7.2 Wave Spectrum Evolution in Water with Ice Cakes
317
As shown in (7.5) and (7.6a), the presence of ice results in diminishing the
wavelength and propagating velocity compared with the case without ice. It
should be noted that the wave number becomes infinitely large in the case of
the wave frequency approaching that of the floe oscillation. This shows the
inapplicability of the traditional approach and the necessity to use a more
precise approximation.
7.2 Wave Spectrum Evolution in Water with Ice Cakes
Spectral formulation of the problem. Now the problem of wave spectrum transformation in water with ice cakes will be considered. Supposing
that the mean ice thickness L(x, y) varies smoothly enough over all space
(i.e. the spatial scale of typical variations of the mean ice thickness is much
larger than the wavelength), then the geometrical optics approximation can
be used for the description of wave packet propagation.
The spectral density balance equation of wave energy evolution is used in
its traditional form:
as as dk as dr as dw _ G
at + ak dt + ar dt + aw dt -
(7.7)
where G is the source function describing physical mechanisms, which form
the wave spectrum in water with ice cakes. The characteristics of (7. 7) are
the solutions to the Hamiltonian equation:
dk
aw
dt
-ar;
dw aw
dt
at '
(7.8)
where the frequency w is written in accordance with the ratio (7.3) as follows:
(7.9)
where a 2 = gk th(kH).
In this section the effects of non-uniform currents and depth variations
are neglected.
As for the source function G, the dissipation connected with the friction
between ice floes is the most typical physical mechanism forming the wave
spectrum in the case of their propagation in water with ice cakes. Wave
generation by wind is less effective in this case, as long as the transfer of
momentum and energy from the atmospheric boundary layer into the ocean
is prevented by ice. The mechanism of on-linear wave energy transfer in the
spectrum is of great interest, so it is thoroughly considered below.
Wave energy dissipation in water with ice cakes. At the first stage
a wave energy dissipation mechanism, connected with the presence of ice
317
As shown in (7.5) and (7.6a), the presence of ice results in diminishing the
wavelength and propagating velocity compared with the case without ice. It
should be noted that the wave number becomes infinitely large in the case of
the wave frequency approaching that of the floe oscillation. This shows the
inapplicability of the traditional approach and the necessity to use a more
precise approximation.
7.2 Wave Spectrum Evolution in Water with Ice Cakes
Spectral formulation of the problem. Now the problem of wave spectrum transformation in water with ice cakes will be considered. Supposing
that the mean ice thickness L(x, y) varies smoothly enough over all space
(i.e. the spatial scale of typical variations of the mean ice thickness is much
larger than the wavelength), then the geometrical optics approximation can
be used for the description of wave packet propagation.
The spectral density balance equation of wave energy evolution is used in
its traditional form:
as as dk as dr as dw _ G
at + ak dt + ar dt + aw dt -
(7.7)
where G is the source function describing physical mechanisms, which form
the wave spectrum in water with ice cakes. The characteristics of (7. 7) are
the solutions to the Hamiltonian equation:
dk
aw
dt
-ar;
dw aw
dt
at '
(7.8)
where the frequency w is written in accordance with the ratio (7.3) as follows:
(7.9)
where a 2 = gk th(kH).
In this section the effects of non-uniform currents and depth variations
are neglected.
As for the source function G, the dissipation connected with the friction
between ice floes is the most typical physical mechanism forming the wave
spectrum in the case of their propagation in water with ice cakes. Wave
generation by wind is less effective in this case, as long as the transfer of
momentum and energy from the atmospheric boundary layer into the ocean
is prevented by ice. The mechanism of on-linear wave energy transfer in the
spectrum is of great interest, so it is thoroughly considered below.
Wave energy dissipation in water with ice cakes. At the first stage
a wave energy dissipation mechanism, connected with the presence of ice