Vicksburg,
Mississippi
Waterways Experiment
Station
u.s.
Arrny Corps of Engineers
Contract Number DACW 39-87-C-0034
Hsieh Wen Shen
Eric Larsen
• ... ..ou- ~ .... v.•
,.
--
'.! ,,
List of Tables
11List of Figures
iii
I.
Introduetion
1
U.
Importanee of This Study
ID.
Data
Souree
2
IV.
Charaeteristics and Statisties for All River Miles
2
V.
Analysis of Bends Suitable for Detailed Study
5
VI.
Classifieation of Bend Types
12
VU.
Behavior of Types
16
VID.
Evolution of Types
18
IX.
Referenees
21
x.
Appendix A. Measurements of Geometrie Properties
22
Table 1.
Characteristics
and Statistics for All River Miles
Table 2
.
Constraint
Related River Miles
Table 3.
Tabulation
of All Cutoff Related River Miles
Table 4.
Tabulation
of All Straight River Miles
Table 5.
Tabulation
of All Divided Channel River Miles
Table 6.
Tabulation
of All IrreguIar River Miles
Table 7.
River Miles Suitable for Detailed Study
Table 8.
Study Bends - Sinuosity and Movement Parameters
Table 9.
Summary
of Curvature
and Movement
Values
for the Average
of All
Bends
Table 10. Study Bends Curvature
Parameters
Table 11. Average Bend Movement
Rates
Table 12. Bends Classified
According to Type - Curvature
Parameters
Table 13. Bends Classified
According to Type - Movement
Parameters
Table 14. Distribution
of Bends By Type
Table IS.
Average
of
All
Bends and
Averages
for
All
Types
- Summary
of
Curvature
Values
Table 16. Average of All Bends and Averages for All Types
Table 17. Bends Sorted According
to Preceding Type
Table 18. Distribution
of Upstream
Bends
Table 19. Distribution
of Upstream
Bends for Each Bend Type
Table 20. Evolution
of Bends Over One Time Step
Table 21. Evolution
of Bends Over Two Time Steps
Table 22. Evolution
of Bends Over Three Time Steps
Figure 1.
Definition of Measurement of Geometrie Parameters and Scale Drawing
of Average Shape and Movement of All Study Bends
Figure 2. Sinuosity versus Time Study Bends
Figure 3. Sinuosity versus Time Study Bends Using Limited Range of Data
Figure 4.
MIL
versus Rate of Change of
MIL
Figure 5.
Sinuosity
(MIL)
versus Average Magnitude
Figure 6. Shapes of Bend Types
I
.
INTRODUCIlON
In
1987, Hsieh Wen Shen was authorized by the Waterways Experiment Station, U.s.
Army Corps of Engineers, to conduct a study titled, "Migration of Meandering Rivers"
under the contract number DACW 39-87-C-0034.
The general scope was to study the migrations of meandering rivers under natural
conditions, and the specific tasks were:
1)
to classify meandering planforms according to potential river evolution from the
analysis of the surveying maps (made at different dates) of the Mississippi
River; and
2)
to investigate the predictability of future meandering evolutions for different
types of meandering planforms, as classified in Task
1.
11.
IMPORT ANCE OF TIllS STUDY
The process of meandering has a powerful influence on rnan's use of rivers and river
valleys. Flood control; navigation; bank protection; water supply projects; agricultural and
urban development; transportation systems; and pipeline adjacent to rivers all require a
sound knowledge of the meandering process, in order to
he
successful, with minimum
expenditure and maximum environmental protection.
There are three basic river patterns:
straight, braided, and meandering.
For yet
unestablished reasons, meandering is the predominant pattern and thus, has received the
most attention.
A straight channel is rather rare, and pure braided rivers usually cannot
provide navigation service. Therefore, this study has addressed meandering rivers only.
In order to effectively study a particular river reach, the river's stability and possible
channel movements must be evaluated.
River engineers and geomorphologists usually
examine both the aerial photographs and surveying maps of a particular river and try to
decide potential channel movements. The results from this study provide useful information
on the determination of projective meander migrations.
2
-ID.
DATA SOURCE
Twelve maps of 842 miles of the lower Mississippi River were taken from "Lower
Mississippi River
Early Stream Channels," prepared
in the Office of
the President,
Mississippi River
Commission, Vicksburg, Mississippi, in August
1938.
