istics were in the middle of the

ranges expected for ice-covered

Number of cross sections

101

rivers at the time when ice jam-

Length step

100 m

ming would be expected. Table 1

Time step

30 s

Channel width

100 m

provides the values of several key

Bed slope

0.0005

parameters chosen for the

Solid ice specific gravity

0.92

baseline configuration. The val-

Ice cover porosity

0.40

ues of the Darcy-Weisbach fric-

Darcy-Weisbach friction factor--bed

0.08

0.12

tion factor were chosen to repre-

0.50

sent a system where the ice jam

λ--coefficient of friction, ice-on-ice

0.65

roughness was significantly

3.85

Theta weighting factor--water (θ)

0.60

greater than the bed roughness.

Theta weighting factor--ice (θi)

0.60

Values of *K*p and were obtained

Maximum number of iterations/time step

4

as explained in the *Laboratory Ex-*

ues and eq 27 leads to *k*0λ = 0.325. By substitution, the values in Table 1 for these

two parameters were chosen.

In addition to the parameters in Table 1, the tolerance for each variable, as

expressed in eq 147, was set to 104 m. The upstream boundary conditions were

specified as water discharge and equilibrium jam thickness by eq 147 with the non-

uniform terms set to zero. The downstream boundary conditions were specified as

normal depth beneath a uniformly thick ice cover and zero ice velocity. The up-

stream water discharge for the baseline testing begins at a steady value of 100

m3/s for 20 minutes, rises to 200 m3/s during the next 20 minutes, and remains at

that level for a total test time of 600 minutes. This upstream discharge hydrograph

is shown in Figure 35. The maximum number of iterations of the Newton-Raphson

solution scheme per time step was set at four. However, rarely more than two itera-

tions were required to reach the specified tolerance.

The initial conditions for the baseline runs were determined by running the model

with a steady upstream water discharge of 100 m3/s and a prescribed uniform

thickness jam that was significantly greater than the equilibrium thickness pre-

dicted from eq 25 to ensure that there would be no further thickening as the water

depths and velocities adjusted to their steady, uniform values. Once the initial con-

ditions for water depth and velocity were determined, the initial uniform ice thick-

ness for the baseline testing was chosen to be only slightly greater than the equilib-

rium value calculated using eq 25. For the baseline runs, this procedure resulted in

the following set of initial uniform values of water depth and velocity, ice thick-

ness, and ice velocity: 1.729 m, 0.578

220

m/s, 1.50 m and 0 m/s, respectively. The

200

equilibrium jam thickness was calcu-

180

lated to be 1.47 m from eq 25.

160

One of the most difficult tasks of mod-

140

eling jam shoving and thickening is the

120

presentation of results. The process is

100

highly unsteady, with large variations in

the values of the dependent variables

80

0

100 200 300 400 500 600 700

over fairly short periods. Also, consid-

Time (min)

ering the interest in all four of the vari-

ables mentioned above means that a

56