2.4

calculated conditions. From eq 118b and

Cr

119, it can be seen that, if θ = 1.0, there is

1

3

no consideration of previous values of

2.2

5

10

the variables or their derivatives in the

12

solution for their values at the new time

2.0

24

step (except in the time derivative).

Cunge et al. (1980) show that while the

1.8

Preissmann scheme becomes second-

order accurate under the special case of

1.6

0

1000

2000

3000

4000

5000

θ = 0.5, it also becomes dispersive due to

x Location (m)

the finite difference approximation of

some of the higher order terms. For open-

water simulations, the dispersion results

in a wavy surface with relatively high-

frequency oscillations. By specifying

2.4

0.5 < θ ≤ 1.0, the solution becomes diffu-

0.55

0.60

sive, smoothing out the high-frequency

2.2

0.66

oscillations and providing a more realis-

1.00

0.80

tic solution. The ideal solution would be

2.0

one in which dispersion and diffusion are

both minimal.

1.8

A separate weighting factor θi was

specified for the ice. The movement of

1.6

the ice can be quite different from that of

0

1000

2000

3000

4000

5000

x Location (m)

the water because it can start and stop

rather suddenly. The effects of using the * Figure 44. Final jam thickness profiles for *θi

current values of ice velocity to calculate *= 1.0 and *θ *= 0.55, 0.6, 0.66, 0.8, and 1.0.*

the new values were unknown. For ex-

ample, if the ice cover was in motion and the new stability check showed that it

should be stable (the new ice velocity, υ = 0), the previous value of ice velocity may

have an undesirable effect on the newly calculated ice thickness if θi < 1.0. Detailed

inspection of the calculated ice velocities (as ice begins moving and as it slows to a

stop), however, showed that absolute changes in ice velocity over a time step were

not large and, thus, the θi-weighted average of ice velocity would provide a proper

solution.

The effects of the water-weighting factor on the solution were investigated by

running the baseline conditions at a variety of θ values for θi = 1.0. Values of θ

tested included 0.5, 0.55, 0.6, 0.66, 0.8, and 1.0. The simulation with θ = 0.5 resulted

in unacceptable oscillations in the water depth and velocity, which, because the

solution is fully coupled, resulted in large oscillations in ice velocity and thickness.

The oscillations in the water variables aggravated the oscillations in the ice vari-

ables and the solution became highly unstable. For instance, the depth would

decrease with an accompanying water velocity increase that increased ice velocity

and thickening, thereby further reducing flow depth. Figure 44 shows the final jam

thickness profile for θ = 0.55, 0.6, 0.66, 0.8, and 1.0. While θ = 0.55 shows a smooth

final profile, it resulted in highly variable water depths, especially at the initial

increase in upstream water discharge. A value of θ = 0.6 provided the most accept-

able solution in terms of both water depth and ice thickness throughout the simu-

lation period.

63

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