These maps
document the Mississippi River from Cairo, lllinois to Baton Rouge, Louisiana .
....The maps are pritued in colors and show early stream channels for
the
[ollowing periods:
1765(Lieut. Ross Survey) in blue; 1820-1830
(U'S,
Land
Office Surveys) in red; 1881-1893 (Mississippi River Commission Survey) in
green; 1930-1932 (Mississippi River Commission Survey) in yellow
.
....The
basic data on the maps shown in black (which
include some
topographie information and developments sueh as cities, roads, railways, and
dikes) were prepared from Mississippi River Commission advance topographic
quadrangle sheets. subject
10correction. surveys 1930-1932.
River distanees
below Cairo gage are shown at 5 mile intervals
.
derived from 1916 published
mileages .
....scale 1:62.500
(abOUIone inch
10one mile ).
The maps are photo offset reproductions printed in colors as noted above. The
size of each sheet is approximately 19-1/2 by 52 inches.
These maps are appropriate for the study of the natural evolution of meander bends
because there was a limited amount of human interference
in the morphology of the
Mississippi River during this time period.
"J
I
A.
Introduetion
IV. CHARACTERISTICSAND STATISTICSFOR ALL RIVER MILES
The Mississippi River between Cairo, Illinois and Baton Rouge, Louisiana, with a total
of 842 river miles, was examined.
The 842 miles were divided into separate river reaches,
corresponding to a ehannel segment between two successive inflection points of a meander
bend.
On the basis of these examinations, eertain characteristics were identified and each
reach was categorized into one of five different categories, as follows. Af ter examining all
bends of all reaches, it was determined that certain reaehes could not be analyzed in depth
because they were eonstraint related, eutoff related, straight, divided, and/or irregular.
1)
Constraint related
2)
Cutoff related
3)
Straight
4)
Divided
5)
Irregular
Four surveys taken during four different years were available for each river reaeh.
Although there were four time periods for each reaeh, each reaeh was assigned to only one
eategory.
If
areach
eould be elassified in only one of the above five categories for at least
three out of four time periods, it was classified in that category.
River lengths as caleulated
from the 1930 maps were
used
to analyze the percentage of certain categories within the
total river reaeh.
Table 1 in the appendix is a tabulation of each reach in the 842 river mlles.
The
river miles, as given on the map of 1930, are given in the columns "Beginning Mile" and
"Ending Mlle." The ehannel centerline length of the river reach
is
given in the next column
entitled "Length."
The
final column, "Comments," briefly describes the eharaeteristics of the
reaeh.
B.
Group I - Reaehes Not Selected for Detailed Analysis
The following paragraphs describe the characteristics of each category.
1.
Constraint Related
Bends were categorized as constraint related
if
the river ehannel bank was directly
constrained.
Bends were
alsocategorized as constraint related if the movement of the bend
in question
was
affected by a constraint, either upstream or downstream.
For
this
study, a constraint
was
considered to be a man-made or natural geologieal
barrier that restrieted the movement of the river bank.
Bank heterogeneities, like the cIay
plugs that are known to exist on the Mississippi River, did not receive special considerations.
In
most cases, the constraint was first identified by observing the movement of the
river over a number of time periods.
Usually, one sector of the bend curve remained
stationary, when experience indicated that it should have moved. In subsequently examining
the original map, the constraint was often identified from features on the map.
In
one case
(mile 120-125), there were no topographic contour lines on the map. and from the restrieted
nature of the movement, a geologie constraint was inferred.
Examples in which the movement pattem
was
influenced by aeonstraint
direetly
upstream or directly downstream are the bends beginning at river miles 437, 604, and 825.
These bends were also categorized as constraint related.
The constraint related reaehes are
listed in Table 2.
2.
Cutoff Related
A reaeh was considered to
he
cutoff related when it underwent a eutoff between two
time periods.
A eutoff occurs when the loop of a meander bend is bypassed by a new
section of straight channel (for example, bends beginning at river miles 592 and
750).
In
addition to the reaeh itself, the adjoining reaehes, both upstream and downstream, were
aften affected in their movement pattems by the eutoff. These were also eonsidered to have
movement pattems whieh were affected by eutoff, and were categorized as eutoff related.
