KNICKPOINT MIGRATION IN BEDROCK STREAMS

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CHAPTER 29 KNICKPOINT MIGRATION IN BEDROCK
STREAMS
View of landscape in New Zealand dominated by
incising bedrock streams with knickpoints. Image
courtesy B. Crosby.
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WHAT IS A KNICKPOINT?
A knickpoint is a point of discontinuity in a
river profile. Knickpoints can be manifested in
terms of either a discontinuity in bed slope S
- ??/?x or bed elevation ?. The second kind of
discontinuity consists of a waterfall.
discontinuity in slope
discontinuity in elevation
Both types of knickpoints are characteristic of
bedrock streams rather than alluvial streams, and
both types tend to migrate upstream. This
chapter is focused on knickpoints of the first
type.
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AN EXAMPLE OF A RIVER PROFILE WITH KNICKPOINTS
The volcanic uplands of the Hawaiian Islands are
dominated by bedrock streams. The top photograph
to the left shows the Mana Plain and the region
of incised uplands behind it, on the island of
Kauai. The bottom image to the left shows the
location of the Kauhao River on northern edge of
the Mana Plain.
The long profile of the Kauhao River below shows
a clear knickpoint (DeYoung, 2000 see also
Chatanantavet and Parker, 2005).
adjacent ridges
channel bed
knickpoint
Plot from DeYoung (2000)
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WHY ARE KNICKPOINTS CHARACTERISTIC OF BEDROCK
STREAMS AND NOT ALLUVIAL STREAMS?
Knickpoints in alluvial streams are generally
transient features that are not self-preserving.
This was shown in Slides 4 and 36 of Chapter 14.
An elevation discontinuity in alluvium created
by e.g. an earthquake sets up erosion upstream
and deposition downstream that causes the
knickpoint to dissipate rapidly. This is due to
the strongly diffusive component to 1D alluvial
morphodynamics discussed in Slide 16 of Chapter
14, which acts to smear out slope
differences. One exception to this rule is a
gravel-sand transition, where a quasi-steady
(arrested) break in slope can be maintained in
alluvium, as discussed in Chapter 27. The
example to the left is that of the Kinu River,
Japan, introduced in Chapter 27.
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EXNER EQUATION OF SEDIMENT CONTINUITY FOR A
BEDROCK STREAM
Since bedrock streams are common (but not
confined to) in regions of active tectonic
uplift, the uplift rate ? is include in the
formulation. Bedrock is assumed to be exposed on
the bed of the river, and the river is assumed to
be incising into this bedrock as sediment-laden
water flows over it. Denoting the incision rate
as vI, the Exner equation of sediment continuity
takes the form where denotes a long-term
average incision rate (rather an instantaneous
incision rate during floods). For many problems
of interest involving bedrock the bed porosity ?p
can be approximated as 0.
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PHENOMENOLOGICAL RELATION FOR INCISION RATE mns
Until recently most treatments of the
morphodynamics of incising bedrock streams have
used the following phenomenological relation for
the incision rate where A is the drainage area
upstream of the point in question, S - ??/?x is
the slope of the river bed at that point, K is a
coefficient and m and n are exponents that may
vary (Kirkby, 1971 Howard and Kerby, 1983). The
relation make physical sense. Drainage basin A
can be thought to be a measurable surrogate for
flow discharge. Larger discharge and larger bed
slope can both be though to enhance incision.
This notwithstanding, the formulation is somewhat
dissatisfying, because as long as m and n are
free variables, the dimensions of the coefficient
K are also free to vary, so defying physical
sense. A more physically based, dimensionally
homogeneous formulation based on the work of
Sklar and Dietrich (1998) is introduced in the
next chapter. Meanwhile, mns provides a quick
way to gain insight on knickpoint migration in
bedrock streams. Stock and Montgomery (1999)
provide a compendium of values of K, m and n. A
fairly standard pair of choices for m and n is m
0.5, n 1.
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THE MORPHODYNAMICS OF INCISION ARE CHARACTERIZED
BY A LINEAR KINEMATIC WAVE EQUATION
Consider as an example the simple choice m 0
and n 1 in the incision relation When
substituted into the Exner equation of Slide 5,
it is found that This is a 1D (kinematic)
wave equation, and c denotes the wave speed of
upstream migration. In the absence of uplift (?
0), the entire bed profile thus migrates
upstream so as to preserve form. That is, if
?i(x) denotes the bed profile at t 0, the
solution to the above equation (with ? 0) is
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GENERALIZATION TO FULL mns 1D NONLINEAR WAVE
EQUATION
The general morphodynamic problem is obtained
from the relations of Slides and 6 as follows
where the absolute value ensures incision
is always associated with a positive bed slope S
- ??/?x. This relation reduces to a kinematic
wave equation in which the speed of upstream
migration c varies nonlinearly with bed
slope. Now the bed profile deforms as it
migrates upstream.
