Transcript Slide 1

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 25:
LONG PROFILES OF RIVERS, WITH AN APPLICATION ON THE EFFECT OF BASE
LEVEL RISE ON LONG PROFILES
The long profile of a river is a plot of bed elevation  versus down-channel
distance x.
The long profile of a river is called upward concave if slope S = -/x is
decreasing in the streamwise direction; otherwise it is called upward convex.
That is, a long profile is upward concave if
S
 2
 2 0
x
x
upward-convex

upward-concave
1
x
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
LONG PROFILE OF THE AMAZON RIVER
The Amazon River shows a rather typical long profile. Note that it is upward
concave almost everywhere. The data are from Pirmez (1994).
Long Profile of the Amazon River
3000
2500
 (m)
2000
1500
1000
500
0
-7000
-6000
-5000
-4000
-3000
x (km)
-2000
-1000
0
2
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TRANSIENT LONG PROFILES
In Chapter 14 we saw that in an idealized equilibrium, or graded state rivers have
constant slopes in the downstream direction, adjusted so that the rate of inflow of
sediment to a reach equals the rate of outflow. When more sediment is fed in than
flows out, the river is forced to aggrade toward a new equilibrium. During this
transient period of aggradation the profile is upward-concave. A sample
calculation showing this (and performed with RTe-bookAgDegNormal.xls) is given
below.
Bed evolution
160
140
120
Elevation in m
Likewise, when more
sediment flows out of
the reach than is fed in,
the river is forced to
degrade toward a new
equilibrium. During this
transient period of
degradation the profile
is upward-convex. (Try
a run and see.)
0 yr
2 yr
4 yr
6 yr
8 yr
10 yr
Ultimate
100
80
60
40
20
0
0
2000
4000
6000
Distance in m
8000
10000
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
QUASI-EQUILIBRIUM LONG PROFILES
The long profiles of long rivers generally approach an upward-concave shape that
is maintained as a quasi-equilibrium form over long geomorphic time. As the word
“quasi” implies, this “equilibrium” is not an equilibrium in the sense that sediment
output equals input over each reach.
Reasons for the maintenance of this quasi-equilibrium are summarized in Sinha
and Parker (1996). Several of these are listed below.
• Subsidence
• Sea level rise
• Delta progradation
• Downstream sorting of sediment
• Abrasion of sediment
• Effect of tributaries
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SUBSIDENCE
As a river flow into a subsiding basin, the river tends to migrate across the surface,
filling the hole created by subsidence. As a result, the sediment output from a
reach is less than the input, and the profile is upward-concave over the long term
(e.g. Paola et al., 1992).
Rivers entering a
(subsiding) graben in
eastern Taiwan.
Image from NASA
website:
https://zulu.ssc.nasa.gov/mrsid/mrsid.pl
5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SEA LEVEL RISE
Rivers entering the sea have felt the effect of a 120 m rise in sea level over about
12,000 years at the end of the last glaciation. The rise in sea level was caused by
melting glaciers. The effect of this sea level rise was to force aggradation, with
more sediment coming into a reach than leaving. This has helped force upwardconcave long profiles on such rivers.
Sea level rise from 19,000
years BP (before present)
until 3,000 years BP
according to the Bard Curve
(see Bard et al., 1990).
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DELTA PROGRADATION
Even when the body of water in question (lake or the ocean) maintains constant
base level, progradation of a delta into standing water forces long-term
aggradation and an upward-concave profile.
Missouri River prograding
into Lake Sakakawea,
North Dakota.
Image from NASA
website:
https://zulu.ssc.nasa.gov/mrsid/mrsid.pl
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DOWNSTREAM SORTING OF SEDIMENT
Rivers typically show a
pattern of downstream
fining. That is,
characteristic grain size
gets finer in the
downstream direction.
This is because in a
sediment mixture, finer
grains are somewhat
easier to move than
coarser grains. Since
finer grains can be
transported by the same
flow at lower slopes, the
result is a tendency to
strengthen the upward
concavity of the profile.
Long profile and median sediment grain
size on the Mississippi River, USA.
Adapted from USCOE (1935) and Fisk
(1944) by Wright and Parker (in press).
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ABRASION OF SEDIMENT
