is the slope of the bed, and (θ + ϕ) is the slope of the water surface. In eq 22, the
terms (reading from left to right) are Fdrag, Fweight, Fstrength, Fbank, F′merge2/b1, and
F″merge2/b1.
For a two-dimensional formulation, the value of bd and bs would need to be
estimated. They can be estimated approximately using eq 4 and 6. With the divid-
ing streamline defined, it is then possible to obtain an approximate estimate of the
additional forces acting to inhibit ice from channel 1 moving into the confluence.
Ice decelerates as it accumulates and congests the confluence. It thereby
increases flow resistance and generates a backwater flow profile through the con-
fluent channels. This process is not taken into account in the foregoing quasi-steady
formulation of ice jams at confluences.
The equilibrium jam formulation given by eq 23, or a two-dimensional equiva-
lent, can be used to estimate the conditions leading to ice-jam formation in conflu-
ent channels. In this regard, it is better suited for delineating the limiting condi-
tions for ice movement through a confluence than is the process model used in the
present study.
2. Jamming due to flow impact
This confluence jam mechanism requires that the flow from one influent chan-
nel be sufficiently large as to block ice movement from the second influent chan-
nel, as shown in Figure 31. The hydrodynamic reaction force generated by the
larger flow, Q1, impacting the ice may be resolved into a component, F′flow2, acting
upstream along the axis of channel 2 and a component, F″flow2, acting normal to
the axis of channel 2. This latter component would result in an additional fric-
tion force, F″flow2, between bank and ice over a short distance, bf, as shown in
Figure 31.
The force components could be estimated as
Ff′ow2 = ρQ1(v11x - v31x )η2b2
′
′
(23)
l
and
Ff′′ w2 = 2ρQ1(v1′ x - v3′1x )(η2b cot α)
′1
′
(24)
lo
where v11x and v31x = lateral velocity components of flow from channel 1 through
the confluence control volume indicated in Figure 31,
v1′1x = the component of v11x acting along the axis of channel 2,
′
v3′ x = the component of v31x acting along the axis of channel 2,
′1
′′
v11x = the component of v11x acting normal to the axis of channel 2,
v3′ x = the component of v31x acting normal to the axis of channel 2,
′1
η2 = the average thickness of the layer of ice moving from chan-
nel 2.
′
The velocity vectors, v11x , etc., can be estimated for specific confluence geome-
tries and flow rates.
An ice jam may develop in channel 2 when
ρ η
(
)
(η2σ x2 ) - y η2τxy 2
+ si 2ρgη2 sin(θ + ϕ)2 -
τi 2 cos i
ρ x2
x
(25)
- ρQ1(v11x - v31x )η2 - 2ρQ1(v1′ x - v3′1x )(η2b cot α) / b2 = 0 .
′
′
′1
′
40
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