Transcript Slide 1

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 31:
EROSIONAL NARROWING AND WIDENING OF A CHANNEL AFTER DAM
REMOVAL
This chapter was written by Gary Parker, Alessandro Cantelli and Miguel Wong
View of a sediment control dam on the Amahata River, Japan.
Image courtesy H. Ikeda.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONSIDER THE CASE OF THE SUDDEN REMOVAL,
BY DESIGN OR ACCIDENT, OF A DAM FILLED WITH SEDIMENT
Before removal
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REMOVAL OF THE DAM CAUSES A CHANNEL TO INCISE INTO THE
DEPOSIT
After removal
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
AS THE CHANNEL INCISES, IT ALSO REMOVES SIDEWALL MATERIAL
sidewall sediment eroded as
channel incises
top of reservoir
deposit
A first treatment of the morphodynamics of this process was given in Chapter 15.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EXNER EQUATION OF SEDIMENT CONTINUITY WITH SIDEWALL EROSION
The formulation of Chapter 15 is reviewed here.
Bb = channel bottom width, here assumed constant
b = bed elevation
t = elevation of top of bank
Qb = volume bedload transport rate
Ss = sidewall slope (constant)
p = porosity of the bed deposit
s = streamwise distance
t = time
Bs = width of sidewall zone
s = volume rate of input per unit length
of sediment from sidewalls
t  b
 Ss
Bs
sidewall sediment eroded as
channel incises
Ss

Q
Bb b   b   s
t
s

  b b
s  2 Bs b  2 t
t
Ss
t
s > 0 for a degrading channel, i.e. b/t < 0
t
b
t
t
b
Bb
Bs
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EXNER EQUATION OF SEDIMENT CONTINUITY INCLUDING SIDEWALL
EROSION contd.
In Chapter 15, the relations of the previous slide
were reduced to obtain the relation:

  b  b
Q
 Bb  2 t

 b
Ss  t
s

or
b
Qb
1

t

t  b  s
 Bb  2

Ss 

That is, when sidewall erosion
accompanies degradation, the sidewall
erosion suppresses (but does not stop)
degradation and augments the
downstream rate of increase of bed
material load.
sidewall sediment eroded as
channel incises
Ss
t
b
t
t
b
Bb
Bs
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ADAPTATION TO THE PROBLEM OF CHANNEL INCISION SUBSEQUENT TO
DAM REMOVAL: THE DREAM MODELS
Cui et al. (in press-a, in press-b) have adapted the formulation of the previous two
slides to describe the morphodynamics of dam removal. These are embodied in the
DREAM numerical models. These models have been used to simulate the
morphodynamics subsequent to the removal of Saeltzer Dam, shown below.
Dam
1200 ft
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Saeltzer Dam, California before its removal in 2001.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE DREAM MODELS
Specify an initial top width Bbt and a
minimum bottom width Bbm.
Bbt
Ss
If Bb > Bbm, the channel degrades and
narrows without eroding its banks.
b
1 Qb

t
Bb s
Bb  Bbt  2Ss (t  b )
If Bb = Bbm the channel degrades and
erodes its sidewalls without further
narrowing.
Bb > Bbm
sidewall sediment eroded as
channel incises
b
Qb
1

