Amanda Brown
ID: 300291567
Geog 319: Essay
1) Discuss the major controls on stream power for bedrock and alluvial rivers with
particular focus on the roles of the exponents m and n in the stream power law.
Introduction
Fluvial erosion has a primary role in altering the environment around us and as
a result of its significance, it is not surprising that Tucker and Whipple (2002)
argue that understanding the dynamics of river erosion is essential to
constructing models of landscape evolution over long time scales. The rate that
channels incise establishes the rate at which the surface landscape evolves and
hence channel incision is the essential link between landscape and tectonics
(Hancock, Anderson and Whipple, 1998).
It is gravitational potential energy that is the driver of the fluvial system, that
gravitational energy is converted into another form and it is this transformed
energy which has the power to erode. Stream power is the transformed energy
and is often thought of as ability of flowing water to carryout geomorphic work.
The main controls on stream power are slope, discharge and bed lithology.
Many equations have been developed in order to calculate the stream power
and it is argued that the most applicable for bedrock incision is the stream
power erosion law due to it being directly related to the physics of erosion
(Howard and Kerby, 1983). For alluvial rivers Seidl and Dietrich (1993) suggest it
is the sediment transport law that is appropriate. As is typical with any equation
the stream power law is generalized. Unfortunately in order to simplify it some
of the complexities of the process mechanics are obscured (Skylar and Dietrich,
2001). As a result a wide range of complex processes are characterised together
under a single coefficient.
Amanda Brown
ID: 300291567
Geog 319: Essay
What is stream power?
Stream power is commonly described as the capacity for flowing water to carry
out geomorphic work, as Bizzi and Lerner (2015) simplified, it offers a measure
of the driving forces within a channel. It is gravitational potential energy that
acts as the driver for the fluvial system. The gravitational potential energy is
converted into kinetic form through the downslope flow of water, where due to
the conservation of energy, any energy lost due to falling must be converted in
to another form (Anderson and Anderson 2010). This transformed energy is
what has the potential to erode channel beds or transport sediments.
As it is stream power that details fluvial erosion, it is therefore unsurprising that
it has a significant effect on the fluvial system with Knighton (1999) arguing that
stream power influences channel characteristics like that of bedform type,
channel pattern and processes like the migration of the channel or the bedload
transport rate.
Stream power law
The development of the stream power equation is necessary because of the
impossibility and expense of collecting all the data needed. By generalizing
equations and using variables that can be interpreted remotely such as drainage
area and slope gradient, calculations can be made for areas where it is either
inaccessible or too expensive to collect data.
The main variables controlling channel erosion and sediment transport rate are
that of slope, discharge and bed lithology. Calculating channel erosion or
transport rates at large spatial and temporal scales, especially due to the
variation of discharge over significant periods of time, can be difficult due to
limited data. Howard and Kerby (1983) suggested that upslope drainage area
can be used as a proxy for that of dominant discharge, Howard, Dietrich and
Seidl (1994) state the surrogate is possible due to the average long term incision
Amanda Brown
ID: 300291567
Geog 319: Essay
rate being proportional to the sheer stress exerted by the dominant discharge
within the channel.
Many equations have been developed to help predict channel incision of a river
into its bed. It is argued by Howard and Kerby (1983) that it is the Stream Power
Law (1), of all that have been proposed, is the most applicable due to it relating
to the physics of erosion. It is also relevant to a wide variety of studies due to its
generality, however as Wipple and Tucker (1999) state, with such a
generalization there are some poorly understood parameters.
The generalized and most frequently used stream power erosion law (1) is
useful because there are few variables and can be measured against
topographical data. (Howard & Kerby, 1983; Skylar & Dietrich, 2001).
E = KAmSn
(1)
This model can also be used to calculate the sediment transport rate (2) which
as Seidl and Dietrich (1993) state is the appropriate form for alluvial channels.
The fundamental parts of the stream power erosion law and the transport law
are quite similar in each equation, with incision being replaces by sediment
transport rate.
qs = KQm Sn
(2)
Geomorphic parts of the equation
E = KAmSn
With E representing the erosion or incision rate, the variable K is as Whipple
(2001) describes a dimensional coefficient of erosion that considers many
factors such as the rock strength and the erosional capabilities of the fluvial
system. A is defined as upstream drainage area which acts as a substitute for
discharge and S represents the slope or stream gradient.
