4.1 Non-Linear Energy Transfer in Wind Wave Spectrum
83
and total momentum:
K = J kN(k) dk.
(4.3c)
It should be noted that these values remain constant in wave evolution.
In spite of the fact that Hasselmann derived the collision integral ( 4.1) for
the first time in the early 1960s, for a long time it was practically impossible
to obtain reliable estimations of this integral in its exact form. The numerical calculation is rather difficult and requires much computing time. This is,
firstly, due to its six-fold form and, secondly, to the very complicated form
of the core function T(k, k 1 , k 2 , k 3 ). The six-fold integral was transformed
to a three-fold form using the delta-functions (4.2), describing the resonance
conditions (Hasselmann & Hasselmann, 1981; Dungey & Hui, 1985; Fox, 1976;
Webb, 1978). However, these procedures resulted in singularities in the integrand, and this fact caused additional difficulties in computations. There
were two ways of solving this problem: either by substituting the variables
and using "stretched" coordinates (Hasselmann&Hasselmann, 1981), or by
estimating the contribution of the singularity area (Masuda, 1981). However, accurate integral computation remained quite a difficult problem. As
a result, some authors (Fox 1976; Longuet-Higgins, 1976; Resio, 1981; Webb,
1978) proposed simplified integral approximations for a narrow spectrum.
Webb (1978), Hasselmann and Hasselmann, (1981) and Masuda (1981)
were among the first who overcame the numerical computation difficulties of
the exact integral expression.
Hasselmann and Hasselmann (1981) proposed an integral calculation
method using the symmetry of its expression. This significantly increased
the calculation rate. In order to optimize the computation they proceeded
from the asymmetric expression (4.1) describing the energy change of the
wave component k (as an interaction result with other components k 1 , k 2
and k 3 ) to the detailed balance property description so that the maximum
symmetry was used.
Subsequent studies (Davidan et al., 1985; Zakharov, 1968; Zakharov & Zaslavskii, 1983a,b; Hasselmann et al., 1973; Komen et al., 1994; Masuda, 1981;
Ocean Wave Modeling, 1985) showed the importance of taking into account
non-linear interaction in the wind wave spectrum and its role in the spectral
maximum shift into the low frequency range with wave evolution.
The necessity for fast calculations of the non-linear interaction integral
in operational wind wave models resulted in different simplified approximations. The most successful was the so-called "discrete interaction approximation" (DIA), suggested by Hasselmann, Hasselmann and Barnett (1985).
This mainly used the symmetry of the integral expression. In spite of simplifications, it retained some principal properties of the initial integral. Later
on, this approximation was used in the WAM model (Komen et al., 1994).
The approximation was written in the following form:
83
and total momentum:
K = J kN(k) dk.
(4.3c)
It should be noted that these values remain constant in wave evolution.
In spite of the fact that Hasselmann derived the collision integral ( 4.1) for
the first time in the early 1960s, for a long time it was practically impossible
to obtain reliable estimations of this integral in its exact form. The numerical calculation is rather difficult and requires much computing time. This is,
firstly, due to its six-fold form and, secondly, to the very complicated form
of the core function T(k, k 1 , k 2 , k 3 ). The six-fold integral was transformed
to a three-fold form using the delta-functions (4.2), describing the resonance
conditions (Hasselmann & Hasselmann, 1981; Dungey & Hui, 1985; Fox, 1976;
Webb, 1978). However, these procedures resulted in singularities in the integrand, and this fact caused additional difficulties in computations. There
were two ways of solving this problem: either by substituting the variables
and using "stretched" coordinates (Hasselmann&Hasselmann, 1981), or by
estimating the contribution of the singularity area (Masuda, 1981). However, accurate integral computation remained quite a difficult problem. As
a result, some authors (Fox 1976; Longuet-Higgins, 1976; Resio, 1981; Webb,
1978) proposed simplified integral approximations for a narrow spectrum.
Webb (1978), Hasselmann and Hasselmann, (1981) and Masuda (1981)
were among the first who overcame the numerical computation difficulties of
the exact integral expression.
Hasselmann and Hasselmann (1981) proposed an integral calculation
method using the symmetry of its expression. This significantly increased
the calculation rate. In order to optimize the computation they proceeded
from the asymmetric expression (4.1) describing the energy change of the
wave component k (as an interaction result with other components k 1 , k 2
and k 3 ) to the detailed balance property description so that the maximum
symmetry was used.
Subsequent studies (Davidan et al., 1985; Zakharov, 1968; Zakharov & Zaslavskii, 1983a,b; Hasselmann et al., 1973; Komen et al., 1994; Masuda, 1981;
Ocean Wave Modeling, 1985) showed the importance of taking into account
non-linear interaction in the wind wave spectrum and its role in the spectral
maximum shift into the low frequency range with wave evolution.
The necessity for fast calculations of the non-linear interaction integral
in operational wind wave models resulted in different simplified approximations. The most successful was the so-called "discrete interaction approximation" (DIA), suggested by Hasselmann, Hasselmann and Barnett (1985).
This mainly used the symmetry of the integral expression. In spite of simplifications, it retained some principal properties of the initial integral. Later
on, this approximation was used in the WAM model (Komen et al., 1994).
The approximation was written in the following form: