Proceedings of the 8th International Conference on Fluid Mechanics (ICFM8)
September 25-28, 2018, Tohoku University, Sendai, Japan

ONE-LINE MODEL FOR
RIVER MEANDERING EVOLUTION
Kazuhiko MATSUNOBU 1,*, Tadaharu ISHIKAWA 2
1

TOKEN C. E. E. Consultants Co., Ltd., 1-15-6, Kita-Otsuka, Toshima Ward, Tokyo 170-0004, Japan;
email: matsunobu-k@tokencon.co.jp
2
Tokyo Institute of Technology, 4259, Nagatsuta Town, Midori Ward, Yokohama City, Kanagawa Pref. 226-8503,
Japan
email: workishikawa0612@yahoo.co.jp

Abstract: Herein, we report on a one-line model for river meandering evolution that was developed by considering
the effects of helical flow. First, flow and riverbed variation in a meandering channel are formulated in one dimension,
after which the model is verified by comparing the numerical computation results with the existing flume experimental
data. Next, the model for describing channel meandering shape deformation is created geometrically. In this model, thevelocity of channel deformations caused by side bank erosion is assumed to be proportional to the linear combination of
the intensity of the main stream lateral shift and secondary circulation. The weights of those two factors are determinedby a comparison between the model response and empirical rules of meandering wavelengths in actual rivers. Finally, thechannel neck cut-off on the channel line deformation area is modeled and the resulting model calculations are comparedwith meandering wavelength data obtained from Google Earth satellite images.

Keywords: River channel meandering, One-dimensional formulation, Secondary flow, Meandering wavelength,
Satellite image*

Corresponding author. Email: matsunobu-k@tokencon.co.jp

1. Introduction
River meandering on alluvial plains is among the more
important research topics in geomorphology and riverengineering, as well as the topic of numerous existing casestudies and experimental flume studies [1]-[4]. Recently,
spurred by advancements in computation technology, anumber of horizontal two-dimensional (2D) or threedimensional (3D) numerical models have been proposed[5]-[8]. Notably, Asahi et al. [5] simulated a channel
meandering development process used for modeling sidebank erosion and deposition by considering the mechanism
outlined in Parker et al. [9]. However, due to various
problems such as high computation loads, those researchers
were unable to simulate long-term and long-reachmeandering channel deformations in 2D or 3D models.
On the other hand, long-time meandering channel
deformation processes have been successfully formulated in
one-dimensional (1D) equations based on research derived
from Ikeda et al. [10]. As an empirical principle, thoseequations begin by assuming that the riverbed height
difference between the left and right banks is proportional
to the curvature of the central river channel axis, assuggested by Engelund [11], and then determine the lateral
stream shift of flow velocity. Next, based on anotherempirical rule [12], they assume that the side bank erosion
velocity is proportional to the lateral stream shift. The
above process led to the creation of a 1D bend equation that
describes the development of a meandering river channel.
Since that time, there have been several numerical research
efforts that applied those equation results to the bend
equation in Ikeda’s study [13]-[15].
Noting that stream shifting in meandering river
channels is affected by cross-sectional secondary flows
caused by the curves of river channel axes, Ishikawa andKim [16] produced a 1D equation for calculating the mainstream shifts and the intensities of the resulting secondary
flows. Liu and Ishikawa [17] then improved that 1D modelby taking the effect of the stream shift mechanism outlined
by Ikeda et al. into consideration using the equation
produced by Ishikawa and Kim, after which they appliedthe resulting model to laminar flows. From that point,
Ishikawa and Liu [18] expanded the theory to coverturbulent flows.
Furthermore, Matsunobu et al. [19] suggested a scheme
for calculating the lateral differences of a channel bed byusing Ishikawa and Liu’s equation without assuming
Engelund’s empirical rule, which is the base of the model
describing the river meandering development process.
In this paper, we report on the construction of a 1D
numerical model of river meandering development thatworks by coupling the computation routine of rivermeandering development produced via side bank erosion tothis meandering channel bed variation model. We thenapply our model to actual free meandering rivers and verify
its accuracy by comparing the output results with thesatellite image data obtained from Google Earth.

2. Modeling of meandering flow and channelbed
In this section, we explain the concepts behind the 1D
meandering flow model produced by Ishikawa and Liu [18]along with the channel bed variation model outlined in
Matsunobu et al. [19]. After that, we explain how wechange and combine these models in order to more easilymodel channel meandering deformation.
The flow model is based on Navier-Stokes equations
and a continuity equation in an orthogonal curving
coordinate system (s, n). As for flow velocity (u, v) andwater depth h, we describe the typical distribution profiles
seen in the meandering channel flow in five mode
functions. Hydraulic quantities, (u, v) and h, are assumed as
Eqs. (1) through (3).
(1)
𝑢𝑢 = 𝑢𝑢0 (1 + 𝛽𝛽1 (𝑠𝑠)𝜂𝜂)(𝜙𝜙(𝑧𝑧) + 𝛽𝛽2 𝜓𝜓(𝑧𝑧))(2)𝑣𝑣 = 𝑢𝑢0 {𝛽𝛽3 (𝑠𝑠)𝜓𝜓(𝑧𝑧) + 𝛽𝛽4 (𝑠𝑠)(1 − 𝜂𝜂 2 )𝜙𝜙(𝑧𝑧)}1ℎ = 𝐻𝐻0 �1 + 𝛽𝛽5 (𝑠𝑠) �𝜂𝜂 − 𝜂𝜂 3 ��
3(3)

where s, n, and z are longitudinal, transverse, and verticaldirection coordinates, respectively, in orthogonal
curvilinear coordinate systems, u and v are flow velocities
in the s and n directions, respectively, h is the water depth,η is a non-dimensional lateral coordinate (=2n/B, B:
channel width, － 1 ≦ η ≦ 1), φ(z) is the vertical mode
function of the main stream in a straight channel, ψ(z) is thevertical mode function of secondary circulation flow anduniformization caused by the secondary circulation, u 0 is
the cross-sectional average flow velocity, and H 0 is the
cross-sectional average water depth.
In Ishikawa and Liu’s model, h is assumed as Eq. (3’).
The reason why it is assumed to have a cubic function asshown in Eq. (3) of this study will be discussed later.

ℎ = 𝐻𝐻0 {1 + 𝛽𝛽5 (𝑠𝑠)𝜂𝜂}(3’)

β 1 to β 5 are the variables that represent the intensity offive typical flow patterns in a meandering channel. They
depend solely on s, which is the longitudinal coordinate in achannel. The flow pattern that corresponds to each variable
is shown in Table 1. By applying Galerkin integration in across section corresponding to the five variables to thebasic equations, the model is formulated in one dimension.
In other words, the independent variable is exclusively s.
Table 1. Variables and correspondent flow pattern
Variable Flow pattern
Main stream shift
β 1 (s)
Uniformization of main flow
β 2 (s)Secondary flow
β 3 (s)Vertically averaged transverse flow
β 4 (s)Transverse deviation of water depth
β 5 (s)Here, the basic equations contain the gradient of z, as
well as the riverbed height in the s and n directions. In
order to apply the equations to the meandering channels
that have the lateral bed slopes, it is necessary to apply
some mode function assumptions, such as flow velocity andwater depth, to the riverbed topography. In this study, thecross-sectional mode is assumed to be the cubic function,
the transverse gradient of which is zero at both sides of thebank, as shown in Eq. (4).
(4)
𝑧𝑧(𝑠𝑠, 𝑛𝑛) = 𝑧𝑧0 (𝑠𝑠) − 𝑎𝑎(𝑠𝑠) ∙ (𝐵𝐵/2)(𝜂𝜂 − 1⁄3 𝜂𝜂 3 )where a(s) is the mode intensity of the cross-sectionalriverbed form assumed in this study. The second term on
the right-hand side in Eq. (4) is the odd function of eta,
which means that if the one side of the bank is eroded,
deposits have formed on the other bank. In addition, inorder to more easily handle the water surface, the water
depth h is assumed in the cubic function of the same form
as Eq. (4), as shown in Eq. (3)In this riverbed model, the time development of the bed
in a meandering channel is computed by combining theriverbed variation to the meandering flow. The 1D bed load
model is based on the study by Matsunobu et al. [19]Eq. (5) is the continuity equation of the bed load.
𝜕𝜕𝜕𝜕1𝜕𝜕𝑞𝑞𝑠𝑠 𝜕𝜕(1 + 𝜎𝜎𝜎𝜎)𝑞𝑞𝑛𝑛(5)(1 + 𝜎𝜎𝜎𝜎)
=−
�
+
�
𝜕𝜕𝜕𝜕
1 + 𝜆𝜆 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕The riverbed is formulated on both sides of the meshthat is divided into two parts in the lateral direction, fromwhich the time development of a, which is the mode
intensity of the riverbed, is computed. The conceptual
figure is shown in Fig. 1.
By substituting Eq. (4) into Eq. (5) and integrating theequation in each side in a transverse direction, the bed loadbalance is described as follows:Table2. Channel
experiment

Channel width
B [cm]
0.3
Cross-sectional
averaged flow
velocityu 0 [cm/s]
32.21

0

configuration
Longitudinalchannel slope
I s00.0141
Cross-sectional
averaged waterdepth
H 0 [cm]
0. 89

inHasegawa’s

Flow discharge
Q [L/s]0.75
Representative
particlediameterd [mm]
0.43

Fig. 1. Conceptual figure of riverbed computation
0

𝜕𝜕𝜕𝜕
1
� (1 + 𝑏𝑏𝑏𝑏𝑏𝑏) � �𝜂𝜂 − 𝜂𝜂3 �� 𝑑𝑑𝑑𝑑
𝜕𝜕𝜕𝜕
3
−1
1
𝜕𝜕𝑞𝑞𝑠𝑠𝑠𝑠 𝜕𝜕𝑞𝑞𝑛𝑛0=�−−�
1 + 𝜆𝜆
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
1
𝜕𝜕𝜕𝜕
1
� (1 + 𝑏𝑏𝑏𝑏𝑏𝑏) � �𝜂𝜂 − 𝜂𝜂3 �� 𝑑𝑑𝑑𝑑
𝜕𝜕𝜕𝜕
3
0
1
𝜕𝜕𝑞𝑞𝑠𝑠𝑠𝑠 𝜕𝜕𝑞𝑞𝑛𝑛0=�−+�
1 + 𝜆𝜆𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕0𝑞𝑞𝑠𝑠𝑠𝑠 = � 𝑞𝑞𝑠𝑠 (𝑠𝑠, 𝑛𝑛)𝑑𝑑𝑑𝑑−11

𝑞𝑞𝑠𝑠𝑠𝑠 = � 𝑞𝑞𝑠𝑠 (𝑠𝑠, 𝑛𝑛)𝑑𝑑𝑑𝑑0

𝑞𝑞𝑛𝑛0 = 𝑞𝑞𝑛𝑛 (𝑠𝑠, 0)

Z[m]

(6)

(7)
Fig. 2. Riverbed height in Hasegawa’s experiment
(8)
(9)
(10)

By observing the difference between Eq. (6) and Eq.
(7), Eq. (11) is obtained and the time development of a canbe calculated.
𝜕𝜕𝜕𝜕
6
𝜕𝜕(𝑞𝑞𝑠𝑠𝑠𝑠 − 𝑞𝑞𝑠𝑠𝑠𝑠 ) 𝑞𝑞𝑛𝑛0
(11)
=−
�
+
�
𝜕𝜕𝜕𝜕
5(1 + 𝜆𝜆)𝜕𝜕𝜕𝜕𝐵𝐵The bed load transport is calculated from the shear
stress observed at the riverbed. The functions are suggested
by numerous existing studies. In this study, Ashida andMichiue’s function [20], as written in Eq. (12), is adopted.

