4.1 Non-Linear Energy Transfer in Wind Wave Spectrum
107
Table 4.1. Values of initial spectrum parameters
I
2s = 2
n13 = 2
n13 = 8
B
Dp
B
Dp
B
Dp
1.0
0.69
0.32
0.69
0.64
0.69
1.16
3.3
0.34
0.32
0.34
0.64
0.34
1.16
7.0
0.25
0.32
0.25
0.64
0.25
1.16
In the process of computation the main parameters are estimated, including the frequency width B, defined as:
B = J S(a) da/S(ap)ap,
(4.25)
1{
where S(a) = J S(a,{3) d{3, and ap is the frequency of the spectrum maxi-n
mum of the time-evaluation spectrum and the directional width D:
D(a) = S(a, {3p)/S(a).
( 4.26)
A series of the initial spectrum parameters"/, nf3, 2s and the initial value B
and D defined in the general direction: Dp = D(ap) = S(ap,{3p)/S(ap) are
shown in the Table 4.1.
Numerical algorithm. The establishment of a self-similar spectral form,
controlled by non-linear energy transfer, demands computation for time scales
larger than the estimation (4.24) at least at one or two orders of magnitude
t = t/T > 10 6 -10 7 . The main problem is that the numerical computation of
the integral ( 4.1) takes a lot of CPU time. This means that an algorithm for
computations of non-linear energy transfer should be very fast. To solve the
evolution equation the integral (4.1) is to be computed thousands of times
to reach the estimated time scale of the spectrum evolution. That is why
the most optimal algorithm is used in the present study, described in the
Sect. 4.1.2.
A numerical solution of the evolution equation (4.1) is carried out using
the semi-implicit method (The WAM Model, 1988; Lavrenov, 1998). It should
be noted that the traditional method of numerical integration usually leads
to numerical instabilities in the high-frequency spectral range. This requires
using a small time step for the numerical integration and applying special
limits on the numerical solution and on the value of the source function
(Lavrenov, 1998; Tolman, 1992). Unfortunately, these methods can disturb
energy conservation, leading to misinterpretation of the physical results. But,
utilization of the semi-implicit numerical method does not produce any instabilities for relatively large time steps of numerical integration (Lavrenov,
1998) and there is no need to use any limits.
107
Table 4.1. Values of initial spectrum parameters
I
2s = 2
n13 = 2
n13 = 8
B
Dp
B
Dp
B
Dp
1.0
0.69
0.32
0.69
0.64
0.69
1.16
3.3
0.34
0.32
0.34
0.64
0.34
1.16
7.0
0.25
0.32
0.25
0.64
0.25
1.16
In the process of computation the main parameters are estimated, including the frequency width B, defined as:
B = J S(a) da/S(ap)ap,
(4.25)
1{
where S(a) = J S(a,{3) d{3, and ap is the frequency of the spectrum maxi-n
mum of the time-evaluation spectrum and the directional width D:
D(a) = S(a, {3p)/S(a).
( 4.26)
A series of the initial spectrum parameters"/, nf3, 2s and the initial value B
and D defined in the general direction: Dp = D(ap) = S(ap,{3p)/S(ap) are
shown in the Table 4.1.
Numerical algorithm. The establishment of a self-similar spectral form,
controlled by non-linear energy transfer, demands computation for time scales
larger than the estimation (4.24) at least at one or two orders of magnitude
t = t/T > 10 6 -10 7 . The main problem is that the numerical computation of
the integral ( 4.1) takes a lot of CPU time. This means that an algorithm for
computations of non-linear energy transfer should be very fast. To solve the
evolution equation the integral (4.1) is to be computed thousands of times
to reach the estimated time scale of the spectrum evolution. That is why
the most optimal algorithm is used in the present study, described in the
Sect. 4.1.2.
A numerical solution of the evolution equation (4.1) is carried out using
the semi-implicit method (The WAM Model, 1988; Lavrenov, 1998). It should
be noted that the traditional method of numerical integration usually leads
to numerical instabilities in the high-frequency spectral range. This requires
using a small time step for the numerical integration and applying special
limits on the numerical solution and on the value of the source function
(Lavrenov, 1998; Tolman, 1992). Unfortunately, these methods can disturb
energy conservation, leading to misinterpretation of the physical results. But,
utilization of the semi-implicit numerical method does not produce any instabilities for relatively large time steps of numerical integration (Lavrenov,
1998) and there is no need to use any limits.