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Physical Geography
Process−form linkages in meander morphodynamics : Bridging theoretical modeling and real
world complexity
Inci Güneralp and Richard A. Marston
Progress in Physical Geography 2012 36: 718 originally published online 31 August 2012
DOI: 10.1177/0309133312451989
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Article
Process–form linkages in
meander morphodynamics:
Bridging theoretical modeling
and real world complexity
Progress in Physical Geography
36(6) 718–746
ª The Author(s) 2012
Reprints and permission:
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DOI: 10.1177/0309133312451989
ppg.sagepub.com
_
Inci
Güneralp
Texas A&M University, USA
Richard A. Marston
Kansas State University, USA
Abstract
Meandering rivers are one of the most dynamic earth-surface systems. They play an important role in
terrestrial-sediment fluxes, landscape evolution, and the dynamics of riverine ecosystems. Meandering rivers
have been of fundamental interest to researchers across a wide range of disciplines, from fluvial geomorphology to fluid mechanics, from river engineering to landscape ecology, owing to the intriguing complexity of
meander morphodynamics. This interest also comes from the socio-economic concerns due to the river
hazards caused by bank erosion, channel change, and flooding, as well as the adverse responses of meandering
rivers to human- and climate-induced changes in the environmental conditions. An in-depth, process-based
understanding of the dynamics of meandering river–floodplain systems is critical in order to investigate the
responses of these systems to the changes in environmental conditions. Over the last few decades, there
have been significant advances in river meandering research, with contributions from both theoretical modeling and experimental and field-based research. This paper presents a detailed overview of river meandering
research, particularly focusing on the advances in the process-based understanding of meander morphodynamics. It also discusses the standing challenges in addressing the dynamics of real meandering rivers and their
floodplain patterns and processes, and potential future directions in river meandering research. The paper
advocates the crucial need for bridging theoretical modeling with field- and laboratory-based research in
order to inform accurate assessments of river-hazard risks and facilitate ecologically sound rivermanagement and restoration practices with the aim of supporting healthy ecosystems.
Keywords
environmental change, floodplain, fluvial geomorphology, meander, modeling, morphodynamics, processbased
I Introduction
Although only about 0.006% of the freshwater
on the Earth is found in rivers, they are one of
the major agents sculpting the Earth’s surface.
Rivers play a vital role in the Earth’s hydrological, biological, and geomorphological processes
Corresponding author:
_Inci Güneralp, Texas A&M University, 810 Eller O&M
Building, College Station, TX 77843, USA
Email: iguneralp@geos.tamu.edu
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Güneralp and Marston
719
Figure 1. Planform of a reach of freely meandering Rio Juruá, Brazil (from 6 45’37.18’’S, 70 49’36.57’’W to
6 41’33.53’’S, 69 53’54.28’’W). The image shows a portion of the reach (obtained from Landsat 7 ETMþ
1999, SWIR, Bands 7,4,2 as RGB). The channel planform is composed of simple bends and complex meander
forms such as compound loops, also called multilobed meanders. The crescent shape lakes seen on the image
are oxbow lakes that are formed by cutoff processes. Flow direction is from left to right. The plot on the left
shows the corresponding planform-curvature series, where simple bends have only one curvature maximum
(e.g. bends 1, 3–4, 7–9) whereas compound loops have multiple distinct curvature maxima (e.g. bends 2, 5–6).
(e.g. Poole, 2002; Ward, 1997). Operating like a
conveyor belt, they transport freshwater as well
as sediment and various important chemical
components collected from the land to the water
reservoirs such as lakes, seas, and oceans. Rivers provide habitat for aquatic plant and animal
species. Most terrestrial species also depend on
a river ecosystem at some point during their life
cycle. Since the dawn of civilization, rivers
have had significance for human settlements,
by offering water for irrigation, households, and
industry, and the fertile land around them for
agriculture, and by serving as transportation
routes and for recreational activities.
Meandering is one of the most common
river-channel patterns (Figure 1). Meandering
rivers are intrinsically dynamic earth-surface
systems. Freely meandering rivers in broad alluvial floodplains rarely exhibit a stability of form
or morphological characteristics; instead, they
progressively evolve by migrating over their
floodplains (Hooke, 1984). The floodplains of
meandering rivers exhibit highly heterogeneous
landscapes, consisting of a variety of channel
morphologies and floodplain landforms, such
as oxbow lakes, meander scars, and scroll bars
(Figure 1), which creates an intricate sedimentary structure and provides diverse in-channel
and floodplain ecosystem habitats.
Meandering rivers have been of fundamental
interest to a wide range of scientific disciplines
for more than a century (Jefferson, 1902;
Thomson, 1876), from fluvial geomorphology
(e.g. Brice, 1974; Dietrich et al., 1979; Hickin,
1974; Hooke, 1984, 2007b; Leopold and Wolman, 1960; Schumm, 1967; Tinkler, 1970) to
fluid mechanics (e.g. Callander, 1978; Einstein
and Shen, 1964; Engelund, 1974; Seminara,
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720
Progress in Physical Geography 36(6)
2006); from river engineering (e.g. Elliot, 1984;
Jansen et al., 1979) to landscape ecology (e.g.
Greco and Plant, 2003; Robertson, 2006; Salo
et al., 1986), to petroleum engineering (e.g.
Henriquez et al., 1990; Swanson, 1993). The
scientific interest to river meandering is mostly
due to the intriguing morphodynamics of meanders and the role of these dynamics in terrestrial
sediment fluxes (Odgaard, 1987; Zinger et al.,
2011), floodplain development and evolution
(Lauer and Parker, 2008; Peakall et al., 2007;
Van De Wiel et al., 2007), and riverineecosystems processes (Amoros and Bornette,
2002; Kalliola and Puhakka, 1988; Nanson and
Beach, 1977; Ward et al., 2002).
River-meandering processes have been
examined across spatial and temporal scales
ranging from small-space scale processes such
as the interactions among flow structure,
sediment transport, and bed morphology in
individual bends (e.g. Dietrich et al., 1984;
Frothingham and Rhoads, 2003) to large-space
scale processes such as the planform evolution
of a series of meander bends (e.g. Gautier
et al., 2010; Güneralp, 2007; Hooke, 2007b);
from short-timescale processes such as
channel-bar dynamics occurring over a few
decades (e.g. Hooke and Yorke, 2011) to longtimescale processes such as landscape evolution
over hundreds to thousands of years (e.g.
Camporeale et al., 2005; Frascati and Lanzoni,
2009; Howard, 1996; Sun et al., 2001b). The
migration dynamics of meandering rivers also
create socio-economic concerns due to the
hazards associated with bank erosion, channel change, and flooding (Girvetz and Greco,
2007; Greco et al., 2008; Güneralp and
Rhoads, 2009a; Kondolf, 2006; Lagasse
et al., 2004; Larsen and Greco, 2002; Piégay
et al., 2005). These concerns may lead to the
development of protective measures against
such hazards (e.g. bank protection against
migration, land protection against flooding
by dam and levee constructions, dredging for
navigational improvement).
Environmental changes and the adverse
responses of meandering river–floodplain systems to these changes also raise concerns.
Migration dynamics are highly influenced by
spatial and temporal variability in environmental conditions. Human-induced environmental
changes (e.g. in-channel and landscape modifications by protective measures, agriculture, and
urbanization on or around floodplain landscapes) and climate change alter flow regime,
floodplain-erodibility characteristics, and
sediment-transport rates, and thus can significantly affect the patterns of channel evolution
and floodplain vegetation patterns and processes. Moreover, the alterations in river–floodplain system functioning can lead to a decrease
in hydrologic connectivity and a degradation of
water quality, which in turn lead to a decline in
the abundance and diversity of riparian and
riverine habitats.
An in-depth, process-based understanding of
the feedbacks between the morphodynamics of
a meandering river and its floodplain patterns
and processes is crucial to determine the
responses of the meandering river–floodplain
system to the variability in environmental conditions. This knowledge allows for identifying
the extent of the influence of specific controlling factors (e.g. the magnitude of variability
in environmental conditions, decisions on river
and land management, and climate change) on
both short- and long-term system trajectories.
Therefore, it is necessary for informing accurate
assessments of river-hazard risks and facilitating the development of effective and ecologically sound management and restoration of
meandering rivers and their riverine landscapes
in a changing environment due to human modifications and future climate change.
