Transcript Slide 1

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 16:
MORPHODYNAMICS OF BEDROCK-ALLUVIAL TRANSITIONS
An alluvial river has a bed that is
completely covered with sediment that the
river can move freely during flood flow.
A bedrock river has patches of bed that
are not covered by alluvium, where
bedrock is exposed. In some bedrock
rivers the bed is almost completely bare of
sediment. This is, however, not the usual
case. In most cases of interest there is a
mixture of patches covered by alluvium
and patches where bedrock is exposed.
A bedrock river in Kentucky
(tributary of Wilson Creek) with
a partial alluvial covering. 1
Image courtesy A. Parola.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE CONCEPT OF TRANSPORT CAPACITY
Equilibrium bedrock streams transport alluvium
under below-capacity conditions, whereas
alluvial streams transport sediment under atcapacity conditions. These concepts can be
explained as follows.
If the sediment supply of an alluvial river is
increased, the bed can be expected to aggrade
toward a new, steeper slope capable of carrying
the extra sediment.
A bedrock stream, on the other hand, may
experience no aggradation when sediment supply
is increased. Instead, the stream responds by
reducing the fraction of the bed covered by
bedrock and increasing the fraction covered by
alluvium. Only when the bed is completely
Big Box Creek, USA, a bedrock
covered with alluvium can the river respond to
river with a stepped profile. 2
increased sediment supply by aggrading.
Image courtesy E. Wohl.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
QUANTIFICATION OF TRANSPORT CAPACITY
The concept of a mobile-bed equilibrium state was outlined in Chapter 14. In the
case of the Chezy resistance relation and the sample sediment transport relation
introduced in that chapter, the governing equations of this equilibrium state take the
following forms:
 C f q2w
H  
 gS
1/ 3



2

 Cf qw
qt  RgD D  t 
g


1/ 3



nt
S  

  c 
RD 



2/3
Now let grain size D, sediment submerged specific gravity R, resistance coefficient
Cf, critical Shields number c* and the parameters g, t and nt be given. The
relations specify two equations in the following four parameters: depth H, bed slope
S, water discharge per unit width qw and volume total bed material sediment
discharge per unit width qt.
Consider a stream with given values of water discharge per unit width qw and bed
slope S. The capacity transport qt is that computed from the above equation.
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BELOW-CAPACITY CONDITIONS
Now suppose that for given values of qw and S, the actual sediment supply qts is
less that the value qt associated with mobile-bed equilibrium, i.e.
2

 Cf qw
qt  RgD D  t 
g


1/ 3



nt
S2 / 3   

  c 
RD 



An alluvial stream would degrade to a lower slope S that would allow the above
equation to be satisfied with qts. A bedrock stream, however, cannot degrade. So in
the event that for given values of qw and S the sediment supply rate qts is less than
the equilibrium mobile-bed value qt, the river responds by exposing bedrock on its
bed instead of degrading. As qts is further reduced the river responds by increasing
the fraction of the bed over which bedrock is exposed (Sklar and Dietrich, 1998). The
river so adjusts itself to transport sediment at the rate qts which is below its capacity qt
for the given values of qw and S. This allows a below-capacity equilibrium.
In the event that the actual sediment supply qts is greater than the capacity transport
rate qt at the given slope S , the river will aggrade to a new, higher slope in
consonance with qts that satisfies the above equation. There is no
4
above-capacity equilibrium.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A SAMPLE CALCULATION
The following values are assumed in the sediment transport relation below: t =
3.97, nt = 1.5, c* = 0.0495 (Wong and Parker, submitted, modification of MeyerPeter and Müller), R = 1.65, g = 9.81 m2/s and Cf = 0.01.
2

