146
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
The value Tph appears to be equal to several spectrum maximum periods.
The condition of applicability of the method can be written as:
(4.73)
It should be noted that in this case the quasi-oscillation period can be
smaller or of the same order with the characteristic time of non-linear wave
evolution TnJ rv T. Using (4.1), the value TnJ can be estimated as (Yuen & Lake
1987):
1
TN 2
TnJ ~ w"(k) (~k) 2 '
(4.74)
Algorithm for numerical solution of (4.71). The equation (4.71), with
its right-hand side term taking into account non-linear energy transfer and an
additional term leading to spectrum oscillation is now solved numerically. It
is necessary to ensure that the temporal step ~t of the numerical integration
is much smaller than the oscillation period T. In this case, the spectral density
quasi-oscillation can be accurately taken into account in the numerical solution. The period T should be less than the specified time of the non-linear
spectrum evolution ~t « T < T 4 . It is necessary to solve (4.71) numerically
for the large time scale ( TrT5 ) using a sufficiently small time step ~t. This
is a rather difficult problem because the numerical computation of the nonlinear energy transfer requires considerable CPU time. This means that the
algorithm has to be efficient. That is why the algorithm, described earlier
in Sect. 4.1, is applied to solve this problem. Using a small number of grid
points the collision integral can be quite accurately calculated, taking little
CPU time.
The equation ( 4. 71) is solved in this case with the help of the two-step
predictor-corrector method similar to that used by Lavrenov (1991b), allowing us to obtain an accurate numerical solution. The numerical integration
time step is equal to 12.5 s, which makes it possible to conduct numerical
integration up to the 10 5 s time interval.
Numerical modelling results. In numerical computation the value of
the parameter JL is assumed to be equal to 0.9, which provides periodic oscillations of spectrum enhancement (4.70) ranging from 1.1 to 9.7 with mean
value "f = 3.3. The oscillation periods are 10, 20 and 40 min respectively. The
results of calculations of the frequency spectra for JL = 0.9 at different moments t (for different values of spectrum peakness .:Y) are shown in Fig. 4.24a.
The spectral density values are normalized by the maximum spectrum value
at .:Y = 3.3.
The non-linear transfer function normalized by its maximum value at
.:Y = 3.3 is shown for different .:Y in Fig. 4.24b. Relative values of the non-linear
transfer function differ significantly compared to the corresponding spectral
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
The value Tph appears to be equal to several spectrum maximum periods.
The condition of applicability of the method can be written as:
(4.73)
It should be noted that in this case the quasi-oscillation period can be
smaller or of the same order with the characteristic time of non-linear wave
evolution TnJ rv T. Using (4.1), the value TnJ can be estimated as (Yuen & Lake
1987):
1
TN 2
TnJ ~ w"(k) (~k) 2 '
(4.74)
Algorithm for numerical solution of (4.71). The equation (4.71), with
its right-hand side term taking into account non-linear energy transfer and an
additional term leading to spectrum oscillation is now solved numerically. It
is necessary to ensure that the temporal step ~t of the numerical integration
is much smaller than the oscillation period T. In this case, the spectral density
quasi-oscillation can be accurately taken into account in the numerical solution. The period T should be less than the specified time of the non-linear
spectrum evolution ~t « T < T 4 . It is necessary to solve (4.71) numerically
for the large time scale ( TrT5 ) using a sufficiently small time step ~t. This
is a rather difficult problem because the numerical computation of the nonlinear energy transfer requires considerable CPU time. This means that the
algorithm has to be efficient. That is why the algorithm, described earlier
in Sect. 4.1, is applied to solve this problem. Using a small number of grid
points the collision integral can be quite accurately calculated, taking little
CPU time.
The equation ( 4. 71) is solved in this case with the help of the two-step
predictor-corrector method similar to that used by Lavrenov (1991b), allowing us to obtain an accurate numerical solution. The numerical integration
time step is equal to 12.5 s, which makes it possible to conduct numerical
integration up to the 10 5 s time interval.
Numerical modelling results. In numerical computation the value of
the parameter JL is assumed to be equal to 0.9, which provides periodic oscillations of spectrum enhancement (4.70) ranging from 1.1 to 9.7 with mean
value "f = 3.3. The oscillation periods are 10, 20 and 40 min respectively. The
results of calculations of the frequency spectra for JL = 0.9 at different moments t (for different values of spectrum peakness .:Y) are shown in Fig. 4.24a.
The spectral density values are normalized by the maximum spectrum value
at .:Y = 3.3.
The non-linear transfer function normalized by its maximum value at
.:Y = 3.3 is shown for different .:Y in Fig. 4.24b. Relative values of the non-linear
transfer function differ significantly compared to the corresponding spectral