The purpose of this paper is to give an overview of the progression of river-meandering
research. The paper specifically focuses on the
advances in the process-based understanding
of meander morphodynamics. First, it briefly
introduces fluid-dynamic and morphodynamic
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Güneralp and Marston
721
Figure 2. Planform characteristics of a meandering river. The inflection points represent the locations where
the direction of the channel curvature reverses.
characteristics of meandering rivers, and then
presents the advances in modeling the process–
form linkages in river meandering. Next, it
discusses the issues and challenges in rivermeandering research in addressing the complexity in the dynamics of meandering rivers. Finally,
it concludes with potential future directions for
research, emphasizing the importance of studies
on the integrated dynamics of meandering rivers
and other earth-surface systems, as well as the
pressing need for developing a framework integrating theoretical modeling and field-based
approaches.
II Toward a theory of the linkages
between meander form and
meander-migration process
Meander planform can be defined quantitatively
by meander sinuosity S (i.e. the ratio of curvilinear length of the river to the valley length),
meander wavelengths and C (i.e. linear and
curvilinear lengths between the apexes or the
first inflection points of successive bends on the
same side of the river, respectively), meander
amplitude 2A (i.e. where A is bend amplitude
and equal to one-half the meander width), and
channel width 2b (Figure 2). Channel curvature
C shows the degree of change in the direction of
a channel along its streamwise axis, whereas an
inflection point marks the location where the
direction of channel curvature reverses (i.e.
transition from one meander bend to the next
occurs) (Figures 1 and 2). Curvature is good
indicator of meander complexity (i.e. the intricacy of the shape of meander bends). Intricate
meander morphologies such as compound loops
(Howard, 1996; Sun et al., 1996), also called
multilobed meanders, are characteristic examples of complex meanders. A compound loop
is an elongated meander with multiple lobes
characterized by multiple absolute maxima in
its planform curvature series (Brice, 1974;
Frothingham and Rhoads, 2003; Hooke and
Harvey, 1983). On the other hand, a simple bend
is a single-lobed meander, which has a single
absolute maximum in its curvature series
(Figure 1).
The planform evolution of meandering rivers
occurs as a result of mutual adjustments
between meandering form and processes. The
interactions among river-flow (Hooke, 1975;
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Progress in Physical Geography 36(6)
Hooke and Harvey, 1983), sediment transport
(Parker and Andrews, 1985), and channel-bed
morphology (Alphen et al., 1984; Dietrich and
Smith, 1983, 1984a; Dietrich et al., 1984; Engelund, 1974) determine the spatial and temporal
patterns of sediment erosion, transportation, and
deposition, and, thus, the evolution of meandering rivers (e.g. Bridge, 1992; Hooke and
Harvey, 1983; Whiting and Dietrich, 1993a,
1993b). River meandering is controlled by two
major flow mechanisms: curvature-driven and
topography-driven secondary flows (Hooke,
1975; Seminara and Tubino, 1989; Solari
et al., 1999; Struiksma et al., 1985; Thompson,
1986; Zimmerman and Kennedy, 1978; Zimmermann, 1977). Helicoidal, curvature-driven
secondary flow is generated by the super elevation of the water surface as water flows through
a bend (Callander, 1978). This secondary circulation creates large cross-stream variation in the
velocity field (Dietrich et al., 1984), which
redistributes the downstream momentum (De
Vriend and Struiksma, 1984). The redistributed
momentum leads to a decrease in the bed-shear
stress along the inner bank of the bend that
activates an inward sediment flux convergence
(i.e. deposition) on the bed, leading to the development of a point bar (Blanckaert and De
Vriend, 2004; Kalkwijk and De Vriend, 1980).
Conversely, the redistributed momentum leads
to a downstream increase in the bed-shear stress
along the outer bank, also called the cutbank
(Figure 3), leading to erosion (Nelson and
Smith, 1989). Moreover, the point bar further
deflects the water flow laterally toward the cutbank by topographic steering (Dietrich and
Smith, 1983, 1984b). This enhances the crossstream variation in the flow velocity and induces
a further outward shift of the downstream flow
toward the cutbank, leading to topographically
driven secondary flow (Kalkwijk and De Vriend,
1980; Seminara, 1998). Type of the sediment
load is also an important factor controlling river
meandering, specifically the wavelength of
meanders (Schumm, 1967). Based on the
Figure 3. Examples of cutbank and point bar, which
are distinct features of meandering rivers, on a reach
on the Little Brazos River, Texas (30 50’23.19’’N,
96 42’15.19’’W).
analysis of the data collected on morphologic,
hydrologic, and sediment characteristics of 36
stable channels, Schumm (1967) concluded that
the rivers transporting mainly sand and gravel
have greater meander wavelengths than those
of similar discharge transporting primarily fine
sediment loads.
A range of geomorphic processes govern the
planform evolution of meandering rivers in
broad, alluvial floodplains. Among them,
meander-migration and cutoff processes are
fundamental for long-term morphological
changes in a river and its floodplain landscape
(Allen, 1965; Camporeale et al., 2005, 2008;
Constantine and Dunne, 2008; Frascati and
Lanzoni, 2008; Hickin and Nanson, 1975;
Hooke, 2007b). During its morphodynamic evolution, a freely meandering river continues to
elongate its meander bends, and thus increases
its meander amplitude and sinuosity. Conversely, cutoffs cause sudden reductions in meander amplitude and sinuosity by shortening the
channel length, and thus increasing the channel
gradient. They create noise, or high-frequency
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Güneralp and Marston
723
Figure 4. (A) Meander bend planimetries simulated with first-order model (Ikeda et al., 1981) which links the
excess near-bank velocity to local and spatially distributed upstream curvature of the channel planform and
can generate the development of fattening and upstream skewness (i.e. Kinoshita curve) – two fundamental
characteristics of meander form. (B) Examples of upstream skewed meanders on real rivers, from left to
right: the Trinity River, Texas, USA (30 7’47.63’’N, 94 48’54.73’’W), the White River, Arkansas, USA
(35 15’36.39’’N, 91 22’30.47’’W), and Rio Purus, Brazil (7 39’12.41’’S, 66 30’8.05’’W). The red dots mark
the inflection points at which the curvature reverses. Flow direction is from left to right.
variations in planform curvature. Numerous
factors including channel width-to-depth ratio,
bend and bank-sediment type, the presence of
bank vegetation (Howard, 1984; Micheli and
Kirchner, 2002a, 2002b; Wynn and Mostaghimi,
2006), and the degree of variability in floodplain
erodibility influence both the characteristics of
meander migration (Güneralp and Rhoads,
2011) and the dominant type of cutoff process
(Constantine et al., 2010; Erskine et al., 1992;
Gagliano and Howard, 1984; Gay et al., 1998;
Lewis and Lewin, 1983).
As the meander bends evolve, they tend to
become round and full, or ‘fat’ (Parker et al.,
1982), often to the point of possessing doublevalued planforms in the Cartesian coordinate
representation (Langbein and Leopold, 1966).
Bends also tend to show a marked asymmetry
or skewness in shape, possessing a convex bank
outline on one side and a concave outline on the
other side (Parker et al., 1982). The fattened and
asymmetrical meander forms are known as
Kinoshita curves (Kinoshita, 1961) and are
commonly observed in nature (Figure 4). The
planform evolution of meander bends can be
explained by three main modes of migration
behavior: (1) extension and/or expansion
through lateral migration; (2) translation (i.e.
downstream migration or, in certain circumstances, upstream migration); and (3) rotation
(Hooke, 1984). Lateral migration results in an
increase in the length of a meandering river
along its streamwise axis (i.e. elongation or fattening) while translation and rotation give the
asymmetry (i.e. skewness) to the meander form.
The mode of meander migration is highly
influenced by upstream and downstream channel morphology such as the variations of channel curvature or channel width – known as
‘morphodynamic influence’ (Seminara, 2010).
For example, planform geometry (characterized
by curvature) has not only a local influence, but
also a cumulative (i.e. spatially extended) influence on the migration rates along the channel.
