BEDROCK INCISION DUE TO WEAR

1
CHAPTER 30 BEDROCK INCISION DUE TO WEAR
This chapter was written by Phairot Chatanantavet
and Gary Parker.
A slot canyon in the southwestern United States
resulting from bedrock incision
2
FROM mns TO A MORE PHYSICALLY-BASED MODEL OF
BEDROCK INCISION
In Chapter 16 Morphodynamics of Bedrock-alluvial
Transitions, it is assumed that the bedrock
platform is fixed in time and is not free to
undergo incision. This is generally true at the
scale of adjustment of alluvial streams, but is
not true in longer geomorphic time. In Chapter
29 Knickpoint Migration in Bedrock Streams, a
formulation for the morphodynamics of bedrock
streams was developed using the following
incision law This relation has provided useful
results, but does not adequately express the
physics of the incisional process. Recently
Parker (2004) has developed a model that
incorporates three mechanisms a) wear caused as
bedload particles strike bedrock (Sklar and
Dietrich, 2004), plucking, by which chunks of
fractured bedrock are torqued out of the bed by
the flow and broken up, and macroabrasion, by
which these chunks are further broken up as
bedload particles strike them (Whipple et al.,
2000). Here a model of incision due to wear
based on Sklar and Dietrich (2004) is developed.
3
OVERVIEW OF BEDROCK INCISION
As noted in the previous slide, aspects of
bedrock rivers were introduced in Chapter 16 and
29. As described in Chapter 16, a bedrock river
has patches of bed that are not covered by
alluvium, and where bedrock is exposed. There
are many ways to cause a river to incise into its
own bedrock. In this chapter, only the process
of wear (abrasion) is considered (e.g. Sklar and
Dietrich, 2004). That is, the bedrock is
gradually worn away as bedrock particles strike
regions of the bed where bedrock is exposed.
A bedrock river in Kentucky (tributary of Wilson
Creek) with a partial alluvial covering. Image
courtesy A. Parola.
A bedrock river in Japan. Image courtesy H.
Ikeda.
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BEDROCK INCISIONAL ZONE
Floor of subsiding graben
Alluvial fan
Uplifting, incising zone
Incisional zone and alluvial fan in Tarim Basin,
China.
Bedrock incision does not need to, but can be
strongly driven by uplift. The Above example
shows incision in an uplifting mountain zone,
with the resulting sediment deposited in an
adjacent subsiding graben.
5
HILLSLOPE DIFFUSION AND LANDSLIDING
Oregon Coast Range USA, Image courtesy Bill
Dietrich
As the channel cuts down in response to uplift,
it causes the adjacent hillslopes to erode by
hillslope diffusion or landsliding.
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WEAR (ABRASION) PROCESS DRIVEN BY COLLISION
The model for incision driven by wear presented
here is similar to that given in Sklar and
Dietrich (2004). Wear or abrasion is the process
by which stones colliding with the bed grind away
the bedrock to sand or silt. Wear is treated in
terms of relations of the same status as those
used to predict gravel abrasion in rivers (e.g.
Parker, 1991). The stones that do the wear are
assumed to have a characteristic size Dw.
Let q(x) denote the volume transport rate of
sediment in the stream per unit width (L2/T)
during the storm events that drive abrasion. Let
the fraction of this load that consists of
particles coarse enough to do the wear be ?. The
volume transport rate per unit width qwear of
sediment coarse enough to wear the bedrock is
then given as
7
WEAR PROCESS DRIVEN BY COLLISION contd.
For simplicity, ? might be set equal to the
fraction of the load that is gravel or coarser.
A more sophisticated formulation might use a
discriminator such as the ratio of shear velocity
to fall velocity, as in Sklar and Dietrich
(2004). Here ? is taken to be a prescribed
constant. Consider the case of saltating bedload
particles. Let Esaltw denote the volume rate at
which saltating wear particles bounce off the bed
per unit bed area L/T and Lsaltw denote the
characteristic saltation length of wear particles
L. It follows from simple continuity that
The mean number of bed strikes by wear particles
per unit bed area per unit time is equal to
Esaltw/Vw, where Vw denotes the volume of a wear
particle. It is assumed that with each collision
a fraction r of the particle volume is ground off
the bed (and a commensurate, but not necessary
equal amount is ground off the wear particle).
