4.4 Influence of Mesoscale Effects on Wind Wave Evolution
149
uously changed at the next time interval. This cannot be explained by the
rough frequency discretization used in the numerical scheme. The frequency
variation of Wmax is much larger than the frequency discretization step. All
in all the evolution of the frequency Wmax is not smooth. The discontinuous
character of the non-linear evolution of the spectrum maximum frequency
reveals that quasi-oscillations can serve as a "starting mechanism" for the
low-frequency spectrum evolution. It occurs only at a specific moment, but
not at every quasi-oscillation period. It takes place in cases when the nonlinear spectrum evolution is accumulated and it changes the spectrum form
in such a way that it is sufficient to make a small push (increasing the spectral density with quasi-oscillations). Thus, the spectral maximum frequency
is transferred to another level.
It should be noted that the value Wmax is not changed with "( = 1 (i.e. the
P-M spectrum) in the case of "oscillation absence".
Non-linear energy transfer causes an average spectrum shift to the lowfrequency range. However, the mean evolution rate of the frequency maximum depends significantly on the given oscillation spectrum peakness.
Thus, in the "oscillation absence" case the spectrum maximum frequency
is monotonically decreased from the initial value of w~ax = 1.88 rad s- 1 to
Wmax = 1.75 rad s- 1 at t = 10 4 s, to Wmax = 1.60 rad s- 1 at t = 3 X 10 4 s,
and to Wmax = 1.47 rad s- 1 at t = 10 5 s. Oscillations result in a faster decrease of the spectrum maximum frequency. The oscillation period (for the
used values) does not significantly influence the general tendency. In the presence of oscillations the spectrum maximum frequency is decreased from its
initial value to Wmax = 1.60 rads- 1 at t = 10 4 s, to Wmax = 1.43 rad s- 1 at
t = 3 X 10 4 s and to Wmax = 1.30 rads- 1 at t = 10 5 s.
In view of the fact that the spectrum maximum frequency is the most
conservative parameter, the comparison of these results indicates a considerable influence of the oscillations on the non-linear spectrum evolution rate. In
this case the average rate of the spectral maximum displacement is increased
more than three-fold. As shown, the average rate of the spectral maximum
displacement depends directly on the quasi-oscillation amplitude of the spectral maximum.
Parameterization of the influence of quasi-oscillation on non-linear
energy transfer.
The approximated frequency-angular spectrum averaged by a period of the order T4 is produced with the help of wind wave mathematical models. In this case the spectrum variations occurring at smaller time
periods T3 are not taken into consideration. There appears the problem of taking into account adequately the quasi-oscillation effect and parameterization
of its corresponding contribution to source functions in the mathematical
models describing the formation of the wind wave spectrum.
It is important to point out that wind wave field fluctuations, observed in
the quasi-stationary state interval, result in significant changes of the spectrum form, its peakness and wave steepness. That is why local variations of
149
uously changed at the next time interval. This cannot be explained by the
rough frequency discretization used in the numerical scheme. The frequency
variation of Wmax is much larger than the frequency discretization step. All
in all the evolution of the frequency Wmax is not smooth. The discontinuous
character of the non-linear evolution of the spectrum maximum frequency
reveals that quasi-oscillations can serve as a "starting mechanism" for the
low-frequency spectrum evolution. It occurs only at a specific moment, but
not at every quasi-oscillation period. It takes place in cases when the nonlinear spectrum evolution is accumulated and it changes the spectrum form
in such a way that it is sufficient to make a small push (increasing the spectral density with quasi-oscillations). Thus, the spectral maximum frequency
is transferred to another level.
It should be noted that the value Wmax is not changed with "( = 1 (i.e. the
P-M spectrum) in the case of "oscillation absence".
Non-linear energy transfer causes an average spectrum shift to the lowfrequency range. However, the mean evolution rate of the frequency maximum depends significantly on the given oscillation spectrum peakness.
Thus, in the "oscillation absence" case the spectrum maximum frequency
is monotonically decreased from the initial value of w~ax = 1.88 rad s- 1 to
Wmax = 1.75 rad s- 1 at t = 10 4 s, to Wmax = 1.60 rad s- 1 at t = 3 X 10 4 s,
and to Wmax = 1.47 rad s- 1 at t = 10 5 s. Oscillations result in a faster decrease of the spectrum maximum frequency. The oscillation period (for the
used values) does not significantly influence the general tendency. In the presence of oscillations the spectrum maximum frequency is decreased from its
initial value to Wmax = 1.60 rads- 1 at t = 10 4 s, to Wmax = 1.43 rad s- 1 at
t = 3 X 10 4 s and to Wmax = 1.30 rads- 1 at t = 10 5 s.
In view of the fact that the spectrum maximum frequency is the most
conservative parameter, the comparison of these results indicates a considerable influence of the oscillations on the non-linear spectrum evolution rate. In
this case the average rate of the spectral maximum displacement is increased
more than three-fold. As shown, the average rate of the spectral maximum
displacement depends directly on the quasi-oscillation amplitude of the spectral maximum.
Parameterization of the influence of quasi-oscillation on non-linear
energy transfer.
The approximated frequency-angular spectrum averaged by a period of the order T4 is produced with the help of wind wave mathematical models. In this case the spectrum variations occurring at smaller time
periods T3 are not taken into consideration. There appears the problem of taking into account adequately the quasi-oscillation effect and parameterization
of its corresponding contribution to source functions in the mathematical
models describing the formation of the wind wave spectrum.
It is important to point out that wind wave field fluctuations, observed in
the quasi-stationary state interval, result in significant changes of the spectrum form, its peakness and wave steepness. That is why local variations of