108
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
The numerical accuracy and the preservation of the main integrals ( 4.3)
are controlled in the process of computation. So, the numerical error of the
total energy conservation does not exceed 10 per cent and the error of the
total momentum estimation does not exceed 25 per cent in comparison to its
initial values.
Numerical results.
Numerical results for the initial condition (4.12)(4.14) with r = 3.3 and n = 2 for four different time moments: i ~ 0, 10 3 ,
10 5 , 10 7 , respectively, are presented in Figs. 4.11-4.14. Numerical simulation
results of the relative frequency-angular spectrum values in a polar coordinate
system, where the radius vector is the relative frequency value a= afap, are
presented in Fig. 4.lla-d. The spectrum time evolution, resulting in a moonlike form, is shown in the figures. The spectrum becomes narrower in the
vicinity of the spectral maximum and wider in the high and low frequency
range.
Two-dimensional non-linear transfer values for the same steps of wave
evolution are presented in Fig. 4.12a-d. The non-linear evolution changes the
spectrum form and shifts the frequency of the spectrum maximum to lower
frequencies. The non-linear energy transfer function is changed to a larger
extent in comparison with the spectrum. It becomes narrower with the concentration intensity in the vicinity of the spectral maximum.
The frequency spectrum for the same evolution time steps is presented in
Fig. 4.13. The spectrum is proportional to,....., a 16 · 3 in the low-frequency range
ap > a. It is proportional to ,....., a- 6 · 1 within the range ap < a < 1.5ap. The
spectrum decreases as , . . . . . , a- 2 · 6 at the larger frequencies a > 1.5ap.
The evolution of the parameter D ( 4.26) for different time moments is
presented in Fig. 4.14. The value of the parameter D becomes larger and
equals 1.33 in the vicinity of the spectral maximum, and it gets smaller in
the low and high frequency ranges. It is evidence of spectral isotropization
within these frequency ranges.
Similar results for the initial frequency-angular spectrum (4.13) and (4.15)
with r = 3.3, 2s = 2 for the following evolution steps i ~ 0, 10 4 , 10 6 ,
10 7 are presented in Figs. 4.15-4.18. It is interesting to note that at times
i ~ 10 4 the frequency-angular spectrum (see Fig. 4.15) varies in such a way
that it becomes more isotropic as mentioned by Lavrenov & Ocampo-Torres
(1999). It also becomes smoother and rounder. But later on (at i ~ 10 5 ) it
is transformed into a more directional form. The two-dimensional spectrum
could not become isotropic, as follows from the total momentum conservation
law (4.3c). So, the previous suggestion about the spectrum becoming isotropic
with the initial angular approximation (4.14) is not proved. The spectrum
becomes narrower in the vicinity of its maximum and it is wider at low and
high frequencies.
An important feature of the non-linear energy transfer (see Fig. 4.16)
is the presence of an area with positive values in the direction opposite to
the main direction of the wave propagation. Due to the action of the non-
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
The numerical accuracy and the preservation of the main integrals ( 4.3)
are controlled in the process of computation. So, the numerical error of the
total energy conservation does not exceed 10 per cent and the error of the
total momentum estimation does not exceed 25 per cent in comparison to its
initial values.
Numerical results.
Numerical results for the initial condition (4.12)(4.14) with r = 3.3 and n = 2 for four different time moments: i ~ 0, 10 3 ,
10 5 , 10 7 , respectively, are presented in Figs. 4.11-4.14. Numerical simulation
results of the relative frequency-angular spectrum values in a polar coordinate
system, where the radius vector is the relative frequency value a= afap, are
presented in Fig. 4.lla-d. The spectrum time evolution, resulting in a moonlike form, is shown in the figures. The spectrum becomes narrower in the
vicinity of the spectral maximum and wider in the high and low frequency
range.
Two-dimensional non-linear transfer values for the same steps of wave
evolution are presented in Fig. 4.12a-d. The non-linear evolution changes the
spectrum form and shifts the frequency of the spectrum maximum to lower
frequencies. The non-linear energy transfer function is changed to a larger
extent in comparison with the spectrum. It becomes narrower with the concentration intensity in the vicinity of the spectral maximum.
The frequency spectrum for the same evolution time steps is presented in
Fig. 4.13. The spectrum is proportional to,....., a 16 · 3 in the low-frequency range
ap > a. It is proportional to ,....., a- 6 · 1 within the range ap < a < 1.5ap. The
spectrum decreases as , . . . . . , a- 2 · 6 at the larger frequencies a > 1.5ap.
The evolution of the parameter D ( 4.26) for different time moments is
presented in Fig. 4.14. The value of the parameter D becomes larger and
equals 1.33 in the vicinity of the spectral maximum, and it gets smaller in
the low and high frequency ranges. It is evidence of spectral isotropization
within these frequency ranges.
Similar results for the initial frequency-angular spectrum (4.13) and (4.15)
with r = 3.3, 2s = 2 for the following evolution steps i ~ 0, 10 4 , 10 6 ,
10 7 are presented in Figs. 4.15-4.18. It is interesting to note that at times
i ~ 10 4 the frequency-angular spectrum (see Fig. 4.15) varies in such a way
that it becomes more isotropic as mentioned by Lavrenov & Ocampo-Torres
(1999). It also becomes smoother and rounder. But later on (at i ~ 10 5 ) it
is transformed into a more directional form. The two-dimensional spectrum
could not become isotropic, as follows from the total momentum conservation
law (4.3c). So, the previous suggestion about the spectrum becoming isotropic
with the initial angular approximation (4.14) is not proved. The spectrum
becomes narrower in the vicinity of its maximum and it is wider at low and
high frequencies.
An important feature of the non-linear energy transfer (see Fig. 4.16)
is the presence of an area with positive values in the direction opposite to
the main direction of the wave propagation. Due to the action of the non-