20
P. Wadhams et al.
are ice and water densities. A solution for $2 is a sum °f potentials of form
<&2 = 2n lAn exp(iknx) + Bn exp(-iknx)] exp(-kny) exp(-iœt).
(6)
A velocity potential of this form satisfies the boundary conditions provided
L kn5 + (pw g - pi h œ2) kn - pw co2 = 0.
(7)
There are three physically feasible roots for kn, of which the real root ko represents a conservative wave propagating through the ice sheet with a dispersion
relation which differs from that of a wave in open water. This new type of wave,
called a flexural-gravity wave, has a dispersion relation which involves a minimum value of phase and group velocity. In continuous ice sheets it can be generated by wind blowing over ice (so long as the wind velocity is sufficient), or by
moving vehicles, while it also represents the mode for long-distance pénétration
of wave and swell energy through ice-covered polar basins.
In the context of scattering by floes, kn defines a set of wave numbers which
give the mismatch from open water propagation which is responsible for reflections at floe edges, and which can be used to predict the energy decay rate [7].
1.3 Wave Propagation in Pancake Ice
In pancake and frazil ice, the floe size is so small in relation to the wavelength that
we can consider the material to be simply a mass of particles floating on the water
surface. If we consider such ice to be a continuum composed of non-interacting
mass points which exerts a pressure upon the water surface but which has no
cohérence or rheological properties as a material, then the relevant équations for
propagation of waves in the ice are those developed above for the wave scattering
model but setting L = 0. This theory was developed First for waves in any kind of
ice [8-11] but in fact it is inapplicable to continuous ice sheets or to marginal ice
zones composed of floes whose diameter is a non-negligible fraction of a wavelength.
Following Wadhams [7], we find that the wave number k; for propagation in ice
is given, setting L to zéro in Eq. (7), by
k, = pw(û2/[pwg-pichw2],
(8)
where h is the thickness of the frazil slurry or pancakes and c is the average ice
concentration. This implies that a wave propagating into the ice from open water
will acquire a reduced wavelength = 2 zc/kj. This has the following implications:
(1) waves incident obliquely on the ice edge are refracted towards the normal,
according to Snell's Law; (2) since the group velocity inside the ice is lower than
in the open water, a wave entering the ice acquires a greater amplitude. This, combined with the reduced wavelength, implies a greater wave steepness; (3) there is
a frequency limit implied in Eq. (8), given by
“c2 = Pw g / Pi c h(9)
P. Wadhams et al.
are ice and water densities. A solution for $2 is a sum °f potentials of form
<&2 = 2n lAn exp(iknx) + Bn exp(-iknx)] exp(-kny) exp(-iœt).
(6)
A velocity potential of this form satisfies the boundary conditions provided
L kn5 + (pw g - pi h œ2) kn - pw co2 = 0.
(7)
There are three physically feasible roots for kn, of which the real root ko represents a conservative wave propagating through the ice sheet with a dispersion
relation which differs from that of a wave in open water. This new type of wave,
called a flexural-gravity wave, has a dispersion relation which involves a minimum value of phase and group velocity. In continuous ice sheets it can be generated by wind blowing over ice (so long as the wind velocity is sufficient), or by
moving vehicles, while it also represents the mode for long-distance pénétration
of wave and swell energy through ice-covered polar basins.
In the context of scattering by floes, kn defines a set of wave numbers which
give the mismatch from open water propagation which is responsible for reflections at floe edges, and which can be used to predict the energy decay rate [7].
1.3 Wave Propagation in Pancake Ice
In pancake and frazil ice, the floe size is so small in relation to the wavelength that
we can consider the material to be simply a mass of particles floating on the water
surface. If we consider such ice to be a continuum composed of non-interacting
mass points which exerts a pressure upon the water surface but which has no
cohérence or rheological properties as a material, then the relevant équations for
propagation of waves in the ice are those developed above for the wave scattering
model but setting L = 0. This theory was developed First for waves in any kind of
ice [8-11] but in fact it is inapplicable to continuous ice sheets or to marginal ice
zones composed of floes whose diameter is a non-negligible fraction of a wavelength.
Following Wadhams [7], we find that the wave number k; for propagation in ice
is given, setting L to zéro in Eq. (7), by
k, = pw(û2/[pwg-pichw2],
(8)
where h is the thickness of the frazil slurry or pancakes and c is the average ice
concentration. This implies that a wave propagating into the ice from open water
will acquire a reduced wavelength = 2 zc/kj. This has the following implications:
(1) waves incident obliquely on the ice edge are refracted towards the normal,
according to Snell's Law; (2) since the group velocity inside the ice is lower than
in the open water, a wave entering the ice acquires a greater amplitude. This, combined with the reduced wavelength, implies a greater wave steepness; (3) there is
a frequency limit implied in Eq. (8), given by
“c2 = Pw g / Pi c h(9)