3/2

𝑞𝑞∗ = 17𝜏𝜏∗

�1 −

𝜏𝜏∗𝑐𝑐
𝜏𝜏∗𝑐𝑐
� �1 − � �
𝜏𝜏∗
𝜏𝜏∗

(12)

where q * is a non-dimensional amount of bed load transport,
τ * is non-dimensional shear stress, and τ *c is nondimensional critical shear stress.
The shear stress is computed by Eq. (13).
𝑓𝑓𝑢𝑢𝑏𝑏 �𝑢𝑢𝑏𝑏2 + 𝑣𝑣𝑏𝑏2
𝑘𝑘𝑓𝑓
𝜏𝜏∗𝑠𝑠
⎛
⎞ + 𝑘𝑘𝑔𝑔 � 𝐼𝐼𝑠𝑠 �𝜏𝜏∗ = �𝜏𝜏 � =𝐼𝐼𝑛𝑛∗𝑛𝑛
𝑅𝑅0 𝑔𝑔𝑔𝑔
2
2
𝑓𝑓𝑢𝑢𝑏𝑏 �𝑢𝑢𝑏𝑏 + 𝑣𝑣𝑏𝑏
⎝
⎠

(13)

where R 0 is the relative specific gravity of sedimentparticles, g is gravity acceleration, d is the representative
diameter of the sediment particles, f is the friction
coefficient, u b and v b are (respectively) the flow velocities

Fig. 3. Velocity vectors in Hasegawa’s experimentin the s and n directions near the river bed, I s , and I n are
the riverbed slope in the s and n directions, respectively(=grad z), k f is the rate of effective shear force and all shear
forces, and k g is the correction coefficient obtained by theshape of the sediment particles.
The first right-hand term in Eq. (13) is the effect ofhydrodynamic force achieved by the flow velocity
determined by Eqs. (1) and (2), while the second term is theeffect of gravity generated by the channel slope determinedby Eq. (4). k f was determined to be 0.44 via trial
computations, and k g is 2/3, which is the value obtained
when the particles were assumed to be spherical.
The models were then verified with Hasegawa’s flume
experimental data [21]. Table 2 shows the configuration of
the channel, while Figs. 2 and 3 show a contour map of theriverbed height and the flow velocity vectors in theequilibrium state, respectively.
Fig. 4 shows the time development of the amplitude ofa, which is computed under the conditions of Hasegawa’s
experiment in this model. First, we can see that the channel
bed variation rapidly becomes bigger, but the change intime becomes smaller, until an equilibrium state is finallyachieved. Fig. 5 compares the longitudinal profile of β 1 and𝛥𝛥𝛥𝛥 = 𝐸𝐸𝐸𝐸𝐸𝐸 ⋅ 𝑢𝑢𝑠𝑠

5.00E-02
0.050
4.50E-02
0.045

where ∆n is the channel axis movement in the normal
direction of the channel, ∆t is the time step in themeandering computation, E is the non-dimensionalcoefficient of side bank erosion, and u s is the flow velocity
that controls the velocity of side bank erosion; and isdetermined by “some pattern of channel flow.”

4.00E-02
0.040
3.50E-02
0.035|amax|

3.00E-02
0.030
2.50E-02
0.025
2.00E-02
0.020
1.50E-02
0.015
1.00E-02
0.010

0.005
5.00E-03

0.000
0.00E+00
0.00E+00 1.00E+03 2.00E+03 3.00E+03 4.00E+03 5.00E+03 6.00E+03
0

3000

6000

Time[s]Fig. 4. Time development of the amplitude of a1

実験データ
■ FlumeExperiment
計算結果― Computation

β1

0.5
0

-0.5
-1

0

90

180
Phase[deg]Phase[deg]

270

360

― Flume Experiment― Computation

0.04

Here, the existing studies that are derived from Ikeda etal. used β 1 , which is the main stream shift intensity.
(15)
𝑢𝑢𝑠𝑠 = 𝛽𝛽1 𝑢𝑢0