 Cf qw
qt  RgD D  t 
g


1/ 3



nt
S2 / 3   

  c 
RD 



Consider a river with D = 20 mm, flood Qw = 90 m3/s and width B = 30 m. The
flood value of qw = Qw/B = 3 m2/s. For any slope S, then, the capacity value of qt
can be computed from the above relation.
Assume that a bedrock river is just barely completely covered with alluvium at the
slope S. How will the river respond if sediment supply qts is reduced or increased?
The following two slides illustrate that the river will aggrade to a new mobile-bed
equilibrium when qts > qt. When qts < qt, the river cannot degrade due to the
presence of bedrock, and instead reaches a below-capacity equilibrium with
exposed bedrock.
5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A SAMPLE CALCULATION contd.
qts = qt(qw,S)
thin covering of
alluvium
bedrock
qts < qt(qw,S)
bedrock exposed
supply
decreased
qts = qt(qw,S)
thin covering of
alluvium
bedrock
bedrock
bed has aggraded
to new mobile-bed
equilibrium
supply
increased
bedrock
6
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A SAMPLE CALCULATION contd.
0.035
above-capacity sediment supply: stream
will aggrade to new alluvial equilibrium
0.03
qts m2/s
0.025
capacity transport curve with
bedrock barely covered by
alluvium:
qts = qt
0.02
below-capacity sediment
supply: stream will
expose bedrock
0.015
0.01
0.005
0
0
0.005
0.01
0.015
0.02
0.025
S
0.03
0.035
0.04
0.045
0.05
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ILLUSTRATION OF BELOW-CAPACITY TRANSPORT OF 7 MM GRAVEL OVER A
BEDROCK BED
The video clip is from the Ph.D. research of Phairot Chatanantavet.
8
rte-bookbelowcaptrans.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
BEDROCK-ALLUVIAL TRANSITIONS: THE “FALL LINE”
The southeastern coastal plain of the
United States is characterized by a
feature called the “Fall Line.”
Upstream (westward) of this line the
streams are in bedrock. Downstream
(eastward) of this line they are in
alluvium. It is of interest to speculate
how the position of the fall line might
respond to changing sea level.
Image of the southeastern coastal
plain of the United States from NASA
https://zulu.ssc.nasa.gov/mrsid/mrsid.pl
9
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM STATE WITH BEDROCK-ALLUVIAL TRANSITION
A bedrock channel has constant slope Sb and carries flood discharge per unit width
qw. Sediment with size D is fed in at the upstream end at rate qts. The at-capacity
slope S consonant with qst, qw and D (as computed, for example, from the transport
relation of Slide 5) is less than Sb. Base level is maintained at some elevation at
the downstream end; this level is higher than the elevation of the bedrock
basement there. An equilibrium bedrock-alluvial transition must occur. To find it,
draw a straight line with slope S and intercept at the point of base level
maintenance, and extend it upstream until it intersects the bedrock profile.
bedrock-alluvial
transition
qts
Sb
base level (bed elevation)
maintained here
alluvium
S  S(qw ,qts,D)  Sb
bedrock
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DYNAMICS OF THE MIGRATION OF BEDROCK-ALLUVIAL TRANSITIONS
Bedrock-alluvial transitions can migrate upstream or downstream due to the effects
of e.g. changing sediment supply from upstream or changing base level
downstream. The figure below shows a case where the alluvial region is (for
whatever reason) aggrading, resulting in an upstream migration of the bedrockalluvial transition.
t1
bedrock
surface
t0
alluvium
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONTINUITY CONDITION AT THE BEDROCK-ALLUVIAL TRANSITION
The elevation profile of the bedrock basement is denoted as base(x); it is assumed
to be unchanging in time. The elevation profile of the alluvial zone is denoted as
(x, t); it can change in time due to aggradation or degradation. The position of the
bedrock-alluvial transition is denoted as x = sba(t). It is a function of time because
the position of the transition can change in time. In order for the bedrock channel
to join continuously with the alluvial channel, the following condition must hold:
( x, t) x s  base ( x ) x s
ba
or
ba
[sba (t),t]  base [sba (t)]
Now take the derivative with respect to time of both
sides of the equation. For example,
t1
bedrock
surface
t0
alluvium
d
[sba (t),t]     dsba    S sba s ba
dt
t sba x sba dt
t sba
where S = -/x denotes the alluvial bed slope
and s ba = dsba/dt denotes the speed of migration of
the bedrock-alluvial transition.
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONTINUITY CONDITION AT THE BEDROCK-ALLUVIAL TRANSITION contd.
Taking the derivative of both sides of the relation
[sba (t),t]  base [sba (t)]
t1
results in:
bedrock
surface

