In this study, the natural process of river meandering is captured in a computational model that considers the effects of bank erosion, the process of land accretion along the inner banks of meander bends, and the formation of channel cutoffs. The methodology for predicting bank erosion explicitly includes a submodel treating the formation and eventual removal of slump blocks. The accretion of bank material on the inner bank is modeled by defining the time scale over which areas that are originally channel become land. Channel cutoff formation is treated relatively simply by recomputing the channel alignment at a single model time step when migrating banks meet. The model is used to compute meandering processes in both steady and unsteady flows. The key features of this new model are the ability (a) to describe bank depositional and bank erosional responses separately, (b) to couple them to bed morphodynamics, and thus (c) to describe coevolving river width and sinuosity. Two cases of steady flow are considered, one with a larger discharge (i.e., “bankfull”) and one with a smaller discharge (i.e., “low flow”). In the former case, the shear stress is well above the critical shear stress, but in the latter case, it is initially below it. In at least one case of constant discharge, the planform pattern can develop some sinuosity, but the pattern appears to deviate somewhat from that observed in natural meandering channels. For the case of unsteady flow, discharge variation is modeled in the simplest possible manner by cyclically alternating the two discharges used in the steady flow computations. This model produces a rich pattern of meander planform evolution that is consistent with that observed in natural rivers. Also, the relationship between the meandering evolution and the return time scale of floods is investigated by the model under the several unsteady flow patterns. The results indicate that meandering planforms have different shapes depending on the values of these two scales. In predicting meander evolution, it is important to consider the ratio of these two time scales in addition to such factors as bank erosion, slump block formation and decay, bar accretion, and cutoff formation, which are also included in the model.
 A model for predicting the evolution of complex planform patterns of natural meandering rivers is presented and tested in this paper. Figure 1 shows a typical example of this morphology in the Omolon River in northeastern Siberia. This image encapsulates both the intricately varied current planform shape of the river, as well as the history of the river form reflected in the patterns of scroll bars and cutoffs. Tracing this history of evolution and understanding the mechanisms that give rise to such elegant river planforms represent problems of importance and interest to both geomorphologists trying to understand such systems and river engineers dealing with practical problems regarding river migration. These issues have been approached in various previous studies of meander migration and the mechanisms which produce meandering.
 The planform shape of meandering rivers is determined by a complex mutual interaction among flow, sediment transport, and both bed and bank morphodynamics. The relationship between flow and bed topography in channel bends was described theoretically by Engelund , but the coupling between the flow and bank morphology was not explicitly considered therein. Subsequently, various efforts have been made to predict bank erosion by coupling it to the local flow field using relationships between bank retreat and near-bank velocity but constraining channel width to take a spatially and temporally constant, user-selected value [e.g., Hasegawa, 1978, 1984; Ikeda et al., 1981]. The role of river bars in meander evolution, as well as the relationship between bars in straight and meandering channels, was further analyzed in terms of an inherent resonance phenomenon between bar and bend instabilities in meandering, mobile bed channels [Blondeaux and Seminara, 1985; Johannesson and Parker, 1989]. However Crosato and Saleh  have shown experimental and numerical evidence indicating that the meanders may form at conditions that are nonresonant. Both bar and bend instabilities play important roles in the evolution of the meandering channels and thus should be incorporated into the analysis of the evolution of meandering.
 Several numerical models of meandering have used the formulation for bank migration of Hasegawa [1978, 1984] and Ikeda et al.  to illustrate model the evolution of complex patterns, including numerous cutoffs [e.g., Howard and Knutson, 1984; Johannesson and Parker, 1989; Sun et al., 2001; Zolezzi and Seminara, 2001; Lanzoni and Seminara, 2006; Crosato, 2008; Frascati and Lanzoni, 2010]. Research to date has not, however, allowed both planform centerline and channel width to evolve in response to the interaction between the concurrent processes of erosion and deposition at riverbanks.
 With the advent of more sophisticated computer technology, several computationally intense numerical models for the analysis of flow and bed topography in meandering channels have been developed. These models can be used to study bed evolution under the assumption of noncohesive bed material, in both straight and meandering channels [e.g., Shimizu et al., 1996; Nagata et al., 2000; Darby et al., 2002]. Validation of these models for bed morphology in the case of fixed banks has been carried out in both laboratory channels and natural rivers [e.g., Darby et al., 2002; Lesser et al., 2004; Legleiter et al., 2011]. In addition to techniques for the prediction of bed topography in channels, several researchers have proposed bank erosion models considering both noncohesive and cohesive material [e.g., Mosselman, 1998; Nagata et al., 2000; Darby et al., 2002; Duan and Julien, 2005]. These developments provide some of the essential building blocks for more advanced models of meander evolution, because they help to define the details of the mechanics governing alluvial channels over relatively short time scales.