The eutoff related reaches are tabulated in Table 3.
3.
Straight
A reach of river miles was considered
to be a straight reach if
the sinuosity was less
than
1.05. Three exceptions to
this
were made in which
bends
of this sinuosity were chosen
as suitable for detaiIed study (Bend 1. 1880; Bend 7. 1765; Bend 44. 1765.) The straight
reaches are tabulated in Table 4.
4.
Divided Channels
This
category describes a channel that is split into two sections with a large island
between the channel divisions.
A channel
in
which the sizes of the two channel divisions
are comparabIe
is
considered to be a divided channel (for example, the bend beginning at
mile 22).
This
is not the same as the classic braided channel but is a distioct separation of
the original single channel into two large separate channels. If the two branch widths were
within 20% of each other, the channel was categorized as divided.
The
divided reaches are
tabulated in Table 5.
5.
Irregular
There were five bends which did not fall strictly into any of the above categories
.
For each of these. it was judged that the bends moved abnormally,
The bends were not
suitable for detailed study. The irregular reaches are tabulated in Table 6.
C.
Group
n -
Reaches Selected for Detailed Analysis
Bends were chosen for detailed analysis if their movement
was
unconstrained natural
movement, free of the influence of cutoff.
These study bends are listed in Table 7
in
the
appendix.
Fifty-seven reaches corresponding to 371 cumulative river miles (or 44% of the
total river miles) were selected for detailed study.
For the detailed study, many auxiliary
tracing maps were prepared.
For each reach, tracings were made of the river channel
planform, outlining the channel bankline, with the average reach of length of approximately
50 river miles.
Each tracing described the bend movements between two successive time
periods and thus, the movement from one time period to another could be examined easily.
These tracings were analyzed quantitatively in detail.
D.
Conclusions
Almost one-half
(44%) of the total river miles were study bends which were
considered suitable for detailed analysis.
Of the remainder of the bends not selected for
study, 25% had some type of restraint which restricted the movement of the river banks;
13% were cutoff
related;
9%
were straight; 6% were divided; and 3% had irregular
movement pattems.
The following factors should be considered in making comparisons with
other rivers.
I)
Channels in other environments may have different degrees of restrictions
.
2)
The rate of change of bend characteristics is a function of hydraulic forces, soil
erodibility, and other factors.
The tendency of the movement, as obtained
in
this
study, may be applicable to other rivers. However, the exact migration rate
cannot be directly applicabie.
3)
A divided channel, as defined
in this study, mayor
may not be generalizabie
to
other rivers.
4)
The
finding that
9%of
all
the river reaches are straight
is
comparabie to the
findings of others on a wide range of rivers.
(Leopold, personal
communica-tion.)
5)
No consideration has been given in this study to the effect of soil type and
therefore the degree of erodibility of the bank.
6)
No consideration has been given in this study to the hydrologie conditions or
geophysical events sueh as earthquakes.
V.
ANALYSISOF BENDS SUITABLE FOR DETAILED STUDY
A.
Introduetion
Fifty-seven reaches representing 44% of the total river miles were ehosen for detailed
analysis. Since records colleeted from four time periods were available for detailed analysis
for 57 reaehes, 228 bend examples were available for analysis.
Out of these, 204 bend
examples were analyzed.
The characteristics
of the bends without subdivisions were
analyzed. The definition and analysis of different
types
of bends is discussed in Section VI.
The following two kinds of bend eharacteristics were measured and analyzed:
I)
Geometrie parameters (size and shape)
2)
Movements
A relationship between the geometrie parameters of a bend and the movement of a
bend was investigated in order to prediet movement, given the geometry of a bend.
B.
Definitions and Methods of Measurement
I.
Definition and Measurement of Geometrie Parameters
In order to characterize the shape of the study bends, the geometrie parameters of
each bend were measured.
All study bends were located between an upstream inflection
point and a downstream inflection point.
These inflection points were defined as the
locations where the eurvature of the channel eenterline changes from convex to concave or
vice versa.
A straight line was drawn between the two inflection points of the bend.
The
downstream direetion of this line was established as the "down-meander" direction, and was
used as the reference direetion for the bend movement over a time interval.