In mountain rivers containing gravel of
relatively weak lithology, the gravel tends
to abrade in the streamwise direction. The
product of abrasion is usually silt with
some sand. As the gravel gets finer, it can
be transported at lower slopes. The result
is tendency to strengthen the upward
concavity of a river profile.
The image shows a) the long profile of the
Kinu River, Japan and b) the profile of
median grain size in the same river. The
gravel easily breaks down due to abrasion.
The river undergoes a sudden transition
from gravel-bed to sand-bed before
reaching the sea.
Image adapted from Yatsu (1955)
by Parker and Cui (1998).
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EFFECT OF TRIBUTARIES
As tributaries enter the main stem of a river, they tend to increase the supply of water
more than they increase the supply of sediment, so that the concentration of
sediment in the main
Sediment Flux in RF1 Rivers
stem tends to decline in
the streamwise
(mt/year)
direction. Since the
same flow carries less
sediment, the result is a
tendency toward an
upward-concave profile.
Image courtesy John
Gray, US Geological
Survey.
Sediment Discharge (mt/year)
0 - 100
100 - 500
500 - 1000
1,000 - 5,000
5,000 - 20,000
20,000 - 160,000
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
UPWARD-CONCAVE LONG PROFILE DRIVEN BY RISING SEA LEVEL
The Fly-Strickland River System in
Papua New Guinea has been profoundly
influenced by Holocene sea level rise.
Fly River
Strickland
River
Image from NASA website:
https://zulu.ssc.nasa.gov/mrsid/mrsid.pl
Fly River
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FORMULATION OF THE
PROBLEM: ASSUMPTIONS
• Downchannel reach length L is specified; x = L
corresponds to the point where sea level is specified.
•The river is assumed to have a floodplain width Bf that
is constant, and is much larger than bankfull width Bbf.
• The river is sand-bed with characteristic size D.
• All the bed material sediment is transported at rate
Qtbf during a period constituting (constant) fraction If of
the year, at which the flow is approximated as at
bankfull flow, so that the annual yield = IfQtbf.
• Sediment is deposited across the entire width of the
floodplain as the channel migrates and avulses. For
every mass unit of bed material load deposited, 
mass units of wash load are deposited in the floodplain.
• Sea level rise is constant at rate  d . For example,
during the period 5,000 – 17,000 BP the rate of rise
can be approximated as 1 cm per year.
• The river is meandering throughout sea level rise,
and has constant sinuosity .
•The flow can be approximated using the normal-flow
assumptions. (But the analysis easily
12
generalizes to a full backwater formulation.)
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BED MATERIAL LOAD AND WASH LOAD
Sea level rise forces a river bed to aggrade. This in turn forces the river to spill out
more often onto the floodplain, and therefore forces floodplain aggradation as well.
Wash load is by definition contained in negligible quantities in the bed of a river, but is
invariably a major constituent of floodplain deposits, and is often the dominant one.
That is, wash load could be more accurately characterized as “floodplain material
load.” In large sand-bed rivers, for example, the floodplain often contains a lower
layer in which sand dominates and an upper layer in which silt dominates.
A precise mass balance for wash load is beyond the scope of this chapter. For
simplicity it is assumed that for every unit of sand deposited in the channel/floodplain
system in response to sea level rise,  units of wash load are deposited, where  is
a specified constant that might range from 0 to 3 or higher. It is assumed that the
supply of wash load from upstream is always sufficient for deposition at such a rate.
This is not likely to be strictly true, but should serve as a useful starting assumption.
In addition, it is assumed for simplicity that the porosity of the floodplain deposits is
equal to that of the channel deposits. In fact the floodplain deposits are likely to have
a lower porosity.
13
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FORMULATION OF THE PROBLEM: EXNER
Sediment is carried in channel but deposited across the floodplain due to
aggradation forced by sea level rise. Adapting the formulation of Chapter 15,
where qtbf denotes the bankfull (flood) value of volume bed material load per unit
width qt, qwbf denotes the bankfull (flood) value of volume wash load per unit width
and  denotes channel sinuosity,

