equal to the change in ice storage over the time interval and considering ρ as con-

stant results in

*t*2

*x*2

∫ (υ*A*i si )x - (υ*A*i si )x *dt *+ ∫ (Ai si )t - (Ai si )t *dx *= 0 .

1

1

(37)

*t*

*x*

2

2

1

1

Further simplifications are made subsequently, such as expressing *A*i in terms of

jam thickness η(*x,t*) .

*Conservation of water momentum*

The analysis examines the control volume for water flow beneath an ice jam

whose channel cross section is prismatic. Figure 25 depicts the forces acting on the

control volume that is bounded by *x*1, *x*2, and the bed and the bottom of the jam.

Conservation of momentum in the *x*-direction requires that the change of

momentum within the control volume between times *t*1 and *t*2 equal the sum of the

net flux of momentum into the control volume and the integral of the external

forces acting on the control volume during the same period. The momentum inside

the control volume at any instant is

*x*2

∫ (ρ*Au*) dx

(38)

*x*1

so the net increase in momentum ∆*M *between times *t*1 and *t*2 is

[(ρ*Au*)

]

*x*2

- (ρ*Au*)t dx .

∆*M *= ∫

(39)

t2

1

*x*1

The net momentum flux *M*f into the control volume between times *t*1 and *t*2 is

(

) - (ρ*Au *)

t2

*M*f = ∫ ρ*Au*2

2

*dt *.

x2

(40)

t1

x1

The external forces acting on the water control volume include: hydrostatic pres-

sure; gravity forces due to the weight of the water, ice, and pore water; and shear

stress at the bed, banks, and jam underside. The hydrostatic pressure forces acting

at sections *x*1 and *x*2 are and as depicted in Figure 25. With the level of the phreatic

surface above the bed denoted as *D(x)*, the vertical distance above the bed as δ(*x*),

and the local width as *b*(δ), for any section *x*

Fg2

*Figure 25. Forces acting on the*

Ffi

*water control volume.*

Fp1''

Fg1

Fp1'

Ffb

x1

x2

32