Transcript Slide 1

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
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FROM m&n’s 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:
m n
vI  KA S
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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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 Japan. Image
courtesy H. Ikeda.
A bedrock river in Kentucky
(tributary of Wilson Creek) with
a partial alluvial covering. 3
Image courtesy A. Parola.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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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 4
sediment deposited in an adjacent subsiding graben.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
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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
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to erode by hillslope diffusion or landsliding.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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WEAR (ABRASION) PROCESS DRIVEN BY COLLISION
Wear (abrasion)
silt
grrrrindch!!
Dust ground off striking rock
and striken bedrock surface
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
qwear  q
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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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
qwear  Esaltw Lsaltw
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
vIw 
Esaltw
rVw
Vw
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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WEAR PROCESS DRIVEN BY COLLISION contd.
from
qwear  Esaltw Lsaltw
It is found that
vIw   w qwear
and
vIw 
, w 
Esaltw
rVw
Vw
r
L saltw
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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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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COVER FACTOR FOR INCISION BY WEAR

q
A “cover function” of the following type is
po  1  wear
proposed by Sklar and Dietrich (2004);
 qwearc
open (exposed) bedrock
no



, therefore
vIw  w qwear po
and finally
1
v Iw
po
0
1
0
qqswear
/qswearc
wear/qwearc

q
  w q1 
 qwearc



no
Wear particles
striking other wear
particles do not
wear the bed
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
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can describe the end point of the incisional zone as well as the incision rate.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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EXPERIMENT ON UNDERCAPACITY
TRANSPORT OF GRAVEL
The image on the left shows an inerodible concretebed 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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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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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
n
qwearc
 b

 RgDw D w  T 
 c 
 RgD w

T
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,
q2f
b  Cf 2
H
where Cf is a friction coefficient, which here is assumed to be constant
for simplicity.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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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)
q2f
b  Cf 2  gHS
H
The bedload transport rate of wear
material is then evaluated as
or
b  C1f/ 3g2 / 3q2f / 3S2 / 3
 b

qwearc  RgDw D w  T 
 c 
 RgD w

nT
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 mobilebed 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
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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.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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
qfBc  iA
or
qf  i

w here
A
Bc
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
qwearc  RgD w Dwqwearc
qwearc
 C1f/ 3i2 / 32 / 3 2 / 3  
 T 
S  c 
1/ 3
 Rg Dw

nT
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
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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
d
qBc   qh
dx
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
qh  vI
dA
dx
A
x
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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SEDIMENT TRANSPORT RATE IN BEDROCK RIVERS contd.
d
qBc   qh
dx
qh  vI
dA
dx
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
d
qBc   vI dA
dx
dx
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
q  vI ,  
A
Bc
Note that the parameter  naturally arises from the formulation.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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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
Bc  b Anb
~
or equivalently Bc  
b
mb
~  1/(1nb ) , m  nb
, 
b
b
b
1  nb
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 Hack’s law (Hack, 1957).
A  Khxnh
Between
w here Kh ~ 6.7 and nh ~ 1.7
d
qBc   vI dA
dx
dx
the following relation is
obtained after some work:
and
~ mb
Bc  
b
mb  1 
q  mb  vImb d
0

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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
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MODIFICATION OF EXNER
If the river is assumed to be morphologically active only intermittently (during
floods), the Exner equation becomes

(1  p )
   IvI
t
where vI denotes the instantaneous incision rate during a flood (rather than the
long-term average value v I 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
N

(1  p )
    Ik vI,k
t
k 1
where Ik = fraction of time the flood flow is in the kth flow range
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Uplift is not really continuous, but it is treated as such here for simplicity.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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FLOW DURATION CURVE
Q = flow discharge, PQ = fraction of time exceeded
Pongema River Papua New Guinea
percent of time exceeded
PQ  100
100
90
80
70
60
50
%
\
40
30
20
10
0
1
10
100
1000
Bedrock!
Q
The fractions Ik can be extracted from a flow duration curve such as
the example given above.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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BEDROCK INCISION MODEL DUE TO WEAR
Summary of the previous results
(1   p )

