single plot of them becomes overly cumbersome to interpret. The interaction
of the dependent variables with and on each other necessitates a form of output
illuminating the changes in all the variables with time. One possible format
is to present the output in the form of profile plots at specific times. While these
sequential time plots can be animated with additional graphic enhancements, they
cannot be presented easily in text form. Also, as the length of the modeled system
increases, changes of values in the vertical become less obvious because of loss in
resolution. Therefore, the water variables (depth and velocity) are plotted on one
set of axes directly above the ice variables' (thickness and velocity) axes. This is
done for several times at points on the inflow hydrograph where there are signifi-
cant changes. Figure 36 presents the results of the baseline model for several time
slices.
Many interesting observations can be made from Figure 36. By 25 minutes (5
minutes after the initial increase in upstream discharge), the water velocity has
increased near the upstream end, but the depth has not changed significantly. The
jam is thickening somewhat near the upstream end, but the ice velocity shows the
most significant change. As the ice moves, the shear stress between the ice and the
water decreases, resulting in less resistance to the water flow and lower water lev-
els. At 35 minutes, ice is moving throughout the system and is by no means uni-
form. The ice velocity plot shows that the jam failure is not simply a progressive or
complete failure as envisioned in the laboratory experiments, but is rather a combi-
nation of these two failure modes, with several areas of instability arising within
the jam. Each of these instabilities moves downstream with time, resulting in local
areas of thickening where ice velocity decreases. The stopping and starting of the
ice movement, coupled with the thickening, results in changing stress levels and
local states of stability of the jam. The local instabilities move downstream, eventu-
ally dying out. At 45 minutes, the ice at the upstream end of the system has ceased
moving, while several instabilities are continuing to move downstream. By 100
minutes, all ice motion has stopped and the final thickness profile prevails. The
water depths and velocities continue to adjust, however, until they reach steady
values.
What becomes most apparent from the output of the baseline runs is the high
degree of interaction between the dependent variables. There is no simple mecha-
nism by which an ice jam fails, moves, and thickens. Shoving and thickening are
dominated by unsteadiness and nonuniformity. Ice velocity depends on the forces
currently exerted against the jam and, thus, is changing constantly as ice comes
into motion and stops. Local changes in ice velocity or thickness cause changes in
water shear on the underside of the jam, which then affect the forces on the jam.
The final thickness profile also points to the importance of ice momentum. The
downstream end of the simulated jam had a boundary condition of zero ice veloc-
ity, i.e., no ice momentum at that location. Inspection of the ice momentum equa-
tion for the last reach, however, shows that ice thickness at the downstream end is
affected by the change in momentum between the last two cross sections. Ice thick-
ness is least at the downstream end, where the effects of ice momentum are least,
and is 1.73 m. This thickness is greater than the equilibrium thickness (η = 1.70 m)
calculated using eq 25, which does not include the effects of ice momentum. The
upper reaches of the jam have greater ice velocities (as evident in the plots at 30 and
35 minutes). The effects of arresting ice momentum are clearly reflected by the much
higher levels of jam thickness in the upstream reaches.
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