RIVER ICE INFLUENCES ON FORT PECK REACH, MISSOURI RIVER
21
position, thalweg sinuosity and channel planform change as the channel slope
changes. Figure 11 indicates that, for a given flow rate and bed sediment size,
channels lengthen or branch into subchannels as the channel slope increases.
Channel lengthening and branching are mechanisms that increase the flow resis-
tance (and thereby the rate of energy use) in an alluvial channel to offset
increased flow energy associated with an increase in channel slope. An ice cover
imposes additional flow resistance, which increases the depth while decreasing
the average flow velocity. The channel effectively conveys a deeper, slower flow
requiring a slower rate of energy dissipation.
When the channel has a free-floating ice cover (Fig. 10b) and Q is constant,
charge may be written as
Qi = Qo = Yi B (8gRiSo / fi)
(3)
where Ri is the hydraulic radius for the covered flow and fi is a composite resis-
tance coefficient for the covered channel. Channel width, B, is taken to be con-
stant. It is assumed here that the overall reach slope, So, does not change. The ice
cover, by reducing the flow velocity, reduces the portion of the flow energy
gradient (or stream power) expended as flow drag along the channel's bed.
For an alluvial channel, a reduction in energy gradient usually implies an
adjustment in planform geometry. Since an ice cover deepens and slows the flow
in a channel, the channel responds as if it were set at a flatter slope. In effect, the
channel responds as if it were conveying an equivalent open water flow whose
cross-sectional area is as shown in Figure 10c but whose energy gradient is
reduced. The effective hydraulic radius, resistance coefficient, and energy gradi-
ent of the equivalent flow are Rio, fio, and Sio, respectively, with Rio ≈ 2Ri and Sio <
So. For this equivalent open water flow,
Qi = Qo = Yi B (8gRioSio / fio).
(4)
Equation 3 and 4 give
2
3
3
Sio fio Yo Ro fio Yo
Yo
=
≈
≈
.
(5)
So fo Yio Rio fo Yio
Yio
Equation 5 assumes the wide-channel approximation Ro ≈ Yo and Rio ≈ Yio,
and, because the bed sediment does not change and if the ice cover is fairly level,
the ratio fio/fo ≈ 1.
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