The rate of bed incision vIw due to wear is then
given as (number of strikes per unit bed area) x
(volume removed per unit strike), or
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WEAR PROCESS DRIVEN BY COLLISION contd.
and
from
It is found that
Here the parameter ?w has dimensions 1/L, and
has exactly the same status as the abrasion
coefficients used to study downstream fining by
abrasion in rivers. This parameter could be
treated as a constant. In so far as Lsaltw
depends on flow conditions and r depends on rock
type and perhaps the strength of the collision,
?w might be expected to vary somewhat with flow
and lithology. The above relation is valid only
to the extent that all wear particles collide
with exposed bedrock. If wear particles
partially cover the bed, the wear rate should be
commensurately reduced. This effect can be
quantified in terms of the ratio qwear/qwearc,
where qwearc denotes the capacity transport rate
of wear particles. Let po denote the areal
fraction of surface bedrock that is not covered
with sediment. In general po can be expected to
approach unity as qwear/qwearc ? 0, and approach
zero as qwear/qwearc ? 1.
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COVER FACTOR FOR INCISION BY WEAR
A cover function of the following type
is proposed by Sklar and Dietrich (2004)
, therefore
and finally
Wear particles striking other wear particles do
not wear the bed
qwear/qwearc
Note that vIw drops to zero when ?q becomes equal
to qwearc, downstream of which a completely
alluvial gravel-bed stream is found. That is,
the above formulation can describe the end point
of the incisional zone as well as the incision
rate.
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EXPERIMENT ON UNDERCAPACITY TRANSPORT OF GRAVEL
The image on the left shows an inerodible
concrete- bed flume with grooves at St. Anthony
Falls Laboratory, University of Minnesota, USA.
The design of the grooves is based on Piccaninny
Creek, Australia (Wohl, 1998). Experiments are
underway to investigate the value and dependence
of the exponent no in the cover function. The
picture below shows a top view of a sample
experimental run with the ratio qwear/qwearc
0.64 also channel slope 2.0, Froude number
1.3, and Shields number ? 0.11. The size of
the gravel is 7 mm. Note that the bed is not
completely covered with gravel.
Flow direction
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CAPACITY BEDLOAD TRANSPORT RATE OF EFFECTIVE
TOOLS FOR WEAR
The parameter qwearc can be quantified in terms
of standard bedload transport relations. A
generalized relation of the form of Meyer-Peter
and Müller (1948), for example, takes the form
where g, ?, and R are given, ?b denotes bed shear
stress, ?c? denotes a dimensionless critical
Shields number, ?T is a dimensionless constant
and nT is a dimensionless exponent. For example,
in the implementation of Fernandez Luque and van
Beek (1976), ?T 5.7, nT 1.5 and ?c? is
between 0.03 and 0.045.
As outlined in Chapter 5, the standard
formulation for boundary shear stress places it
proportional to the square of the flow velocity U
qf/H where qf denotes the flow discharge per
unit width and H denotes flow depth. More
precisely,
where Cf is a friction coefficient, which here is
assumed to be constant for simplicity.
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CAPACITY BEDLOAD TRANSPORT RATE contd.
For the steep slopes of bedrock streams, the
normal flow approximation, according to which the
downstream pull of gravity just balances the
resistive force at the bed, should apply, so that
momentum balance takes the form (Chapter 5)
or
The bedload transport rate of wear material is
then evaluated as
The concept of below-capacity conditions is
reviewed in Chapter 16. Briefly described here,
an alluvial stream that is too steep relative to
its sediment supply rate of wear material qwear
would degrade to a lower slope S that would allow
the above equation to transport wear material at
the rate qwear. A bedrock stream, however,
cannot degrade (without bedrock incision). So if
for given values of qf and S it turns out that
the sediment supply rate qwear is less than the
equilibrium mobile-bed value qwearc, the river
responds by exposing bedrock on its bed instead
of degrading. As qwear is further reduced the
river responds by increasing the fraction of the
bed over which bedrock is exposed (Sklar and
Dietrich, 1998). The bedrock river so adjusts
itself to transport wear sediment at a rate qwear
which is below its capacity qwearc for the given
values of qf and S.
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CAPACITY BEDLOAD TRANSPORT RATE contd.