This influence decays with increasing streamwise distance from the channel location, for
which migration rate is determined. The rate
of the decay in the influence depends on flow
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Progress in Physical Geography 36(6)
resistance and controls its spatial extent (i.e. the
spatial memory of the river) (Furbish, 1991;
Güneralp and Rhoads, 2009b; Howard and
Knutson, 1984; Smith and McLean, 1984). In
fact, a slow decay (i.e. long morphodynamic
influence or spatial memory) in the upstream curvature influence caused by a low flow resistance
leads to an increase in the downstream translation and upstream skewness of meander bends
(Güneralp and Rhoads, 2009b). In other words,
the spatial extent of the influence in the upstream
direction determines the delay in the spatial
response (i.e. migration) of the river that imparts
a downstream component to the migration pattern (Howard, 1984; Parker et al., 1982; Zolezzi
and Seminara, 2001). This spatially delayed
response is the reason for bank erosion becoming
concentrated against the cutback downstream of
the bend apex where downstream increases in
bed-shear stress near the cutbank are the greatest
(Whiting and Dietrich, 1993a, 1993b).
The knowledge on the physical processes
leading to the occurrence of cutoffs is limited
due to their infrequent nature and the wide range
of hydro-geomorphic conditions governing
floodplain patterns and processes (Hooke,
1995, 2004). A cutoff process is generally
defined as neck cutoff when the meander neck
is shorter than one channel width. The neck of
an elongated meander may become increasingly
narrow due to progressive migration of the
upstream and downstream bends and doubles
back upon itself leading to neck cutoff (Allen,
1965; Erskine et al., 1992; Fares, 2000; Hooke,
1995). Conversely, a chute cutoff develops
through the incision of a new channel across the
neck of a meander bend. Chute cutoffs commonly develop in the presence of remnants of
old abandoned channels on the floodplain (e.g.
meander bends with ridge and swale topography), which can provide flow-routing paths and
channelize the overbank flow (Bridge et al.,
1986; Hickin and Nanson, 1975); during overbank flows capable of cutting a new, shallow
side channel across the neck of a meander bend
(Hooke, 1995; Lewis and Lewin, 1983); or via
the downstream extension of an embayment
(i.e. an indentation) at the upstream part of a
meander bend along a large river with uniform
floodplain topography (Constantine et al., 2010).
A cutoff meander bend, produced by either
neck or chute cutoff, is not abandoned immediately; it stays connected to the main channel fully
or partially until sediment deposition closes off
its entrance and exit, creating an oxbow lake
(Figure 1). The infilling of an oxbow lake is
controlled by the character of the sediment
transported during high flows, the magnitude and
frequency of these flows, and the distance
between the abandoned and present channels
(Erskine et al., 1992). Over time, oxbow lakes are
filled mostly with fine sediment deposits and
form sediment plugs (i.e. clay plugs), which are
highly resistant to erosion (Hooke, 2004).
A dynamic meandering river oscillates
between meander migration and cutoff events
and maintains a statistically steady-state (Camporeale et al., 2005; Frascati and Lanzoni, 2010;
Hooke, 2004; Stølum, 1996). Through planform
migration and cutoff processes, meandering
rivers mobilize stored floodplain sediment and
continually rework it. Reworking of the sediment gives rise to the segregated and complicated structure of the sedimentary deposits
with different erodibility characteristics. Both
the high-frequency variations in channel curvature and the variability in floodplain erodibility
can contribute to the emergence of complex
meander morphologies and cause increasing
irregularity in planform geometry (Güneralp
and Rhoads, 2011; Howard, 1992, 1996;
Howard and Knutson, 1984; Sun et al., 1996,
2001a, 2001b).
III Modeling the morphodynamics
of meandering rivers
Empirical studies have contributed to the
general theory of meander migration, which
provided the basis for the development of
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725
process-based mathematical models of meander
morphodynamics. Over the last four decades,
there has been a growing interest in modeling
the morphodynamics of meandering rivers. The
motivation for modeling has primarily been to
deepen the understanding of morphodynamic
processes of meandering and also to improve
our predictive capabilities of meander evolution
with the aim of decreasing hazard risks associated with bank erosion and channel migration.
Meandering planforms were initially characterized as regular meander paths composed of
symmetrical bends. The simplest approaches
to the geometric characterization of meander
planforms involved the fitting of sine waves or
the sequence of joined circular arcs with constant curvature (Chitale, 1970, 1973; Leopold
and Wolman, 1960), and the representation of
the bends with linearly varying curvature using
Fargue’s spiral (Ferguson, 1973; Leliavsky,
1955). These approaches were followed by the
characterization of meander planforms as regular waveforms using sine-generated curves. In
the characterization using sine-generated
curves, the channel is defined by a sine function,
oscillating sinusoidally with distance along
the channel (Langbein and Leopold, 1966). The
maximum angle of the streamwise axis from the
mean valley direction determined the ‘fatness’
of bends. The morphometric studies showing
that there is no single characteristic scale of
directional oscillation (Speight, 1965, 1967)
supported the idea that meander paths in fact are
not regular. This idea gave rise to an alternative
characterization of meander planforms as
irregular paths resulting from purely random
changes in channel direction (Langbein and
Leopold, 1966; Scheidegger, 1967; Thakur and
Scheidegger, 1968, 1970). For example,
Langbein and Leopold (1966) defined the
meander planform geometry in terms of random
walk whose most frequent form minimizes the
sum of the squares of the changes in direction
in each unit length (Von Schelling, 1951). In a
following study, Ferguson (1976) reconciled the
regular and random approaches in a disturbedperiodic model which takes into account the
influence of varying degrees of irregularity on
meander geometry characterized as regular
paths (Ferguson, 1976).
During the 1970s and 1980s, the relationship
between meander form, namely radius of curvature, and the morphodynamic evolution of
meandering rivers received particular attention.
The view that planform curvature has a major
influence on local migration rates, and thus on
meander-migration patterns along a channel,
emerged through field observations (Hickin,
1974; Hickin and Nanson, 1975; Nanson and
Hickin, 1983; Odgaard, 1987). Hickin (1974)
performed empirical analysis of the migration
patterns recorded by scroll bars of the meander
bends of the Beatton River, British Columbia,
Canada. This study demonstrated that local
bend curvature is an important parameter controlling both the direction and rate of lateral
migration (Figure 5). The study also showed
that the relationship between the dimensionless
radius of local channel curvature (i.e. the ratio
of the radius of bend curvature r to channel width
2b) and dimensionless migration rate (i.e. the
ratio of migration rate M to channel width 2b)
is non-linear. r/2b ratio characterizes the sharpness of the bend. Dimensionless migration rate,
M/w, along a meander bend increases until r/2b
ratio reaches a critical value of 2.0 < r/2b <
3.0, but then decreases with increasing value of
the ratio (Hickin and Nanson, 1984; Nanson and
Hickin, 1983). These findings challenged the traditional notion that bend evolution exhibits
asymptotically stable behavior whereby meander
bends converge on stable geometries over time
(Keller, 1972; Langbein and Leopold, 1966).
Early modeling efforts have involved the
development of kinematic models (e.g. Ferguson,
1984) which came out of the ideas developed
through the characterization of meander planform geometry in the 1960s and 1970s (Ferguson,
1973, 1976; Langbein and Leopold, 1966). Kinematic models are based on the empirical relations
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726
Progress in Physical Geography 36(6)
Figure 5. Influence of local curvature on migration rates. Development of typical meander bends on the
Beatton River, Canada: upstream skewed bends (1–3), downstream skewed bends (4–6), compound loops
or multilobed meander (2–3 and 5–6). Dashed line shows the previous channel position whereas arrowed
paths represent the direction of channel migration (i.e. erosional pathlines).
Source: Modified from Hickin (1974). Reproduced with permission from the American Journal of Science.
between meander form and migration obtained
through the pioneering field evidence by Hickin
and Nanson (Hickin, 1974; Hickin and Nanson,
1975, 1984; Nanson and Hickin, 1983). Existing
kinematic models characterize bank retreat as a
non-linear function of channel curvature with a
spatial lag that takes into account downstream
migration of meanders (Ferguson, 1984). The
model of Ferguson (1984) is able to reproduce the
evolution of fundamental meander forms including bend asymmetry and compound looping, and
the results are in overall agreement with the data
obtained from real rivers (Gilvear et al., 2000;
Hooke, 2003).
