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1. EARTH SURFACE PROCESSES AND LANDFORMS, VOL. 20,661-670 (1995)
A REVIEW OF MATHEMATICAL MODELS OF
RIVER PLANFORM CHANGES
ERIK MOSSELMAN zyxwvut
Devt Hydraulics, PO Box 152, 8300 A D Emmeloord, The Netherlands
Received 12 May 1994
Accepted 2 March 1995
ABSTRACT
Different mathematical models of river planform changes exist or are being developed.They are reviewed here by discuss-
ing a two-dimensional depth-averaged model, various meander models and a model for the braided Brahmaputra-Jamuna
River in Bangladesh. Much emphasis is placed on topics where further research is needed. It is concluded that the models
help in understanding the underlying processes, but cannot yet be considered generally valid and easy-to-use software
packages. In the hands of experienced geomorphologists or river engineers, however, some of the models do already
form valuable tools which allow better predictions of future river planforms. zyxwv
KEY WORDS river morphology; bank erosion;meandering; mathematical modelling
INTRODUCTION
The elegant windings of a meandering river and the rhythmic patterns of a braided river please the eye and
the formation and changes of these planforms have been intriguing their beholders for centuries. Nowadays
different mathematical models for river planform changes exist or are being developed. These models are
study tools to increase the understanding of the processes involved as well as prediction tools to estimate
future changes of particular rivers.
There are two ways in which mathematical models increase understanding. Firstly, they force the modeller
to identify the relevant underlying processes and to describe them properly. This steers the observations in
the field and may pin-point which processes are to be investigated in laboratory experiments under con-
trolled conditions. Secondly, they allow analytical deductions and numerical experiments to test hypotheses
and to carry out sensitivity analyses.
As prediction tools, the models are needed in, for instance, land-use planning in alluvial river valleys and
investigating the choice of locations for bridges and other hydraulic structures. In particular, they are poten-
tially important for the prediction of responses to major interventions, when simple extrapolations of past
behaviour are inadequate.
This paper has two purposes. Firstly, it is intended to provide an overview of currently existing math-
ematical models and their state of development. This is examined by discussing a two-dimensional,
depth-averaged model, various meander models and a model for the braided Brahmaputra-Jamuna River
in Bangladesh. Secondly, the paper is intended to provide justification for future research, which includes
mathematical analysis and rigorous testing against field observations as well as further geomorphological
investigations of the underlying processes. Emphasis is placed on the need for future research rather than
on model applications, which have been reported elsewhere.
A TWO-DIMENSIONAL, DEPTH-AVERAGED MORPHOLOGICAL MODEL
The bed topography in alluvial rivers with arbitrary geometries can be computed using two-dimensional z
01995by John Wiley zyxwvutsrqp
& Sons, Ltd.
CCC 0197-9337/95/070661-
10

2. 662 zyxwvutsrqp
E. MOSSELMAN zyxwvuts
depth-averaged morphological models. An example for single-thread rivers is the model of Olesen (1987).
Mosselman (1992) extended this model by adding a mechanism for the erosion of cohesive banks, partly
using a time-averaged description of the model of Osman and Thorne (1988). Osman and Thorne model
the retreat of cohesive banks as a discontinuous sequence of mass failures induced by erosion at the toe
of the bank. However, the time-average behaviour can be modelled as an immediate response to toe erosion.
Toe erosion is divided into lateral fluvialentrainment of cohesive bank material and near-bank degradation
of the non-cohesive bed (Figure 1). The retreat due to lateral entrainment is determined with a simple, but
generally used, relation for the erosion of cohesive soils:
= O zyxwvutsrq
at
for rwHrwc
in which zyxwvuts
ann/& zyxwvutsrqpo
= rate of bank retreat, E = erodibility coefficient,rw= flow shear stress on the bank and
rwc
critical shear stress below which no bank erosion occurs. The near-bank bed degradation results from
gradients of sediment transport capacity as expressed by the continuity equation for sediment. The corre-
sponding bank retreat follows from Figure 1:
in which zb = bed level, t = time and 4 = bank slope angle. This contribution to bank retreat vanishes for
vertical banks (4= 90').
In Equations 1 and 2, bank retreat immediately followstoe erosion. The equations are considered to be the
outcome of an integration over an erosion cycle in which bank geometry and the rate of bank retreat fluc-
tuate. Toe erosion decreasesbank stability by increasing the slope and the height of the bank at the beginning
of the cycle, after which the portion of the bank above the toe retreats more dramatically by mass failure.