   Iv I
t

q
v I   w q1 
 qwearc



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.
no
mb  1 
q  mb  vImb d
0

C i 

qwearc  RgDw Dw  T 
S2 / 3  c 
 Rg Dw

1/ 3 2 / 3
f
1/ 3
2/3
nT
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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BEDROCK INCISION MODEL DUE TO WEAR contd.
From

q
vI   w q1 
 qwearc
no



and
mb  1 
q  mb  vImb d
0


 mb  1 

mb



v

d


 
I

mb


0


m 1 

obtain

vI   w  bmb  vImb d  1  
0

qwearc
 
 





To solve this equation, introduce the new variable
1 dWd

v

I
mb
from which
Wd  vImb d
d

0
no

and then

 mb  1  
  mb Wd  



m 1 
dWd

 mb  w  bmb Wd  1  

d
qwearc
 
 





no
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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GOVERNING EQUATION ANDUPSTREAM BOUNDARY CONDITION
This equation is a first-order

 mb  1  
ordinary differential equation
  mb Wd  





dWd
m

1

(ODE). After obtaining one
 mb  w  bmb Wd  1  

d
qwearc
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
d
The appropriate
flow dominated zone (x = 0 to xb).
qBc   vI dA
form of
dx
dx
at the channel head (x = xb) is
x=0
no
qBc b  vIA b
x = xb
results in
the relation
Substituting into
vIb 
qwearcb
b
or
qb  vIbb

q
vI   w q1 
 qwearc
no



  1 1/ n0  where again the subscript
  “b” denotes the channel
1  
  w b   head
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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UPSTREAM BOUNDARY CONDITION contd.
Equate
to obtain
or
qb  vIbb
qb 
and
mb  1
Wdb
mb
b
vIbmbb 1
Wdb 
mb  1
1 / n0

 1  
qwearcb 
 
1  
Wdb 
mb  1   w b  


mb
b
  b
at
which is the boundary condition for the first order ODE below.

 mb  1  
  mb Wd  





dWd
m

1

 mb  w  bmb Wd  1  

d
qwearc
 
 





Note that qwearcb denotes the value of qwearc at  = b.
no
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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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,
  b
~

L  b
=b
  L  b ~
  b
 = L
or
d   L   b d~

Thus the ODE becomes

dWd
mb  1Wd
 L  b w mb  1Wd 1 
mb
~
~
d












qwearc

L
b
b



n0
This is solved numerically to obtain Wd, i.e. by the Runge-Kutta method with
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the previously derived boundary condition.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
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FINAL EQUATIONS IN BEDROCK INCISION MODEL

dWd
mb  1Wd












m

1
W
1

L
b
w
b
d 
m
d~

  b  b qwearc
 L  b ~
Wd ~0
qwearc
1 / n0

 1  
qwearcb 
 
1  
 Wdb 
mb  1   w b  


mb
b
 C1f/ 3i2 / 32 / 3 2 / 3

 RgDw Dw  T 
S


c
1/ 3
 Rg Dw

m 1
q  bmb Wd


q
vI   w q1 
 qwearc
nT
  b
~

L  b

S
x
no






n0
(1  p )

   Iv I
t
If bed elevation is held constant at the downstream end, the downstream boundary
condition on the Exner equation becomes
 x L   ~ 1  const (e.g. 0)
Not too difficult to model in any program
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
NUMERICAL MODEL: USING RUNGE-KUTTA TO SOLVE FIRST ORDER O.D.E.

