Geoscience Reference
In-Depth Information
o
h
u
ðÞ
¼0
;
o
A
o
t
þ
o
o
t
þ
o
u
ðÞ
¼0
ð
0
A
1
Þ
ð
5
:
47b
Þ
o
x
o
x
If the landfast ice condition is broken, ice drift commences. Without loss of generality
we can assume that the forcing is toward the positive x-axis,
s
a
q
hg
rn
[
0
. The free
drift solution is directly obtained as:
r
s
a
q
hg
r
n
q
w
C
w
u
F
¼
þ
U
w
ð
5
:
48
Þ
flow free toward left in the channel.
Therefore it is assumed that ice velocity is positive and thus directed toward the rigid
boundary at the channel end. The steady-state solution of Eq. (
5.47a
,
b
) shows that
If the ice velocity were negative, the ice would
uA
and
uh
are constant across the ice
field, and thus, if the ice
field is in contact with a rigid
u ≡
boundary, the steady state velocity is
0.
For the remainder of this section, we assume that the ice
field is in contact with the
rigid boundary and vertical average water velocity is zero. If the ice is moving, it will drag
surface water with it, but then there is a return
flow at a deeper layer because of the
continuity; a zero reference current
water stress is therefore a reasonable
assumption. The solution of the quasi-steady-state momentum equation is formally:
for
ice
-
s
1
q
w
C
w
s
a
þ
d
r
dx
u ¼
ð
5
:
49
Þ
where the expression under square root must be positive,
˄
a
≥
-
d
σ
/dx. At the end of the
channel the ice must stop and there
˄
a
=
d
σ
/dx. In plastic
flow, the stress is given by the yield
-
level
). The surface pressure gradient is zero as shown in the next paragraph.
The steady-state solution of the one-dimensional coupled lake ice
σ
=
σ
(
A
,
h
water body model
for constant forcing is straightforward. In the channel of depth H, closed at one end, the
coupled system is
-
o
r
o
x
þ s
a
þ q
w
C
w
U
w
u
j
j
U
w
u
ð
Þ q
hg
rn
¼
0
ð
:
Þ
5
50a
q
w
C
w
U
w
u
j
j
U
w
u
ð
Þ q
w
C
b
U
w
j
j
U
w
q
w
Hg
rn
¼ 0
ð
5
:
50b
Þ
where U
w
is chosen as the vertical average mean current. The wind stress on the system is
balanced by the internal friction of ice, sea surface tilt, and bottom friction. At the steady
state for the ice thickness and water level elevation, u
≡
0 and U
w
≡
0, and then we must
have
. Thus,
in the steady state internal ice stress totally removes the surface pressure gradient, and the
∇ʾ
= 0. In the absence of ice, the equilibrium condition would be
˄
a
=
ˁ
Hg
∇ʾ
Search WWH ::
Custom Search