Debris from mass failure may accumulate at the toe, thus protecting or even buttressing the bank. It
must be removed by the flow before further toe erosion can take place. During some time interval within
the erosion cycle, higher and steeper banks are more likely to collapse than lower and less steep ones. The
rate of bank retreat can, hence, be expected to be correlated with bank height and slope when looking at
time scales that are smaller than the period of the erosion cycle. A relation with bank height is postulated as:
!E= G ( ~ )
H - Hc
at
for H 3 H c
for H < H,
(3)
Figure 1 Erosion of a cohesiveriver bank by lateral fluvial entrainment, An,,and near-bank bed degradation, Azb both inducing mass
failure

3. RIVER PLANFORM CHANGES 663 zy
where G zyxwvutsrqp
= erodibility coefficient, zyxwvutsr
H = total bank height and Hc= critical bank height below which no bank
erosion occurs. The correlation with zyxwvu
4 could be postulated in a similar way, but is disregarded here, The total
bank height is given by: zyxwvutsr
H = hw +Hfi (4)
in which hw = near-bank water depth and Hfi = freeboard, that is, the differencebetween the water surface
elevation and the top of the bank.
For the parameter ranges where all contributions are non-zero, the relations for bank erosion are com-
bined into:
This superposition is not justified formally. The terms with zyxwv
rw and zb stem from a time averaged description
in which the term with H is supposed to vanish. The equation must be viewed as merely a general expression
for which it is still to be decided which terms must be omitted in a specific application. Mosselman (1992)
uses the full equation in one-dimensional analysis, but does not include the contribution from Equation 2
in the two-dimensional model.
Bank erosion products contribute to the sediment balance. Treating the input of bank erosion products as
a boundary condition for the transverse sediment transport at the banks is not compatible with the notion
that the sediment transport (bed load) is determined by the local flow field and bed topography. The bank
erosion products are, therefore, treated as a source term in the continuity equation for sediment.
The computation of river planimetry can be uncoupled from the coml--itation of bed topography when
bank retreat is much slower than degradation or aggradation of the bed. This is a reasonable assumption
for cohesive banks. Furthermore, the flow is assumed to be quasi-steady, which means that the computation
of bed topography can be uncoupled from the flow computation as well. The equations are hence separated
into a steady-flow model, a bed deformation model and a bank migration model. The corresponding com-
putational procedure consists of three steps. In the first step, the flow field is computed while keeping the bed
and bank configuration fixed. Sediment transport rates and bank migration rates are calculated from the
flow field. In the second step, bed level changes are computed from the sediment transport gradients and
the input of bank erosion products. Finally, bank-line changes are calculated from the bank migration rates
in the third step.
A channel-fitted coordinate systemis used, so that all points of the computational grid correspond to loca-
tions in the river, and boundary conditions can be imposed conveniently on grid lines. After a certain
amount of bank migration a new grid must be generated, which is adapted to the new river planform.
Bank height and values of bank erodibility parameters are attributed to individual bank points of the
grid. In order to retain the bank properties at the proper locations, bank points are not allowed to shift along
the banks when generating a new grid after bank migration. They only move perpendicularly to the local
bank lines. This restriction implies that it is not possible to generate a purely orthogonal grid and the model
is, therefore, formulated in such a way that it can account for non-orthogonality.
Some results
Numerical experiments were carried out for hypothetical cases of bank erosion in order to investigate the
behaviour of the model. Figure 2 shows one of the hypothetical channels used. The channel is straight but
with a small curve near the entrance to trigger meander initiation. Pertinent data are given in Table I.
Bank erosion was modelled by Equation 1 alone. All computations were started from a flat bed. The for-
mula of Engelund and Hansen (1967) was used to calculate sediment transport rates. The results in Figure 3
straight channel
Figure 2. Initial planform of hypothetical channel for numerical experiments

4. 664 zyxwvutsrqp
E. MOSSELMAN
Table I. Parameters of numerical experiment
Quantity Symbol Value
Channel width
Channel length
Channel gradient
Chezy roughness coefficient
Discharge
Median sediment grain size
Bank erodibility coefficient
Critical bank shear stress zyxwvuts
B
L
C zyxwv
1 zyxwvutsrqponm
Q zyxwvu
D
5
0
E zyxwvu
Twc
2.44m
50m
0.95m km-l
37.6ml’’ s-’
0.154m3s-I
0.3 mm
1.0 Pa
1 lop7 zyxw
s-l
show the influence of the time step of bank migration. The rates of bank erosion adapted to the changing
flow field during wideningwhen the time step was small. The larger time step led to incorrect over-migration.
