cos(2ntjT),
(4.71)
where h(w) = J-lf(w) ln("f) 2rr/T, and Gni(S) is the non-linear interaction
integral in the wind wave spectrum relative to the spectral energy density
S(w,cp).
It should be noted that in case Gn,(S) is equal to zero, the expression
(4.70) is the solution to equation (4.71). On the other hand if oscillations are
excluded (h(w) = 0) the traditional non-linear spectrum evolution can be
described by (4.71).
Starting with fundamental papers by Hasselman (1962) and Zakharov
(1969), where the right-hand term of (4.71) contains only the non-linear interaction term Gn1(S), (4.71) is traditionally applied in studying the nonlinear wave spectrum evolution. But, unlike the aforementioned papers, the
right-hand term of (4.71) includes an additional component, periodic in time,
not used in the wind wave models. The second component of the right-hand
term ( 4. 71) may be assumed to be an approximation of the wind wave energy input and energy dissipation. But the problems of wave development
under the wind effect and dissipation are not discussed here. An attempt
to describe the spectrum evolution due to non-linear energy transfer and
spectrum quasi-oscilation is undertaken below.
In order to solve the problem correctly the quasi-oscillation period should
be sufficiently large for the waves "to have time" to interact, the latter being
determined by the characteristic time of phase intermixing (Yuen & Lake,
1987):
_!____ ~ w"(k) (~k) 2 ,
Tph
( 4. 72)
where t1k is the wave spectrum width.