106
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
the integral spectral parameters vary within narrow bands for a large time
scale of the non-linear evolution. The effect is considered to establish one of
the self-similar spectral forms and is confirmed by the analytical estimations
(Zaslaskii, 2000).
The problem of a self-similar solution seems to have been solved. But in
the paper by Komatsu & Masuda (1996) the numerical solutions differ from
the results of Polnikov (1990) and Zaslavskii (2000). The difference is not
only in another spectral tail frequency dependence, but also in the integral
parameters. Thus, the numerical results lead to the following approximation
S(a) rv a- 4 (for a > 1.5ap). The problem remains unsolved and can be
formulated this way: is there an exact self-similar form of the spectrum? And
then: if the answer is positive, what are the values of its parameters?
An attempt is undertaken to solve the problem in this section.
Time scale of establishing a self-similar solution. The time scale T
of the spectral non-linear evolution, on which the non-linear energy transfer
exchanges fully a spectral form, is considered in this chapter. The spectrum
should not depend on the initial details. It is possible to estimate the time
scale with the help of the conservation law of the total wave action:
aN(k)
.
~ + d1vkF n = 0,
( 4.21)
where F n is a vector of the action flux. According to Polnikov (1999) the value
T can be estimated with the help of the ratio of the spectral maximum NP
to the maximum of the non-linear transfer (aN I at) p:
( 4.22)
In order to get the estimation T one can transfer from the wave action spectrum N(k) to the frequency-angular spectrum S(a,/3), using the formula:
1
g3
N(k) dk ex -;;S(k) dk ex 2 0' 4 S(a, /3) da d/3,
( 4.23)
where a 2 = gk is the deep-water dispersion relation. Application of the spectrum S( a, /3) gives a reliable estimation so long as its form is well known. For
the typical values O'p = 1 rads-1, Smax = 0.2 m 2 s it is possible to obtain:
(4.24)
where T = 2njap is the period of the initial spectral maximum.
Thus, a non-linear evolution establishing a self-similar spectral form happens at a time scale larger than the estimation ( 4.24) at least at one or two
orders of magnitude: t = tjT >> TjT.
Non-linear evolution of JONSWAP spectrum. The non-linear energy
spectrum evolution is computed for the initial frequency-angular JONSWAP
spectrum (4.12)-(4.16).
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
the integral spectral parameters vary within narrow bands for a large time
scale of the non-linear evolution. The effect is considered to establish one of
the self-similar spectral forms and is confirmed by the analytical estimations
(Zaslaskii, 2000).
The problem of a self-similar solution seems to have been solved. But in
the paper by Komatsu & Masuda (1996) the numerical solutions differ from
the results of Polnikov (1990) and Zaslavskii (2000). The difference is not
only in another spectral tail frequency dependence, but also in the integral
parameters. Thus, the numerical results lead to the following approximation
S(a) rv a- 4 (for a > 1.5ap). The problem remains unsolved and can be
formulated this way: is there an exact self-similar form of the spectrum? And
then: if the answer is positive, what are the values of its parameters?
An attempt is undertaken to solve the problem in this section.
Time scale of establishing a self-similar solution. The time scale T
of the spectral non-linear evolution, on which the non-linear energy transfer
exchanges fully a spectral form, is considered in this chapter. The spectrum
should not depend on the initial details. It is possible to estimate the time
scale with the help of the conservation law of the total wave action:
aN(k)
.
~ + d1vkF n = 0,
( 4.21)
where F n is a vector of the action flux. According to Polnikov (1999) the value
T can be estimated with the help of the ratio of the spectral maximum NP
to the maximum of the non-linear transfer (aN I at) p:
( 4.22)
In order to get the estimation T one can transfer from the wave action spectrum N(k) to the frequency-angular spectrum S(a,/3), using the formula:
1
g3
N(k) dk ex -;;S(k) dk ex 2 0' 4 S(a, /3) da d/3,
( 4.23)
where a 2 = gk is the deep-water dispersion relation. Application of the spectrum S( a, /3) gives a reliable estimation so long as its form is well known. For
the typical values O'p = 1 rads-1, Smax = 0.2 m 2 s it is possible to obtain:
(4.24)
where T = 2njap is the period of the initial spectral maximum.
Thus, a non-linear evolution establishing a self-similar spectral form happens at a time scale larger than the estimation ( 4.24) at least at one or two
orders of magnitude: t = tjT >> TjT.
Non-linear evolution of JONSWAP spectrum. The non-linear energy
spectrum evolution is computed for the initial frequency-angular JONSWAP
spectrum (4.12)-(4.16).