t

t  b  s
 Bbm  2

Ss 

But Bbm must be user-specified.
Ss
8
Bb = Bbm
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SUMMARY OF THE DREAM FORMULATION
narrowing without
sidewall erosion
when Bb > Bbm
trajectories of left
and right bottom
bank position
top of deposit
Ss
no narrowing and
sidewall erosion
when Bb = Bbm
increasing
time
But how does the process really work? Some results from the experiments
of Cantelli et al. (2004) follow.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EROSION PROCESS VIEWED FROM DOWNSTREAM
Double-click on the image to see the video clip.
10
rte-bookdamremfrontview.mpg: to run without relinking, download to same folder as PowerPoint presentations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NOTE THE TRANSIENT PHENOMENON OF
EROSIONAL NARROWING
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EROSION PROCESS VIEWED FROM ABOVE
Double-click on the image to see the video clip.
12
rte-bookdamremtopview.mpg: to run without relinking, download to same folder as PowerPoint presentations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EVOLUTION OF CENTERLINE PROFILE
UPSTREAM (x < 9 m) AND DOWNSTREAM (x > 9 m) OF THE DAM
Upstream degradation
Downstream aggradation
Former dam location
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHANNEL WIDTH EVOLUTION UPSTREAM OF THE DAM
The dam is at x = 9 m downstream of sediment feed point.
Note the pattern of rapid channel narrowing and degradation, followed
by slow channel widening and degradation. The pattern is strongest
near the dam.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REGIMES OF EROSIONAL NARROWING AND EROSIONAL WIDENING
0
Progress in time
Water Surface Elevation (cm)
-1
The dam is at x = 9 m
downstream of sediment feed
point.
-2
First 4.3 minutes of run: period
of erosional narrowing
The cross-section is at x = 8.2
m downstream of the sediment
feed point, or 0.8 m upstream of
the dam.
-3
-4
-5
Subsequent 16.0 minutes of run:
period of erosional widening
-6
20
21
22
23
Water Surface Width (cm)
24
25
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SUMMARY OF THE PROCESS OF INCISION INTO A RESERVOIR DEPOSIT
trajectories of left
and right bottom
bank position
incisional narrowing
suppresses sidewall
erosion
incisional widening
enhances sidewall
erosion
top of deposit
Ss
rapid
incision with
narrowing
slow
incision with
widening
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CAN WE DESCRIBE THE MORPHODYNAMICS OF RAPID EROSIONAL
NARROWING, FOLLOWED BY SLOW EROSIONAL WIDENING?
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PART OF THE ANSWER COMES FROM ANOTHER SEEMINGLY UNRELATED
SOURCE: AN EARTHFLOW IN PAPUA NEW GUINEA
The earthflow is caused by the dumping of large amounts of waste rock
from the Porgera Gold Mine, Papua New Guinea.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE EARTHFLOW CONSTRICTS THE KAIYA RIVER AGAINST A VALLEY
WALL
The Kaiya River must somehow “eat” all the sediment delivered to it by the
earthflow.
Kaiya River
earthflow
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE DELTA OF THE UPSTREAM KAIYA RIVER IS DAMMED BY THE
EARTHFLOW
The delta captures all of the load from upstream, so downstream the Kaiya River
eats only earthflow sediment
earthflow
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE EARTHFLOW ELONGATES ALONG THE KAIYA RIVER, SO MAXIMIZING
“DIGESTION” OF ITS SEDIMENT
A downstream constriction (temporarily?)
limits the propagation of the earthflow.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE VIEW FROM THE AIR
Kaiya River
The earthflow encroaches on the river, reducing width, increasing bed shear
stress and increasing the ability of the river to eat sediment!
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE BASIS FOR THE SEDIMENT DIGESTER MODEL
(Parker, 2004)
Upstream dam created by earthflow
Earthflow
River
Sediment taken sideways into stream
• The earthflow narrows the channel, so increasing the sidewall shear stress and
the ability of the river flow to erode away the delivered material.
• The earthflow elongates parallel to the channel until it is of sufficient length to
be “digested” completely by the river.
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This is a case of depositional narrowing!!!
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
GEOMETRY
H = flow depth
n = transverse coordinate
nb = Bb = position of bank toe
Bw = width of wetted bank
nw = Bb + Bw = position of top
qˆ ne
of wetted bank
Ss = slope of sidewall (const.)
b = elevation of bed
= volume sediment input per unit
streamwise width from earthflow
H Bw  Ss
•
•
•
•
•
inerodible valley wall
nw
river
earthflow
qˆ ne
H
n
Ss
nb
Bb
b
Bw
b+H
The river flow is into the page.
The channel cross-section is assumed to be trapezoidal.
H/Bb << 1.
Streamwise shear stress on the bed region = bsb = constant in n
Streamwise shear stress on the submerged bank region = bss = bsb = constant
in n,  < 1.
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• The flow is approximated using the normal flow assumption.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EXNER EQUATION OF SEDIMENT BALANCE ON THE BED REGION
Local form of Exner:
(1   p )
q
q

  bs  bn
t
s
n
inerodible valley wall
where qbs and qbn are the streamwise and
transverse volume bedload transport rates
per unit width.
nw
river
(1   p )
0
/t(sediment in bed region)
qˆ ne
H
n
Integrate on bed region with qbs = qbss, qbn
= 0;
nb
earthflow
Ss
nb
Bb
nb q
nb q

bs
bn
dn   
dn  
dn
0
0
t
s
n
b
Bw
b+H

transverse input from wetted bank region
differential steamwise transport
b
qbsb qˆ bns
(1  p )


t
s
Bb
,
qˆ bns  qbn n
b
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EXNER EQUATION OF SEDIMENT BALANCE ON WETTED BANK REGION
Integrate local form of Exner on wetted
bank region with region with:
qbs = qbss for nb < n < nb + Bw
qbn = - qˆ neat n = nt where q denotes the
volume rate of supply of sediment
per unit length from the
earthflow
inerodible valley wall
nw
river
earthflow
qˆ ne
H
n
Ss
nb
Geometric relation:
  b  Ss (n  Bb )
Result:
(1   p )
nw
nb

