| Longitudinal profiles of natural streams are known to show concave forms. Saying A as
drainage area, channel gradient S can be expressed as the power-law, S/A-θ (Flint, 1974),
which is one of the scale-invariant features of drainage basin. According to literature, θ of
most natural streams falls into a narrow range (0.4 < θ < 0.7) (Tucker and Whipple, 2002). It
leads to fundamental questions: “Why does θ falls into such narrow range?” and “How is
this related with other power-law scaling relationships reported in natural drainage
basins?”
To answer above questions, we analytically derive θ for a steady-state drainage basin
following Lane’s equilibrium (Lane, 1955) throughout the corridor and named this specific
case as the ’critical concavity’. In the derivation, sediment transport capacity is estimated by
unit stream power model (Yang, 1976), yielding a power function of upstream area. Stability
of channel at a local point occurs when incoming flux equals outgoing flux at the point.
Therefore, given the drainage at steady-state where all channel beds are stable, the exponent
of the power function should be zero. From this, we can determine the critical concavity.
Considering ranges of variables associated in this derivation, critical concavity cannot be
resolved as a single definite value, rather a range of critical concavity is suggested. This
range well agrees with the widely reported range of θ (0.4 < θ < 0.7) in natural
streams.
In this theoretical study, inter-relationships between power-laws such as hydraulic
geometry (Leopold and Maddock, 1953), dominant discharge-drainage area (Knighton et al.,
1999), and concavity, are coupled into the power-law framework of stream power sediment
transport model. This allows us to explore close relationships between their power-law
exponents: their relative roles and sensitivity. Detailed analysis and implications will be
presented.
References
Flint, J. J., 1974, Stream gradient as a function of order, magnitude, and discharge, Water
Resources Research, 10, 969-973.
Knighton, A. D., 1999, Downstream variation in stream power, Geomorphology, 29,
293-306.
Lane, E. W., 1955, The importance of fluvial morphology in hydraulic engineering,
American Society of Civil Engineers, Proceedings, 81, 1-17
Leopold, L. B., Maddock, T., 1953, The hydraulic geometry of stream channels
and some physiographic implications, United States Government Printing Office,
1953.
Tucker, G. E., Whipple, K. X., 2002, Topographic outcomes predicted by stream erosion
models: Sensitivity analysis and intermodel comparison, Journal of Geophysical Research,
107(B9), 2179, doi:10.1029/2001JB000162, 2002.
Yang, C. T., 1976, Minimum unit stream power and fluvial hydraulics, Journal of
Hydraulics Division, ASCE 102, 919-934. |