The reason for this choice is that the 1D model
formulated by Ikeda et al. is based on shallow waterequations. Therefore, their model had more limited
parameters than the model used in this study. Morespecifically, they did not use the two parameters, intensityof secondary flow β 3 and the uniformization of the mainstream, in the vertical direction caused by secondary flow
β2.
On the other hand, in an actual meandering channel,
downward and upward flows are generated in the outer andinner bank, respectively. These are caused by the secondary
flow that is considered in Ishikawa and Liu’s model. This
results from the fact that the outer bank of the channel
erodes, while deposits form on the inner bank, as pointed
out by Sekine [22].
Therefore, when computing the development ofmeandering channel form, better results can potentially beobtained by considering the side bank erosion produced by
the secondary flow intensity, β 3 .

0.08

a

(14)

0

However, the most appropriate method for combining

β 1 (main stream shift) and β 3 (secondary flow intensity) is
-0.04
-0.08

0

90

180

270

360

Phase[deg]Fig. 5. Longitudinal profile of β 1 and aa between the experimental data and computational results.
Here, it can be seen that the computational results roughly
agree with those of the flume experiment.

3. Modeling of channel deformation by rivermeandering
In this section, we formulate meandering river channel
deformation based on the riverbed variation model
constructed in the previous section.
Meandering channel development is determined by thecentral axis movement of the channel and is the same as theexisting 1D model. The deformation velocity is
proportional to the intensity of “some pattern of channel
flow” in the equilibrium riverbed as seen in Eq. (14).

not yet clear. In this study, we assumed the linear
combination as shown in Eq. (16) in order to grasp thegeneral influence trends produced by the change of those
two factors.
(16)
𝑢𝑢𝑠𝑠 = {𝛾𝛾𝛽𝛽1 + (1 − 𝛾𝛾)𝛽𝛽3 }𝑢𝑢0
where γ is an empirical constant for the combination ratio,
and when γ = 1, Eq. (16) is equal to Eq. (15).

The movement of the node points (x i , y i ) in Cartesian
coordinates that is caused by the channel deformation is
described in Eq. (17) and is based on ∆n obtained by Eq.
(14).
𝑥𝑥𝑖𝑖 (𝑡𝑡 + 𝛥𝛥𝛥𝛥) = 𝑥𝑥𝑖𝑖 (𝑡𝑡) − 𝛥𝛥𝛥𝛥 ∙ sin𝜃𝜃𝑖𝑖
𝑦𝑦𝑖𝑖 (𝑡𝑡 + 𝛥𝛥𝛥𝛥) = 𝑦𝑦𝑖𝑖 (𝑡𝑡) + 𝛥𝛥𝛥𝛥 ∙ cos𝜃𝜃𝑖𝑖
(17)
𝑛𝑛
𝑛𝑛
𝑠𝑠𝑖𝑖 = � 𝛿𝛿𝑠𝑠𝑖𝑖 = � �𝛿𝛿𝑥𝑥𝑖𝑖 2 + 𝛿𝛿𝑦𝑦𝑖𝑖 2
𝑖𝑖=0

𝑖𝑖=0

where θ i is the local angle of the channel from the x axis,(δx i / δs i , δy i / δs i ) = (cos θ i , sin θ i ), δx i = x i - x i-1 , δy i = y i y i-1 , δs i = s i - s i-1 , and s i is the longitudinal coordinate s of
the i-th node point.
Each time a new channel form is determined, the modeintensity parameters in the equilibrium riverbed of thechannel condition, including β 1 and β 3 , are computed. By
using them, the new channel form is calculated from Eqs.
(14), (16), and (17). Then, by repeating the routine, the timedevelopment of river channel meandering can be
reproduced.

section of the Madeira River, which belongs to the Amazon
River system. In this figure, the traced channel line isshown with a red line and the central axis of themeandering is shown with a blue line. Note that the wave
half-length is the distance between the two green points thatare next to each other.
Fig. 9 is a histogram of L/B, which is the wavelength

4. Model verification

L[m]Next, we attempted the following numericalexperiment aimed at estimating the value of γ in Eq. (18).
We began by noting that the wavelength that appears whenthe meandering deformation of a straight river channel
begins is almost the same with that of an alternating bar. In
addition, the wavelength L is roughly proportional tochannel width B, and L/B is almost equal to 10, as shown in
Fig. 6.