Amanda Brown
ID: 300291567
Geog 319: Essay
qs = KQm Sn
Where qs represents the sediment transport rate and K is once again the
dimensional coefficient. Q is the water discharge and S represents the stream
slope.
Stock and Montgomery (1999) identify the exponent m as the representation of
the discharge drainage area interaction which is weighted by the importance of
discharge on incision and could vary between river basins due to variations in
hydrological processes. Little information is available on the slope exponent n,
perhaps due to the process involved not being completely understood.
Although Whipple (2001) suggests it depends on the dominant erosion process.
The n exponent is said to be directly associated to the degree of nonlinearity in
the link between erosion rate and stream power (Whipple and Tucker, 1999).
The exponent is also said to be important in determining the landscape
response time.
Very few studies have reported the values for the exponents m and n. Howard
and Kirby (1983) conducted a study in the anthropogenic Virginian Badlands to
determine that m=≈1/3 and n=≈2/3 (assuming erosion rates are linearly
proportional to bed shear stress).
The ratio m/n has been called the concavity index as it influences the speed
with which the gradient changes as area increases (Anderson and Anderson,
2011). This reflects the shape a rivers longitudinal profile takes. Whipple and
Tucker (1999) argued that the ratio is independent of the dominant erosion
process and dependant on only the relative rates of increase of discharge. It is
thought by many studies that the ratio has been shown to significantly influence
the shape of stream profiles.
Amanda Brown
ID: 300291567
Geog 319: Essay
Whipple (2001) states that the m/n ratio is geometric term as a result of it being
influenced by the downstream rates of increase in discharge and channel width
and thereby does not change considerably in response to tectonic or climatic
disturbances. This means that the ratio is predicted to fall between 0.35 and 0.6
which conforms to values calculated from field data. Larger values of m/n
typically mean the river profile has greater concavity.
However Stock and Montgomery (1999) conducted a study in order to calibrate
the equation (1), their results showed that there was no individual set of results
for m and n that could be applied universally to all channels studied and that
boundary conditions had a significant influence on the parameters.
Stream Power Law Assumptions
Skylar and Dietrich (2001) state the simplicity of the stream power equation
comes at the expense of obscuring the complexities of the process mechanics.
This means that a wide range of complex processes are put together under the
same coefficient. Individually these processes would not usually be
characterised together by a common stream power parameter (Howard et at.,
1994). As a result the equation carries quite a few assumptions, the most
significant being that parameters are kept constant over time and space. This is
especially the case with K, which assumes that lithology and precipitation are
unvarying and implies that slope is the only variable that may adjust in response
to change in boundary conditions (Whipple and Tucker 1999). As a result the
simple act of changing the channel from bed rock to that of alluvial cover would
violate these assumptions.
In fact Seidl and Dietrich (1992) state that the exponent m and the coefficient K
are particularly sensitive to climate variations. For example changes in the
temperature, distribution and amount of precipitation affect the drainagedischarge area relationship which influences m.
Amanda Brown
ID: 300291567
Geog 319: Essay
The presence or absence of alluvium cover on a bed is mostly determined by
the gradient (Howard and Kerby, 1983). Alluvial rivers traditionally have a lesser
gradient and have an inverse dependence upon drainage area. Bedrock rivers
however, are usually steeper lacking a sediment cover. It is not surprising that
there is a difference in stream power between the two channel types when
slope or gradient is a large control upon the equation. With a steeper gradient
the stream power is likely increased. A channel bed with no alluvial cover,
where sediment supply is limited, stream power is likely to concentrate on
incision. However for an alluvial channel with an excess supply of sediment, a
transport limited channel, stream power is instead likely to be used for
transport.
Conclusion
Fluvial erosion and therefore stream power plays an important role in the
evolution of the landscape. Many authors have developed equations in order to
calculate stream power and the one argued to be the most applicable to
bedrock incision is the Stream Power Erosion model (1) due to it relating to the
physics of erosion. For alluvial channels it is the sediment transport equation (2)
that is the most applicable.
