The momentum equation in the x-direction for the
Ku = Ks+Ki
Ks = conveyance of seepage flow qs
under-ice water layer is
Ki = conveyance of flow carried by ice.
ql2
qlxqly
qlx
1
+ ( x )+ (
) = (τsx - τ bx ) +
ρ
x H′
y H′
t
The conveyance
Hu
η
Ks = λ
= λ(η - η′)
Tyx
1 T
- Mex
+ ( xx +
) - gH ′
1- N
(6)
ρ
x
y
x
where λ is a seepage coefficient defined by (Bear 1972)
and for water in the upper ice layer it is
p3
λ=
k
gds
1- p
2
quxquy
qux
qux
+ (
)+ (
)=
t
x Hu
y Hu
in which
ds = 6/M
η
1
s
τix - gHu
=
+ Mex
p = porosity
(7)
ρ
x
Ms = average surface area per volume
k = dimensionless empirical coefficient.
where
Hu = (η η′)(1 N) = net water depth in the ice
For randomly placed square plastic blocks (5 5
layer
0.6 cm and 10 10 1.3 cm), Beltaos and Wong (1986)
found k = 0.70. Applications to the Credit River ice
attributable to ice
jams by Beltaos (1993a) gave an average λ value of
qlj
qlk
Tjk = ε jk (
+
)
1.6 m/s. For the Restigouche and Rushoon Rivers,
xk
xj
εjk = generalized eddy viscosity coefficients
Beltaos (1993a) used 2.5 and 1.0 m/s. In this study a
value of λ =1.0 m/s is used for freezeup conditions.
j and k = the two coordinate directions
→
τ s = wind drag on the water surface or the ice
The conveyance Kl is
drag on the water surface on the underside
2
of the ice layer
H ′(α h H ′)
→
3
Kl =
.
nb
Mex = momentum exchange at the interface of the
The factor αh is the fraction of the total water flow depth
upper ice and lower water layer,
which is
affected by the bed friction. Shen et al. (1990) discussed
η′
η′
the distribution of shear stresses on the channel bed
Mex = wwx (-
- wwx
-
and the interface between the moving ice layer and the
t
x
water current underneath it, and derived the expression
η′
+ wwz )
- wwy
for the coefficient αh as
(8)
y
1
1
αh =
=
where, wwx, wwy, and wwz are components of water
(10)
A
3
ni2 N (Vw -u)2 4
velocity at the interface between ice and water layers.
1+ i
Ab 1 +
When eq 6 and 7 are combined and the unit-width
2
2
nb
Vw
discharges are expressed in terms of hydraulic
conveyance, i.e., q = KSf1/2, the momentum equation
in which
Ai
=
flow area affected by ice resistance
becomes
Ab
=
flow area affected by bed resistance
qt2 Kl2 K u
2
qtxqty Kl2
qtx
ni
=
Manning's coefficient for ice
+
+
)+
+
x(
(
x K 2 H ′ Hu
y K2 H′
nb
=
Manning's coefficient for the bed
t
t
t
Vw
=
depth-averaged current velocity
2
(9)
1
K
+ u ) = (τix + τsx - τ bx )
u
=
ice velocity.
ρ
Hu
η 1 Txx
Ki is estimated as
Tyx
+ (
+
- g( Hu + H ′)
)
qi
x ρ
x
y
qt (Kl + Ks )
Ki =
where
q
1- qi
Kl = conveyance of the lower water layer
t
→
→
Ku = conveyance of the upper ice layer
where q i = V i Hu .
Kt = total conveyance
5
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