6.2 Weak Non-Linear Wave Interaction in Shallow Water
261
It is obvious that there is no need for the isobath to be rectilinear for turning rays. Entrapment of waves near the shore can occur at any frequency, for
which the corresponding conditions exist for the appearance of a caustic at
some distance from the shore. The waves can be trapped in a similar way
in the absence of a shore by submerged topographical bottom features, for
example, by underwater ridges under conditions that caustics exist on both
sides of this feature. A number of similar cases for geophysical waves of a different nature is given in the papers by Dobrokhotov & Zhevandrov, 1988a,b;
Le Blond & Maysek, 1981; Rabinovich, 1993. They describe the entrapment,
resonance and radiation of waves in the shelf ocean zone, resulting in the
appearance of seiche oscillations in gulfs, bays and harbours.
6.2 Weak Non-Linear Wave Interaction
in Shallow Water
Statement of the problem.
Weak non-linear interaction of waves is
one of the main physical mechanisms determining wind wave formation and
evolution. That is why the estimation of its effect on the wave spectrum
transformation in shallow water is of some definite interest. The numerical
calculation of the collision integral describing weak non-linear energy transfer
is very complicated and demands much computing time (see Sect. 4.1) even
for the condition of infinitely deep water. For finite depth water, calculation
of the collision integral is even more complicated.
At the same time, it was shown (Hasselmann&Hasselmann, 1981) that
the effectiveness of non-linear wave interaction is increased with diminishing water depth. The applicability of the weak turbulent description of this
mechanism is disturbed at sufficiently small depths. The wave movements in
a free water surface are usually described within the framework of non-linear
dispersion equations of Boussinesq or Korteveg de Vriz, whose studies are
often quoted in scientific papers. In this connection the study of non-linear
evolution of the wave spectrum in the intermediate case is of some interest.
This means that, on the one hand, the water is not deep enough for the considered wavelengths and, on the other, it is not so shallow as to disturb the
applicability of the weak turbulent theory. A similar situation can be seen in
shallow water or in the case of wave propagation from deep sea to a coastal
area.
Formulation of the problem. In the spatially uniform case, with the
right-hand side of the kinetic equation (5.1), describing the non-linear interaction, the formulation of the spectrum evolution problem can be presented
in the form:
dN
dt = Gnl·
(6.17)
261
It is obvious that there is no need for the isobath to be rectilinear for turning rays. Entrapment of waves near the shore can occur at any frequency, for
which the corresponding conditions exist for the appearance of a caustic at
some distance from the shore. The waves can be trapped in a similar way
in the absence of a shore by submerged topographical bottom features, for
example, by underwater ridges under conditions that caustics exist on both
sides of this feature. A number of similar cases for geophysical waves of a different nature is given in the papers by Dobrokhotov & Zhevandrov, 1988a,b;
Le Blond & Maysek, 1981; Rabinovich, 1993. They describe the entrapment,
resonance and radiation of waves in the shelf ocean zone, resulting in the
appearance of seiche oscillations in gulfs, bays and harbours.
6.2 Weak Non-Linear Wave Interaction
in Shallow Water
Statement of the problem.
Weak non-linear interaction of waves is
one of the main physical mechanisms determining wind wave formation and
evolution. That is why the estimation of its effect on the wave spectrum
transformation in shallow water is of some definite interest. The numerical
calculation of the collision integral describing weak non-linear energy transfer
is very complicated and demands much computing time (see Sect. 4.1) even
for the condition of infinitely deep water. For finite depth water, calculation
of the collision integral is even more complicated.
At the same time, it was shown (Hasselmann&Hasselmann, 1981) that
the effectiveness of non-linear wave interaction is increased with diminishing water depth. The applicability of the weak turbulent description of this
mechanism is disturbed at sufficiently small depths. The wave movements in
a free water surface are usually described within the framework of non-linear
dispersion equations of Boussinesq or Korteveg de Vriz, whose studies are
often quoted in scientific papers. In this connection the study of non-linear
evolution of the wave spectrum in the intermediate case is of some interest.
This means that, on the one hand, the water is not deep enough for the considered wavelengths and, on the other, it is not so shallow as to disturb the
applicability of the weak turbulent theory. A similar situation can be seen in
shallow water or in the case of wave propagation from deep sea to a coastal
area.
Formulation of the problem. In the spatially uniform case, with the
right-hand side of the kinetic equation (5.1), describing the non-linear interaction, the formulation of the spectrum evolution problem can be presented
in the form:
dN
dt = Gnl·
(6.17)