Mapping the Thickness of Pancake Ice Using Océan Wave Dispersion...
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1.2 Wave Propagation in Ice
If pancake ice is not présent, then when waves enter the ice edge from the open
sea they first encounter an outermost fringe of wave-broken floes of finite diameter, the marginal ice zone (MIZ), before reaching the interior of the polar ice
cover, which can be thought of in a simplified way as a continuons elastic sheet.
Extensive field observations of wave decay in MIZ régions hâve been carried out
during the past three décades, with some of the more recent measurements being
described by Wadhams et al. [5, 6]. The main conclusions from the observations
are that the atténuation of waves with distance into the pack takes a négative
exponential form, with an atténuation coefficient which decreases with increasing
wave period over most of the spectral range. In heavy compact ice (e.g. East
Greenland) the energy atténuation coefficient typically varies from 2 x 10'4 m 1 for
the longest swell to 8 x 10'4 m'1 for 8 to 9 s waves, corresponding to e-folding distances of 5-1.2 km.
A simple scattering model (described in full in [7]) was developed to explain
these findings. It treats each floe as an elastic floating raft, solving for velocity
potentials at the leading and trailing edges of the raft. Within the raft itself energy propagates as a flexural-gravity wave with an altered dispersion relation. The
treatment yields an energy reflection coefficient from which an atténuation rate
can be derived. Floe diameter is a more critical parameter than ice thickness. The
model agréés well with experimental results at normal and long wave periods,
although it does not predict an observed increase in directional spread as waves
pass into the ice. The basis for the model is that the scattering is due to a mismatch between the mode of propagation under an elastic raft and under an open
water surface.
In open water a wave of angular frequency œ (= 2 7t / T, where T is wave period) follows the familiar dispersion relation
k = (û2/g
(1)
if water depth is great compared to the wavelength X (= 2 7t / k). Under ice, however, with p(x,t) being the pressure just below the water-ice interface, we hâve
p(x,t) - Ld4q2 / dx4 = p; h d2q2 / dt2
(équation of motion),
(2)
p(x,t) = -pw [g q2 + d$2 / dt |y=0]
(Bernoulli équation),
(3)
- dq2 / dt = d2 / dy |y=0
(boundary condition).
(4)
Here L is the flexural rigidity of the ice, given by
L = E h3 / 12 (1 - v2)
(5)
where E is Young’s modulus,h the ice thickness and v Poisson’s ratio. 2 is a velocity potential for a wave of amplitude q2 propagating under the ice in the positive
x-direction with the y-axis extending vertically down from the surface; p, and pw