Loading...

PPT – LONG PROFILES OF RIVERS, WITH AN APPLICATION ON THE EFFECT OF BASE LEVEL RISE ON LONG PROFILES PowerPoint presentation | free to download - id: 10e293-ZDc1Z

The Adobe Flash plugin is needed to view this content

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

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).

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.)

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

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

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).

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

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).

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).

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.

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

- 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

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.

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,

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

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.

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.

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

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.

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

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

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.

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

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 .

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.

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.

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.

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.

SAMPLE CALCULATION SLOPE PROFILE

SAMPLE CALCULATION DEVIATORIC BED ELEVATION

PROFILE

SAMPLE CALCULATION BED ELEVATION PROFILES

SAMPLE CALCULATION PROFILES OF BANKFULL WIDTH

AND DEPTH

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

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

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.

CALCULATIONS WITH RTe-book1DRiverwFPRisingBaseLeve

lNormal.xls.

Input to the calculation is as specified below.

Up to 250 years

Up to 250 years

Bankfull Width m

Up to 250 years

Up to 2500 years steady state achieved!

Up to 2500 years steady state achieved!

Bankfull Width m

Up to 2500 years steady state achieved!

Sea level rise is halted in year 2500 by year

5000 the bed slope is evolving to a constant

value.

Sea level rise is halted in year 2500 by year

5000 channel width is evolving to a constant

value.

Bankfull Width m

Sea level rise is halted in year 2500 by year

5000 channel depth is evolving to a constant

value.

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.