Choosing too large a time step can even cause widening beyond the situation in which the maximum shear
stress on the banks is equal to the critical bank shear stress. Unlike the evolution of the bed, in which the
sediment balance can restore over-degradation by sedimentation, the evolution of bank-line geometry based
on Equation 1is irreversible. Over-migration is not restored by a counteracting mechanism. This would still
hold if bank accretion were to be included in the model, because the accretion would occur at other bank
sections. The time step of bank migration should, therefore, not be too large. The irreversibility of bank ero-
sion also has a bearing on the predictability of river planforms, as it implies that, for instance, a local bank
failure or a single, exceptional flood can cause a lasting change in planform.
Field verification by applying the model to the River Ohie in Bohemia in the former state of Czechoslo-
vakia did not yield satisfactory results. This could be ascribed, however, to the neglect of grain sorting, which
hampered a correct representation of the topography of the gravel bed, and to the absence of a submodel for
bank accretion. Hence, notwithstanding the insight gained from constructing the model and conducting
numerical experiments, no decisive conclusions on the predictive power of this approach to bank erosion
could be drawn.
Some areas needingfurther research
Verijication. Obviously the extension of two-dimensional, depth-averaged morphological models with
mechanisms for the erosion of cohesive banks still awaits proper verification. Better results are expected
when the model is applied to sand-bed rivers or when submodels for grain sorting and armouring are added.
Bunk accretion. Mechanisms of bank accretion have received much less attention than mechanisms of
bank erosion and thus form the weakest part of present morphological models for rivers with mobile
banks. Banks advance by near-bank overloading of suspended sediment, modelled mathematically by
Parker (1978), or by the development of a point bar during high discharges and subsequent emergence
from the water level when the discharges are low. In addition to these mechanisms of fluvial deposition,
the redistribution of sediment by wind and the trapping of fine material by vegetation may be important.
The flat slopes of accreting point bars make the locations of lateral model boundaries very sensitive to
point-bar elevations as well as to the discrete discharge values that are taken to represent the discharge
7 x zyxwvutsrqpon
0.63 years 1 x 4.4 years
Figure 3. Resulting planforms after 4.4years, computed with different time steps of bank migration (seven steps of 0.63 years versus one
step of 4.4years)

5. RIVER PLANFORM CHANGES zyxwvu
665
hydrograph in a morphological computation. This might be a serious problem because the locations of the
boundaries make up the river planform and the results of computations of flow and bed topography strongly
depend on the geometry of that planform. zyxwvuts
Mathematical analysis of alluvial bars. There is a strong interrelation between bar development and
planform changes. For instance, incipient meandering can be understood as a positive feedback between
a bed deformation and a planform deformation which are operating at the same wavelength. Blondeaux
and Seminara (1985) reveal this resonance phenomenon theoretically. Incipient braiding by mid-channel
bar formation and associated widening (Leopold et al., 1964) is a similar positive feedback, but related to
a higher Fourier mode in the transverse direction.
Simply discretizing mathematical models for river morphology into a numerical computer model is not
enough. The mathematical models should also be analysed theoretically. It is true that a numerical solution
using a computer requires fewer assumptions and simplifications than an analytical solution, but the numer-
ical solution produces other inaccuracies because the internal representation of numbers involves rounding
errors and the discretization of equations generates truncation errors. It is not always clear whether, for
instance, a certain oscillation in the results from a numerical model represents real river behaviour or is
due to numerical effects. Often, model results are then judged by the ‘principle of minimum astonishment’,
which can be quite dangerous when trying to predict the response to imposed changes. Mathematical ana-
lysis of the model reveals the essential behaviour of the system and, thus, helps in making a good interpreta-
tion of computer results by reducing the astonishment when the results represent reality. In addition,
mathematical analyses are necessary for the choice of the numerical scheme and the solution procedure.
Mathematical analyses of alluvial bars, such as the work of Blondeaux and Seminara (1985), Struiksma
et al., (1985), Tubino and Serninara (1990) and Schielen et al. (1993), definitely deepen our understanding
but, nevertheless, more work remains to be done.
Link with soil mechanics. The present submodel for bank erosion is very simple. It could be improved by
including details of soil mechanics, for instance by using the mass-wasting algorithms developed by Darby
(1993).
MEANDER MODELS
Several meander models currently exist with different levels of complexity. They can be divided into kine-
matic models, which are basically induced from observations of meander migration, and dynamical models,
which are primarily deduced from the fundamental laws of physics for small volumes of water and sediment.