Now let i denote the precipitation rate (L/T),
Bc(x) denote channel width, and A(x) denote the
drainage basin area upstream of the point at
distance x from a virtual origin near the
headwater of the main-stem stream . Assuming no
storage of water in the basin, the balance for
water flow is
The parameter ? L is a surrogate for
down-channel distance x. It will appear
naturally in the model. Also, note that hydrology
now enters into the model through the rainfall
rate. The capacity bedload transport rate of
effective tools for wear is then given as
14
SEDIMENT TRANSPORT RATE IN BEDROCK RIVERS
A routing model is necessary to determine the
volume sediment transport rate per unit width q,
and thus qwear. The equation of sediment
conservation on a bedrock reach can be written as
where qh denotes the volume of sediment per unit
stream length per unit time entering the channel
from the hillslopes (either directly or through
the intermediary of tributaries). Several models
can be postulated for qh depending on hillslope
dynamics. For simplicity, it is assumed that the
watershed consists of easily-weathered rocks that
are rapidly uplifted, so that bed lowering by
channel incision results in hillslope lowering at
the same rate. In this case
15
SEDIMENT TRANSPORT RATE IN BEDROCK RIVERS contd.
Note that in the latter equation, vI is the total
incision rate and not just that due to wear.
Note that the latter equation is just an example
that must later be generalized to forms including
e.g. hillslope diffusion, hillslope relaxation
due to landslides driven by e.g. earthquakes or
saturation in the absence of uplift, etc. The
above two equations lead to
The above relation can be used in the case of
weak deviation from steady-state incision. In the
case of steady-state incision in response to
spatially uniform (piston-style) uplift, it
reduces to qBc vIA, or thus
Note that the parameter ? naturally arises from
the formulation.
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SEDIMENT TRANSPORT RATE IN BEDROCK RIVERS contd.
To obtain an approximate treatment of the case of
deviation from steady-state incision in response
to piston-style uplift, it is useful to postulate
the structure relation
or equivalently
In general, ?b 0.02 and nb 0.3 to 0.5
(Montgomery and Gran, 2001 Whipple,
2004). Drainage area A can be written in the
function of down-channel distance x in terms of
Hacks law (Hack, 1957).
and
Between
the following relation is obtained after some
work
17
MODIFICATION OF EXNER
If the river is assumed to be morphologically
active only intermittently (during floods), the
Exner equation becomes where vI denotes the
instantaneous incision rate during a flood
(rather than the long-term average value used
in Chapter 29) and

• ? uplift rate
• ?p porosity of bedrock ( 0)
• I intermittency of large flood events
  (fraction of time)
• In the case of a more general hydrologic model

• where Ik fraction of time the flood flow is in
  the kth flow range
• Uplift is not really continuous, but it is
  treated as such here for simplicity.

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FLOW DURATION CURVE Q flow discharge, PQ
fraction of time exceeded
PQ ? 100
Bedrock!
The fractions Ik can be extracted from a flow
duration curve such as the example given above.
19
BEDROCK INCISION MODEL DUE TO WEAR
Summary of the previous results
The sediment transport rate and the incision
rate talk to each other. The incision rate at a
point is a function of all incision upstream.
20
BEDROCK INCISION MODEL DUE TO WEAR contd.
and
From
obtain
To solve this equation, introduce the new variable
from which
and then
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GOVERNING EQUATION ANDUPSTREAM BOUNDARY CONDITION
This equation is a first-order ordinary
differential equation (ODE). After obtaining one
boundary condition, it can be solved numerically,
i.e. by the Runge-Kutta method.
It is assumed that the channel begins at x xb,
upstream of which is a debris flow dominated zone
(x 0 to xb).
The appropriate form of
at the channel head (x xb) is
or
Substituting into
where again the subscript b denotes the channel
head
results in the relation
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UPSTREAM BOUNDARY CONDITION contd.
and
Equate
to obtain
or
at
which is the boundary condition for the first
order ODE below. Note that qwearcb denotes
the value of qwearc at ? ?b.
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MAKING THE PARAMETER ? DIMENSIONLESS
To solve the O.D.E. numerically, ? is first
recast into a dimensionless parameter varying
from 0 to 1. Where ?L denotes the value of ? at
the downstream end of the basin, where x L,
or
? ?L
Thus the ODE becomes
This is solved numerically to obtain Wd, i.e. by
the Runge-Kutta method with the previously
derived boundary condition.