○●

Data from studies on
flume experimentand actual rivers
Data from anotherexperimental study

Data fromfield surveys
in 8 JapaneseriversHere, the value of γ when the meandering wavelength
in the channel meets the condition of L/B=10 under theinfinitesimal amplitude is verified with the numericalsimulation as described below.
The channel is configured as shown in Table 3, which is thevirtual conditions on a scale of actual rivers where thetypical free meandering on an alluvial plain can befrequently seen. The initial deformation is given in the formof a long straight channel by a sine-generated curve, as seenin the black line in Fig. 7. The computation results forvarious values of γ are also shown in the figure. The smaller
the values of γ are, more specifically, the bigger the
influence of β 3 is, the shorter the length of the meanderingwave is, and the bigger the amplitude becomes. It can beseen that when gamma is 0.2, L/B approaches 10.
Next, using satellite images of actual rivers obtained
from Google Earth, the central axis lines of 30 meandering
sections of rivers around the world, with a variety of riverwidths, were digitized and the central axes of theirmeandering waves were estimated. Then, the wavelengthsof each of those rivers were computed and compared withthe computational results.

L/B=101
1

B[m]Fig. 6. Relationship between L and B in many actual
rivers (Committee on hydraulics and hydraulicengineering, JSCE) [23]Table 3. Channel configuration in the computation ofmeandering development
Longitudinal
Flow
Channel widthchannel slope
discharge
B [m]
I s0
Q [m3/s]
300
1/30003600
Cross-sectional
Cross-sectional
Representative
averaged flow
averaged waterparticlevelocity
depth
diameteru 0 [m/s]
H 0 [m]d [mm]
1.577
7.608
0.768

As one example, Fig. 8 shows a portion of the upper

800― Initial deformation― γ=0.1― γ=0.2― γ=0.5― γ=1.0

y[m]

400
0

0

5000

10000

-400
-800

15000
L/B=10x[m]Fig. 7. Development of channel meandering for various values of γ

20000

Fig. 8. Example of the digitization of satellite images by Google Earth

The cut-off phenomenon is modeled geometrically as
follows: In each step, after the computation of channel
deformation, the distance between two nodes (i and j: theirindices) are computed to produce a judgment, and cut-off
occurs if the two nodes meet the conditions of Eqs. (18) and(19).
𝐿𝐿𝑖𝑖𝑖𝑖 < 𝑘𝑘1 𝐵𝐵(18)
𝑆𝑆𝑖𝑖𝑖𝑖 < 𝑘𝑘2 𝐿𝐿𝑖𝑖𝑖𝑖(19)

where L ij is the straight distance between the two nodes andcan result in a cut-off when this value approaches the riverwidth B. However, if you judge whether channel cut-off
occurs only by Eq. (18), it is judged that the cut-off will
occur in two nodes which are next to each other. Obviously,that makes no sense either in computations or in actual
phenomena. So, to avoid that, the conditions of Eq. (19) areapplied. S ij is the distance along the channel axis between
these nodes. By Eq. (19), channel cut-off will occur only
when the amplitude of channel meandering becomes large
to some extent.
Here, k 1 and k 2 are the coefficients in the value of one
or more, and were determined as k 1 = 2.0 and k 2 = 3.8 via
trial computations. Finding a more reasonable way todetermine these values is a future task. The channels aretied in a cubic Bezier curve that is sequential in angle after
the cut-off has occurred. Fig. 10 shows an example of the20

Frequency rate[%]in each wave divided by the representative river width. In
this figure, the mode value of L/B is 10, which agrees with
the infinitesimal amplitude analysis used in the previous
discussion. However, the profile is distorted and the meanvalue of L/B is about 19. This suggests that the wavelength
becomes bigger as the meandering develops. One possible
cause of this is that the number of waves is reduced by the
cut-off that results from the increasing amplitude and theapproach of side banks in a channel.

