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where
τ b is the local bed shear stress, determined by
τ b = γ
hS , with h being the local
flow depth;
c is the critical shear stress, which is given zero in the case of deposition;
m is an exponent; y is the cross-stream coordinate; and B is the channel width at the
water surface.
The value of m is generally between 0 and 1; it essentially affects the pattern
of bed change distribution. A small value means a fairly uniform distribution of
τ
z b along the cross-section, while a larger value gives a less uniform distribution
of
z b . In Chang's model, the value of m is determined at each time step, such
that a correction in the channel bed profile will result in the most rapid movement
toward uniformity in power expenditure, or linear water surface profile, along the
channel.
Eq. (5.167) is only applied in straight channels. For curved channels, the following
curvature-weighting relation is used to adjust the cross-section:
m
τ
)
/
r
b
c
B b τ c )
z b
=
r
A b
(5.168)
m
y
/
where r is the coordinate along the radius of channel bend.
Similar relations can be obtained by replacing the excess shear stress
τ b τ
c in
Eqs. (5.167) and (5.168) with the excess velocity U
U c .
A simplification can be made by setting
τ
=
0 and S to be constant along the
c
cross-section. Thus, Eq. (5.167) becomes
h m
B h m
z b =
y
A b
(5.169)
A more complicated method for lateral allocation of bed change is the stream tube
model proposed by Yang et al . (1998). The entire cross-section is divided into several
stream tubes, and a 1-Dmodel is adopted to simulate the flow, sediment transport, and
bed change in each stream tube. This technique is more like a quasi-two-dimensional
approach. The shape of the cross-section is adjusted according to the assumption of
minimum stream power.
In addition, the change in bed elevation due to the consolidation of cohesive bed
material needs to be considered. This is discussed in Section 11.1.6.
5.3.6 1-D simulation of bank erosion and channel
meandering
5.3.6.1 1-D bank erosion model
Stream bank erosion occurs due to channel degradation, toe erosion, mass failure,
seepage flow, weathering, etc. Channel bed degradation increases bank heights, and
lateral erosion undercuts bank toes. Both processes make banks steeper and more
unstable. Seepage flow and weathering may aggravate these processes. Once the
stability criterion is exceeded, a bank mass failure event occurs and the bank top
retreats. The failed bank material is first piled on the bed near the bank toe and then
washed away by flow. Thus, bank erosion can significantly affect sediment balance
and channel morphology in rivers.
 
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