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FINAL EQUATIONS IN BEDROCK INCISION MODEL
If bed elevation is held constant at the
downstream end, the downstream boundary condition
on the Exner equation becomes
Not too difficult to model in any program
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NUMERICAL MODEL USING RUNGE-KUTTA TO SOLVE FIRST
ORDER O.D.E.
or summarizing
INPUT Initial values , Wdb, step size h,
and M(1/h) OUTPUT Approximation Wdn1 to the
solution
where n0,1, M-1 For n0, 1,
, M-1 do
subject to the b.c.
at
End
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NUMERICAL MODEL DISCRETIZATION
upstream
downstream
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INTRODUCTION TO RTe-bookBedrockIncisionWear.xls
The program computes the time evolution of the
long profile of a bedrock river with incision due
to wear (abrasion). The output also includes
plots of sediment transport a, slope, incision
rate vI and areal fraction of bed exposure po as
they vary in time. A generalized relation of the
form of Meyer-Peter and Müller (1948) relation is
used to compute bedload transport capacity.
Resistance is specified in terms of a constant
Chezy coefficient Cz. The flow is calculated
using the normal flow (local equilibrium)
approximation. The drainage area is computed by
using Hack's law and the river has varying width
by the relation The basic input parameters are
nb, i, I, bw, Dw, Cz, a, Sinit, xb, L, N, dt, and
au. The auxiliary parameters include gt, nt, tc,
R, lp, no, Kh, nh, Kb, nb, and h. Note that the
value of the initial slope Sinit must be
sufficiently high so that the lowest value of
sediment transport, which is at the headwater,
exceeds zero.
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INTRODUCTION TO RTe-bookBedrockIncisionWear.xls
contd.
An estimat of the minimum initial slope (Sinit)
for each set of inputs is also shown at the
bottom of the page Calculator. This estimation
is calculated by fitting a line to the lower
bound of a band given in Sklar and Dietrich
(1998) dividing alluvial coarse bed streams from
bedrock streams. The relation so obtained is
where drainage basin area A is measured in
km2.
Figure from Sklar and Dietrich (1998)
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INTRODUCTION TO RTe-bookBedrockIncisionWear.xls
contd.
The final set of input includes the reach length
L, the number of intervals N into which the reach
is divided (so that ?x L/N), the time step ?t,
the upwinding coefficient ?u , and two parameters
controlling output the number of time steps to
printout Ntoprint and the number of printouts
Nprint. A value of ?u 0.25 is recommended for
stability in this program. The basic program in
Visual Basic for Applications is contained in
Module 1, and is run from worksheet
Calculator. In any given case it is necessary
try various values of the parameter N (which sets
?x) and the time step ?t in order to obtain good
results. For any given ?x, it is appropriate to
find the largest value of ?t that does not lead
to numerical instability. The program is
executed by clicking the button Click to run the
program from the worksheet Calculator.
Outputs are given in numerical form in worksheet
ResultsofCalc and in graphical form in four
worksheets beginning with the word Plot. Some
sample calculations are as follows.
30
A SAMPLE CALCULATION BEDROCK INCISION IN RIVERS
DUE TO WEAR
31
A SAMPLE CALCULATION BEDROCK INCISION contd.
The results in the next slide (Slide 32) were
generated the following input parameters uplift
rate ? 5 mm/yr, initial river bed slope Sinit
0.006, effective rainfall rate i 25 mm/hr,
flood intermittency ? 0.002, wear coefficient
bw 0.0001 m-1, effective size of particles that
do the wear Dw 50 mm, fraction of load
consisting of sizes that do the wear ? 0.05,
bed friction coefficient Cf 0.01, and value xb
at the channel head 1500 m. The total river
length is L 10 km. The total time of calculation
is 7200 years. The results produce an autogenic,
upstream-propagating knickpoint. Slide 32
explains how such a knickpoint, which is not
forced by such factors as base level drop, is
formed. The results in Slide 34 have the same
input parameters as in Slide 32 except that the
rock is rendered weaker by increasing the wear
coefficient ?w to 0.0002 m-1. The results show
that no autogenic knickpoint produced by the
model in this case. Slide 34 shows results for
the case of a sudden base level fall. The input
parameters are the same as those of Slide 32
except that at in the first year there is a base
level fall of 30 m at the downstream end. The
results manifest a knickpoint propagating
upstream as well but here, but this time
allogenically induced by base level fall.
