348
8 Conclusion
Using the geometrical optics approximation method, combined with the
wave action density balance equation, the most adequate description of wave
behaviour is obtained. This approach excludes solution singularities, appearing around caustics. It gives the possibility of obtaining detailed results of
wave parameter transformation in a non-uniform current with different ways
of determining the problem parameters. The combined effect of non-uniform
current and bottom unevenness is shown to cause such wave field variations,
which are impossible to describe by simple superposition of current and depth
effects taken separately.
A drawback of the method is that the traditional spectral approach does
not allow the description of the wave shape, because the wave phase is not
taken into consideration. But the phase data are very important, for example,
for the explanation of a wave form in the ocean. Also, some singularities of
the solution can arise with the frequency angular spectrum becoming very
narrow, i.e. for monochromatic waves. That is why the asymptotic estimations
obtained in the diffraction approximation are given simultaneously with the
solution of the spectral problem.
An attempt is undertaken in this monograph to describe the principal
physical mechanisms forming the wave spectrum in sea with marginal ice.
There are the following mechanisms: wave energy scattering, wave refraction,
wave energy dissipation due to friction between ice floes and weak non-linear
transfer in the wave spectrum. These results are supposed to be the basis for
elaborating a mathematical model of wave spectrum evolution in seas covered
with marginal ice. In particular, it can be used for numerical modelling of
natural phenomena such as "ice storms", which can be generated by storm sea
waves in the marginal ice zone. This is an extremely dangerous phenomenon
for oil platforms and other hydraulic engineering structures in Arctic seas.
3. With the exception of the aforesaid problems, some physical mechanisms forming the wave spectrum in deep water are investigated. Thus,
one of the most effective algorithms for calculating the collision integral describing non-linear interaction in the wind wave spectrum is proposed. The
reliable calculation results are obtained with the help of numerical integration methods of the highest accuracy. In particular, some fine effects, such
as non-linear energy transfer to spectral components propagating against the
wind are described, explaining the physics of the generation of these waves.
The non-linear evolution of the wave spectrum is estimated for large time
periods and a self-similar solution is found.
At the same time some important questions still remain unsolved, in
particular, the investigation of wind wave energy dissipation. The absence
of a well-grounded theory of wave dissipation allows various groundless hypotheses concerning this mechanism. The investigation shows a principally
different character of dissipation mechanisms within different frequency spectral ranges. Thus, as for the low-frequency ranges, the dissipation is caused
by interaction of waves and turbulence. As for the high-frequency ranges, it
is connected with wave breaking, being essentially a non-linear mechanism.
8 Conclusion
Using the geometrical optics approximation method, combined with the
wave action density balance equation, the most adequate description of wave
behaviour is obtained. This approach excludes solution singularities, appearing around caustics. It gives the possibility of obtaining detailed results of
wave parameter transformation in a non-uniform current with different ways
of determining the problem parameters. The combined effect of non-uniform
current and bottom unevenness is shown to cause such wave field variations,
which are impossible to describe by simple superposition of current and depth
effects taken separately.
A drawback of the method is that the traditional spectral approach does
not allow the description of the wave shape, because the wave phase is not
taken into consideration. But the phase data are very important, for example,
for the explanation of a wave form in the ocean. Also, some singularities of
the solution can arise with the frequency angular spectrum becoming very
narrow, i.e. for monochromatic waves. That is why the asymptotic estimations
obtained in the diffraction approximation are given simultaneously with the
solution of the spectral problem.
An attempt is undertaken in this monograph to describe the principal
physical mechanisms forming the wave spectrum in sea with marginal ice.
There are the following mechanisms: wave energy scattering, wave refraction,
wave energy dissipation due to friction between ice floes and weak non-linear
transfer in the wave spectrum. These results are supposed to be the basis for
elaborating a mathematical model of wave spectrum evolution in seas covered
with marginal ice. In particular, it can be used for numerical modelling of
natural phenomena such as "ice storms", which can be generated by storm sea
waves in the marginal ice zone. This is an extremely dangerous phenomenon
for oil platforms and other hydraulic engineering structures in Arctic seas.
3. With the exception of the aforesaid problems, some physical mechanisms forming the wave spectrum in deep water are investigated. Thus,
one of the most effective algorithms for calculating the collision integral describing non-linear interaction in the wind wave spectrum is proposed. The
reliable calculation results are obtained with the help of numerical integration methods of the highest accuracy. In particular, some fine effects, such
as non-linear energy transfer to spectral components propagating against the
wind are described, explaining the physics of the generation of these waves.
The non-linear evolution of the wave spectrum is estimated for large time
periods and a self-similar solution is found.
At the same time some important questions still remain unsolved, in
particular, the investigation of wind wave energy dissipation. The absence
of a well-grounded theory of wave dissipation allows various groundless hypotheses concerning this mechanism. The investigation shows a principally
different character of dissipation mechanisms within different frequency spectral ranges. Thus, as for the low-frequency ranges, the dissipation is caused
by interaction of waves and turbulence. As for the high-frequency ranges, it
is connected with wave breaking, being essentially a non-linear mechanism.