dWd
mb  1Wd












m

1
W
1

L
b
w
b
d 
m
d~

  b  b qwearc
 L  b ~
or summarizing
dWd
~
~

f
[

,
q
(
), Wd ]
wc
~
d
0 , Wdb, step size h, and M(=1/h)
INPUT: Initial values ~
OUTPUT: Approximation Wdn+1 to the solution Wd (~
n 1 )
~
~
where n=0,1, … M-1
at n 1  0  (n  1)h
For n=0, 1, …, M-1 do:
k1  h * f ~
n, Wdn 
subject to the b.c.
1
1 

k2  h * f  ~
n  h, Wdn  k1 
2
2 

1
1 

k3  h * f  ~
n  h, Wdn  k 2 
2
2 

k  h * f ~
  h, Wd  k 
1 / n0
qwearcb mbb   1  
 
1  
Wdb 
mb  1   w b  


at



n0
~
0
4
n
n
3
~
n1  ~
n  h
End
Wdn1  Wdn 
1
k1  2k 2  2k 3  k 4 
6
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
NUMERICAL MODEL: DISCRETIZATION
S1  1  2  / x
x 
upstream
SN1  N  N1  / x
i=1
L
x  ( i  1)  x , i  1 ..N  1
i
N
x
2
downstream
3
N -1
N
i = N+1
L
(i1  i )
(i  i1)
Si  u
 (1  u )
x
x
qwearc ,i
 C1f/ 3i2 / 3 i 2 / 3 2 / 3


 RgDw Dw  T 
Si   c 
1/ 3
 Rg Dw




q
i

vI,i   w qi 1 

 qwearc ,i 
nT
no
i t t  i t 
mb  1
qi  mb Wd,i
i
t
  I vI,i 
(1  p )
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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
Bc  b Anb
The basic input parameters are b, i, I, w, Dw, Cz, , Sinit, xb, L, N, dt, and u.
The auxiliary parameters include t, nt, c*, R, p, 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.
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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
S  0.03A0.5
where drainage basin area A is measured in km2.
Figure from Sklar and
Dietrich (1998)
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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
29
“Plot”. Some sample calculations are as follows.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
A SAMPLE CALCULATION: BEDROCK INCISION IN RIVERS DUE TO WEAR
Input Parameters
(ub)
(irain)
(Inter)
(betaw)
(Dw)
(Cz)
(alpha)
(Sinit)
(xu)
(L)
(etaend)
(N)
(dx)
(dt)
(Ntoprint)
(Nprint)
(au)
b
i
I
w
Dw
Cz

Sinit
xb
L
end
N
dx
dt
Ntoprint
Nprint
u
5
25
0.002
0.0001
50
10
0.05
0.006
1500
1.36E+00
10000
0
100
100
4
300
6
0.25
7200
mm/yr
mm/hr
m-1
mm
m
m
m
m
yr
uplift rate
Click to run the program
effective precipitation rate
Intermittency
Wear coefficient
Effective size of particles that do the wear
Chezy bed friction coefficient
Fraction of load consisting of sizes that do the wear
Ab
Initial bed slope (See the suggestion below)
Bcb
Value of x at channel "head"
Choose xb so that this number, i.e. d1 = wb exceeds unity
b
Reach length
Downstream elevation
Number of spatial intervals
Input Cell
Spatial step length
Time step length
Information Cell
Number of time steps to printout
Number of printouts
Here 1 = full upwind, 0.5 = central difference (0.25 is recommended in this program)
Total time of calculation, years
IMPORTANT : 1) Be sure to lower the value of dt if the code fails due to numerical instability.
2) Be sure that the initial bed slope (Sinit) is higher than the minimum value below for a bedrock stream.
Min. Sinit
4.10E-03
Sinit should also not exceed about 0.2, a value which indicates transition to a debris flow regime
2
1.68E+06 m
6.18E+00 m
2.72E+05 m
30
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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 I = 0.002, wear coefficient w =
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 31
induced by base level fall.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
SAMPLE RESULTS WITH WEAR COEFFICIENT  = 0.0001 m-1
Bed Evolution
Sediment Transport Per Unit Width
80
0.45
0.4
70
Elevation (m)
60
50
40
30
20
0.35
1200 yr
2400 yr
3600 yr
4800 yr
6000 yr
7200 yr
0 yr
0.3
q(m2/s)
0 yr
1200 yr
2400 yr
3600 yr
4800 yr
6000 yr
7200 yr
0.25
0.2
0.15
0.1
10
0.05
0
0
0
2000
4000
6000
8000
10000
12000
14000
0
Distance (m)
2000
4000
6000
8000
10000 12000 14000
Distance x (m)
Bedrock Incision Rate
Areal Fraction of Surface Bedrock not Covered with Wear Particles
25
1
0.8
1200 yr
2400 yr
3600 yr
4800 yr
6000 yr
7200 yr
0 yr
15
10
5
1200 yr
2400 yr
3600 yr
4800 yr
6000 yr
7200 yr
0 yr
0.6
po
vIw (mm/year)
20
0.4
0.2
0
0
2000
4000
6000
8000
Distance x (m)
10000 12000 14000
0
0
2000
4000
6000
8000
Distance x (m)
10000
12000
14000
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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
  2 vI 
  S 
   2 
 2vI
   I  2 
 2    I 2
or
t  x 
 x 
t x
x