1 Process-based mathematical models of
meander morphodynamics
Following the development of kinematic models, a significant emphasis has been given on
improving the understanding of the linkages
among physical processes governing meander
dynamics, the character of morphodynamic
influence, and the evolution of channel planform. This resulted in the development of
process-based mathematical models of river
meandering (e.g. Ikeda et al., 1981; Johannesson and Parker, 1989; Parker and Andrews,
1986; Parker et al., 1983; Seminara and Tubino,
1989; Zolezzi and Seminara, 2001). The mathematical models have two main components: one
characterizing fluid dynamic–morphodynamic
interactions and the other characterizing bankerosion process. The representation of fluid
dynamic–morphodynamic interactions is based
on fluid-mechanics principles (i.e. momentum
and mass conservation principles of the fluids).
The process-based models of meander morphodynamics vary in their mathematical sophistication; they can be divided into two main
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Güneralp and Marston
727
groups of analytical and numerical models. To
make the solution analytically tractable, analytical models employ simplified assumptions
regarding river morphology (i.e. steady-state
flow conditions and various geometric constraints such as invariable channel width, mild
bend curvatures, and slowly varying bed topography). Analytical models can be linear or
non-linear. In linear models (Ikeda et al.,
1981; Johannesson and Parker, 1989; Parker
and Andrews, 1986; Zolezzi and Seminara,
2001), the solution of the governing equations
is obtained analytically through linearization.
Thus, flow non-linearities are neglected
although they might be important in certain
cases (Camporeale et al., 2007; Pittaluga et al.,
2009). Fluid dynamic–morphodynamic component includes two-dimensional (2D) depthaveraged flow-field equations (i.e. St Venant
equations) for shallow flow, a sedimentcontinuity equation (i.e. Exner equation), and
empirical sediment-transport formulae. Based
on these equations, the model calculates the
flow field, bed deformation, and free-surface
elevation along a channel at any spatial coordinate for given hydraulic conditions and planform configuration. Then it determines the
excess flow velocity near the outer bank of a
bend (i.e. the difference between the depthaveraged near-bank velocity and the crosssectionally averaged velocity) that occurs due to
velocity distortions resulting from the variations
in channel curvature and bed topography. Thus,
excess near-bank velocity reflects the influence
of planform curvature as well as in-channel
geometry (i.e. bankfull width, channel depth, bed
morphology, and longitudinal slope), flow
characteristics (i.e. bankfull discharge), bed
material (i.e. sediment size) and bed-form
structure (Bridge, 1976, 1977, 1992; Bridge and
Jarvis, 1976; De Vriend and Struiksma, 1984;
Keller, 1972; Odgaard, 1987; Parker and Johannesson, 1989; Seminara and Tubino, 1989;
Struiksma et al., 1985; Termini, 2009; Thompson, 1986; Whiting and Dietrich, 1993a, 1993b).
The numerical models of meander morphodynamics include both non-linear versions of
linear analytical models (Camporeale et al.,
2007) and fully numerical models (Blanckaert
and De Vriend, 2003; Darby et al., 2002; Duan
et al., 2001; Imran et al., 1999; Mosselman,
1998; Nelson and Smith, 1989). Non-linear analytical models have been extended from their
corresponding linear versions using a nonlinear iterative procedure (e.g. Imran et al.,
1999) to solve the sediment mass continuity
equation. On the other hand, in fully numerical
models the governing equations are not
obtained analytically, rather they are solved
numerically. Numerical solution routines allow
for fewer geometric restrictions and provide a
much better solution for 3D helicoidal flow;
however, they are computationally very expensive. Advances in computing technologies have
started permitting detailed simulations of meander morphodynamics with the use of 2D and 3D
computational fluid-dynamic (CFD) models
(e.g. Duan et al., 2001; Olsen, 2003; Ruther and
Olsen, 2007).
Nevertheless, the selection of the type of
model to be used in the analysis should be based
on the specific characteristics of the problem.
Analytical models are proven to be successful
for rapid assessments of channel change and for
studies on the large-space scale and long-term
behavior of meandering rivers (Camporeale
et al., 2005, 2007). On the other hand, numerical
models may be more useful in the analysis of
short-term evolution such as in determining
geomorphic response of a river reach to human
modifications. The numerical models can be
used to determine the migration rates more
accurately at specific locations within a real
river with an irregular channel planform by
inputting detailed information on flow, bank
and channel geometry, and bed- and bankmaterial characteristics. For instance, when the
channel curvature is very high (i.e. bend is
sharp) or the width-depth ratio b (¼ b/H, where
b is one-half the channel width and H is flow
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Progress in Physical Geography 36(6)
depth) is very low, the analytical modeling
scheme cannot fully characterize the flow field.
In such cases, the use of numerical models may
become necessary.
2 Bank erosion model
The most widely used bank erosion submodel,
on the other hand, is a semi-empirical model
that assumes a linear relationship between
excess near-bank velocity and bank-erosion
(i.e. migration) rate along the channel with an
empirically derived bank-erosion (i.e. erodibility) coefficient (Ikeda et al., 1981):
ðsÞ ¼ E0 ub ðsÞ
ð1Þ
where E0 is bank-erosion coefficient, which
depends on the soil-mechanical properties of the
material along the bank (Parker and Andrews,
1986); ub is the excess near-bank velocity and
is the migration rate along the channel axis
s. The bank erosion coefficient E0 represents the
hydraulic or the fluvial erosion (i.e. the erosion
of soil particles due to the shear forces exerted
on the bank by the flow velocity ub). In other
words, the rate of bank erosion depends on the
rate of material removal from the base of the
bank (Thorne and Lewin, 1982). In addition,
the bank-erosion model assumes a bank erodibility which is homogeneous across the floodplain.
Although the validity of E0 has been supported
by field observations (Hasegawa, 1989; Micheli
and Kirchner, 2002a, 2002b; Odgaard, 1987; Pizzuto and Meckelnburg, 1989; Wallick et al.,
2006), it is important to note that it represents a
bank-erosion process where the primary mechanism is fluvial erosion, the failed blocks of material are easily eroded, and the banks are tall and
vegetation-free.
The linear bank-erosion model has been
employed in numerous studies on the longterm evolution of meandering rivers (e.g. Ikeda
et al., 1981; Lancaster and Bras, 2002; Parker
and Andrews, 1986) as well as in predictive
models (e.g. Larsen and Greco, 2002). If the
purpose is to predict migration rates along a
channel, then, typically, bank-erosion coefficient E0 is calibrated for historical planform
changes. A recent study shows the potential for
defining bank erosion E0 from field measurements of bank material properties, and thus the
ability to determine E0 without calibration
(Constantine et al., 2009). Such a capability
would allow for predicting migration rates in the
cases of limited or no historical planform-data
availability or in the presence of changing flow
and bank conditions. In reality, bank-erosion
rate strongly depends on the properties of
the bank (i.e. bank height, slope, and bankmaterial composition) (e.g. Hasegawa, 1989;
Wallick et al., 2006; Wynn and Mostaghimi,
2006) and the channel (i.e. width and slope). It
also depends on the presence of in-channel and
floodplain vegetation (e.g. Micheli and Kirchner, 2002a, 2002b; Micheli et al., 2004; Wynn
and Mostaghimi, 2006), the influence of which
may not be easy to specify with an empirical
coefficient. All of these factors may result in
variability in erodibility E0 values, and thus in
migration rates. For example, recently performed
physical laboratory experiments demonstrate
that processes governing erosion and failure in
riverbanks composed of coarse-grained material
are significantly different than those in riverbanks composed of fine-grained material (Nardi
et al., 2012).