B
 b

 Ss b
t
t
t
n w q
n w q

bs
bn
dn   
dn  
dn
nb
nn
t
s
n
/t(sediment in wetted
bank region)
differential
steamwise transport
Bb
b
Bw
b+H

transverse input
from earthflow
transverse output
to bed region
B 
1 
 
qbssH  qbss Bb  qˆ bns  qˆ be
(1  p )Bw  b  Ss b   
t 
Ss s
s
 t
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUATION FOR EVOLUTION OF BOTTOM WIDTH
inerodible valley wall
Eliminate b/t between
b
qbsb qˆ bns
(1  p )


t
s
Bb
and
nw
river
earthflow
qˆ ne
H
n
Ss
nb
Bb
b
Bw
b+H
B 
1 
 
qbssH  qbss Bb  qˆ bns  qˆ be
(1  p )Bw  b  Ss b   
t 
Ss s
s
 t
to obtain
Bb
qbsb
qbss H qbss Bb
Bw  Bb
H qbss
1
ˆ
(1  p )Ss




 qbns

qˆ be
t
s
SsBw s
SsBw s Bw s
BwBb
Bw
Note that there are two evolution equations for two quantities, channel bottom
elevation b and channel bottom width Bb. To close the relations we need to
have forms for qbsb, qbss and qˆ bns . The parameter qˆ ne is specified by the motion
of the earthflow.
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FLOW HYDRAULICS
Flow momentum balance: where S =
streamwise slope and Bw = H/Ss,

Ss  1  S2s H 

1 H


bbBb 1  
 gHSB b  Bb 

Ss
Bb 
2
S
B
s
b 



inerodible valley wall
nw
Flow mass balance
river

1 H

Q w  UHB b 1 
2
S
B
s
b 

earthflow
qˆ ne
H
n
Ss
nb
Bb
Manning-Strickler resistance relation
b
Bw
b+H
1/ 6
H
2
1/ 2
b  Cf U , Cf  r  
, k s  nkD
k
 s
Here ks = roughness height, D = grain size, nk = o(1) constant. Reduce under the
condition H/Bs << 1 to get:
 k 1s/ 3Q 2w 
H   2 2 
  r gBb S 
3 / 10
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Shields number on bed region:


bb
bb
1  k1s/ 3Q2w



RgD RD   r2gBb2



BEDLOAD TRANSPORT CLOSURE RELATIONS
3 / 10
S7 / 10
where R = (s/ - 1)  1.65. Shields
number on bank region:

  bs   bb
RgD

bs
  k1s/ 3Q2w
 2 2

RD   r gBb



inerodible valley wall
3 / 10
S
nw
7 / 10
river
 
qbsb  RgD D s bb

c
1  
 bb



Ss
nb
Bb
4 .5
,
qˆ ne
H
n
Streamwise volume bedload transport rate
per unit width on bed and bank regions is
qbsb and qbss, respectively: where s = 11.2
and c* denotes a critical Shields stress,
1.5
earthflow

qbss  RgD D s  bb
b

1.5
Bw

c
1  
 bb
b+H



4.5
(Parker, 1979 fit to relation of Einstein, 1950). Transverse volume bedload
transport rate per unit width on the sidewall region is qbns, where n is an orderone constant and from Parker and Andrews (1986),
qˆ bns  qbn B  qbss  n
b
c
Ss 

 bb

RgD D s 

 1.5
bb

c
1  
  bb
4.5

c
  n
Ss

 bb

29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SUMMARY OF THE SEDIMENT DIGESTER
Equation for evolution of bed elevation
Equation for evolution of bottom width
(1  p )Ss
(1  p )
b
q
qˆ
  bsb  bns
t
s
Bb
The earthflow encroaches on the channel
Bb
q
q
B  Bb
H qbss
H qbss Bb
1
  bsb 
 bss

 qˆ bns w

qˆ be
t
s
SsBw s
SsBw s Bw s
BwBb
Bw
Hydraulic relations
 k 1s/ 3Q 2w 
H   2 2 
  r gBb S 
3 / 10