 S s s ba  Sb s s ba
ba
ba
t sba
t0
alluvium
where Sb = -base/x = the slope of the bedrock channel. Reducing, the following
cute little relation is obtained:


t s
sba 
S

 bs S 
s 

(Parker and Muto, 2003). Now since x = sba denotes a bedrock-alluvial transition, it
can always be expected that the bedrock slope Sb exceeds the alluvial slope S there.
So the continuity condition says simply:
If the bed aggrades, the transition moves upstream;
13
and if the bed degrades the transition moves downstream.
ba
ba
ba
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MOVING-BOUNDARY FORMULATION FOR RIVER MORPHODYNAMICS WITH A
BEDROCK-ALLUVIAL TRANSITION
The downstream end of the reach is located at the constant value x = sd, where
base level is maintained. The bedrock-alluvial transition is located at x = sba(t) < sd.
The goal is to describe the morphodynamics of the evolution of the stream so as to
obtain both the change in the alluvial profile (x,t) as a function of time and the
trajectory sba(t) of the transition as a function of time.
To this end we introduce the coordinate transformation
x
x  sba (t)
,
sd  sba (t)
tt
Note that the bedrock-alluvial transition is located at x  0 , and the downstream
end of the reach is located at x  1 .
Using the chain rule,
 t  x 


t t t t x
,

t  x 


x x t x x
14
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TRANSFORMATION OF THE EXNER EQUATION TO MOVING-BOUNDARY
COORDINATES
x  sba (t)
x
,
sd  sba (t)
tt

 t  x


t t t x t
,

 t
 x


x t x x x
Evaluating the derivatives,
t
1 ,
t
t
0 ,
x
x
s ba

(1  x ) ,
t
sd  sba
x
1

x sd  sba
Transforming the Exner equation of sediment continuity
qt

(1   p )
 -If
t
x
to the moving-boundary coordinate system results in the form
  (1  x )s ba  
If
qt
  
(1  p ) 
sd  sba x 
sd  sba x
 t
15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TRANSFORMATION OF THE CONTINUITY CONDITION TO MOVING-BOUNDARY
COORDINATES
Now from Slide 12 and the moving-boundary coordinate transformation.

s ba 

t sba
S
 b sba

s ba   
  
  
 

t
s

s
  x 0
d
ba  x  x 0

(Sb x 0  S )
 S 
x 0
sba 
However slope is given as
S

1 

x
sd  sba x
Between these two relations,
s ba  
1
Sb x 0

t x 0
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHARACTER OF THE MORPHODYNAMIC PROBLEM
 (1  x )s ba 
If
qt


t
sd  sba x
sd  sba x
s ba  
1
Sb x 0

t x 0
There is one more variable to solve than before, i.e. the speed s ba of the moving
boundary, but there is one more equation as well. Further reducing the continuity
condition with Exner,
s ba  
 
1
qt
 s ba
 If
Sb x 0 (sd  sba )  x x 0
x
or thus
s ba 
(1 
1
S x 0
Sb x 0
qt
x x  0
Sb x 0 (sd  sba )


x 0 
If
)
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DISCRETIZATON FOR NUMERICAL SOLUTION
The domain from x  0 to x  1 (x = sba to x = sd) is discretized into M intervals
bounded by M+1 nodes. The node i = 1 denotes the bedrock-alluvial transition and
the node i = M+1 denotes the point where base level is maintained.
xx
i=1
2
3
M -1
M
i = M+1
M+1
1
The sediment feed rate qtf during floods is specified at the node i = 1; no ghost
node is needed in this formulation. The Exner equation discretizes to
i t  t
 (1  xi )s ba 
If
qt 
 i  

 t
 sd  sba x xi sd  sba x xi 
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DISCRETIZATION FOR NUMERICAL SOLUTION
As noted in the previous slide, the Exner equation discretizes to
i t  t
 (1  xi )s ba 
If
qt 
 i  

 t
 sd  sba x xi sd  sba x xi 
The derivatives discretize to the forms
q  qt,i
qt
 t,i1
x i
x
 2  1
, i 1
 