Some recent models couple fluid dynamic–
morphodynamic models of meandering with
detailed mechanistic models of bank erosion
(Darby et al., 2002; Duan and Julien, 2005;
Duan et al., 2001; Motta et al., 2012). Mechanistic models relate the bank erosion to the physical
processes controlling bank retreat. Thus, they
avoid the need for model calibration to determine
bank-erodibility parameters. Mechanistic characterization can account for bank-failure
mechanisms (e.g. cantilever, planar, rotational,
and seepage induced failures) as well as hydraulic erosion. It also allows for better representation of natural bank profiles. For example,
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Güneralp and Marston
729
using a bank-erosion model derived from the
mass-conservation law for near-bank cells and
a 2D depth-averaged meander morphodynamics
model, Duan et al. (2001) showed that the rate
of bank erosion is a function of longitudinal
gradient in sediment transport, the strength of
secondary flow, and mass wasting from bank
erosion. Similarly, Darby et al. (2002) replaced
the existing bank-erosion model of a 2D depthaveraged numerical model (RIPA) of flow and
bed topography for single-thread rivers with
irregular planform (Mosselman and Crosato,
1991) with a more mechanistic model (Osman
and Thorne, 1988). The modified model can
simulate the basal erosion of cohesive bank
material (mainly composed of silt and clay) and
subsequent bank failure as well as transport and
deposition of eroded bank material. In another
study, Duan and Julien (2005) followed an
approach that separates the calculation of bank
erosion and the migration of banks. The model
calculates mass wasting from bank failure using
a simple parallel bank-failure model for noncohesive bank material (mainly composed of
sand and gravel). In a recent study, Motta
et al. (2012) replaced the semi-empirical bankerosion model of 2D depth-averaged meandermigration model (Abad and Garcia, 2006) with
a mechanistic bank-erosion model developed
based on the CONCEPTS (CONservational
Channel Evolution and Pollutant Transport System) model (Langendoen and Alonso, 2008;
Langendoen and Simon, 2008). This mechanistic model relates bank erosion to hydraulic erosion and mass failure and accounts for natural
bank profile and multiple parallel layers of soil
in the vertical profile with different erodibility
or shear-strength characteristics.
In spite of their advantages, the mechanistic
bank-erosion models are rather specific to the
bank material type. Thus, they are not as widely
applicable as the semi-empirical model
(equation 1), which can represent the overall
bank-erosion behavior resulting from the combination of various factors (e.g. bank erodibility
characteristics and governing failure mechanisms of bank material, sediment removal processes, etc.) (Constantine et al., 2009). The
model of Duan et al. (2001), for example, does
not account for the geotechnical bank failure
of cohesive bank material following basal erosion. In the model of Darby et al. (2002), the
only failure mechanism of bank migration is
planar failure. Equally importantly, overall
results obtained from these studies point out the
importance of taking into account (1) aggradation and degradation processes that occur along
the channel; (2) floodplain heterogeneity and
variability in environmental factors; and (3) the
feedback between sediment-transport and bankerosion processes in order to accurately characterize bank-erosion processes of meandering
rivers.
IV Simulating meander
morphodynamics with linear
analytical models
The linear analytical models produce qualitatively the same behavior of their non-linear
counterparts; they have proven to be successful
in reproducing the long-timescale planform
evolution of meandering rivers (Camporeale
et al., 2005, 2007). A range of linear models
with various orders, from first to fourth order,
have been developed. Model order reflects the
number of curvature-related convolution terms
in the model that depends on the sophistication
of the mathematical representation of the interactions between excess near-bank flow and
planform curvature (Camporeale et al., 2007;
Seminara et al., 2001; Zolezzi and Seminara,
2001). Regardless of the degree of sophistication in the fluid dynamic–morphodynamic component, nearly always, these models employ the
semi-empirical linear model of bank erosion
(i.e. constant bank-erosion coefficient, equation 1). In addition, notably important is that these
models are deterministic (i.e. the model produces
the same output from a given set of initial
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730
Progress in Physical Geography 36(6)
conditions or initial state; in other words, no
randomness is involved in the evolution of the
meandering-river system).
In their pioneering analytical model, Ikeda
et al. (1981) showed that meandering occurs
as a result of an inherent instability between
river flow and a mobile channel boundary due
to the asymmetries induced by channel curvature. According to this theory, both bar and bend
instabilities that result from the infinitesimal
curvature-induced perturbations of the flow
field and bed topography are sufficient for the
development of meanders (Ikeda et al., 1981;
Parker et al., 1982). The model of Ikeda et al.
(1981) is a first-order model represented by a
single, ordinary-differential equation (ODE)
characterizing the fluid dynamic–morphodynamic process:
du
dC
þ 21 u ¼ 1 A0 C ds
ds
ð2Þ
where u is streamwise velocity; C is the curvature along the streamwise axis s; 1 ¼ bCf
where b is the width-depth ratio defined as
¼ b=H (where b is one-half the channel width
and H is flow depth) and Cf is the friction coefficient; A0 is the scour factor. In equation 2, Cf is
characterized with no spatial variation. The
solution of equation 2 is in the form of a single
convolution integral, which links the excess
near-bank velocity to the planform curvature
(Ikeda et al., 1981; Sun et al., 1996):
ub ðsÞ ¼ bUC þ
1
ð
2Cf
0
UbCf 2
½F þ A0 þ 2
H
e H ðs Þ Cðs s0 Þds0
ð3Þ
0
where ub(s) is excess near-bank velocity along
the streamwise axis s, which promotes cutbank
erosion; b, U, H, and F are, respectively, onehalf the channel width, the depth-averaged
streamwise velocity, mean velocity, average
depth, and the Froude number; Cf is the friction
factor defined as gHI/U2; g is the gravitational
acceleration and I is the valley slope. The effect
of secondary circulation on sediment transport
and the coupling between flow field and bed
topography has been taken into account through
a theoretical relationship (Engelund, 1974),
ðnÞ
0
0
h0 ¼ A C, where A is the scour factor, (n)
is the bed topography elevation relative to any
constant level, and n is the transverse coordinate
(normal to the streamwise axis). The convolution integral characterizes a morphodynamic
influence on the local migration rate that decays
exponentially in the upstream direction (Furbish, 1991; Güneralp and Rhoads, 2009b).
In spite of its simplicity, the model of Ikeda
et al. (1981) suitably reproduces the fundamental characteristics of freely meandering rivers
including fattening and upstream skewness of
meander bends (e.g. Kinoshita curves, Figure 4)
supporting the basic validity of the form of the
convolution integral. This success resulted in
widespread use of the model in studies of
long-term dynamics of meandering rivers,
including the numerical investigations of floodplain sedimentation, the influence of sedimentary
and geological constraints on meander migration
(Howard, 1992, 1996; Sun et al., 1996, 2001a,
2001b), cutoff processes (Camporeale et al.,
2008; Howard, 1984; Stølum, 1996; Sun et al.,
1996) (Figure 6; Figure 1.4 of Howard 1992),
and the influence of meandering on riparianvegetation establishment (Perucca et al.,
2006). It was also used in environmental applications such as river restoration (Abad and
Garcia, 2006) and by the oil and gas industry
in the investigations of hydrocarbon reservoirs
within sedimentary deposits of meanderingriver floodplains (Henriquez et al., 1990;
Swanson, 1993). Despite its widespread
implementation, before cutoffs occur, the
first-order model fails to reproduce complex
meander forms such as downstream-skewed
meanders and compound loops or multilobed
meanders and the irregular planform patterns.
Therefore, the emergence of complex meanders and planform irregularity is attributed to
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Güneralp and Marston
731
Figure 6. Long-term evolution of a simulated meandering river, and its associated sediment size distribution on
the surface of the floodplain for characteristic parameters of 2b/H ¼ 20 and I ¼ 0.00067, where b is half channel
width, H is the average flow depth, and I is the valley slope. The channel is represented by green curve; the clay plugs
are represented by yellow curves. (See colour version of this figure online). Sediment size ranges from 4.78 mm
(red) to 10.58 (blue). Source: Sun et al. (2001b). Reproduced with permission from Water Resources Research.
the influence of high-frequency variations in
channel curvature resulting from cutoff processes (Figure 1 versus Figure 4A) (Camporeale
and Ridolfi, 2007; Güneralp and Rhoads,
2009b; Seminara et al., 2001). However, the
emergence of complex meanders is commonly
observed in rivers in nature before the occurrence of cutoffs (Hooke, 1984, 2007a, 2007b).
Following the first-order model of Ikeda et al.