sB f (1   p )x v  sIf Bbf (qtbf  qwbf ) x  sIf Bbf (qtbf  qwbf ) x  x
t
x
  , Qtbf  Bbf qtbf
x v
xv
x
B
xv
(1  p )
Bf
, Qwbf  Bbf qwbf
Qwbf 

I  Q
  f  tbf 

t
Bf  x
x 
14
xv+xv
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FORMULATION OF THE PROBLEM: EXNER contd.
It is assumed that for every one unit of bed material load deposited  units of
wash load are deposited to construct the channel/floodplain complex;
Q wbf
Q tbf

x
x
Thus the final form of Exner becomes
xv
x
B
xv

If (1   ) Qtbf
(1  p )
 
t
Bf
x
Bf
15
xv+xv
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FORMULATION OF THE PROBLEM: MORPHODYNAMIC EQUATIONS
Relation for sediment transport:
Using the formulation of Chapter 24 for sand-bed streams,
Bˆ 

Cf
EH 

form

2 .5
ˆ
Q
t
,
R 3 / 2C1f/ 2
S
EH form
ˆ
Q
t
ˆ
Q

2

(

)
ˆ  EH form
, H
RC f 1/ 2
ˆ
Q
ˆ
Q
t
Expressing the middle relation in dimensioned forms and solving for Qtbf as a
function of S and Qbf,
Qtbf
EH form

Qbf S
1/ 2
RC f
Note that according to this relation the bed material transport load Qtbf is a linear
function of slope.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION OF THE EXNER EQUATION
The Exner equation can be expressed as

If (1  ) Qtbf

t
(1  p ) Bf x
Reducing with the sediment transport relation
Qtbf
EH form

Qbf S
1/ 2
RC f
it is found that

 2
 d 2
t
x
,
If (1 ) EHformQbf
d 
(1 p )Bf
RC1f/ 2
where d denotes a kinematic sediment diffusivity. Note that the resulting form is a
linear diffusion equation.
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DECOMPOSITION OF THE SOLUTION FOR BED ELEVATION
The bed elevation at specified point x = L is set equal to sea level elevation, so
that where do denotes sea level elevation at time t = 0,
(L, t)  do   dt
Bed elevation (x,t) is represented in terms of this downstream elevation
approximated by sea level and the deviation dev(x,t) =  - (L,t) it, so that
  do   dt  dev ( x, t)
The problem is solved over reach length L where x = 0 denotes the upstream
length and x = L is the point where the river meets the sea. From the above
relations, then,
dev (L, t )  0
Substituting the second of the above relations into the Exner formulations of the
previous page yields the forms
dev 
I (1 ) Qtbf
 d   f
t
(1 p )Bf x
or
dev 
 2dev
 d  d
t
x 2
,
If (1 ) EHformQbf
d 
(1 p )Bf
RC1f/ 2
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BOUNDARY CONDITIONS
The upstream boundary condition is that of a specified sediment feed rate Qtbf,feed
(during floods) at x = 0. That is,
Qtbf (0, t)  Qtbf ,feed
The downstream boundary condition, i.e.
dev (L, t )  0
is somewhat unrealistic in that L is a
prescribed constant. In point of fact,
rivers flowing into the sea end in
deltas. The topset-foreset break of