x

x i  i1  i1 , i  2..M
 2x
, i  1..M
Derivatives need not be evaluated at i = M+1 because bed elevation M+1 is held
constant. The shock condition discretizes to
sba t  t  sba  s ba t
, s ba 
(1 
1
S i1
Sb i1
qt
x i1
Sb i1(sd  sba )
If
)
, S i1  
1 
sd  sba x i1
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTRODUCTION TO RTe-bookBedrockAlluvialTrans.xls
The treatment of bedrock-alluvial transitions is implemented in RTebookBedrockAlluvialTrans.xls.
The formulation uses the Engelund-Hansen (1967) total bed material transport
relation for sand-bed streams. (Could you modify the code for gravel-bed
streams?) The downstream end of the reach is located at the fixed point sd. The
bedrock channel is assumed to have constant slope Sb. (Could you modify the
code to let Sb vary in space?)
Initially the alluvial zone has bed slope Sinit and length sd, so the bedrock-alluvial
transition is located at sba = 0. The water discharge per unit width qw and volume
sediment bed material feed rate per unit width qtf are specified along with the grain
size D of the bed material and the constant Chezy resistance coefficient Cz.
The flow is computed using the normal (steady, uniform) flow approximation.
(Could you change it to use a backwater formulation?). The values of submerged
specific gravity R and bed porosity p are set equal to 1.65 and 0.4, respectively, in
“Const” statements in the code. They can be changed by modifying the
20
relevant “Const” statements.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A SAMPLE CALCULATION: UPSTREAM MIGRATION OF THE BEDROCKALLUVIAL TRANSITION TO A NEW, STABLE POSITION
The initial bed slope of the alluvial region Sinit = 0.00009 is too low for the specified
water discharge, sediment feed rate (qtf = 0.0006 m2/s) and grain size. So the bed
must aggrade to a final equilibrium slope Sfinal, which is less that the bedrock slope
Sb. The result is that the bedrock-alluvial transition must move upstream to a new
equilibrium from its initial point at sba = 0. The results of the computation follow.
Input cell
Information cell: do no
Input
qw
If
qtf
D
Cz
Sb
Sinit
sd
M
dt
Mtoprint
Mprint
6
0.1
0.0006
0.5
15
0.00035
0.00009
50000
20
0.005
10000
20
5.02E+03
1000
m2/s
m2/s
mm
m
years
Water discharge/unit width during floods
Intermittency factor for floods: 0 < I  1
Volume sediment feed rate/width at upstream end during floods
Grain size of alluvium
Chezy resistance coefficient
Slope of bedrock basement
Initial slope of alluvial region
Position of the downstream end of the reach, = initial length of alluvial region
Number of spatial intervals
Time step
Number of steps to printout
Number of printouts
Bed material feed rate in metric tons/meter/year
21
Time of calculation in years
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Evolution of Alluvial River Profile with Bedrock-Alluvial
Transition
20
18
16
eta m
14
12
10
8
6
4
2
0
-60000
-40000
-20000
0
xm
20000
40000
60000
0 year
50 year
100 year
150 year
200 year
250 year
300 year
350 year
400 year
450 year
500 year
550 year
600 year
650 year
700 year
750 year
800 year
850 year
900 year
950 year
1000 year
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
0
20
-5000
18
-10000
16
14
-15000
12
-20000
sba
etaup
-25000
10
8
-30000
6
-35000
4
-40000
2
-45000
0
1200
0
200
400
600
time, years
800
1000
Elevation up of bedrockalluvial transition, m
Position sba
of the bedrock-alluvial
transition, m
Trajectory of the bedrock-alluvial transition
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A SAMPLE CALCULATION: DOWNSTREAM MIGRATION OF THE BEDROCKALLUVIAL TRANSITION TO A NEW, STABLE POSITION
The initial bed slope of the alluvial region Sinit = 0.0002 is too high for the specified
water discharge, sediment feed rate (qtf = 0.0001 m2/s) and grain size. So the bed
must degrade to a final equilibrium slope Sfinal, which is less that the bedrock slope Sb.
The result is that the bedrock-alluvial transition must move downstream to a new