(1981), higher-order (i.e. second- to fourthorder) models have been developed (Johannesson and Parker, 1989; Zolezzi and Seminara,
2001). These models incorporate additional
details of flow–sediment transport–bed morphology interactions reflected in the increased
sophistication of differential equations. In fact,
they consider the presence of alternate bars (also
known as free bars) and their influence on
the morphodynamic processes (Ikeda, 1989;
Tubino and Seminara, 1990; Whiting and
Dietrich, 1993a, 1993b). Alternate bars are
induced by bottom flow instability and migrate
along the channel. They are highly common in
many meandering rivers, especially in those
with low or high sinuosity (Seminara and
Tubino, 1989). On the other hand, point bars,
also called forced bars, which are produced by
curvature-driven secondary flows, are stationary. In meandering rivers with intermediate
sinuosity, the interactions between alternate
bars and point bars can inhibit the growth of
alternate bars (Tubino and Seminara, 1990).
Among high-order models, second-order
models assume that alternate bars do not have
a systematic effect on average bank-erosion
rates (Johannesson and Parker, 1989). The main
reason behind this assumption is that, in most
cases, the wavelength of meanders and that of
alternate bars are not the same.
If the wavelength of alternate bars is the same
as the wavelength of the meanders, a ‘resonance
condition’ can occur. In their theoretical work,
Zolezzi and Seminara (2001) and Seminara
et al. (2001) demonstrated the resonance condition by examining the influence of the interactions between the planform morphology and
the large-scale variations in bed topography on
meander migration. For this purpose, they
developed a fourth-order analytical model of
meander morphodynamics (Zolezzi and
Seminara, 2001: equation 5.15):
d 4 um
d 3 um
d 2 um
dum
þ
þ
þ 1
3
2
4
3
2
ds
ds
ds
ds
6
j
X
dC
jþ1 j
þ 0 um ¼ Am
ds
j¼0
4
ð4Þ
where C is the curvature ofP
1the channel axis s;
um sinðMnÞ with
streamwise velocity u ¼
m¼0
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732
Progress in Physical Geography 36(6)
M ¼ (2mþ1)/2; um is the mth Fourier mode and
Am ¼ 8(-1)m[(2mþ1)]2 where m = 0, 1, 2, . . . .
All the other parameters and the solution of the
equation are given in detail in Zolezzi and Seminara (2001).
Based on the solution of their model, they
showed that resonance occurs for a threshold
value (bR) which characterizes force-free spatial
modes of the river system consisting of migrating alternate bars. At this threshold value, the
alternate bars can neither grow nor decay in
time or in space. In other words, they do not
have a morphodynamic influence on the migration behavior. On the other hand, the conditions
where b value is lower or higher than bR are
defined, respectively, as sub- and super-resonant conditions. In channels characterized by
the sub-resonant condition (b < bR), the migration rate is influenced by upstream planform
morphology, whereas for those characterized
by the super-resonant condition (b > bR), it is
influenced by downstream planform morphology. In contrast to the model of Ikeda et al.
(1981) which can only simulate upstreamskewed meanders – a case similar to that which
occurs in the sub-resonant condition – the
Zolezzi and Seminara (2001) model can successfully simulate the evolution of meanders
from simple bends to compound loops or multilobed meanders in pre-cutoff timescales. Under
sub-resonant conditions, the morphodynamic
influence is in the downstream direction, contributing to downstream migration and
upstream skewness of meanders (Figure 1).
On the other hand, under super-resonant conditions, it is in the upstream direction, causing
upstream migration, and thus the development
of downstream-skewed meanders. Temporal
shift from upstream- to downstream-skewed
meanders may also be possible under superresonant conditions. Moreover, superresonance combined with sub-resonance can
lead to the development of compound loops.
The detailed characterization of fluiddynamic and morphodynamic processes in the
fourth-order model (equation 4) allows the
influence of high-frequency variations in planform curvature to be taken into account (Zolezzi
et al., 2009). In fact, the solution of the fourthorder model yields several convolution integrals
linking excess near-bank velocity not only to
local and upstream curvature, but also to downstream curvature (Camporeale et al., 2007;
Zolezzi and Seminara, 2001). In addition, the
spatial-weighting distributions characterizing
these convolution integrals present a morphodynamic influence on migration rates that is much
more complicated than simple exponential decay
(Camporeale et al., 2007; Zolezzi and Seminara,
2001) yielded from the first-order model. The
model of Zolezzi and Seminara (2001) also
shows that the spatial extent (i.e. decay rate or
spatial memory) of the morphodynamic influence depends on the value of the width-depth
ratio b and typically displays spatial oscillations
on a scale of the order of meander wavelength.
Notably important, however, is that, similar to
the first-order models, high-order models also
assume a constant bank erodibility which implies
homogeneous floodplain conditions.
V Empirical evidence for the linear
analytical models and further
insights on process–form linkages
Although significant attention has been given to
the mathematical models of meander migration,
the qualitative or visual comparisons of meander forms and planform patterns simulated by
the models to those in nature have been viewed
as adequate to judge the validity of process
characterization. To date only few studies have
rigorously and quantitatively evaluated the
validity of the meander morphodynamics
obtained from the solutions of these models
using real river data (Furbish, 1991; Güneralp
and Rhoads, 2009b, 2010). In an early study
on the evaluation of the linear models, Furbish
(1991) examined the influence of upstream
planform curvature on local migration rates of
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Güneralp and Marston
733
the Beatton River in British Columbia, Canada
(Hickin, 1974; Hickin and Nanson, 1975) to
determine whether an empirically obtained planform curvature–migration relation conforms to a
pure exponential-decay form of upstream curvature effect yielded by the first-order model. The
study concluded that the model can adequately
capture the morphodynamics of the compound
loops.
Recent empirical work examined further in
detail the influence of planform morphology
on the development of different meander
forms including simple bends and multilobed
meanders in several freely meandering rivers
(Güneralp, 2007; Güneralp and Rhoads,
2009b, 2010). The framework for this study was
to view an active meandering river as a linear
dynamical system (Franklin et al., 2002) where
a mapping function (i.e. spatial relation) characterizes the influence of the system input (i.e. the
morphodynamic influence in the form of cumulative planform curvature) on its output (i.e. the
local migration rate ). The spatial relations
were empirically determined using Discrete
Signal Processing techniques (Franklin et al.,
2002; Kamen and Heck, 1997). The findings
of the study show that planform geometry
strongly influences meander morphodynamics.
The influence of planform curvature on meander migration has multiple behavior modes,
characterized by exponential decay and oscillatory patterns. In other words, the morphodynamic influence is characterized by multiple
curvature-related convolution terms. Moreover,
the relative importance of different behavior
modes, and thus the spatial weighting distribution of planform-curvature influence, changes,
depending on the initial planform morphology
of the migrating channel (Güneralp and Rhoads,
2009b).
Güneralp and Rhoads (2009b) also demonstrated that the evolution of simple bends including Kinoshita curves (Kinoshita, 1961) can
adequately be characterized by the exponentialdecay mode of downstream morphodynamic
influence. In contrast, the emergence of compound loops or multilobed meanders can only
be described by a damped-oscillatory mode
(i.e. exponential decay superimposed on oscillations). Remarkably, empirically derived exponential decay mode corresponds to the
planform influence yielded by the first-order
models of meander morphodynamics (Ikeda
et al., 1981). On the other hand, the dampedoscillatory mode is associated with the solution
of higher-order models that can simulate the
emergence of compound loops (Camporeale
et al., 2007; Zolezzi and Seminara, 2001). These
findings imply that the morphodynamic influence characterized by damped-oscillatory mode
acts like a high-pass filter that is much more sensitive to high-frequency variations in planform
morphology than pure exponential-decay structure yielded by the first-order model. The
damped-oscillatory mode reflects the changing
influence of evolving-bed morphology (e.g. the
formation of multiple point bars) on planform
evolution as the channel evolves from simple
bends to complex meanders (Güneralp and
Rhoads, 2009b, 2010). All of these findings
provide empirical support for the high-order
mathematical models characterizing the feedbacks between fluid-dynamic and morphodynamic processes governing the dynamics of
river meandering. A recent experimental study
on a high-amplitude meander bend in a flume has
revealed that changing water-surface slope
induced by the channel curvature causes the
development of damped-oscillatory flowvelocity variations and bed deformations along
the bend, which may result in the development
of multiple bar forms and pools (Termini, 2009).
Multiple point bars and pools along a meander
bend may then be attributed to the river’s morphological response to the changing channel curvature
(De Vriend and Struiksma, 1984; Parker and
Johannesson, 1989; Struiksma et al., 1985) that
creates flow instability along the meander bend.