The angle
omega
(w)
for the entranee and exit angles was measured as defined in Figure 1. The angle
is always with respect to the down-meander direction as drawn.
The curvature of a typical bend changes throughout the bend.
Curvatures were
measured at three locations: upstream, apex, and downstream. The apex was defined as the
point of maximum eurvature of the eenterline of the ehannel.
Upstream curvature was
defined as the curvature measured at a location halfway betwen the apex and the upstream
inflection point
Downstream eurvature was defined as the curvature measured at a location
halfway between the apex and the downstream infleetion point (see Figure I).
1.1
The Radius of Curvature
to Width Ratio
The radius (R/W)
ratio has been used extensively
and defined differently
by various
researchers.
Indeed,
even using the same definition,
the measurement
of R/W is still open
to personaI judgment.
For this study,
the radius of curvature
to width
ratio (R/W) at each of the three
locations was measured
in the following
way.
The radius of curvature
was measured
by
fitting an arc of a circle which subtends an angle of 45° to the curvature
at the centerline
.
The local average width of the river channel was determined
by taking three measurements
of width in the vicinity
of the curvature
measurement
For the average width of a reach,
an average
was
taken of the local average widths.
The radius of curvature to width ratio is the ratio of local channel centerline curvature
to local average width.
It was measured at a point as described above
.
Since the curvature
was
constantly changing, the most appropriate value of R/W ratio for a complete bend
required further investigation
.
Initial investigation showed that the curvature at the apex did
not completely represent the nature of the curvature throughout the entire bend
.
Therefore,
the curvature values were averaged throughout the bend with more emphasis given to the
apex, and a composite radius of curvature to width ratio as defined below was used in this
study. The apex reach was given twice the weight as upstream and downstream limbs.
composite R/W
=[R/W(upstream)
+
2 x R/W (apex)
+
R/W(downstream»)/4
This
method worked well for low sinuosity bends.
When this method was used for
hi&!!.ersinuosity bends, it resulted in composite R/W ratios which did not indicate the large
R/W ratio that is intuitively associated with higher sinuosity bends.
The large numerical
influence of the long straight upstream and downstream reaches gave an R/W number
unrepresentative of the characteristic curvature.
However, for this study, this composite
R/W ratio as defined above was used. Later in the analysis of bends, meander bends were
classified into four types based on sinuosity. Types land 11were lower sinuosity bends, and
Types III and IV were of higher sinuosity, Since Type III bends were not weIl characterized
by the composite radius of curvature, the R/W of the apex only was used without any
weight from upstream or downstream for Type III bends. Type IV bends were not analyzed
with respect to R/W.
1.2
Sinuosity (Are Length to Wave Length) as a Characterizing Parameter
The sinuosity (M/L) is commonly defined as the ratio of the are distance of the
centerline of the channel between the two inflection points (M) to the straight line between
the same two inflection points (L).
It
is the most commonly used parameter to describe the
sh~e of meandering bends. The determination of sinuosity is much less subjective than the
R/W ratio, and is easy to measure. The sinuosity indicates a dimensionless shape of a bend.
The shape of the bend directly determines the pattern of the flow lines through the bend.
In turn, the direction of flow through the bend directly determines the erosion-deposition
and subsequent shape of the bend.
Therefore, the M/L ratio may be correlated with the
erosion-deposition pattern and subsequent migration of a river bend.
2.
Definition and Measurement of Movement Parameters
Two superimposed mappings of a single meander bend from two different
time
periods were used
toinvestigate the movement of
this
bend.
The
bend as it existed at the
beginning of the time interval
was
referred
toas the "old" bend.
The bend as it existed at
the end of the time period was referred to as the "new" bend.
The process for measuring the movement parameters
was
as follows:
a)
The inflection points of the old bend and the new bend were located
.
b)
The
apices of the okt and the new bends were located.
e)
The
movements (of the inflection points and the apex in question) from the okt
bend to the new bend were measured by drawing the veetors from the okt
points to the corresponding new points. These veetors of movement were then
decomposed into the vector components in the down-meander direction called
tangentlal movement, and the vector component normal
tothe down-meander
direction ca1led normal.
Normal movement
was
defined to be positive toward
the direction of the concave eurve of the bend.
All the movements were
measured by this method, and the results are tabulated in Table 8.
c.