In the kinematic models, meander migration depends on geometrical properties of the river, notably a
weighted average of local and upstream channel curvatures. In the dynarnical models,’meander migration
depends on excess shear stresses or excess velocities of the near-bank flow, which are obtained from a linear
computation of the flow field and the bed topography in the curved channel. The models of Ferguson (1983)
and Howard and Knutson (1984) are kinematic and the models of Ikeda et al. (1981),Johannesson and Par-
ker (1989) and Crosato (1990) are dynamical. In all these models, bank accretion is simply taken to balance
the erosion of the opposite bank by assuming that the width remains constant. It should be noted, though
that sometimes bank accretion rather than erosion governs the rate of channel migration (Nanson and
Hickin, 1983; Howard and Knutson, 1984; Ikeda, 1989). Under these circumstances the ‘bank erosion
formula’ in the models may actually be a representation of accretion rather than erosion. Meander migration
models are now in operational use, but there are still questions concerning their capabilities and limitations.
Some examples are given below.
Some areas needingfurther research
Comparative study of meander models. Parker (1983) argues that Howard’s kinematic meander model is
equivalent to the bend equation of Ikeda et al. (1981). A difference is that Howard’s model contains an
additional factor based on Hickin and Nanson’s (1984) non-linear relationship between migration and
channel curvature, which might correspond to the deviations between the first-order model of Ikeda et al.,
and the more complete second-order models of Johannesson and Parker (1989) and Crosato (1990).

6. 666 E. MOSSELMAN zyxwvuts
Figure zyxwvutsrqp
4. Sucessive centre-lines in a numerical simulation of river meandering by Howard and Knutson (1984)
A comparitive study would reveal the differencesin performance between the different classes of meander
models. This would show the effects of the various simplifications and assumptions. It would also provide
valuable insight into the applicability of each of the models to specific cases where a certain accuracy for
a given prediction span is required. zyxwvuts
Sensitivity to initial and boundary conditions. The non-periodicity of meanders which develop from a
straight alignment in the numerical simulations of Howard and Knutson (1984) and Crosato (1990)
suggest chaotic behaviour, that is, a sensitive dependence on initial and boundary conditions. Typical
non-periodic shapes are shown in Figures 4 and 5. Figure 5 shows that despite identical configurations at
the start of the simulation, the two bends above the horizontal axis develop differently and, at the end of
the simulation, they are not identical. This chaotic behaviour bears on the predictability of river
meandering and needs to be investigated further.
Maximum amplitude of meanders. Modern ecological restoration projects for rivers often involve the
transformation of a straightened channel into a meandering stream. One of the first questions which
arises is then how wide the associated natural meander belt will be or, in other words, how large the
maximum amplitude of the meanders will be. River meanders do not grow to infinity, but reach a finite
amplitude due to non-linear effects, bend cut-offs, topographic confinement or a decrease in stream
power as increasing sinuosity reduces channel slope for a given valley slope. Parker et al. (1982) showed
that geometrical non-linearities produced skewing and fattening of meander bends and Parker et al.
(1983) indicated that skewing allowed equilibrium planforms in which the meanders have a finite zy
/ -
< .
'
. -_
-'
/
.-/-'
Figure 5. Successive centre-lines in a numerical simulation of river meandering by Crosato (1990)

7. RIVER PLANFORM CHANGES zyxwvu
667
amplitude and do not extend further. They added, however, that while the equilibrium planforms were not
stable for small amplitudes, it was unclear whether they could be stable for larger amplitudes. Later, Parker
(1990)concluded that the equilibrium planforms are always unstable and that the maximum amplitude is not
determined by geometrical non-linearities. Seminara and Tubino (1992) indicated that, apart from
geometrical non-linearities, flow non-linearities also damp meander growth. They suggested that it is
possibly the flow non-linearities which lead to a finite amplitude. This is still to be investigated.
The longitudinal slope of a river decreases as sinuosity increases during meander development, provided
that meander migration operates at smaller time scales than adaptation of the longitudinal profile and pro-
vided that the river width remains constant. This implies a decrease in stream power or equivalently (for con-
stant width and roughness), in the words of mathematicians, a decrease in the zero-order flow velocity. The
associated decrease in bank erosion also offersan explanation for the river attaining a finiteamplitude. Ikeda zy
et al. (1981), Parker et zyxwvutsrq
al. (1982, 1983),Johannesson and Parker (1989), Crosato (1990) and Seminara and
Tubino (1992) do not account for this effect in their dynamical models. Some of the researchers do model a
decrease in zero-order flow velocity as sinuosity increases but, at the same time, they take the critical flow
velocity for bank erosion to remain equal to the zero-order flow velocity. As a result, this critical flow velo-
city decreases as well. In reality, it should remain unchanged, however, because it is a function of the soil
properties alone and these are independent of river sinuosity. The effect of this shortcoming in present dyna-
mica1 meander models is a topic worthy of further research.
