Introduction
5
Studies of non-linear wave interaction in a spectrum were made in the
following directions: analytical estimation and parameterisation of the collision integral; elaboration of numerical algorithms and calculation programs;
and solution of the non-linear evolution of the spectrum. It was interesting
from the theoretical point of view to find the stationary forms of the spectra
that made the collision integral equal to zero. K. Hasselmann (1965) showed
that the Rayleigh-Jeans distribution was isotropic, transforming to zero not
only the collision integral, but also the integrand. Other stationary distributions were obtained (Zakharov & Smilga, 1981; Zakharov & Zaslavskii 1982,
1983a,b). Attempts to obtain non-stationary analytical solutions were also
undertaken (Zaslavskii, 1989a,b; 1997, 2000).
In view of the importance of the non-linear energy redistribution mechanism in the wave spectrum and the need to take it into account in wind
wave models for operational use, attempts were undertaken to approximate
the collision integral by simpler analytical ratios than the accurate integral
expression. For example, first attempts at approximating the collision integral can be found in the models of Barnett (1968); Ewing (1971); Theoretical
Bases and Methods for Wind Sea Calculation (1988); S. Hasselmann and
K. Hasselmann (1981). Among the most accurate approximations the results
of Hasselmann et al. (1985); Polnikov (1988); and Zakharov & Pushkarev
(1999), should be mentioned. They give a description of the non-linear energy exchange between wind waves and swell.
Among various results in determining the wave spectrum transformation
in a non-uniform medium, representing a horizontal-non-uniform current and
an uneven seabed, the best were obtained in the framework of the geometrical optics approximation. Wave element evolution, reflection and blocking
in a current were investigated (Basovich & Talanov, 1977; Basovich et al.,
1982; Hurghes & Grant, 1978; Theoretical Bases and Methods for Wind Sea
Calculation, 1988; Phillips, 1980; Pokazeyev & Rosenberg, 1983). A review
of these studies was made by D. Peregrine (1976), G. Kantargi, N. Tsivtsivadze and Kh. Akmuratov (1984). Further research in this field was continued by Dreyzis et al. (1986); Lavrenov (1984, 1985b, 1986, 1988a,b, 1989a,b,
1991a,b); Lavrenov & Ryvkin (1986, 1990); and Lavrenov et al. (1992), where
the frequency-angular spectrum evolution at horizontally non-uniform nonstationary current at a non-uniform depths was described in the geometrical
optics approximation.
Wind wave and ice interaction is one of the problems discussed in this
monograph. It is rather complicated to describe theoretically wind waves in
water partially covered by ice, as the dynamic processes in the "water-air-ice"
three-phase medium in the case of the presence of ice should be taken into
account. Both random wave movements and the probabilistic distribution
of the characteristics of the ice cover make the solution rather complicated.
The theory of this problem remains less researched, although nowadays some
intensive studies have been done in this field (Bukatov & Bukatova, 1993;
Lavrenov & Novakov, 2000; Liu et al., 1991; Marchenko, 1988; Masson &
5
Studies of non-linear wave interaction in a spectrum were made in the
following directions: analytical estimation and parameterisation of the collision integral; elaboration of numerical algorithms and calculation programs;
and solution of the non-linear evolution of the spectrum. It was interesting
from the theoretical point of view to find the stationary forms of the spectra
that made the collision integral equal to zero. K. Hasselmann (1965) showed
that the Rayleigh-Jeans distribution was isotropic, transforming to zero not
only the collision integral, but also the integrand. Other stationary distributions were obtained (Zakharov & Smilga, 1981; Zakharov & Zaslavskii 1982,
1983a,b). Attempts to obtain non-stationary analytical solutions were also
undertaken (Zaslavskii, 1989a,b; 1997, 2000).
In view of the importance of the non-linear energy redistribution mechanism in the wave spectrum and the need to take it into account in wind
wave models for operational use, attempts were undertaken to approximate
the collision integral by simpler analytical ratios than the accurate integral
expression. For example, first attempts at approximating the collision integral can be found in the models of Barnett (1968); Ewing (1971); Theoretical
Bases and Methods for Wind Sea Calculation (1988); S. Hasselmann and
K. Hasselmann (1981). Among the most accurate approximations the results
of Hasselmann et al. (1985); Polnikov (1988); and Zakharov & Pushkarev
(1999), should be mentioned. They give a description of the non-linear energy exchange between wind waves and swell.
Among various results in determining the wave spectrum transformation
in a non-uniform medium, representing a horizontal-non-uniform current and
an uneven seabed, the best were obtained in the framework of the geometrical optics approximation. Wave element evolution, reflection and blocking
in a current were investigated (Basovich & Talanov, 1977; Basovich et al.,
1982; Hurghes & Grant, 1978; Theoretical Bases and Methods for Wind Sea
Calculation, 1988; Phillips, 1980; Pokazeyev & Rosenberg, 1983). A review
of these studies was made by D. Peregrine (1976), G. Kantargi, N. Tsivtsivadze and Kh. Akmuratov (1984). Further research in this field was continued by Dreyzis et al. (1986); Lavrenov (1984, 1985b, 1986, 1988a,b, 1989a,b,
1991a,b); Lavrenov & Ryvkin (1986, 1990); and Lavrenov et al. (1992), where
the frequency-angular spectrum evolution at horizontally non-uniform nonstationary current at a non-uniform depths was described in the geometrical
optics approximation.
Wind wave and ice interaction is one of the problems discussed in this
monograph. It is rather complicated to describe theoretically wind waves in
water partially covered by ice, as the dynamic processes in the "water-air-ice"
three-phase medium in the case of the presence of ice should be taken into
account. Both random wave movements and the probabilistic distribution
of the characteristics of the ice cover make the solution rather complicated.
The theory of this problem remains less researched, although nowadays some
intensive studies have been done in this field (Bukatov & Bukatova, 1993;
Lavrenov & Novakov, 2000; Liu et al., 1991; Marchenko, 1988; Masson &