Transcript Slide 1

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
knickpoint
knickpoint

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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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 Kaua’i. 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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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
 

(1  p )
     vI
 t

where v I 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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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PHENOMENOLOGICAL RELATION FOR INCISION RATE: m&n’s
Until recently most treatments of the morphodynamics of incising bedrock streams
have used the following phenomenological relation for the incision rate:
vI  KAmSn
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, m&n’s
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. 6
A fairly standard pair of choices for m and n is m = 0.5, n = 1.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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:
vI  KS
When substituted into the Exner equation of Slide 5, it is found that


K
c
 , c
t
x
(1  b )
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
(x, t)  i (x  ct)
profile at time t
cSDt
cDt
profile at time t+Dt
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
GENERALIZATION TO FULL m&n’s: 1D NONLINEAR WAVE EQUATION
The general morphodynamic problem is obtained from the relations of Slides and 6
as follows:
n1

vI  KA
x
m

x
 

m 
(1  p )
    KA
x
 t

n1

x
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.


K

c
 , c
Am
t
x
(1  p )
x
Now the bed profile deforms as it migrates upstream.
n1
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 v I everywhere perfectly
balances the uplift rate :
KAmSn  
For a constant uplift rate  the steady-state long profile of river slope is then given as
1/ n

S    Am / n
K 
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

S    A 0.5
K 
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 9
S decreasing in the downstream direction.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
knickpoint
sk(t)

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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
u [sk (t),t]  d [sk (t ),t]
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.
d
d
u [sk (t), t ]  d [sk (t), t ]
dt
dt
results in:
u
u
d
d


 sk

 sk
t sk
x sk
t sk
x sk
where s k = dsk/dt denotes the migration speed of the knickpoint.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
Sku  
u

, Skd   d
x sk
x sk
Reducing the equation at the bottom of the previous slide with the above
definitions results in:
u
d

 skSku 
 s kSkd
t sk
t sk
Further reducing the above relation with the Exner equation of Slide 6 results in
the following relation for s k :
sk   vI ( Ak ,Sku )  vI ( Ak ,Skd )
1  p Sku  Skd 
,
Ak  A(sk )
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTERPRETATION
sk   vI ( Ak ,Sku )  vI ( Ak ,Skd )
1  p Sku  Skd 
,
Ak  A(sk )

First consider the case Sku > Skd, i.e. a sudden drop in
slope in the streamwise direction. This is the case
illustrated for the Kauhao River, Kaua’i in Slide 3. As
long as vI is an increasing function of S (n > 0 in the
incision relation v I = K AmSn), it follows that v I(Ak, Sku)
> v I (Ak, Skd), so that s k  0 and the knickpoint
migrates upstream.
Next consider the case Sku < Skd. The same reasoning
coupled with the assumption the m > 0 also yields s k  0
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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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:
~
x
x
sk ( t )
~
t t
Domain
downstream of
knickpoint:
xˆ 
x  sk ( t )
s f  sk ( t )
ˆt  t
x goes from 0 to 1 over the upstream reach and xˆ goes from 0 to
Note that ~
1 over the downstream reach. The Exner equation
 

m 
(1  p )     KA
x
 t

n1

x
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is then transformed to the moving boundary coordinates.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COORDINATE TRANSFORMATION FOR THE UPSTREAM REACH
~
x
x
~
, t t
sk ( t )
~
~
x 1
t
~
x
x 
s k
~

,
 0,
  2 sk   x ,
x sk
x
t
sk
sk
~
 ~
x  t 
1 



x x ~
x x ~t sk ~
x
~
 ~
x  t 
s k 

~


 x
 ~
~
~
~
t t x t  t
s x  t
~
t
1
t
k
Upstream form for Exner:
n1
m

u sk ~ u
K
A u u

x


~ s
n
~
~
~
x (1  p ) sk x
x
t
k
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COORDINATE TRANSFORMATION FOR THE DOWNSTREAM REACH
x  sk ( t )
xˆ 
,
sf  sk ( t )
xˆ
1

,
x s f  sk
ˆt  t
ˆt
0
x
 1
xˆ
x  sk  
s k
 

s  (1  xˆ )
,
2 k
t
( s f  sk )
 s f  sk ( s f  sk ) 

xˆ  ˆt 
1




x x xˆ x ˆt (s f  sk ) xˆ
ˆt
1
t
 xˆ  ˆt 
s k




 (1  xˆ )

t t xˆ t ˆt
(s f  sk ) xˆ ˆt
Downstream form for Enxer
d
s k
d
K
Am
d
ˆ

(1  x )

ˆ
ˆ
x (1  p ) (s f  sk )n xˆ
t (s f  sk )
n1
d

ˆ
x
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SETUP FOR NUMERICAL SOLUTION
The two kinematic wave equations
n1
u s k ~ u
K
Am u u


~  s x ~
n
~
~
x
(
1


)
s

x

x
t
k
p
k
d
s k
d
K
Am
d

(1  xˆ )

xˆ (1  p ) (s f  sk )n xˆ
ˆt (s f  sk )
n1
d

ˆ
x
and the relation for migration speed
m
n
n
K
(
A
)
[(
S
)

(
S
)
]
k
ku
kd
s k  
1 p Sku  Skd 
,
Ak  A(sk )
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
d xˆ 1  base (t) , u ~x 1  d xˆ 0
The numerical implementation, which is not given here, follows the outline of
Chapter 16 for a bedrock-alluvial transition.
17
1D SEDIMENT
TRANSPORT
MORPHODYNAMICS
1D SEDIMENT
TRANSPORT
MORPHODYNAMICS
with applications to
with applications to
RIVERS AND TURBIDITY CURRENTS
RIVERS
AND
TURBIDITY
CURRENTS
© Gary Parker
November,
2004
REFERENCES FOR CHAPTER 29
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.
DeYoung, N.V, 2000, Modeling the geomorphic evolution of western Kaua’i, Hawai’i; 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.
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