equilibrium from its initial point at sba = 0. The results of the computation follow.
Input cell
Information cell: do no
Input
qw
If
qtf
D
Cz
Sb
Sinit
sd
M
dt
Mtoprint
Mprint
6
0.1
0.0001
0.5
15
0.00035
0.0002
50000
20
0.005
2500
20
8.36E+02
250
m2/s
m2/s
mm
m
years
Water discharge/unit width during floods
Intermittency factor for floods: 0 < I  1
Volume sediment feed rate/width at upstream end during floods
Grain size of alluvium
Chezy resistance coefficient
Slope of bedrock basement
Initial slope of alluvial region
Position of the downstream end of the reach, = initial length of alluvial region
Number of spatial intervals
Time step
Number of steps to printout
Number of printouts
24
Bed material feed rate in metric tons/meter/year
Time of calculation in years
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Evolution of Alluvial River Profile with Bedrock-Alluvial
Transition
12
10
eta m
8
6
4
2
0
0
10000
20000
30000
xm
40000
50000
60000
0 year
12.5 year
25 year
37.5 year
50 year
62.5 year
75 year
87.5 year
100 year
112.5 year
125 year
137.5 year
150 year
162.5 year
175 year
187.5 year
200 year
212.5 year
225 year
237.5 year
250 year
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Trajectory of the bedrock-alluvial transition
12
10
20000
8
15000
sba
etaup
6
10000
4
5000
2
0
0
50
100
150
time, years
200
250
Elevation up of bedrockalluvial transition, m
Position sba
of the bedrock-alluvial
transition, m
25000
0
300
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOME COMMENTS FOR FUTURE WORK
The simple analysis presented in this chapter invites a wide range of extensions. A
few are suggested below.
• How does sea level change affect the position of the fall line on coastal plains of the
southeastern United States?
• When a dam is placed on a bedrock stream, how far upstream does the alluvialbedrock transition created by deltaic deposit behind the dam migrate upstream?
• What happens to the position of a bedrock-alluvial transition if climate changes,
resulting in changes in flood sediment supply qtf, flood water discharge qw and flood
intermittency If?
• How could the analysis be generalized to gravel-bed streams and sediment
mixtures?
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NOTE IN CLOSING: BEDROCK INCISION
In the analysis described here it is assumed that the bedrock platform is fixed in
time, and is not free to undergo incision. This assumption is correct on the time
scale of many alluvial problems. It is erroneous, however, to assume that bedrock
channels cannot incise their beds over sufficiently long time scales (e.g. Whipple et
al., 2000). The ability of a river to incise through bedrock is amply illustrated by the
image below. Bedrock incision is considered in more detail in Chapters 29 and 30.
Image of a Bolivian river from NASA
https://zulu.ssc.nasa.gov/mrsid/mrsid.pl
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 16
Engelund, F. and E. Hansen, 1967, A Monograph on Sediment Transport in Alluvial Streams,
Technisk Vorlag, Copenhagen, Denmark.
Parker, G. and Muto, T., 2003, 1D numerical model of delta response to rising sea level, Proc.
3rd IAHR Symposium, River, Coastal and Estuarine Morphodynamics, Barcelona, Spain, 1-5
September.
Sklar, L., and W. E. Dietrich, 1998, River longitudinal profiles and bedrock incision models:
Stream power and the influence of sediment supply, in Rivers Over Rock: Fluvial Processes
in Bedrock Channels, Geophys. Monogr. Ser., vol. 107, edited by K. J. Tinkler and E. E.
Wohl, pp. 237–260, AGU, Washington, D. C.
Whipple, K. X., G. S. Hancock, and R. S. Anderson, 2000, River incision into bedrock: Mechanics
and relative efficacy of plucking, abrasion, and cavitation, Geol. Soc. Am. Bull., 112, 490–
503.
Wong, M. and Parker, G., submitted, The bedload transport relation of Meyer-Peter and Müller
overpredicts by a factor of two, Journal of Hydraulic Engineering, downloadable at
http://cee.uiuc.edu/people/parkerg/preprints.htm .
29