the delta, where x = L, can move
seaward as the delta progrades at
constant water surface elevation, and
can move seaward or landward under
conditions of rising or falling sea level.
These issues are examined in more
detail in a future chapter.
topset-foreset break
L
x
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOLUTION FOR STEADY-STATE AGGRADATION IN RESPONSE TO SEA LEVEL
RISE
The case illustrated below is that of steady-state aggradation, with every point
aggrading at the rate  d in response to sea level rise at the same constant rate. In
such a case dev becomes a function of x alone, and the problem reduces to
dev 
I (1  ) dQtbf
 d   f
t
(1  p ) Bf dx
Qtbf
x 0
 Qtbf ,feed
dev (L, t )  0

 d t
 d t
L
x
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOLUTION FOR STEADY-STATE AGGRADATION IN RESPONSE TO SEA LEVEL
RISE contd.
The Exner equation thus reduces to the form
(1 p )Bf 
dQtbf

d
dx
If (1 )
which integrates with the upstream boundary condition of the previous page to
Qtbf  Qtbf ,feed
(1 p )Bf 

d x
If (1  )
That is, the bed material load decreases linearly down the channel due to steadystate aggradation forced by sea-level rise. The sediment delivery rate to the sea
Qtbf,,sea is given as
Qtbf ,sea  Qtbf ,feed
(1  p )Bf 

dL
If (1  )
21
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOLUTION FOR STEADY-STATE AGGRADATION IN RESPONSE TO SEA LEVEL
RISE contd.
Further reducing,
Qtbf
 1  xˆ ,
Qtbf ,feed
x
xˆ 
L
, 
(1  p ) Bf  dL
If (1   )Qtbf ,feed
Between this relation and the load relation
Qtbf
EH form

Qbf S
1/ 2
RC f
it is seen that
RC1f/ 2 Qtbf ,feed
Su 
EHform Qbf
,
S
 (1 xˆ )
Su
where Su denotes the upstream slope at x = 0. That is, slope declines
downstream, defining an upward concave long profile.
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOLUTION FOR STEADY-STATE AGGRADATION IN RESPONSE TO SEA LEVEL
RISE contd.
Now S = - d/dx = - ddev/dx . Making dev and x dimensionless with L as follows
ˆ 
dev
L
results in the equation given below for elevation profile.
dˆ
 Su (1   xˆ )
dxˆ
Integrating subject to the boundary condition dev(L) = 0, or thus
ˆ xˆ 1  0

the following parabolic solution for long profile is obtained;
1
1


ˆ  Su (1  )  xˆ  xˆ 2 
2
2


23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REVIEW OF THE STEADY-STATE SOLUTION
  do   dt  dev ( x)
ˆ 
dev
L
,
xˆ 
x
L
1
1 2

ˆ
ˆ  Su (1  )  x  xˆ 
2
2


RC1f/ 2 Qtbf ,feed
Su 
EHform Qbf

(1  p ) Bf  dL
If (1  )Qtbf ,feed
The parameter EH = 0.05 for the Engelund-Hansen relation, and for sand-bed
rivers form* can be approximated as 1.86. Reach length L, bed porosity p,
floodplain width Bf, channel sinuosity , intermittency If friction coefficient Cf and the
ratio  of wash load deposited to bed material load deposited must be specified.
The steady-state long profile can then be calculated for any specified values of
24
upstream flood bed material feed rate Qtbf,feed and rate of sea level rise  d .
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REVIEW OF THE STEADY-STATE SOLUTION contd.
The predicted streamwise variation in channel bankfull width Bbf and depth Hbf are
given from the relations
Bˆ 

Cf
EH form

2 .5
ˆ
Q
t

2

(

)
ˆ  EH form
, H
RC f 1/ 2
ˆ
Q
ˆ
Q
t
or in dimensioned terms
Bbf
Cf

D
EH form


2.5
Qtbf
RgD D2
,
Hbf EH ( form )2 Qbf

D
C1f/ 2
Qtbf
In the absence of tributaries, decreasing bed material load in the streamwise
direction causes a decrease in bankfull width Bbf and an increase in bankfull depth
Hbf.
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHARACTERISTICS OF THE STEADY-STATE SOLUTION
The parameter  has a specific physical meaning. The mean annual feed rate of bed
material load Gt,feed available for deposition in the reach is given as
Gt,feed  (R  1) Qtbf ,feed If
When wash load is included, the mean annual rate Gfeed available for deposition becomes
Gfeed  (1 )Gt,feed  (1 )(R  1) Qtbf ,feed If
Valley length Lv is given as
Lv  L 
The tons/s of sediment Gfill required to fill a reach with length Lv and width Bf with sediment
at a uniform aggradation rate  d in m/s is given as
Gfill  (R  1)(1 p )BfLv  d
It follows that