The major controls on the equation are that of slope, discharge/upstream area
and erodibility of the rock. The parameters m and n are not understood
comprehensively, although m is thought to represent the drainage discharge
area interaction and n is said to be directly associated to the nonlinear link
between the erosion rate and stream power. N is also thought to be significant
in determining the landscape response time. Howard and Kirby’s (1983) study
of the Virginian Badlands determined that m=≈1/3 and n=≈2/3, although no
single set of numbers can be applied universally to all rivers.
Amanda Brown
ID: 300291567
Geog 319: Essay
The ratio of m/n defines the longitudinal profile of a channel and is often called
the concavity index. The ratio is thought to be particularly robust when facing
climate variations and is predicted to fall between 0.35 and 0.6 which conforms
to values calculated from field data. Often the larger the m/n ratio number is
the more concave a channels profile is.
Due to the simplified nature of the stream power equations many complex
processes are grouped together under a single parameter. This means that
there some significant assumptions in order for the equation to hold true. An
example is the assumption that the parameters such as dimensional coefficient
is held spatially constant over time. It simply takes the widening of the channel
or the addition of alluvial cover to void this assumption. The parameters K and
m seem to particularly vulnerable to climate and simply takes a change in
precipitation or temperature for the discharge area relationship to be
influenced.
The stream power law is a powerful tool in order to understand the processes
behind fluvial interaction with the landscape and provides a cheaper alternative
to the labour intensive gathering of field data. However further study into
understanding the constants K, m and n is needed due to assumptions being
easily violated.
Amanda Brown
ID: 300291567
Geog 319: Essay
References
Book
Anderson, R. S., & Anderson, S. P. (2010) Geomorphology: The
mechanics and chemistry of landscapes. Cambridge, New York: Cambridge
University Press.
Howard, A. D. (1980) Thresholds in river regimes. In D. R. Coates & J. D.
Vitek (Eds.), Thresholds in Geomorphology (pp. 227-258). Boston, London: Allen
and Unwin.
Journal
Bizzi, S., & Lerner, D. N. (2015). The use of stream power as an indicator
of channel sensitivity to erosion and deposition processes. River Research and
Applications, 31(1), 16-27.
Hancock, G. S., Anderson, R. S., & Whipple, K. X. (1998). Beyond power:
Bedrock river incision process and form. Rivers over rock: Fluvial processes in
bedrock channels, 35-60.
Howard, A. D. (1980). Thresholds in river regimes. Thresholds in
geomorphology, 227-258.
Howard, A. D., & Kerby, G. (1983). Channel changes in badlands. GSA
Bulletin, 94(6), 739.
Howard, A. D., Dietrich, W. E., & Seidl, M. A. (1994). Modeling fluvial
erosion on regional to continental scales. Journal of Geophysical Research - Solid
Earth, 99(B7), 13971. doi:10.1029/94JB00744
Seidl, M. A., & Dietrich, W. E. (1993). The problem of channel erosion
into bedrock. Catena supplement, 23, 101-101.
Stock, J. D., & Montgomery, D. R. (1999). Geologic constraints on
bedrock river incision using the stream power law. Journal of Geophysical
Research: Solid Earth (1978–2012), 104(B3), 4983-4993.
Tucker, G. E., and K. X. Whipple, Topographic outcomes predicted by
stream erosion models: Sensitivity analysis and intermodel comparison, J.
Geophys. Res., 107(B9), 2179, doi:10.1029/2001JB000162, 2002.
Whipple, K. X. (2001). Fluvial landscape response time: how plausible is
steady-state denudation?. American Journal of Science, 301(4-5), 313-325.
Amanda Brown
ID: 300291567
Geog 319: Essay
Whipple, K. X., & Tucker, G. E. (1999). Dynamics of the stream‐power
river incision model: Implications for height limits of mountain ranges,
landscape response timescales, and research needs. Journal of Geophysical
Research: Solid Earth (1978–2012), 104(B8), 17661-17674.