A MODEL FOR THE BRAHMAPUTRA-JAMUNA RIVER IN BANGLADESH
Planform changes are particularly dramatic in the Brahmaputra-Jamuna River in Bangladesh. This large,
braided, sand-bed river has a total width of the order of 10 to 15km. Individual channels within its braid
belt are up to 2km wide. Locally, bank erosion rates can be as high as 1000m per year. The intertwining
of several channels makes the resulting rapid planform changes very complex and, therefore, they present
a stiff challenge for morphological models. The braid belt as a whole is widening and migrating westward.
ISPAN (1993) concluded from examination of maps dating back to 1830 and satellite images from 1973
onwards that this general trend produces an average retreat of the west bank of about 100m per year.
Planforms changes in the Brahmaputra-Jamuna River operate on different scales of time and space, each
scalerequiring a different modelling approach. The question as to whether a certain location will be attacked
by bank erosion 10 or 20 years into the future can only be answered in a statistical sense by extrapolating zy
(a) Bank erasion and
related channel migration zyxwvutsrq
(b) Changes in witdh due (c) cut-offs zyxwv
to behaviaur of
bifurcated chonnels
legend
erosion
sedimentation
(d) mid-channel bar formation
Figure 6. Aggregated mechanisms of planform change in a braided river

8. 668 zyxwvutsrq
E. MOSSELMAN
current trends. This is elaborated by ISPAN (1993) and Thorne and Russell (1993). Predictions for time
spans of several months to a few years can partly be based on submodels of the underlying fluvial mechan-
isms. Klaassen zyxwvutsrqp
et al. (1993) follow this approach in their planform model for the Brahmaputra-Jamuna
River. The underlying mechanisms in this model are not the fundamental laws of physics for small volumes
of water and sediment, but fluvial mechanisms on a higher level of aggregation, such as width adjustment,
channel migration, island formation,channel creation and channel abandonment. Some examples are shown
in Figure 6.
The submodels for the aggregated mechanisms are induced from observations, which usually have limited
accuracy, do not provide complete information and do not allow a clear distinction to be made between the
contributions from each of the mechanisms separately. One should be reminded, however, that detailed
models based on fundamental laws of physics would also have a limited predictability due to a sensitive
dependence on initial and boundary conditions, the possible occurrence of extreme events (e.g. major earth-
quakes) and the interactions with other dynamical systems with their own limited predictability (meteorol-
ogy, tectonics). The submodels must, therefore, be formulated in a stochastic way and the overall model in
which the submodels are integrated must be probabilistic. Klaassen et al. (1993)construct their model for the
Brahmaputra-Jamuna River accordingly. They explain that thresholds in the morphological behaviour dic-
tate that the probabilistic elements cannot be dealt with by a parameterization of stochastic distributions,
but must be included through Monte Carlo simulation.
The usual opinion is that large-scale migration of the Brahmaputra-Jamuna River is primarily controlled
by tectonics while local pattern changes within the braid belt can be understood from the dynamics of water
and sediment alone. There are indications, however, that this view is too simplistic. Thorne et al. (1993) pro-
pose that the general westward migration might also be explained as the development of a very large mean-
der, which is plausible as a residual effect from all the westward and eastward migrations of the individual
channels within the curved braid belt. An analysis of lineaments on satellite images by the University of
Amsterdam (Hartmann et al., 1993)reveals correlations between local channel patterns and geological fault
lines. Hence, there may be fluvially derived elements as well as tectonic controls of large-scale migration and,
conversely, tectonic influences as well as fluvial controls on local pattern changes.
Some areas needingfurther research
Further development and testing. The modelling approach of Klaassen et al. (1993) is now operational as a
formal evaluation procedure. Calculations are performed by hand, with submodels for the aggregated
mechanisms used to form a tree of possible developments and associated probabilities of occurrence.
