8

thickness calculated using eq 25.

Initial WSL

6

Final WSL

Many calculation schemes are sensi-

4

Initial Ice Bottom

tive to the computational time and space

intervals (∆*t*, ∆*x*) used. The sensitivities

2

Final Ice Bottom

for hyperbolic problems are usually dis-

Bed

cussed in terms of the Courant number

0

0

1000

2000

3000

4000

5000

x Location (m)

∆*t u *+ *c*

(151)

∆*x*

where *c *is the gravity wave speed

through the fluid. Because of the as-

1.8

sumption that ice floats on water and is

free to move up and down in the verti-

cal direction in accordance with the dic-

1.6

1.3 m Initial

tates of buoyancy, gravity wave speed

beneath a jam would be equivalent to

1.45 m Initial

1.4

that of open water, i.e.

(152)

1.2

0

1000

2000

3000

4000

5000

x Location (m)

The Courant number expresses the

ratio of the distance traveled by a dis-

turbance in one time step to the length

of a computational distance step. For the

simplest Method of Characteristics, the Courant number must be less than or equal

to unity so as to ensure that the solution remains within the computational do-

main. Implicit finite difference methods, as well as interpolated-grid Methods of

Characteristics, relax this requirement somewhat. Care must be taken, however, to

make sure that disturbances do not travel too far over a time step, thereby resulting

in loss of resolution or too much smoothing. For a given computational length step,

larger Courant numbers imply larger time steps and less total computation time,

with concomitant loss of temporal resolution.

Equations 151 and 152, applied to the baseline parameters in Table 1 and the

initial conditions for depth and water velocity, yield *Cr *= 2.82. By calling this value

nominally *Cr *= 3, runs were then made with *Cr *= 1, 5, 10, 12, and 24. The Courant

number was varied in these tests by adjusting the time step and leaving the length

step at 50 m. The boundary conditions file was modified, however, to retain the

same shape and timing of the inflow hydrograph. Figure 43 shows the final jam

thickness profile for the runs with *Cr *= 1, 3 (standard), 5, 10, 12, and 24. The plot

shows that there is little difference in results with *Cr *values up to 12, and that some

diffusion or smoothing is evident for *Cr *= 24.

The theta-weighting factors for the water θ and for the ice θi are presented in the

averaging between the known conditions at the current time step and the future,

62