eq 153 would no longer be in balance and would represent a condition of greater

jam stability. In this case, the *a/c *ratio decreases because the *a *term remains con-

stant. As water discharge increases, the jam should remain stable for a longer

period prior to shoving and thickening, thus resulting in lower ice velocities and

less of an effect of ice momentum on the jam thickness. Also, for a given initial jam

thickness (whether at the limit of stability or not), a smaller discharge rise has less

effect on jam thickness than does a larger one. Finally, a larger relative discharge

increase has more effect than does a smaller one. For instance, ice momentum would

be expected to influence the final jam thickness more for an increase from 100 to

200 m3/s than an increase from 200 to 300 m3/s.

To express these trends, a dimensionless parameter was developed that includes

initial jam conditions (indicating how close the jam is to the limit of stability), as

well as the relative increase in discharge expected. This number is

∆*Q *

*a * ∆*Q *

Ω =

=

*c * *Q*in 8*gs*i (1 - *p*)(1 - *s*i )k0λ*K*pη2 *Q*in

(155)

where *Q*in is initial water discharge and ∆*Q *is the expected change in discharge.

This dimensionless parameter is the product of the initial state of stability of the

jam and the relative discharge increase applied to cause an instability.

Several runs were made with the fully coupled model using an inflow hydrograph

that rose at the same rate as the baseline inflow hydrograph, but with ten different

Runs were made for eight different bed slopes of 0.00005, 0.00008, 0.0001, 0.00025,

0.0005, 0.00075, 0.001, and 0.0025. The final jam thickness profile for each of these

calculated by eq 25. Average values of jam thickness η were calculated for each

profile. The values of Ω were plotted against η / ηeq in Figure 58. The data points

delineate a line, showing combinations of channel and flow conditions, or Ω val-

ues, for which ice momentum significantly affects jam thickness (from the η / ηeq

value). It is clear that ice momentum is very important for low bed slope values,

because water shear stress engages the greater portion of the jam strength for these

cases. The scatter in the data at higher values of η / ηeq is most likely attributable

to the highly nonuniform jam thickness profiles for those cases. Figure 59 shows a

plot of the final jam thickness profiles for cases of relatively small, medium, and

large ice momentum effects ( η / ηeq ). A smaller ice momentum effect results in a

significantly more uniform thickness profile.

∆Q *(m*3/s)

∆Q/Q

1

100

112.5

12.5

0.125

2

100

125

25

0.25

3

100

150

50

0.50

4

100

175

75

0.75

5

100

200

100

1.00

6

100

250

150

1.50

7

100

300

200

2.00

8

112.5

200

87.5

0.78

9

125

200

75

0.60

10

150

250

100

0.67

73