 However, there are also other longer-term factors that should be considered in creating a model of river meandering. The relevant time scales are associated with annual or longer variations in flow, as well as factors that influence medium- and long-term bank stability. For example, bank erosion is typically not a slow, consistent process; it tends to occur episodically during large flood events. Rinaldi et al.  pointed out using the example of the Cellina River (Italy) that bank retreat at river bends often occurs after and not during peak discharges. On the other hand, vegetation that can provide bank and bar stability tends to become established at more common lower flows, but may be removed during rare, high flows. Therefore, it is necessary to quantify the degree of growth of vegetation in terms of the time required for the growth of vegetation over bare sediment and conditions for which this vegetation is flushed out by subsequent flooding. Tsujimoto  studied the effect of alternating low and high flows on colonization by vegetation and bed erosion. Vegetal encroachment leads to bank advance in the direction of channel centerline, as noted by Allmendiger et al.  and Gurnell et al. . Perruca et al.  showed that spatial and temporal scales of vegetation growth and decay also play important roles in the evolution of river planform. Though the overall effect of vegetal encroachment on meandering planform shape is clear, the mutual processes between bed topography, flow, and vegetation process still are open questions in that they depend on the type of vegetation, climate, hydrologic regime, etc. These factors complicate the development of advanced numerical models of meander evolution.
 In this paper, an approach that allows consideration of both the details of short-term processes and key elements of longer time scale effects is proposed to investigate meandering. The short-term process of flooding is captured directly in hydrograph modeling, because high flow is found to play an important role in regard to the evolution of bed topography, bank erosion, and cutoff formation. On the other hand, the low flow period is parametrically compressed in hydrograph modeling, because the effect of low flow period to the river shape can be considered indirectly. In this study, a time scale parameter which indicates leading time to advance an inner bank is used in the modeling. Using the modeling, the computation hydrograph can be shorter than in the field by using the hydrograph modeling. Our model suggests that a resolution of both relatively short and relatively long temporal scales provides a more realistic and comprehensive description of the coevolution of channel planform and width.
 The goal of this approach is the modeling of the long-term evolution of river channel and planform. The key factor is the relationship between the time required for the vegetal stabilization of newly deposited bare, exposed sediment, resulting in land accretion, and the flood return time. In order to examine these factors in the simplest possible way, the model is implemented at small, experimental scale. Modeling at field scale can proceed after the relevant parameters have been quantitatively evaluated for one or more rivers.
2 Description of the Meandering Model
2.1 Flow Model
 Two-dimensional depth-averaged continuity and momentum equations are used in the flow model. Those equations are transformed into a moving boundary-fitted coordinate (MBFC) system (Figure 2a) and are quantified using the following continuity and momentum equations:
where t is time; x, y are axes of an orthogonal coordinates system; ξ, η are axes of a nondimensional generalized coordinate system which is defined as (0 ≤ ξ ≤ 1, 0 ≤ η ≤ 1); ξt, ξx, ξy, and ηt, ηx, ηy are differential metric coefficients between the x, y and ξ, η coordinates; J is the Jacobian determinant for the coordinate transformation, which is defined as J = ξxηy − ξyηx; uξ, uη are velocity components in the ξ and η directions, respectively; h is water depth; H is water surface elevation; g is acceleration due to gravity (=9.8 m/s2); and Cd is a bed friction coefficient given by following equation: Cd = gnm2/h1/3. Here nm is Manning's roughness coefficient given by [Kishi and Kuroki, 1973], where d is the diameter of the bed material. In equations ((2)) and ((3)), Dξ, Dη are terms describing the diffusion of momentum in the ξ and η directions, respectively. The parameters α1–α6 are given as follows:
 Here we consider the relation between the dimensioned and dimensionless coordinate system. Let and denote dimensioned versions of ξ and η parameters. These have the same direction axes as ξ and η (Figure 2b). The parameters ξr and ηr are defined using local grid width as follows (Figure 2b):
 Using these factors, the physical parameters in equations ((1))–((3)) can be transformed from the nondimensional general coordinates system (ξ, η) into the dimensioned coordinate system () as follows:
where are depth-averaged flow velocity components in the and directions, respectively. Also, the parameters Dξ and Dη can be described as follows:
where νt is a coefficient of eddy viscosity given as follows:
 Here κ is the Karman coefficient (=0.4), u* is the shear velocity, and u, v are flow velocities in the x and y directions, which are given by the following equations,
respectively [Jang and Shimizu, 2005]. This model, known as Nays2D, is available in the public domain as part of the International River Interface Cooperative (iRIC) software package. It can be downloaded at http://i-ric.org.
2.2 Bed Evolution Model
 While in principle Nays2D can handle multiple grain sizes, here it is applied to the case of a single grain size for simplicity. Also, considering that bed evolution typically occurs much more slowly than flow evolution over that bed (quasi-steady assumption), morphodynamic bed evolution in the MBFC approach is calculated by means of the following Exner equation of sediment conservation:
where zb is riverbed elevation, λ is porosity of the bed material (=0.4 here), and are bed load sediment volume transport rate per unit width in the ξ and η directions, respectively. These components depend on near-bed velocity and transverse bed slope as follows:
where are near-bed velocities in the and directions; Vb is the resultant velocity of the near-bed velocities, i.e., , γ is a correction coefficient to account for the bed slope (described below); and θ is the intersection angle between the and axes. In addition, qb is the magnitude of the volume bed load sediment transport rate per unit width. Here qb is given by the relation of Ashida and Michiue  which was developed for, and is applicable to, the transport of sand and fine gravel:
where sg is the submerged specific weight of a sediment grain in water (=1.65 for quartz), τ* is the nondimensional shear stress (Shields number), τ* c is the critical Shields number for the onset of sediment motion, and u* c is the corresponding critical bed shear velocity. The Shields number is given as
 Both the critical Shields number and the critical shear velocity are computed using the formula of Iwagaki .
