Geology Reference
In-Depth Information
half the product of mass and velocity, so for a stream
it may be defined as
expressed in the oft-reproduced Hjulstrøm diagram
(Figure 3.11), cover a wide range of grain sizes and flow
velocities. The upper curve is a band showing the criti-
cal velocities at which grains of a given size start to erode.
The curve is a band rather than a single line because
the critical velocity depends partly on the position of the
grains and the way that they lie on the bed. Notice
that medium sand (0.25-0.5 mm) is eroded at the low-
est velocities. Clay and silt particles, even though they
are smaller than sand particles, require a higher velocity
for erosion to occur because they lie within the bottom
zone of laminar flow and, in the case of clay particles,
because of the cohesive forces holding them together.
The lower curve in the Hjulstrøm diagram shows the
velocity at which particles already in motion cannot be
transported further and fall to the channel bed. This is
called the fall velocity. It depends not just on grain size
but on density and shape, too, as well as on the vis-
cosity and density of the water. Interestingly, because
the viscosity and density of the water change with the
amount of sediment the stream carries, the relationship
between flow velocity and deposition is complicated. As
the flow velocity reduces, so the coarser grains start to fall
out, while the finer grains remain in motion. The result
is differential settling and sediment sorting . Clay and
silt particles stay in suspension at velocities of 1-2 cm/s,
which explains why suspended load deposits are not
dumped on streambeds. The region between the lower
curve and the upper band defines the velocities at which
particles of different sizes are transported. The wider the
gap between the upper and lower lines, the more con-
tinuous the transport. Notice that the gap for particles
larger than 2 mm is small. In consequence, a piece of
gravel eroded at just above the critical velocity will be
deposited as soon as it arrives in a region of slightly lower
velocity, which is likely to lie near the point of erosion. As
a rule of thumb, the flow velocity at which erosion starts
for grains larger than 0.5 mm is roughly proportional to
the square root of the grain size. Or, to put it another
way, the maximum grain size eroded is proportional to
the square of the flow velocity.
The Hjulstrøm diagram applies only to erosion, trans-
port, and deposition in alluvial channels. In bedrock
channels, the bed load abrades the rock floor and causes
vertical erosion. Where a stationary eddy forms, a small
mv 2 /2
=
E k
where m is the mass of water and v is the flow velocity.
If Chézy's equation (p. 71) is substituted for velocity, the
equation reads
=
E k
( mCRs )/2
This equation shows that kinetic energy in a stream is
directly proportional to the product of the hydraulic
radius, R (which is virtually the same as depth in large
rivers), and the stream gradient, s . In short, the deeper
and faster a stream, the greater its kinetic energy and
the larger its potential to erode. The equation also con-
forms to the DuBoys equation defining the shear stress
or tractive force,
τ
(tau), on a channel bed:
τ = γ
ds
where
(gamma) is the specific weight of the water
(g/cm 3 ), d is water depth (cm), and s is the stream
gradient expressed as a tangent of the slope angle.
A stream's ability to set a pebble in motion - its
competence - is largely determined by the product of
depth and slope (or the square of its velocity). It can
move a pebble of mass m when the shear force it creates
is equal to or exceeds the critical shear force necessary for
the movement of the pebble, which is determined by the
mass, shape, and position of the pebble in relation to
the current. The pebbles in gravel bars often develop an
imbricated structure (overlapping like tiles on a roof ),
which is particularly resistant to erosion. In an imbri-
cated structure, the pebbles have their long axes lying
across the flow direction and their second-longest axes
aligned parallel to the flow direction and angled down
upstream. Consequently, each pebble is protected by its
neighbouring upstream pebble. Only if a high discharge
occurs are the pebbles set in motion again.
A series of experiments enabled Filip Hjulstrøm
(1935) to establish relationships between a stream's flow
velocity and its ability to erode and transport grains of a
particular size. The relationships, which are conveniently
γ
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