15
10
5
0

10

20

30

40

50

60

L/B

Fig. 9. Histogram of L/B in Google Earth images

channel deformation created by a cut-off.
In eight of the 30 rivers that we analyzed using
Google Earth satellite images, we added partial disturbanceinto the straight channels and then simulated thedevelopment and propagation of channel meandering. The
red markers in Fig. 11 show the mean wavelength in thecomputational results of each river, while the green markers
show the values obtained by analyzing Google Earthimages. Note that the black dots are values extracted fromthe images of the 22 unexamined rivers, and the blue
markers show the computational results obtained via themodel set forth in Ikeda et al. That is to say, using Eq. (15)instead of Eq. (16). Here, it can be seen that thecomputation result of this model agrees more closely withthat of the image analysis.
Fig. 12 shows a histogram containing channel
information obtained via numerical computations using thesame process as seen in Fig. 10. Here, we see that the shapeof the Fig. 12 histogram is a little sharper than that seen

100000

Before cut-off■ image analysis◆ computation(old)● computation(new)

L[m]

10000

1000

After cut-off
100

10Fig. 10. Example of channel cut-off

As a specific channel form example, Fig. 13 compares
the satellite image and the channel determined by our
computational results for a section of the Purus River,which is part of the Amazon River system. Here, we notethat qualitatively similar irregularities exist in both channel
axes.

1000

10000

B[m]

Fig. 11. Wavelengths comparison
20

Frequency rate[%]in Fig. 9. In fact, even though the mode value of L/B in Fig.
13 is about 11, which nearly matches that seen in Fig. 9, themean value is 15, which is smaller than the Fig. 9 value. In
addition, a second peak can be seen at nearly L/B=20. One
possible reason for these tendencies is that the conditions
that exist when channel cut-off occurs are determined in
one pair of values, k 1 and k 2 in Eqs. (18) and (19), eventhough the cut-off phenomenon is essentially a probabilistic
event. However, the primary characteristics of thehistogram profile, mode, and mean values seen in Fig. 12
are similar to those seen in Fig. 9.

100

15
10
5
0

10

20

30

40

50

L/B

Fig. 12. Histogram of L/B in computation results

5km

There are two mechanisms involved in these irregularities.
One relates to the fact that two important parameters (β 1and β 3 ), which are used in this study for calculating channel
deformation (as shown in Eq. (16)) have different
characteristics, including the phase used. The other is thechannel cut-off repetition.

5. Conclusions
The conclusions of this study are as follows:
1.

A 1D meandering flow and riverbed variation model
was constructed based on existing models whileassuming the lateral riverbed height will be used as a
cubic function.

2.

This model, which can describe the development of ameandering channel via an examination of side bankerosion, is coupled to the flow and riverbed model. As
a new scheme, the velocity of side bank erosion isassumed to depend not only on the main stream shift,
but also on the secondary flow intensity.

3.

The computations were attempted for the various
values of γ, which is the weighted parameter of thelinear combination of these two flow patterns. Whenγ = 0.2, the ratio of the meandering wavelength and

5km

Fig. 13. Comparison of channel meandering shapechannel width (L/B) obtained from the computations
were in good agreement with the observation data.

4.

The meandering model was then applied to several
actual rivers, after which the channel cut-off
mechanism was built into the model. The primarycharacteristics seen in the histogram of L/B in
computation results agreed well with an analysis ofsatellite image data obtained from Google Earth. In
addition, irregularities in the actual river images canalso be seen in the computation results.

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