(1   p ) Bf  dL
If (1   )Qtbf ,feed

Gfill
Gfeed
That is if  > 1 then the there is not enough sediment feed over the reach to fill the space
created by sea level rise, and the sediment transport rate must drop to zero before the
26
shoreline is reached. If  < 1 the excess sediment is delivered to the sea.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHARACTERISTICS OF THE STEADY-STATE SOLUTION contd.
As a result of the above arguments, meaningful solutions are realized only for the
case   1. In such cases there is excess sediment to deliver to the sea. In
actuality, part of this sediment would be used to prograde the delta of the river, so
increasing reach length L. Delta progradation is considered in more detail in a
subsequent chapter.
If for a given rate of sea level rise  d it is found that  > 1 for reasonable values of
reach length L, floodplain width Bf and sediment feed rate Gfeed, no steady state
solution exists for that rate of sea level rise.
The implication is that the entire profile, including the position of the delta, must
migrate upstream, or transgress. A model of this transgression is developed in a
subsequent chapter.
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION
The calculation is implemented in the spreadsheet workbook
RTe-bookSteadyStateAg.xls. The following sample input parameters are used in
the succeeding plots. It should be noted that a sea level rise of 10 mm/year forces
a rather extreme response.
  10 mm / year
d
L  160 km
B f  20 km
D  0.25 mm
R  1.65
G t,feed  12 Mt / year (annual average)
  2 .5
Qbf  5000 m3 / s
If  0.1
Cz  25
 p  0 .4
  1 .5
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION: SLOPE PROFILE
Long Profile of Bed Slope
3.E-04
Bed slope S
2.E-04
2.E-04
1.E-04
5.E-05
0.E+00
0
20
40
60
80
100
Streamwise distance km
120
140
160
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION: DEVIATORIC BED ELEVATION PROFILE
Long Profile of Deviatoric Bed Elevation
Deviatoric bed elevation, m
25
20
15
10
5
0
0
20
40
60
80
100
Streamwise distance km
120
140
160
30
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION: BED ELEVATION PROFILES
Long Profiles of Bed Elevation
160
0 yr
1000 yr
2000 yr
3000 yr
4000 yr
5000 yr
6000 yr
7000 yr
8000 yr
9000 yr
10000 yr
11000 yr
12000 yr
Bed elevation m
140
120
100
80
60
40
20
0
0
20
40
60
80
100
Streamwise distance km
120
140
160
31
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION: PROFILES OF BANKFULL WIDTH AND DEPTH
Downstream Variation in Bankfull Width and Depth
700
25
600
20
Bbf, Hbf m
500
15
400
Bbf m
Hbf m
300
10
200
5
100
0
0
20
40
60
80
100
120
Downstream distance km
140
0
160
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CAN THE WIDTH DECREASE SO
STRONGLY IN THE DOWNSTREAM
DIRECTION?
The Kosi River flows into a zone of rapid
subsidence. Subsidence forces a
streamwise decline in the sediment load in
a similar way to sea level rise, as will be
shown in a subsequent chapter. Note how
the river width decreases noticeably in the
downstream direction.
This notwithstanding, a sea level rise of 10
mm/year forces a rather extreme response.
Kosi River and Fan, India (and adjacent
countries).
Image from NASA;
https://zulu.ssc.nasa.gov/mrsid/mrsid.pl
33
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MORPHODYNAMICS OF THE APPROACH TO STEADY-STATE RESPONSE TO
RISING SEA LEVEL
Recalling that
  do   dt  dev ( x, t)
the governing partial differential equation is
dev 
 2dev
 d   d
t
x 2
,
If (1 ) EHformQbf
d 
(1 p )Bf
RC1f/ 2
subject to the boundary conditions