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STEADY STATE RIVER PROFILE FOR WHICH INCISION
BALANCES UPLIFT
The nonlinear kinematic wave equation of the
previous page admits the following solution for
the steady state case for which the incision rate
everywhere perfectly balances the uplift rate
? For a constant uplift rate ? the
steady-state long profile of river slope is then
given as Various researchers (e.g. Whipple
and Tucker, 1999) have used this relation and
measured values of S and A to deduce parameters
in the incision relation of Slide 6, and in
particular the ratio m/n. For the fairly
standard values m 0.5, n 1, the above
relation gives Since drainage area A typically
increases in the downstream direction, the above
relation typically predicts a steady state
profile that is upward-concave (with slope S
decreasing in the downstream direction.
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MIGRATION OF KNICKPOINTS CONSISTING OF SLOPE
DISCONTINUITIES
The focus of this chapter is neither the full
morphodynamics of bedrock incision described by
the relations of Slide 8, nor the morphodynamics
of the steady state. It is rather on knickpoint
migration in bedrock streams. A knickpoint
consisting of a slope discontinuity is a kind of
shock that applied mathematicians call a front.
At a front the parameter in question (bed
elevation) is continuous, but its derivative (bed
slope) is not. Now let sk(t) be the position of
the knickpoint, i.e. the moving boundary between
the regimes upstream and downstream of it
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CONDITION OF ELEVATION CONTINUITY AT THE
KNICKPOINT
Let ?u(x, t) denote the bed profile upstream of
the knickpoint and ?d(x, t) denote the profile
downstream. The condition of elevation
continuity at the knickpoint requires that The
migration speed of the knickpoint can be obtained
in the same way that the migration speed of a
bedrock-alluvial transition was determined in
Chapter 16. Taking the derivative of both sides
of the equation with respect to t,
i.e. results in where dsk/dt
denotes the migration speed of the knickpoint.
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RELATION FOR MIGRATION SPEED OF THE KNICKPOINT
Now let Sku be the bed slope just upstream of the
knickpoint and Skd be the bed slope just
downstream of the knickpoint, so that Reducing
the equation at the bottom of the previous slide
with the above definitions results
in Further reducing the above relation with
the Exner equation of Slide 6 results in the
following relation for
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INTERPRETATION
First consider the case Sku gt Skd, i.e. a sudden
drop in slope in the streamwise direction. This
is the case illustrated for the Kauhao River,
Kauai in Slide 3. As long as vI is an
increasing function of S (n gt 0 in the incision
relation K AmSn), it follows that (Ak,
Sku) gt (Ak, Skd), so that and the
knickpoint migrates upstream. Next consider the
case Sku lt Skd. The same reasoning coupled with
the assumption the m gt 0 also yields and
thus the knickpoint again migrates upstream! The
implication is that knickpoints in bedrock
characterized by slope discontinuities always
migrate upstream, at least in the context of the
analysis presented here.
h
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OUTLINE OF A MOVING BOUNDARY FORMULATION FOR
INCISION WITH KNICKPOINT MIGRATION
Let x 0 denote the upstream end of the domain,
x sk(t) denote the position of the knickpoint
and x sf denote the downstream end of the
calculational domain, a point that is taken as
fixed. Introduce the following coordinate
transformations.
Domain upstream of knickpoint
Domain downstream of knickpoint
Note that goes from 0 to 1 over the upstream
reach and goes from 0 to 1 over the
downstream reach. The Exner equation is then
transformed to the moving boundary coordinates.
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COORDINATE TRANSFORMATION FOR THE UPSTREAM REACH
Upstream form for Exner
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COORDINATE TRANSFORMATION FOR THE DOWNSTREAM REACH
Downstream form for Enxer
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SETUP FOR NUMERICAL SOLUTION
The two kinematic wave equations and the
relation for migration speed are solved
subject to a downstream boundary condition and a
continuity condition. The downstream boundary
condition is one of prescribed variation in base
level (i.e. constant base level or base level
fall at a prescribed rate), and the continuity
condition is one of matching elevations. Thus
where ?base(t) denotes the time variation in
downstream base level, the boundary conditions
take the form The numerical implementation,
which is not given here, follows the outline of
Chapter 16 for a bedrock-alluvial transition.
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REFERENCES FOR CHAPTER 29
Chatanantavet, P. and Parker, G., 2005, Modeling
the bedrock river evolution of western Kauai,
Hawaii, by a physically-based incision model
based on abrasion, River, Coastal and Estuarine
Morphodynamics, Taylor and Francis, London,
99-110. DeYoung, N.V, 2000, Modeling the
geomorphic evolution of western Kauai, Hawaii
a study of surface processes in a basaltic
terrain, M.S. thesis, Dalhousie University, Nova
Scotia, Canada. Kirkby, M.J., 1971, Hillslope
process-response models based on the continuity
equation, Slopes Form and Process, Spec.
Publication 3, Institute of British Geographers,
London, 15-30. Howard, A.D. and Kerby, G., 1983,
Channel changes in badlands, Geological Society
of America Bulletin, 94(6), 739-752. Stock, J.D.,
and Montgomery, D.R., 1999, Geologic constraints
on bedrock river incision using the stream power
law, Journal Geophysical Research, 104, 4983
4993. Whipple, K.X., and 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,
104, 17,661 17,674.