However, the complexity of the river makes the hand calculations very cumbersome and hence the
evaluation is only feasible in a highly simplified form. Automation of the procedure in an integrated
computer model is therefore required. The functional design of this computer model for the
Brahmaputra-Jamuna River was completed in 1993. The next phases of model development are technical
design, coding, testing and verification. Meanwhile, further studies with more data are needed to improve
the submodels.
Stability of bifurcated channels. Channel abandonment is one of the aggregated fluvial mechanisms for
which a submodel is induced from observations. This mechanism is related to the stability of bifurcated
channels, which also forms a problem in one-dimensional morphological models deduced from the
fundamental laws of physics. The silting of bifurcating channels depends on the distributions of water
and sediment at the point of bifurcation upstream. The assessment of the distribution of sediment is more
demanding than that of the distribution of flows, because the morphological development depends on the
extent to which the distribution of sediment deviates from the flow distribution. This residual effect is
governed by the local three-dimensional flow pattern, which strongly depends on the geometry of the
bifurcation. Three-dimensional flow effects, however, are not included in one-dimensional mathematical
models for river morphology. In these models, the sediment transport distribution at bifurcations is
usually described by empirical node relations. Flokstra (1985), Wang et zyxw
uf. (1993) and Fokkink and
Wang (1993) show that the model behaviour of bifurcated channels is very sensitive to these node
relations. A small difference in the value of an exponent in the node relation can make the difference

9. RIVER PLANFORM CHANGES 669
between a stable, self-restoring bifurcation and a bifurcation which is bound to disappear due to the
abandonment of one of its branches. The present state-of-the-art is that detailed morphological
predictions of rivers with bifurcations always require either two- or three-dimensional numerical models
or physical models. This may change eventually, however, if a systematic programme of laboratory
experiments were performed to obtain accurate node relations for various bifurcation geometries. zyx
Tectonics. The influence of tectonics on channel patterns within the braid belt of the Brahmaputra-
Jamuna River should be further investigated by verifying in the field that lineaments identified on satellite
images do correspond to active faults, and by quantifying the vertical movements of the terrain
associated with these faults. In a two-dimensional, depth-averaged model, one can describe the influence
of tectonics mechanistically by adding a source term to the sediment balance. This source term represents
an extra increase or decrease in bed level which is independent of erosion or deposition by the flow. zy
As
yet it is not clear, however, how the influence of tectonics could be included in meander models or the
model for the Brahmaputra-Jamuna River. Possibly, approaches from models of quantitative dynamic
stratigraphy (Cross, 1989)could be used. zyxwvut
Radar remote sensing. The model for the Brahmaputra-Jamuna River in Bangladesh is largely based on
remotely sensed observations of planform changes. Optical and infrared bands of the electromagnetic
spectrum have routinely been used until now, but these bands do not penetrate clouds or turbid water.
This restricts their use in Bangladesh to the dry season, whereas the most significant morphological
changes occur during the summer monsoon. Radar remote sensing does not suffer from these restrictions.
It penetrates clouds directly and, moreover, it allows the observation of bathymetry under clear or turbid
water in an indirect way. Bathymetry assessment by radar remote sensing is now more or less proven
technology for marine applications (e.g. Vogelzang et al. 1992) and is perhaps, despite some fundamental
differences, also applicable to large rivers. Further development of this technology would allow much
more detailed study of the morphological processes in the Brahmaputra-Jamuna River.
CONCLUSION
The models of river planform changes that currently exist help in understanding how river planforms evolve
as a result of interactions of flow, sediment transport, erosion and sedimentation. There are still many topics
which require further research and so far none of the models has reached the level of being a generally valid
and easy-to-use software package. In the hands of experienced geomorphologists or river engineers, how-
ever, operational models such as those for meander evolution and the morphology of the Brahmaputra-
Jamuna River do form valuable tools which allow better prediction of future planforms.
ACKNOWLEDGEMENTS
Funding from the Institute of British Geographers and the Department of Geography, University of Not-
tingham, to present this paper at the IBG Annual Conference in Nottingham on zyxw
5 January 1994is gratefully
acknowledged. The two-dimensional depth-averaged model was developed during my employment at Delft
University of Technology with financial support from the Centre for Civil Engineering Research, Codes and
Specifications(CUR). The model for the Brahmaputra-Jamuna River is being developed at Delft Hydraulics
for FPCO (Flood Plan Coordination Organization, Bangladesh) with funding from KfW (Kreditanstalt fur
Wiederaufbau, Germany) and CFD (Caisse FranGaise de Dkveloppement, France). I sincerely thank the
many people who provided useful information or entered into fruitful discussions with me during prepara-
tion of this paper.
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