32
SAMPLE RESULTS WITH WEAR COEFFICIENT ? 0.0001
m-1
33
HOW CAN AN AUTOGENIC KNICKPOINT FORM?
The process can be briefly explained as follows.
Consider the Exner equation of Slide 19. Taking
the second derivative in x and assuming a
constant uplift rate results in
or
Now consider the plot of incision rate in the
previous slide at year zero. Note that the shape
of the curve of the incision rate vIw changes
from concave-upward upstream to convex-upward
downstream at a point near 4000 m. Thus the term
changes from positive to
negative near this point. Considering the above
equations, a stream with such a shape of the
incision curve must gradually form an autogenic
knockpoint such that the term has a
sign opposite to . This results
in an elevation curve that changes from upward
convex in the upstream reach to upward concave in
the downstream reach. The inflection point
sharpens to a knickpoint and migrates upstream.
The size of an autogenic knickpoint can vary
depending on the input parameters. The next slide
shows a case without an autogenic knickpoint.
Note that the shape of the curve of the incision
rate at the initial year is convex-upward
everywhere.
34
SAMPLE RESULTS WITH WEAR COEFFICIENT ? 0.0002
m-1
35
SAMPLE RESULTS SUDDEN BASE LEVEL FALL
36
SOME COMMENTS FOR THOUGHTS

• In Chapter 16 alluvial river profiles were found
  to change over time scales of a few hundred
  years. Here bedrock rivers are seen to evolve
  over time scales of thousands of tens of
  thousands of years. The assumption of Chapter
  16, then, that the bedrock platform is fixed over
  characteristic time scales for alluvial
  adjustment is thus justified. At longer time
  scales incision cannot be ignored.
• The results presented in this chapter support the
  idea that knickpoints can form due to autogenic
  processes, in addition to allogenic forcing such
  as base level drop. In some cases, then, bedrock
  incision by knickpoint migration may thus simply
  be a consequence of the normal abrasion process.
  More details concerning this can be found in
  Chatanantavet and Parker (2005).

37
REFERENCES FOR CHAPTER 30
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. Hack, J.T., 1957, Studies of longitudinal
stream profiles in Virginia and Maryland. Prof.
Paper 294-B, US Geological Survey,
45-97. Montgomery, D.R. Gran, K.B. 2001.
Downstream variations in the width of bedrock
channels. Water Resources Research, 37, 6,
1841-1846. Parker, G. 1991. Selective sorting and
abrasion of river gravel I Theory, Jour. of
Hydraulic Eng. 117, 2, 131-149. Parker, G., 2004,
Somewhat less random notes on bedrock incision,
Internal Memorandum 118, St. Anthony Falls
Laboratory, University of Minnesota, 20 p.,
downloadable at http//cee.uiuc.edu/people/parkerg
/reports.htm . Sklar, L.S. Dietrich, W.E.,
1998, River longitudinal profiles and bedrock
incision models Stream power and the influence
of sediment supply, in River over rock fluvial
processes in bedrock channels. Rivers over Rock,
Geophysical Monograph Series, 107, edited by
Tinkler, K. and Wohl, E.E., 237 260, AGU,
Washington D.C. Sklar, L.S. Dietrich, W.E,
2004, A mechanistic model for river incision into
bedrock by saltating bed load, Water Resources
Research, 40, W06301, 21 p.
38
REFERENCES FOR CHAPTER 30 contd.
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, Jour. of Geophysical Res., 104,
B8, 17661-17674. Whipple, K.X., Hancock, G.S.
and Anderson, R.S., 2000, River incision into
bedrock Mechanics and relative efficacy of
plucking, abrasion, and cavitation, Geological
Society of America Bulletin, 112,
490503. Whipple, K.X., 2004, Bedrock rivers and
the geomorphology of active orogens, Annual
Review Earth and Planetary Sciences, 32,
151-185. Wohl, E. E., 1998, Bedrock channel
morphology in relation to erosional processes,
Rivers over Rock, Geophysical Monograph Series,
107, edited by Tinkler, K. and Wohl, E.E., 133
151, AGU, Washington D.C.