4.1 Non-Linear Energy Transfer in Wind Wave Spectrum
105
of calculating the collision integral. The problem was not only in the complexity of the traditional calculation of the non-linear interaction integral,
but in the fact that its real contribution to this component was considerably
less than the corresponding value of the wave components directed along
the wind. The numerical methods for the calculation of the collision integral
must be sufficiently accurate to distinguish this effect from the numerical error background, which could be quite considerable. Asshown in this section,
such calculations have become possible nowadays only by using the accurate
algorithm presented in this monograph.
4.1.4 Numerical Study of Non-Stationary Solution
of the Hasselmann Equation
In spite of the fact that the main integral features have been investigated (Hasselmann & Hasselmann 1981, 1985; Komen et al., 1994; Komatsu
& Masuda, 1996; Masuda, 1981; Lavrenov, 1991a,b, 1998, 2000; Lavrenov
& Ocampo-Torres, 1999; Polnikov, 1989, 1990, 1993; Resio & Perrie, 1991;
Snyder et al., 1993; Webb, 1978 etc.), not enough attention was paid to the
problem of the analytical and numeric study of the kinetic equation ( 4.1).
Among the stationary analytical solutions, the following one is known as
"thermodynamic":
N= (a+bw+ck)- 1 ,
(4.20)
where a, band care constants. The solution (4.20) reduces the integral value
to zero. However, the solution (4.20) is of no great physical importance, because a small disturbance produces a divergence of the integral (4.1).
Investigation of the solution to (4.1) remains an important and difficult
problem. The main progress in its analytical study was achieved by Zakharov
et al. (1966, 1981, 1982, 1983a,b). The solution of the stationary analytical
spectra of (4.1) is derived for the isotropic angle case and infinite frequency
range [O,oo]. Analytical methods of solution were developed further by Zaslavskii (1989a,b, 2000), based on "the narrow directional approximation".
According to Zaslavskii (2000), the evolution equation ( 4.1) can be presented
as self-similar solutions. The spectral frequency dependence is found to be
equal to S(a) "' a- 13 1 2 (for a > ap, where ap is the peak spectrum frequency), the evolution of the spectral frequency maximum is estimated as
ap "'r 1 1 11 , the angular narrowness in the vicinity of the spectral maximum
is Dp ~ 1. But, unfortunately, the confidence equations used by Zaslavskii
(2000) do not provide a complete study of the whole evolution equation. His
self-similar spectrum approximation does not satisfy the energy conservation
law.
There are not so many papers devoted to the problem of numerical simulation of the evolution equation (4.1). In the papers by Polnikov (1990, 1999)
it is shown that the spectral form does not depend on the initial spectrum and
105
of calculating the collision integral. The problem was not only in the complexity of the traditional calculation of the non-linear interaction integral,
but in the fact that its real contribution to this component was considerably
less than the corresponding value of the wave components directed along
the wind. The numerical methods for the calculation of the collision integral
must be sufficiently accurate to distinguish this effect from the numerical error background, which could be quite considerable. Asshown in this section,
such calculations have become possible nowadays only by using the accurate
algorithm presented in this monograph.
4.1.4 Numerical Study of Non-Stationary Solution
of the Hasselmann Equation
In spite of the fact that the main integral features have been investigated (Hasselmann & Hasselmann 1981, 1985; Komen et al., 1994; Komatsu
& Masuda, 1996; Masuda, 1981; Lavrenov, 1991a,b, 1998, 2000; Lavrenov
& Ocampo-Torres, 1999; Polnikov, 1989, 1990, 1993; Resio & Perrie, 1991;
Snyder et al., 1993; Webb, 1978 etc.), not enough attention was paid to the
problem of the analytical and numeric study of the kinetic equation ( 4.1).
Among the stationary analytical solutions, the following one is known as
"thermodynamic":
N= (a+bw+ck)- 1 ,
(4.20)
where a, band care constants. The solution (4.20) reduces the integral value
to zero. However, the solution (4.20) is of no great physical importance, because a small disturbance produces a divergence of the integral (4.1).
Investigation of the solution to (4.1) remains an important and difficult
problem. The main progress in its analytical study was achieved by Zakharov
et al. (1966, 1981, 1982, 1983a,b). The solution of the stationary analytical
spectra of (4.1) is derived for the isotropic angle case and infinite frequency
range [O,oo]. Analytical methods of solution were developed further by Zaslavskii (1989a,b, 2000), based on "the narrow directional approximation".
According to Zaslavskii (2000), the evolution equation ( 4.1) can be presented
as self-similar solutions. The spectral frequency dependence is found to be
equal to S(a) "' a- 13 1 2 (for a > ap, where ap is the peak spectrum frequency), the evolution of the spectral frequency maximum is estimated as
ap "'r 1 1 11 , the angular narrowness in the vicinity of the spectral maximum
is Dp ~ 1. But, unfortunately, the confidence equations used by Zaslavskii
(2000) do not provide a complete study of the whole evolution equation. His
self-similar spectrum approximation does not satisfy the energy conservation
law.
There are not so many papers devoted to the problem of numerical simulation of the evolution equation (4.1). In the papers by Polnikov (1990, 1999)
it is shown that the spectral form does not depend on the initial spectrum and