 dev
x
x 0
RC1f/ 2 Qtbf ,feed
 Su 
EHform Qbf
dev (L, t )  0
and the initial condition
dev ( x,0)  dev ,I ( x)
34
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MORPHODYNAMICS OF THE APPROACH TO STEADY-STATE RESPONSE TO
RISING SEA LEVEL contd.
Two scientific questions
Consider the case analyzed in Slide 28, but now consider the approach to steady state.
Suppose sea level rise is sustained at a rate of 10 mm/year for 2500 years. How close
does a given reach approach steady-state aggradation by 2500 years?
Suppose sea level is held steady for the next 2500 years. How much of the signal of
steady-state aggradation is erased over this time span?
These questions can be answered with the following Excel workbook:
RTe-book1DRiverwFPRisingBaseLevelNormal.xls. This workbook implements the
formulation of the previous slide to describe the evolution toward steady-state
aggradation. The treatment allows for both sand-bed and gravel-bed rivers, as outlined
in Chapter 24.
35
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATIONS WITH RTe-book1DRiverwFPRisingBaseLevelNormal.xls.
Input to the calculation is as specified below.
Input Parameters
(Qf)
Qbf
5000
(Qtbffeed) Qtbf,feed
1.4349
(lambig)

2.5
(Ifl)
If
0.1
4.20E+07
(D)
D
0.25
(Rr)
R
1.65
(L)
L
400000
(Bf)
Bf
8000
(Sinu)

