(41)

0

For a rectangular channel, *b *is constant in the vertical and equal to the top width *B*.

Therefore

(42)

0

[

]

(43)

Thus, the time integral for the net pressure *F*p1 is

(

)

′

″

∫ *F*p1dt = ∫ *F*p1 - *F*p1 dt = *g *∫ (ρ*I*1)x - (ρ*I*1)x *dt*

2

(44)

2

1

1

1

1

where

[

]

(45)

Two gravity forces act vertically on the water control volume. The first acts on

the bottom surface of the control volume (the bed). It is attributable to the combined

weight of water, ice, and pore water above. The second acts on the upper surface of

the control volume (the bottom of the jam). It is attributable to the weight of the ice

and pore water above. The horizontal component of the first gravity force is

(

)

(46)

where *S*o is bed slope

.

(47)

For the period *t*1 to *t*2

∫ *F*g1dt = ∫ ∫ ρ*g*(A + *A*i si )Sodxdt .

(48)

The gravity force attributable to the weight of ice and pore water acting on the

upper surface of the control volume (in the *x*-direction) is

[

]

(49)

where *S*ib is the slope of the jam underside

(yb + *d*) = -

*x *+ *x * = *S*o - *x *.

(50)

Substituting eq 50 into 49 and integrating over the period *t*1 to *t*2 gives

∫ *F*g2dt = ∫ ∫ ρ*gA*i si *S*o - *dxdt *.

(51)

33