bb
1  k1s/ 3Q2w


RD  r2gBb2



3 / 10
Sediment transport relations
 
qbsb  RgD D s bb
1 .5



qˆ bns  RgD D s  bb



4 .5
4 .5

c 
1   
  bb 
4. 5
1.5 
c 
1     n
  bb 

qbss  RgD D s  bb

c
1  
 bb
1 .5
S7 / 10
bs  bb
As the channel narrows the Shields
number increases
A higher Shields number gives higher
local streamwise and transverse
sediment transport rates.
Higher local streamwise and transverse

c
30
S s sediment transport rates counteract

 bb
channel narrowing
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM CHANNEL
Equilibrium channels that transport bedload without eroding their banks can be
created in the laboratory (Parker, 1979). The image below shows such a channel
(after the water has been turned off). The image is from experiments conducted by
J. Pitlick and J. Marr at St. Anthony Falls Laboratoty, University of Minnesota.
31
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM CHANNEL SOLUTION
As long as  < 1, the formulation allows for an equilibrium channel without
widening or narrowing as a special case (without input from an earthflow).
c
 
bb

qbss  RgD D s  bb

qˆ bns  RgD D s bb


1.5
1.5

c
1  
  bb

c
1  
 bb
 c
Qb  RgD D s 

1 .5






4.5
0
Streamwise sediment transport on wetted bank
region = 0
4.5

c
 n
Ss  0

bb

1  4.5 Bb
 k 1s/ 3Q 2w 
H   2 2 
  r gBb S 
bb
Choose bed shear stress so that bank shear stress
= critical value
 bs  bb  c
Transverse sediment transport on
wetted bank region = 0
Total bedload transport rate
3 / 10
c
bsb
1  k1s/ 3Q2w




 RgD RD   r2gBb2



3 / 10
S7 / 10
Three equations; if any two of Qw, S, H,
Qb and Bb are specified, the other three
can be computed!!
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ADAPTATION OF THE SEDIMENT DIGESTER FOR EROSIONAL NARROWING
• As the channel incises, it leaves exposed sidewalls below a top surface t.
• Sidewall sediment is eroded freely into the channel, without the
external forcing of the sediment digester.
• Bb now denotes channel bottom half-width
• Bs denotes the sidewall width of one side from channel bottom to top
surface.
• The channel is assumed to be symmetric, as illustrated below.
nt
t  b
 Ss
Bs
Ss
river
H
n
nb
Bb
b
Bs
t
33
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTEGRAL SEDIMENT BALANCE FOR THE BED AND SIDEWALL REGIONS
On the bed region, integrate Exner from n = 0 to n = nb = Bb to get
b
qbsb qˆ bns
(1  p )


t
s
Bb
On the sidewall region, integrate Exner from n = nb to n = nt under the conditions
that streamwise sediment transport vanishes over any region not covered with
water, and transverse sediment transport vanishes at n = nt
nt
t  b
 Ss
Bs
Ss
river
H
n
nb
Bb
b
Bs
t
34
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTEGRATION FOR SIDEWALL REGION
Upon integration it is found that
B 
1 
 
qbssH  qbss Bb  qˆ bns
(1  p )Bs  b  Ss b   
t 
Ss s
s
 t
or reducing with sediment balance for the bed region,
(1  p )Ss
Bb
q
B  Bb
H qbss qbss H qbss Bb
  bsb 


 qˆ bns s
t
s
SsBs s
SsBs s Bs s
BsBb
nt
t  b
 Ss
Bs
Ss
river
H
n
nb
Bb
b
Bs
t
35
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTEGRAL SEDIMENT BALANCE: SIDEWALL REGION
For the minute neglect the indicated terms:
B 
1 
 
qbssH  qbss Bb  qˆ bns
(1  p )Bs  b  Ss b   
t 
Ss s
s
 t
The equation can then be rewritten in the form:
B 
 
qˆ bns  (1  p )Bs   b  Ss b 
t 
 t
As the channel degrades i.e. b/t < 0, sidewall material is delivered to the
channel.
Erosional narrowing, i.e. Bb/t < 0 suppresses the delivery of sidewall
material to the channel.
36
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTEGRAL SEDIMENT BALANCE: SIDEWALL REGION contd.
trajectories of left
and right bottom
bank position
incisional narrowing
suppresses sidewall
erosion
top of deposit
Ss
incisional widening
enhances sidewall
erosion
rapid
incision with
narrowing
slow
incision with
widening