Analysis of Study Bend Data
In
the first part of the analysis of the study bend data, arithmetie averages were found
for
all
the measured parameters and the parameters derived from them.
These values are
reported and examined below. First the geometrie parameters were studied. These reflected
the relative shape of all study bends.
Next the movement parameters were studied.
These
indicated the migration of the meander bend.
FinaUy, the correlation of the geometrie
parameters and the movement parameters was studied.
I.
Geometrie Parameters
l.l
Average Values of the Geometrie Parameters
An arithmetic average was made of the values of eaeh of the geometrie parameters of
the study bends. Table 9 gives the average value (ave.) and standard deviation (s.d.) of eaeh
of the measured geometrie parameters for all study bends.
Figure 1 shows a scale drawing
representing the shape as defined by the average values presenred below.
The average entrance angle (74°) was slightly greater than the average exit angle (670).
This is significant and reflects the tendeney for bends of high sinuosity
tobe asymmetrie.
The average sinuosity
(MIL)
was
found to be 1.8.
The average straight line distance
between inflection points or half wave length
(L/2)
was 3.93 miles. The average ehannel
width was
0.82
miles.
These values correspond to a meander wave length (see Leopold,
1960, for definition) of 9.59 ehannel widths, whieh falls within the range of 7 to 12 as
determined by various researchers
.
(Riehards, 1982; Leopold, Wolman, and Miller, 1964)
Although
the radius of curvature
to
width ratio of a bend has traditionally been
used
to characterize the geometry of a bend, there is no standard method of measuring this value.
A composite radius of curvature
to
width ratio
was
defined in
this
study, because it better
characterizes the bend curvature
than
the apex radius of curvature
to
width ratio. Af ter
all
the analyses in this study were done, it was found that the apex R/W ratio
was
the simplest
to use and the most consistent with other researchers
.
The average values of apex R/W and
aIso
of the composite R/W are both reported.
The arithmetic average of the apex radius of curvature to width ratio for 204
examples was found
to be
2.1. The arithmetic average of the composite R/W ratio for 204
examples was computed
and
found to be 2.9, with values ranging from a low of 0
.
3
to
a
high of 9.3. These figures compared well with a median value of 2.7 within a normal range
of 2-3 for a large number of rivers, as determined by Leopold and Wolman (1960). In their
studies the apex radius of curvature to width ratio
was
used. The apex was usually smaller
than
the corresponding value for the inflection points.
The complete data for the average
calculated in this study is included in Tables 8 and 10.
1.2
Relationship Between Sinuosity and Radius of Curvature to Width Ratio
The correlation between the apex R/W ratio and the sinuosity, and the composite R/W
ratio and sinuosity did not yield a satisfactory relationship.
1.3
Change in Sinuosity With Time
The change of sinuosity with time was studied by constructing a graph
to
indicate the
change in sinuosity with elapsed time for all study bends. This graph began at time equals
0.0 and sinuosity equals l.O.
First, a graph was made in which the average relationship for sinuosity versus time
was determined for all study bends with the initial sinuosity equal
to
l.O. Then, a second
graph
was
made for all study bends with the initial sinuosity equal
to 1.1.
The second graph
was
then superimposed on the f'irst graph, starting the second graph at the point where the
sinuosity on the first graph equaled
1.1,
thereby developing a total graph of sinuosity versus
time for sinuosity ranging from 1.0 to 2.8.
Most
of the data are for the sinuosity range
between 1.0 and 1.9. The results using all the data are shown in Figure 2, and the results
using this limited range of data are shown in Figure 3.
The total time interval in the study was from the years 1765 to 1930.
In
1765, each
bend had a sinuosity not necessarily equal to l.O.
The increment of time elapsed was
significant in this study, rather than specific years.
Therefore, separate graphs were made
of sinuosity versus increment of time elapsed, and they were joined appropriately.
An
analysis of Figure 2 reveals the following results:
1)
Sinuosity tends
to
increase with time.
2)
The average rate of change of sinuosity between 1.0 and 2.8 was about 0.003
(MIL)
per
year,
3)
Figure
'2
suggested that the time scale for the sinuosity of a meander bend to
develop from 1.0 to 2.8 was 600 years.