1.5
(lamp)
p
0.4
(Cz)
Cz
25
(SfbI)
SfbI
0.0001
(etaddot) dd/dt
10
Yearstart
0
Yearstop
2500
M
50
x
8000
t
0.5
Mtoprint
2000
Mprint
5
5000
266666.67
The cells:
m /s
Flood discharge
are computed for you.
3
m /s
Upstream bed material sediment feed rate during floods
Units of wash load deposited in the fan per unit bed material load deposited
Intermittency
tons/year Annual sediment supply to river (bed material load + wash load)
mm
Grain size of bed material
Submerged specific gravity of sediment (e.g. 1.65 for quartz)
m
Reach length (downchannel distance)
m
Floodplain width
Channel sinuosity
Bed porosity
Chezy resistance coefficient
Initial fluvial bed slope
mm/year Rate of rise of downstream base level (should be positive)
years
Year base level change starts
years
Year base level change stops
Number of intervals
Cf
0.0016
m
spatial step
form*
1.86
year
time step
Hn
3.7672486
Number of time steps to printout
Sn
0.0002037
Number of printouts
Bn
611.93144
years
Duration of calculation
Frn
0.3567766 36
m
Reach length (downvalley distance)
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Up to 250 years
Elevation Profiles
60
Elevation m
50
0 yr
50 yr
100 yr
150 yr
200 yr
250 yr
40
30
20
10
0
0
50000
100000
150000
200000
250000
300000
Downchannel distance m
350000
400000
37
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Bankfull Width Profiles
Up to 250 years
700
0 yr
50 yr
100 yr
150 yr
200 yr
250 yr
0 yr
50 yr
100 yr
150 yr
200 yr
250 yr
Elevation
m m
Width
Bankfull
600
500
400
300
200
100
0
0
50000
100000
150000
200000
250000
300000
Downchannel distance m
350000
400000
38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Bankfull Depth Profiles
Up to 250 years
14
Bankfull depth m
12
10
0 yr
50 yr
100 yr
150 yr
200 yr
250 yr
8
6
4
2
0
0
50000
100000 150000 200000 250000 300000 350000 400000
Downchannel distance m
39
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Up to 2500 years:
steady state achieved!
Elevation Profiles
80
70
Elevation m
60
0 yr
500 yr
1000 yr
1500 yr
2000 yr
2500 yr
50
40
30
20
10
0
0
50000
100000
150000
200000
250000
300000
Downchannel distance m
350000
400000
40
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Bankfull Width Profiles
Up to 2500 years:
steady state achieved!
700
Elevation
m m
Width
Bankfull
600
500
0 yr
500 yr
1000 yr
1500 yr
2000 yr
2500 yr
400
300
200
100
0
0
50000
100000
150000
200000
250000
300000
Downchannel distance m
350000
400000
41
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Bankfull Depth Profiles
Up to 2500 years:
steady state achieved!
Bankfull depth m
25
20
0 yr
500 yr
1000 yr
1500 yr
2000 yr
2500 yr
15
10
5
0
0
50000
100000 150000 200000 250000 300000 350000 400000
Downchannel distance m
42
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Sea level rise is halted in year
2500: by year 5000 the bed
slope is evolving to a constant
value.
Elevation Profiles
100
90
Elevation m
80
70
0 yr
1000 yr
2000 yr
3000 yr
4000 yr
5000 yr
60
50
40
30
20
10
0
0
50000
100000
150000
200000
250000
300000
Downchannel distance m
350000
400000
43
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Bankfull Width Profiles
700
Sea level rise is halted in
year 2500: by year 5000
channel width is evolving
to a constant value.
Elevation
m m
Width
Bankfull
600
500
0 yr
1000 yr
2000 yr
3000 yr
4000 yr
5000 yr
400
300
200
100
0
0
50000
100000
150000
200000
250000
300000
Downchannel distance m
350000
400000
44
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Bankfull Depth Profiles
Bankfull depth m
25
Sea level rise is halted in
year 2500: by year 5000
channel depth is evolving
to a constant value.
20
0 yr
1000 yr
2000 yr
3000 yr
4000 yr
5000 yr
15
10
5
0
0
50000
100000 150000 200000 250000 300000 350000 400000
Downchannel distance m
45
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 25
Bard, E., Hamelin, B., and Fairbanks, R.G., 1990, U-Th ages obtained by mass spectrometry in
corals from Barbados: sea level during the past 130,000 years, Nature 346, 456-458.
Fisk, H.N., 1944, Geological investigations of the alluvial valley of the lower Mississippi River,
Report, U.S. Army Corp of Engineers, Mississippi River Commission, Vicksburg, MS.
Pirmez, C., 1994, Growth of a Submarine meandering channel-levee system on Amazon Fan,
Ph.D. thesis, Columbia University, New York, 587 p.
Paola, C., P. L. Heller and C. L. Angevine, 1992, The large-scale dynamics of grain-size variation
in alluvial basins. I: Theory, Basin Research, 4, 73-90.
Parker, G., and Y. Cui, 1998, The arrested gravel front: stable gravel-sand transitions in rivers.
Part 1: Simplified analytical solution, Journal of Hydraulic Research, 36(1): 75-100.
Parker, G., Paola, P., Whipple, K. and Mohrig, D., 1998, Alluvial fans formed by channelized
fluvial and sheet flow: theory, Journal of Hydraulic Engineering, 124(10), pp. 1-11.
Sinha, S. K. and Parker, G., 1996, Causes of concavity in longitudinal profiles of rivers, Water
Resources Research 32(5),1417-1428.
USCOE, 1935., Studies of river bed materials and their movement, with special reference to the
lower Mississippi River, Paper 17 of the U.S. Waterways Experiment Station, Vicksburg, MS.
Wright, S. and Parker, G, submitted, Modeling downstream fining in sand-bed rivers I:
formulation, Journal of Hydraulic Research.
Yatsu, E., 1955, On the longitudinal profile of the graded river, Transactions, American
Geophysical Union, 36: 655-663.
46