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 concaveupward upstream to convex-upward downstream at a point near 4000 m.
Thus the term 2vI x 2 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  2 x 2
has a sign opposite to 2vI x.2This 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
33
year is convex-upward everywhere.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
SAMPLE RESULTS WITH WEAR COEFFICIENT  = 0.0002 m-1
Bed Evolution
Sediment Transport Per Unit Width
0.5
70
0.45
60
Elevation (m)
40
30
20
0.4
1200 yr
2400 yr
3600 yr
4800 yr
6000 yr
7200 yr
0 yr
0.35
q(m2/s)
0 yr
1200 yr
2400 yr
3600 yr
4800 yr
6000 yr
7200 yr
50
0.3
0.25
0.2
0.15
0.1
10
0.05
0
0
2000
4000
6000
8000
10000
12000
0
14000
0
Distance (m)
4000
6000 8000 10000 12000 14000
Distance x (m)
Areal Fraction of Surface Bedrock not Covered with Wear Particles
Bedrock Incision Rate
18
2000
1
16
0.8
1200 yr
2400 yr
3600 yr
4800 yr
6000 yr
7200 yr
0 yr
12
10
8
6
4
1200 yr
2400 yr
3600 yr
4800 yr
6000 yr
7200 yr
0 yr
0.6
po
vIw (mm/year)
14
0.4
0.2
2
0
0
0
2000
4000
6000 8000 10000 12000 14000
Distance x (m)
0
2000
4000
6000
8000
Distance x (m)
10000
12000
14000
34
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
SAMPLE RESULTS: SUDDEN BASE LEVEL FALL
Bed Evolution
Sediment Transport Per Unit Width
80
1
70
0.9
0.8
0 yr
200 yr
400 yr
600 yr
800 yr
1000 yr
1200 yr
50
40
30
20
0.7
q(m2/s)
Elevation (m)
60
200 yr
400 yr
600 yr
800 yr
1000 yr
1200 yr
0.6
0.5
0.4
0.3
0.2
10
0.1
0
0
0
2000
4000
6000
8000
Distance (m)
10000
12000
14000
0
Bedrock Incision Rate
2000
4000
6000 8000 10000 12000 14000
Distance x (m)
Areal Fraction of Surface Bedrock not Covered with Wear Particles
250
1
0.9
0.8
200 yr
400 yr
600 yr
800 yr
1000 yr
1200 yr
150
100
50
0.7
200 yr
400 yr
600 yr
800 yr
1000 yr
1200 yr
0.6
po
vIw (mm/year)
200
0.5
0.4
0.3
0.2
0.1
0
35
0
0
2000
4000
6000 8000 10000 12000 14000
Distance x (m)
0
2000
4000
6000
8000
Distance x (m)
10000
12000
14000
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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).
36
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
REFERENCES FOR CHAPTER 30
Chatanantavet, P. and Parker, G., 2005, Modeling the bedrock river evolution of western
Kaua’i, Hawai’i, 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.
37
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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, 490–503.
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.
38