Basic Concepts and Management Issues of Rivers - River Dynamics and Integrated River Management - page 20

channel slope, such as a waterfall or a quake lake, resulting from differential rates of erosion above and
below the knickpoint. Differential rates of erosion can be resulted from a change in the lithology of the
river channel and bed structures. The river, having gained more potential energy due to gravity, will then
proceed to work the knickpoints out of its system by either erosion (in the case of waterfalls) or
deposition (in the case of quake lakes) in order for the river to reattain its smooth concave graded profile.
In the southwesten China almost all streams have knickpoints, which are resulted mostly from landslide
dams. The knickpoints consist of not bed rocks but step-pool system.
Longitudinal bed profile of alluvial rivers —For alluvial rivers the bed materials consist mainly of
sand (or silt). The longitudinal profile is usually concave upward. As described previously in the discussion
of dynamic equilibrium, streams adjust their profiles and patterns to try to minimize the time rate of
expenditure of potential energy, or stream power, present in flowing water. The concave upward shape of
a stream's profile appears to be due to adjustments a river makes to help minimize stream power in the
downstream direction. Stream power has been defined as the product of discharge and slope. Since stream
discharge typically increases in the downstream direction, slope must decrease in order to minimize stream
power. The decrease in slope in the downstream direction results in the concave up longitudinal profile.
According to the minimum stream power theory, the morphology of fluvial rivers develops to reach
the minimum stream power (Yang, 1983). This can be described by the following equation:
d

P d d s d Q

J §

·

( J sQ ) Q s 0 (1.23)

¨ ¸

dd x x d d x x

© ¹

in which P is the stream power, J is the specific weight of water, s is the riverbed slope, and x is the
distance along the river course. For most rivers, the discharge increases along its course due to the inflow
from tributaries; thus, the term s d Q /d x is positive. According to Eq. (1.23), the term Q d s /d x must be negative,
or the slope of the riverbed decreases along its course; so that rivers exhibit concave riverbed profiles.
Most longitudinal profiles of streams are concave upward. As described previously in the discussion of
dynamic equilibrium, streams adjust their profiles and patterns to minimize the time rate of expenditure
of potential energy, or stream power, present in flowing water. The concave upward shape of a stream's
profile appears to be due to adjustments a river makes to help minimize stream power in a downstream
direction. The decrease in slope in a downstream direction results in the concave longitudinal profile.
Figure 1.18 shows the distributions of annual average discharge and average water stage along the course
of the Yangtze River. Following the increase in the discharge, the slope of the water surface, which in
fact represents the average bed slope, becomes gentler. The bed profile is then a concave shape.

Fig. 1.18 Distribution of annual average discharge and average stage, which represents the longitudinal bed profile
of the river, along the course of the Yangtze River