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
fiu2B
Ω =
=
c Qin 8gsi (1 - p)(1 - si )k0λKpη2 Qin
(155)
where Qin 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.
Table 3. Characteristics of various inflow hydrographs.
∆Q (m3/s)
∆Q/Q
Initial Q (m3/s)
Final Q (m3/s)
Hydrograph type
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
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