Changing spatial structure of the morphodynamic influence on migration rates with increasing
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734
Progress in Physical Geography 36(6)
Figure 7. A reach of Rio Jutaı́, Brazil, representing an irregular planform pattern composed of highly elongated, irregular, and multilobed meander bends. Flow direction is from left to right (4 48’29.72’’S,
68 31’44.74’’W).
planform irregularity also suggests that the
relationship between planform morphology
and migration rates may be non-linear and
spatially variable (Güneralp and Rhoads,
2009b). Phase-space diagrams, which provide
a visual representation of the evolution of a
dynamical system, reveal non-linear oscillatory trajectories of the spatial structure of
morphodynamic influence as opposed to a
linear trajectory that corresponds to a pure
exponentially decaying influence (Güneralp
and Rhoads, 2010). Oscillatory trajectories
may reflect the damped-oscillatory mode
characterizing the emergence of compound
loops and multilobed meanders. Alternatively, they may indicate the non-linear character of migration response. Such non-linear
response is also suggested by various theoretical, experimental, and field observations
(Hooke, 2003, 2004, 2007a, 2007b; Nanson
and Hickin, 1983; Seminara and Tubino,
1992; Seminara et al., 2001) and is potentially
a critical factor in the development of
compound loops or multilobed meanders.
Further work is needed to confirm that these
non-linear trajectories are characteristic of
complex meander morphologies and to provide a process-based explanation for such
non-linear trajectories.
VI From deterministic models to
real-world complexity
Advances in the process-based mathematical
modeling of river meandering (Ikeda et al.,
1981; Johannesson and Parker, 1989; Parker
and Andrews, 1986; Zolezzi and Seminara,
2001) have provided critical insights into the
feedbacks between fluid dynamic–morphodynamic processes and planform evolution. However, regardless of the degree of sophistication,
in pre-cutoff timescales, planforms simulated
by mathematical models are highly regular in
shape and size (Figure 4) compared to complicated meander morphologies and irregular patterns of real rivers (Figures 1 and 7) (Güneralp
and Rhoads, 2009b; Howard and Hemberger,
1991). Potential reasons for the discrepancies
between simulated meander patterns and the
real rivers may be attributable to the simplified
assumptions – such as homogeneous erodibility
of the floodplain, constant flow discharge, and
uniform sediment size – made in the characterization of autogenic processes. Alternatively,
they may result from the inadequate characterization of bank-erosional resistance and/or of
bank-erosion and bank-deposition processes.
Moreover, until recently, in morphodynamic
modeling, meandering rivers and their floodplain
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Güneralp and Marston
735
Figure 8. Examples of heterogeneous floodplains composed of erodibility patch mosaics. Patch size ranges
as a function of l along valley (left-right) axis and of A along cross-valley (up-down) axis, where l and A are,
respectively, the linear wavelength and the bend amplitude of initial meander bends used in the simulations.
Hot (cold) colors represent high (low) erodibility. (See colour version of this figure online). The mosaics of
patches represent the spatial arrangements of environmental factors that influence migration and that vary
with spatial scale, including patches of vegetation (e.g. forested versus non-forested areas, low versus high
biomass density), floodplain sedimentological complexity (e.g. clay plugs versus sandy alluvium), and human
activities (e.g. cropland versus managed forest).
landscapes have typically been seen as separate
entities although the interactions between them
play a significant role in biomorphodynamic evolution of riverine landscapes as well as in riverineecosystem dynamics (Van De Wiel et al., 2007;
Ward et al., 2002). Moreover, the role of spatial
and temporal variability in the environmental
conditions on the biophysical feedbacks between
meander morphodynamics and floodplain patterns and processes remains to be identified.
Past numerical work based on the processbased mathematical models of meandering
examined the influence of spatial variability in
sedimentary processes and geological conditions, such as erodible point bar deposits versus
resistant clay plugs and valley walls (Howard,
1996; Sun et al., 1996) and vegetation density
and patterns (Perucca et al., 2007), on the meander migration patterns. These studies showed that
meandering processes are highly influenced by
spatial and temporal variability in the floodplain
soil erodibility. Similarly, through a preliminary
analysis, Sun et al. (1996) suggested that a
spatially uncorrelated stochastic variability in
soil erodibility can contribute to the development
of irregularities in meander planform.
In a recent study, Güneralp and Rhoads (2011)
have systematically examined and quantified the
dependence of meander morphodynamics on the
spatial scale of heterogeneity (i.e. patch size), the
magnitude of variability, and the stochasticity in
floodplain erodibility. For this purpose, the study
used a first-order model of river meandering
(Ikeda et al., 1981) and stochastically simulated,
heterogeneous floodplains composed of patches
of differential erodibility with different scales of
patchiness. Patch size in the valley direction
ranged from 2 to /16 and in the cross-valley
direction from 4A to A/4, where and A are,
respectively, the linear wavelength and bend
amplitude of initial meander bends used in the
simulations. The patch mosaics of erodibility
reflect the spatial arrangements of environmental
factors influencing the magnitude and scale of
variability such as vegetation, soil properties,
topography, and human activities (Figure 8). The
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Progress in Physical Geography 36(6)
Figure 9. Influence of patch size of differential erodibility (e.g. Figure 8) on planform evolution. Red and blue
arrows represent the direction of increasing maximum cross-valley extent of the meander belt and increasing
variability in bend amplitude, respectively. The area marked by the dashed line identifies the patch structures
resulting in highly complex meander morphologies including upstream- or downstream-skewed, irregular,
elongated bends, and compound loops or multilobed meanders. It also corresponds to the patch sizes for
which the occurrence of cutoffs in shorter timescales is more common than that for the other patch sizes.
Source: Modified from Güneralp and Rhoads (2011). Reproduced with permission from Geophysical Research
Letters.
findings of the study show that meanders simulated with heterogeneous landscapes evolve into
complex morphologies with remarkable similarities to those of freely meandering rivers (Figure 9)
as opposed to highly regular meanders simulated
with homogeneous floodplains (Figure 4A).
The patch size substantially influences the
spatial characteristics (e.g. shape and size) of
meander bends. Specifically, meander bends
simulated with floodplains composed of patches
larger than the initial size of the meanders (e.g.
2l, 4A) are highly elongated and upstreamskewed resembling distorted forms of Kinoshita
curves (Kinoshita, 1961), whereas other bends
are compressed, leading to an irregular planform pattern with high variability in bend
amplitudes and cross-valley extent (Figure 9).
As the patch size along the valley axis decreases
for large patch sizes along the cross-valley axis
(e.g. l/16, 4A), elongated compound loops
develop. On the other hand, as cross-valley heterogeneity increases, the maximum cross-valley
extent of the meander belt decreases (e.g. 2l, A/4)
and compound loops develop (e.g. l/4, A). The
smallest patch size or the highest heterogeneity
(i.e. l/16, A/4) results in the lowest variability in
bend amplitudes although meander bends have
morphologies still much more complicated than
those for homogeneous floodplain. This confirms the strong influence of small, high-frequency variations in planform curvature on
morphodynamic evolution of meanders (Camporeale et al., 2007; Güneralp, 2007; Güneralp
and Rhoads, 2009b, 2010; Zolezzi and Seminara, 2001). Moreover, the sensitivity of autogenic meandering processes to stochastic
variability in the environment leads to different
patterns of meander evolution even in landscapes with the same patch sizes and magnitudes of erosional variability (Güneralp and
Rhoads, 2011). In summary, Güneralp and
Rhoads (2011) show the strong influence of
floodplain heterogeneity in the morphodynamic
evolution of meandering rivers.
In another recent study, Parker et al. (2011)
have introduced a new framework for the
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Güneralp and Marston
737
modeling of meander migration, called the
‘moving-boundary framework’, in order to
address the limitations of the analytical models
related to bank-erosion and bank-deposition
processes. According to this framework, the
bank-erosion process, which is characterized
in the analytical models as a linear relation
between excess near-bank velocity – a simple
aspect of fluid dynamics – and migration rate,
is not completely satisfactory. The framework
also addresses another simplified assumption
that the migration due to the deposition on the
inner bank of a meander is equal to the migration due to the erosion on the opposite bank,
leading to a constant bankfull width as the channel migrates. By incorporating the role of
slumping at the eroding banks and of vegetation
both in erosion and deposition processes, the
new framework attempts to replace the current
assumptions and provide a more physically
based explanation of channel migration. Such
an approach would allow for considering the
dynamic interaction between erosion and
deposition processes in determining the morphodynamic evolution of meandering rivers.