B
ˆqbns  (1  p )Bs   b  Ss b 
t 
 t
37
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTERPRETATION OF TERMS IN RELATION FOR EVOLUTION OF HALFWIDTH
Bb
qbsb
Bs  Bb
H qbss qbss H qbss Bb
ˆ
(1  p )Ss




 qbns
t
s
SsBs s
SsBs s Bs s
BsBb
Auxiliary streamwise terms
This term causes narrowing
whenever sediment transport is
increasing in the streamwise
direction.
This term always causes
widening whenever it is
nonzero.
But this is exactly what we
expect immediately upstream of
a dam just after removal:
downward concave long
profile!
38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION FOR CRITICAL CONDITION FOR INCEPTION OF EROSIONAL
NARROWING
Where NS and NB are order-one parameters,
Bb
qbsb S
qbsb Bb
Bs  Bb
ˆ
(1  p )Ss
 NS
 NB
 qbns
t
S s
Bb s
BsBb
Narrows if
slope
increases
downstream
Either way
Widens
At point of width minimum Bb/s = 0
39
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION FOR CRITICAL CONDITION FOR INCEPTION OF EROSIONAL
NARROWING contd.
Where Ns and Nb are order-one parameters,
Bb
qbsb S
qbsb Bb
Bs  Bb
ˆ
(1  p )Ss
 NS
 NB
 qbns
t
S s
Bb s
BsBb
After some reduction,
Bb S
B  Bs
M b
S s
Bs
c
Ss

bb
where M is another order-one parameter.
That is, erosional narrowing can be expected if the long profile of the river is
sufficiently downward concave, precisely the condition to be expected
immediately after dam removal!
40
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NUMERICAL MODELING OF THE MORPHODYNAMICS OF EROSIONAL
NARROWING AND WIDENING
Wong et al. (2004) used the formulation given in this chapter to numerically model
one of the experiments of Cantelli et al. (2004). The code will eventually be made
available in this e-book. Meanwhile, some numerical results are given in the next
two slides. The reasonable agreement was obtained with a minimum of
parameter fitting.
41
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COMPARISON OF NUMERICAL MODEL WITH EXP. 5 OF CANTELLI et al.
(2004): EVOLUTION OF LONG PROFILE
Water surface elevation (m)
0.41
0.39
0.37
0.35
0.33
0.31
0.29
0.27
2.20
initial profile
calc
meas
3.20
4.20
5.20
6.20
7.20
8.20
Distance in the downstream direction (m)
Calculated and measured long profile 1200 seconds after commencement of
experiment.
42
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COMPARISON OF NUMERICAL MODEL WITH EXP. 5 OF CANTELLI et al.
(2004): EVOLUTION OF CHANNEL WIDTH
Channel width at water surface elevation (m)
0.29
0.27
0.25
0.23
calc
meas
0.21
0.19
0.17
0.15
0
200
400
600
800
1000
1200
Time (seconds)
Calculated and measured water surface width 0.9 m upstream of original
position of dam.
43
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 31
Cantelli, C. Paola and G. Parker, 2004, Experiments on upstream-migrating erosional narrowing
and widening of an incisional channel caused by dam removal, Water Resources Research,
40(3), doi:10.1029/2003WR002940.
Cui, ,Y., Parker, G., Braudrick, C., Dietrich, W. E. and Cluer, B., in press-a, Dam Removal
Express Assessment Models (DREAM). Part 1: Model development and validation, Journal
of Hydraulic Research, preprint downloadable at:
http://cee.uiuc.edu/people/parkerg/preprints.htm .
Cui, Y., Braudrick, C., Dietrich, W.E., Cluer, B., and Parker, G, in press-b, Dam Removal
Express Assessment Models (DREAM). Part 2: Sample runs/sensitivity tests, Journal of
Hydraulic Research, preprint downloadable at:
http://cee.uiuc.edu/people/parkerg/preprints.htm .
Einstein, H. A., 1950, The Bed-load Function for Sediment Transportation in Open Channel
Flows, Technical Bulletin 1026, U.S. Dept. of the Army, Soil Conservation Service.
Parker, G., 1979, Hydraulic geometry of active gravel rivers, Journal of Hydraulic Engineering,
105(9), 1185-1201.
Parker, G., 2004, The sediment digester, Internal Memorandum 117, St. Anthony Falls
Laboratory, University of Minnesota, 17 p, downloadable at:
http://cee.uiuc.edu/people/parkerg/reports.htm .
Wong, M., Cantelli, A., Paola, C. and Parker, G., 2004, Erosional narrowing after dam removal:
theory and numerical model, Proceedings, ASCE World Water and Environmental
Resources 2004 Congress, Salt Lake City, June 27-July 1, 10 p., reprint available at: 44
http://cee.uiuc.edu/people/parkerg/conference_reprints.htm .