4)
The average increment
of time for a bend to increase in sinuosity
from
1.2 to
1.6 was 155 years. The average increment of time
to
increase in sinuosity from
1.6
to
2.6 was 285 years.
1.4
Time Rate of Change of Sinuosity Versus Sinuosity
The time rate of change of sinucsity was analyzed in another way by correlating it
with the mean sinuosity.
Mean sinuosity was defined as the arithmetic average of the
sinuosity of two time periods. The rate of change of sinuosity was determined by taking the
ratio of change in sinuosity divided by the change in time in
years,
This relationship is
shown in Figure 4.
This
correlation showed an increase in the time rate of change of sinuosity with an
increase in mean sinuosity for the interval of sinuosity between 1.0 and 1.5 or 1.6 and
appeared to reach a maximum somewhere near
l.s.
(See
Figure 4.) For values of sinuosity
greater than this, the pattems were unclear, partially due to a limited number of data points.
The time rate of change of sinuosity increased from about 0.001 (MIL) per year
to
about
0.005 (MIL) per year for the interval of sinuosity betwen J.O and
l.s.
A sinuosity of 1.5 or 1.6 was the sinuosity with the greatest rate of change, and
suggested that the shape represented by this sinuosity was the shape with the greatest rate of
migration.
2.
Movement Properties
2.1
Average Movement Over Time
An average bend tended to migrate in the downstream direction and in the normal
direction at different rates, as predicted by the average values of movement. In general, the
sinuosity tended
to
increase.
An average angle between baselines of two consecutive bends
was
determined to be 15° and this showed that, on the average, two consecutive bends were
not symmetrical.
22
The Average Magnitude of Movement
The average magnitude of movement for a single bend in a single time step was
defined as the arithmetic average of the magnitudes of movement (between two periods of
time) of the upstream inflection point, the apex, and the downstream inflection point. This
definition produced an average magnitude of migration.
2.3
The Average of the Average Magnitudes
of Movement
The average
for all study
bends of all the average
magnitudes
of movements
was
found to be 0.87 miles over an average time period of SS
years.
This
represents an average
channel movement of 84 feet per year, or expressed nondimensionally, 0.019 channel widths
per year. Two other rivers were examined and the results are given in Tab
l
e 11
.
Hickin and Nanson (1983) report values of rate magnitude of movement ranging from
.
002 to .020 channel widths per year for the Beatton River.
See
Hickin and Nanson (1975)
for the definition of the average of movement used in their study.
Unpublished surveys made of Pole Creek, Wyoming over a span of 22 years by
Leopold (survey made in 1964) and by Larsen (survey made in 1986) indicate an average
rate of the average magnitude of movement (as defined in this study) of 0.016 channel
widths per year
.
Asshown in Table 11, the average widths of these rivers are given in
order to provide a scale by which to compare the rivers in these three cases.
D
.
Correlations of Geometry and Movement
1.
The Average Magnitude of Movement Correlated with Sinuosity
The average magnitude of movement for all study bends was correlated with the
sinuosity for all study bends in three ways.
The relationship was plotted on an arithmetic
scale, with aregression
line drawn, as shown in Figure 5.
The average magnitude of
movement appeared to decrease with increasing sinuosity of a bend.
Hickin and Nanson (1975) have reported that, as the R/W
ratio increases, the
migration rate of a bend increases, reaches a maximum value sornewhere between 2 and 3,
and then decreases. Their definition of movement
is
based on the lateral migration (1975)
and may be comparable to the movement of the apex as used in this study.
However,
results froll!_this study did not indicate the same trend between the average apex movement
and the R/W ratio, although data scattering may obscure the trend.
To f'urther examine the maximum movement related to a certain curvature value, the
sinuosity was first correlated with the magnitude of movement of the apex
.
The sinuosity
was then correlated with the component of the apex movement tangential to the baseline,
and finally, the sinuosity was correlated with the component of the apex movement normal
to the baseline. The correlation of the sinuosity with the magnitude of the apex movement
was weak and unclear.
When the sinuosity was correlated with the component of movement
tangential to the baseline, there was a tendency for the movement to deercase with sinuosity.
When the sinuosity was correlated with the component of movement normal to the baseline,
there
was
a tendency for the movement to increase in magnitude up to a value of sinuosity
=
