t2
t2 x2
∫ Fisdt = gsi (1 - si )(1 - p) ∫ ∫ ρI4dxdt
(94)
t1
t1 x1
where
I4 = k0λKpη2 .
(95)
Therefore, the full conservation of momentum equation for the ice cover
becomes
t2
t2
t2
t2
t2
∆M - Mf = ∫ Fp2dt + ∫ Fg3dt + ∫ Ffidt + ∫ Fppdt - ∫ Fisdt
(96)
t1
t1
t1
t1
t1
or
(
) (
)
x2
t2
(s ρυA ) - (s ρυA ) dx - ρυ2 A s
- ρυ2 Ai si dt
∫ i
∫
it
it
i
ii
1
x1
t1
x2
x1
2
t2 x2
t2
d
(ρI ) - (ρI ) dt + g
∫ ∫ ρAi si So - dxdt
=g∫
2 x
x
2 x
t1
2
1
t1 x1
t2 x2
t2
fP
+ ∫ ∫ ρ i i (u - υ) dxdt + gsi (1 - si ) ∫ (ρI3 )x - (ρI3 )x dt
2
2
(97)
t1
8
1
t1 x1
t2 x2
- gsi (1 - si )(1 - p) ∫ ∫ ρI4dxdt.
t1 x1
With ρ being a constant
(
) - (υ A s )
x2
t2
(s υA ) - (s υA ) dx - υ2 A s
2
∫ i it
∫
dt
x2
it
i
ii
ii
x
1
t
x1
2
1
1
t2 x2
t2 x2
t2
(I ) - (I ) dt + g ∫ ∫ A s S - d dxdt + ∫ ∫ fi Pi (u - υ)2dxdt
=g∫ 2 x
i i o
2 x
x
2
t1 x1 8
1
t1 x1
t1
(98)
t2
t2 x2
+ gsi (1 - si ) ∫ (I3 )x - (I3 )x dt - gsi (1 - si )(1 - p) ∫ ∫ I4dxdt.
t
2
1
t x
1
1 1
The integral relations given by eq 35, 37, 66, and 98 are valid for a channel of
constant rectangular cross section. Substituting rectangular channel cross-section
relationships for Sf, I1, I2, I3, and I4 (i.e., A = Bd and Ai = Bη), then canceling out
common terms, produces the conservation of mass and momentum equations in
their integral form.
Conservation of water mass is
[
]
[(d)
]
t2
x2
∫ (ud)x2 - (ud)x1 dt + ∫
- (d)t dx = 0.
(99)
t2
1
t1
x1
Conservation of ice mass is
[
]
[(η)
]
t2
x2
∫ (υη)x2 - (υη)x1 dt + ∫
- (η)t dx = 0.
(100)
t2
1
t1
x1
Conservation of water momentum is
41
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