There has been an increasing emphasis on
advancing the understanding of integrated
dynamics of river meandering with landscape
dynamics, particularly with riparian-vegetation
dynamics (Greco and Plant, 2003; Marston
et al., 2005; Perucca et al., 2006, 2007). The
interactions between meandering rivers and
their surrounding riparian vegetation involve
both hydraulic and ecological processes
(Hughes et al., 2005; Hupp and Osterkamp,
1996). The hydrological, hydraulic, and geomorphological characteristics of a meandering
river and environmental conditions of its floodplain control the availability of water, sediments, nutrients, and seeds for the riparian
environment (Bendix and Hupp, 2000; Malanson, 1993; Salo et al., 1986). On the other hand,
vegetation biomass density and patterns affect
the river channel by changing its flow field and
morphology through its effect on the erosion
along the cutbank and the deposition of the sediment on the point bar (Gurnell et al., 2006; Masterman and Thorne, 1992).
The variability in soil properties (i.e. soil type
and sedimentary structure resulting from erosion, deposition, and hydrological processes)
and floodplain topography and the land change
due to human activities influence the spatial patterns of vegetation succession in riparian zones
(Nanson and Beach, 1977). Newly formed point
bars by channel migration serve as possible sites
for vegetation establishment (Everitt, 1968;
McKenney et al., 1995), whereas eroded cutbanks remove the vegetation cover. Cutoff
channels such as oxbow lakes also significantly
contribute to the complex mosaics of morphological and ecological habitats of floodplain landscapes. Oxbow lakes increase the subsurface
hydrologic connectivity between the river and
floodplain (Amoros and Bornette, 2002) and
provide suitable environment for the colonization of certain vegetation species (Stella et al.,
2011). Segregated and complicated sedimentary
patterns of floodplain landscapes created by
meander morphodynamics also control the texture of floodplain soils, which in turn influences
the soil moisture balance (Rodriguez-Iturbe and
Porporato, 2005; Rodriguez-Iturbe et al., 1999)
and the establishment of riparian vegetation
(Nanson and Beach, 1977; Piégay et al., 2000;
Robertson and Augspurger, 1999). Moreover,
subtle topographic variations in the floodplain
can affect the connectivity of surface waters
(Jones et al., 2008; Poole, 2002) and thus fluvial, hydrologic, and ecological processes.
Environmental heterogeneity and stochastic
variability of floodplain landscapes, in turn, influence the morphodynamics of meandering rivers
by creating non-uniform bank erodibility as well
as sediment deposition patterns on the point bars
and surrounding floodplain (Constantine et al.,
2009; Hudson and Kesel, 2000; Wynn and
Mostaghimi, 2006).
Recent numerical work (Camporeale and
Ridolfi, 2010; Van De Wiel and Darby, 2004;
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Progress in Physical Geography 36(6)
Perucca et al., 2006, 2007), also supported by
field observations (Meitzen, 2009; Nanson and
Beach, 1977; Robertson, 2001, 2006), has highlighted the role of meander dynamics in the
development of vegetation patterns observed
in riparian landscapes as well as the importance
of vegetation-succession processes on the planform evolution of river meanders. In particular,
this work emphasized that variability in bank
erodibility induced by vegetation patterns and
density may yield different meander shapes
from the usual upstream-skewed ones generated
by the morphodynamic models for sub-resonant
rivers. These findings provide the first numerical evidence for the feedbacks between meander
morphodynamics and floodplain-vegetation
dynamics. The floodplain landscapes of freely
meandering rivers, however, are still much
more heterogeneous than their simulated counterparts. Identifying the biophysical feedbacks
between the morphodynamics of meandering
and floodplain patterns and processes and the
influence of spatial and temporal variability
in the environmental conditions on these
feedbacks is critical for process-based understanding of the evolution of meandering river–
floodplain systems and the response of such
coupled systems to changes in environmental
conditions such as the hydrological regime or
land use/land cover (Güneralp et al., 2012).
VII Concluding remarks
Changes in meandering river channels reflect
the adjustments of these rivers to their intrinsic
dynamics as well as to variability in environmental conditions such as soil and geological
characteristics of the floodplain, vegetation patterns and processes, land use, hydrologic
regime, tectonic activity, and climate. Where
human activity encroaches on meandering rivers, the morphodynamics of meandering rivers
are often viewed as a sign of river instability and
raise societal concerns due to the hazards associated with bank erosion and channel change
(Hooke, 1995; Lawler, 1993; Piégay et al.,
2005). Due to the rapid increase in demand for
freshwater and land, humans have been modifying meandering rivers and their floodplains both
directly (e.g. dams, diversions, channelization,
bank protection, and mining) and indirectly
(e.g. by changing the sediment load and flow
regime through urbanization, agriculture, deforestation, and mining). Such changes in environmental conditions can influence the inherently
intricate meander morphodynamics and may
result in complicated migration responses to
these changes. In many cases, this complexity
makes it difficult to perform a long-term prediction of the future states of morphologies in
meandering rivers in nature.
There is a critical need for more viable solutions to the problems created by bank erosion
and channel migration (Brookes and Shields,
1996; Kondolf, 2006; Larsen and Greco,
2002). This need calls for a comprehensive and
predictive understanding of both process–form
feedbacks in meander morphodynamics –
particularly the feedbacks governing the integrated dynamics of river meandering and their
floodplain landscapes – and the influence of environmental variability on these feedbacks. Driven
by such a need, there have been exciting developments in river-meandering research; however, to
date, most research efforts have focused mainly
on either theoretical and numerical modeling or
field-based studies, or laboratory experiments
(Güneralp et al., 2012; Hooke et al., 2011). The
importance of bridging the theoretical modeling
and real world complexity has been recognized
since the first symposium on river meandering
in 1983 (Elliot, 1984). Shen (1984) highlighted
this issue in his keynote for the session ‘Present
Knowledge and Future Directions’:
It is extremely difficult to develop theoretical models to describe the occurrence of meandering and the
flow characteristics in stream bends because these
models will ultimately provide us with long-term
solutions. However, on the other hand, researchers
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Güneralp and Marston
739
must attempt to bridge the large gaps between theoretical solutions for highly idealized stream bends
and practical problems. (Shen, 1984: 1011)
The success of the management and restoration
strategies (e.g. solutions to bank erosion and
river channel adjustments, flooding hazards,
and the management of in-channel and floodplain habitats) requires a framework that explicitly integrates theoretical and numerical
approaches with field-based studies and laboratory experiments (Güneralp, 2007; Güneralp
et al., 2012; Van De Wiel et al., 2011).
Recent advances in computer technology
have increased our capabilities for developing
more detailed, computationally intensive models of meander morphodynamics. Similarly,
advances in GIScience and geospatial technologies and field-measurement equipment and
techniques have provided extensive and highly
accurate data, and allowed for modeling, analysis, and monitoring of meandering dynamics
over various spatial and temporal scales. This
integrated approach also necessitates interdisciplinary research that integrates the expertise of
various disciplines from geomorphology to engineering, hydrology, ecology, and GIScience.
Such an integrated framework would help
improve the theoretical models of meander morphodynamics and allow for performing more
extensive testing and validation of these models.
Thus, it would allow for comprehensive understanding of the process–form linkages that govern the dynamics of meandering in the presence
of spatial and temporal environmental variability. An in-depth understanding of the impact of
environmental change on the morphodynamics
of meandering rivers would then help us understand how human activities interact with these
environmental factors to influence the patterns
and processes of meandering rivers. Such
knowledge would ultimately inform accurate
assessments of river-hazard risks and facilitate
ecologically sound river-management and
restoration practices with the aim of supporting
healthy river ecosystems.
Funding
This research received no specific grant from any
funding agency in the public, commercial, or notfor-profit sectors.
Acknowledgements
The helpful comments and suggestions of two anonymous reviewers are gratefully acknowledged.
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