Geoscience Reference
In-Depth Information
annually covered by ice, and the maximum annual ice thickness is 50
70 cm. Mechanical
ice breakage, displacements and formation of small ridges is a common phenomenon, but
the length scale of lateral displacements is limited (Alestalo 1980).
Drift ice physics is conveniently presented in right-hand co-ordinate system (x, y, z) with
positive directions east, north and up, respectively. The drift ice medium is described by the
state variables, which contain the properties needed to model the internal stress. The
-
rst
approximation is a two-level ice state J ={A, h}, where A is ice compactness and h is mean
ice thickness (Doronin 1970). If necessary, this can be re
ned by using the ice thickness
distribution for the ice state with several thickness categories (Thorndike et al. 1975). For a
region
ʩ
, the distribution function is de
ned by
Z
1
Pð1Þ
Pð
h
Þ
¼
Hh
fðxÞ
½
d
X
ð
5
:
34
Þ
X
where h and
ʶ
denote the thickness of ice so that h is the thickness distribution argument
and
ʶ
is the actual thickness in points
ˉ ∈ ʩ
, and H is the Heaviside function H(x) = 0 (1)
for x
(h) is simply the normalized area of ice with thickness less than or
equal to h in a region
≤
0(x > 0);
ʠ
ed as a histogram by the set of classes
J ¼
fP ðÞ; P
h
ðÞP ðÞ; P
h
ðÞP
h
ð ; ...;
1
P
h
ðÞg
ʩ
. Ice state is speci
, where h
k
'
s are n + 1
xed
thickness levels.
For a two-dimensional medium, the stress
σ
is obtained from the three-dimensional stress
by integration:
r
¼
R
h
0
σ
are the freeboard and draft of the ice sheet,
respectively. Drift ice rheology investigations have been made for sea ice and river ice but
the results are applicable for lake ice as well, provided the broken structure of the ice cover
prevails (see Leppäranta 2011). The rheological law of drift ice is in general form
h
00
r
dz
, where h
′
and h
″
r
¼
rð
J
; e; eÞ
ð
5
:
35
Þ
known as the free drift. The main
drawback of this model is that mechanical ice accumulation is not limited by any resisting
force, and unrealistic ice thickness
The simplest rheology is the no-stress case
ðr
0
Þ
fields may result. Nevertheless, free drift is applicable
for A < 0.8, when stress levels are very small and convergence/divergence in ice drift just
arranges distances between ice
oes.
Realistic rheologies of compact drift ice (A > 0.8) have the following general
properties: (1) The strength is sensitive to ice compactness in 0.8
≤
A
≤
1, (2) Yield
strength
0, (4) Tensile strength
is small, and (5) No memory. The two-dimensional ice stress is the three-dimensional
stress integrated through the thickness of ice; its dimension is therefore force per length.
The low tensile strength is due to that not much force is needed to drive ice
≫
0 for A
≈
1, (3) Compressive strength > shear strength
≫
floes further
away from each other. At high ice compactness, a plastic
flow results with yield strength
increasing with ice thickness (Coon et al. 1974). The plasticity is not perfect but the ice
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