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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
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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).
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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.
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.)
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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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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
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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
upward-concave 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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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//zul
u.ssc.nasa.gov/mrsid/mrsid.pl
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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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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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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 stem tends to decline in the
streamwise 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.
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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
Fly River
Image from NASA website https//zulu.ssc.nasa.gov
/mrsid/mrsid.pl
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• 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 . 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
• generalizes to a full backwater formulation.)

FORMULATION OF THE PROBLEM ASSUMPTIONS
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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.
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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,
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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 Thus the final form of Exner becomes
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FORMULATION OF THE PROBLEM MORPHODYNAMIC
EQUATIONS
Relation for sediment transport Using the
formulation of Chapter 24 for sand-bed
streams, Expressing the middle relation in
dimensioned forms and solving for Qtbf as a
function of S and Qbf, Note that according to
this relation the bed material transport load
Qtbf is a linear function of slope.
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REDUCTION OF THE EXNER EQUATION
The Exner equation can be expressed
as Reducing with the sediment transport
relation it is found that where ?d denotes
a kinematic sediment diffusivity. Note that the
resulting form is a linear diffusion equation.
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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, 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 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, Substituting the
second of the above relations into the Exner
formulations of the previous page yields the forms
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BOUNDARY CONDITIONS
The upstream boundary condition is that of a
specified sediment feed rate Qtbf,feed (during
floods) at x 0. That is,
The downstream boundary condition, i.e.
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.
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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 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
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SOLUTION FOR STEADY-STATE AGGRADATION IN RESPONSE
TO SEA LEVEL RISE contd.
The Exner equation thus reduces to the
form which integrates with the upstream
boundary condition of the previous page to
That is, the bed material load decreases
linearly down the channel due to steady-state
aggradation forced by sea-level rise. The
sediment delivery rate to the sea Qtbf,,sea is
given as
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SOLUTION FOR STEADY-STATE AGGRADATION IN RESPONSE
TO SEA LEVEL RISE contd.
Further reducing, Between this relation and
the load relation it is seen that where Su
denotes the upstream slope at x 0. That is,
slope declines downstream, defining an upward
concave long profile.
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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 results in the
equation given below for elevation profile.
Integrating subject to the boundary condition
?dev(L) 0, or thus the following parabolic
solution for long profile is obtained
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REVIEW OF THE STEADY-STATE SOLUTION
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
upstream flood bed material feed rate Qtbf,feed
and rate of sea level rise .
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REVIEW OF THE STEADY-STATE SOLUTION contd.
The predicted streamwise variation in channel
bankfull width Bbf and depth Hbf are given from
the relations or in dimensioned terms 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.
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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 When wash load is included, the mean
annual rate Gfeed available for deposition
becomes Valley length Lv is given as The tons/s
of sediment Gfill required to fill a reach with
length Lv and width Bf with sediment at a uniform
aggradation rate in m/s is given as It
follows that That is if ? gt 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 shoreline is reached. If ? lt 1 the excess
sediment is delivered to the sea.
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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
it is found that ? gt 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.
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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.
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SAMPLE CALCULATION SLOPE PROFILE
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SAMPLE CALCULATION DEVIATORIC BED ELEVATION
PROFILE
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SAMPLE CALCULATION BED ELEVATION PROFILES
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SAMPLE CALCULATION PROFILES OF BANKFULL WIDTH
AND DEPTH
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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.nas
a.gov/mrsid/mrsid.pl
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MORPHODYNAMICS OF THE APPROACH TO STEADY-STATE
RESPONSE TO RISING SEA LEVEL
Recalling that the governing partial
differential equation is subject to the
boundary conditions and the initial condition
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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-book1DRiverwFPRising
BaseLevelNormal.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.
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CALCULATIONS WITH RTe-book1DRiverwFPRisingBaseLeve
lNormal.xls.
Input to the calculation is as specified below.
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Up to 250 years
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Up to 250 years
Bankfull Width m
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Up to 250 years
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Up to 2500 years steady state achieved!
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Up to 2500 years steady state achieved!
Bankfull Width m
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Up to 2500 years steady state achieved!
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Sea level rise is halted in year 2500 by year
5000 the bed slope is evolving to a constant
value.
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Sea level rise is halted in year 2500 by year
5000 channel width is evolving to a constant
value.
Bankfull Width m
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Sea level rise is halted in year 2500 by year
5000 channel depth is evolving to a constant
value.
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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.