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 Fundamentals of Fluvial Geomorphology and Channel Processes
dimensions (slope, width, depth), vegetation (type and density), and climate. Dependent
variables include: channel slope, depth, width, and planform.
A channel that has adjusted its dependent variables to accommodate the basin inputs
(independent variables) is said to be stable. Mackin (1948) gave the following definition of
a graded stream:
A graded stream is one in which, over a period of years, slope is
delicately adjusted to provide, with available discharge and with prevailing
channel characteristics, just the velocity required for the transportation of
the load supplied from the drainage basin. The graded stream is a system
in equilibrium.
Mackin did not say that a stream in equilibrium is unchanging and static. A more commonly
used term today for this type of stability is dynamic equilibrium. A stream in dynamic
equilibrium has adjusted its width, depth and slope such that the channel is neither aggrading
nor degrading. However, change may be occurring in the stream bank, erosion may result,
and bank stabilization may be necessary, even on the banks of a stream in dynamic
equilibrium.
The equilibrium concept of streams discussed above can also be described by various
qualitative relationships. One of the most widely used relationships is the one proposed by
Lane (1955) which states that:
QS % QsD50
where Q is the water discharge, S is the slope, Qs is the bed material load, and D50 is the
median size of the bed material. This relationship, commonly referred to as Lane's Balance,
is illustrated in Figure 2.14. Mackin's concept of adjustment to changes in the controlling
variables is easily illustrated by Lane's balance (Figure 2.14) which shows that a change in any
of the four variables will cause a change in the others such that equilibrium is restored. When
a channel is in equilibrium, it will have adjusted these four variables such that the sediment
being transported into the reach is transported out, without significant deposition of sediment
in the bed (aggradation), or excessive bed scour (degradation). It should be noted that by
this definition of stability, a channel is free to migrate laterally by eroding one of its banks and
accreting the one opposite at a similar rate.
Meandering can be thought of as nature's way of adjusting its energy (slope) to the
variable inputs of water and sediment. Cutoffs (oxbow lakes) and abandoned courses in the
floodplain attest to the dynamic behavior of rivers. Oftentimes the engineer or scientist draws
the erroneous conclusion that a dis-equilibrium condition exist because natural cutoffs are
occurring. However, this type of dynamic behavior is quite common in rivers that are in a
state of dynamic equilibrium. In this situation, as natural cutoffs occur, the river may be
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