→
→
τ(y- w) = ρcw V i - V w (v - Vwy )
i
Similarly, the y-component of the momentum
(17)
s
equation is
2
in which
qtxqty K l2 K u
2
K l2 K u
2
qty
qty
+
+
)+
+
)=
(
(
x K 2 H ′ Hu
y K 2 H ′ Hu
t
→
→
t
t
q
Vw =
= water current velocity
H′
1
(τ iy + τsy - τ by )
(11)
→
→
→
ρ
V i = u i + v j = ice velocity
cw = water drag coefficient on ice, which
η 1 Txy
Tyy
+ (
+
- g(H u + H ′)
) .
varies with ice concentration and ice
y ρ
x
y
floe geometry.
→
The flow model solves for components of q t and the
This coefficient can be related to Manning's coefficient
water depth H using eq 3, 9, and 11. A finite-element
of the underside of the cover as
model with the lumping technique and leapfrog time
2
integration (Connor and Brebbia 1978, Wake and Xiao
ni g
cw =
.
(18)
1989) is used. The bed shear stresses can be expressed
1
[(1 - α h )H ′]
3
as
1
qx (qx + qy )
2
2
For a partially ice-infested water surface, the surface
2
τ bx = cf ρ
(12)
shear stress is assumed to be a linear combination of
2
H′
τ(a-w) and τ(i-w)
s
s
1
qy (qx + qy )
2
2
2
τ by = cf ρ
→
→
→
(13)
τ s = (1 - N ) τ s (a-w ) + N τ s (i-w ) .
2
H′
(19)
The drag of the seepage flow on ice is
terms of Manning's coefficients of the bed and the shear
qsqsx
stress distribution coefficient αh as
τix = -ρgHu
(20)
2
Ks
2
nb
cf =
g
qsqsy
τiy = -ρgHu
.
1
α hH ′
.
(21)
3
2
Ks
On the open water surface, the surface shear stress
attributable to wind effect can be expressed as
Ice dynamic model
The momentum equation of the surface ice can be
τ(a-w) = ρa γ 2W 2 cos θa
(14)
sx
written in the Lagrangian form as (Shen et al. 1990)
→
τ(a-w)
DV i → → →
→
= ρa γ W sin θa
2
2
(15)
= R+ F a + F w + G
sy
(22)
Mi
Dt
in which
in which
→
DV i
γ2 = wind drag coefficient (Wu 1973)
= acceleration of ice
Dt
= ρiNti = ice mass per unit area
W = wind velocity at 10 m above the water
M
→i
surface
R
= internal ice resistance
→
ρa = density of air
Fa
= wind drag
→
θa = angle between the wind direction and the
Fw
= water drag
→
x-axis.
G = gravitational force
ρi = density of ice
For a fully ice-covered water surface, the surface
N = concentration of ice
shear stress components on the water can be written as
ti = thickness of ice.
→
→
τ(x- w) = ρcw V i - V w (u - Vwx )
Force terms in the momentum equation can be
i
(16)
s
expressed in two-dimensional forms as follows.
6
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