net force applied to the ice cover element may be less than that "allowed" by the
passive pressure limit. Application of the equation in this case would result in
either ice motion upstream (negative ice momentum) or thinning of the jam to
achieve a balance. In natural systems, thinning or upstream movement of ice rarely
occurs. The forces exerted on a thickened, but static, section of jam are reduced.
The jam simply remains in place with no change in thickness. In such a case when
the forces exerted lead to a stable jam (i.e., below the passive pressure limit), the
conservation of ice momentum equation must be adjusted so that negative ice
velocities or thinning do not take place.
The computational procedure commences at the upstream end of the system,
determining the stability of each reach at the beginning of each time step. The forces
acting on the jam and water in the reach are assessed and compared with the maxi-
mum resistive force at the downstream end of the reach. The assumption of passive
pressure limits the net force at the downstream end of each reach to that given by
eq 89. If the sum of the forces (normal force at the upstream end of the reach, ice
momentum, hydrostatic pressure, gravity, shear on the underside of the cover, and
friction at the banks) is less than the passive pressure limit, the reach is deemed
stable. The stability check continues in the downstream direction with the normal
force at the upstream end of the next reach set equal to the net force at the down-
stream end of the current reach. The stability check assesses the stability of each
cross section by comparing the net force and passive pressure resistance.
Modifications to the conservation equations of ice mass and ice momentum
applied to each reach are necessary, depending on the stability conditions at the
ends of each reach. One of the upstream boundary conditions of the system pro-
vides a relationship for the upstream ice thickness of the first reach. Thus, the con-
servation of ice mass equation provides a relationship for the upstream ice velocity
of a reach. The conservation of ice momentum provides a relationship for the down-
stream ice thickness for a reach. One of the downstream boundary conditions of
the system describes a relationship for the downstream ice velocity at the last reach.
If a cross section is deemed to be stable, ice velocity in it will be zero, and the
conservation of ice mass equation reverts to
υn+1 = 0.
Several combinations of upstream and downstream stability are possible for a
reach (Fig. 32). For the first case, both upstream and downstream ends are unstable
and the full equations are used. In the next case, both ends are deemed stable and,
thus, the ice velocity equals zero at both ends, and eq 148 is used. Because there is
no ice movement, the ice thickness must remain constant. Consequently, the con-
servation of ice momentum equation reverts to
ηn+1 - ηn 1 = 0.
In the third case of Figure 32, the downstream end is stable and, thus, ice velocity is
equal to zero. But the upstream end is unstable and the full ice-continuity equation
is required. Ice moves into the reach, thickening at the downstream end, and
requiring use of the full ice momentum equation. The last case shown occurs when
the ice is stable at the upstream end, but unstable at the downstream end. Again, eq
148 is used for the ice continuity equation, but the full ice momentum equation is
required for defining the downstream thickness.