 The near-bed velocity in the direction of the vertically averaged flow is given as follows [Engelund, 1974]:
where is the near-bed velocity in streamwise direction (Figure 3); V is the resultant velocity of the depth averaged velocities, i.e., ; β is a parameter obtained by assuming a parabolic distribution of velocity in the vertical direction; σ = 3/(φκ + 1); and φ is velocity factor given as φ = V/u*. In order to include the effect of curvature-induced secondary flow, the near-bed velocity component perpendicular to the direction of the vertically averaged flow, is estimated as follows (Figure 3):
where N* is a coefficient of the strength of the secondary flow taking the constant value N* = 7.0 [Engelund, 1974] and rs is the radius of curvature of the streamline in question. Here rs is given as follows:
 In addition, can be obtained as follows:
where cos θs = u/V and sin θs = v/V, respectively.
 According to Hasegawa , the parameter γ in ((12)) and ((13)) is given as
where μs is the static coefficient of Coulomb friction (=1.0) and μk is the dynamic coefficient of Coulomb friction (=0.45). While the simple treatment of secondary flow used here is based on fully developed bend flow and precludes the use of varying roughness, it has been used successfully for predictions of bed morphology in channel bends in the past [Jang and Shimizu, 2005] and is thus assumed to be adequate for the present purposes.
2.3 Bank Erosion Model
 An equation for bank migration equation can be obtained by integrating the sediment continuity equation in the near-bank region in the cross-sectional direction [Parker et al., 2011]. Defining θBc to be the angle of incipient collapse of the bank, zBL and zBR can be written as follows:
where zBL and zBR are the local elevations of the left and right banks, respectively, and are the corresponding bed elevations at the base of the corresponding bank, and , , , are the values of the transverse coordinate at the bottom and top of both banks (Figure 4).
 Integration of the Exner equation of sediment continuity over each bank region yields, in combination with ((23)) and ((24)), the following relations for bank migration:
where and denote the widths of left and right bank regions, respectively. The parameters and are calculated along each riverbank at each time step. The calculation grid is then shifted using those values.
 In natural meandering rivers, it is quite common for a noncohesive layer to be overlain by a cohesive, root-rich cap emplaced via floodplain deposition (Figure 5). The protection of the noncohesive part of the bank from erosion due to the presence of slump blocks obtained from this cohesive layer is considered using the methodology of Parker et al. . The riverbank is treated as a two-layer system with a cohesive layer above a noncohesive layer (Figure 6). The bed load sediment transport rate in the direction is found by considering the armoring effect of the slump blocks [Parker et al., 2011]. Thus, where is the transverse transport rate of noncohesive material in the presence of slump block armoring and is the corresponding value in the absence of armoring, then
where 0 ≤ K ≤ 1 is an armoring coefficient given as
where Dchunk is characteristic size of slump block, and B* is the length BL or BR (Figure 4). In addition, Achunk is the volume per unit area of slump block material, here estimated as follows [Parker et al., 2011]:
where qchunk is the volume rate of production per unit streamwise distance of slump block material and Tchunk is the characteristic lifetime of a chunk before it breaks down. Here qchunk is given as
where c denotes the transverse speed of bank migration and Hc denotes the thickness of the cohesive layer (Figure 6). The characteristic lifetime of a slump block Tchunk is estimated herein as
where Es is an entrainment rate of cohesive material (loss of slump block volume per unit surface area). Nishimori and Sekine  proposed the following equation using experimental results for the Es (cm/s) in terms of shear stress, moisture content, and water temperature:
where Rwc is the moisture content of the slump block and α (s2/cm2) is a dimensioned coefficient varying with water temperature T (°C).
2.4 Land Accretion Model
 Figure 7 shows the time evolution of the inner bank of the Nakashibetsu River in Hokkaido, Japan, from November 2003 to August 2006. In the picture taken at the beginning of this period, i.e., November 2003, a relatively low point bar without vegetation can be seen along the inner bank. Following flows which inundated the bar, it subsequently became higher, as can be seen in the picture taken in May 2005. As the bar became higher, it spent less time submerged, resulting in the vegetal encroachment seen in the photo from August 2006. This sequence of point bar growth, stabilization, and accretion to the floodplain is an essential element of the meandering process. The case of the Nakashibetsu River illustrates how the inner bank of a meandering channel tends to be progressively less frequently submerged and more vegetated as it builds. After vegetation is established, less frequent high flows over the bar top result in further sedimentation of relatively fine material due to the increase in roughness and the sediment trapping effect of the vegetation. This further increases the elevation of the sandbar, as illustrated in Figure 8. Ultimately, the elevation of the point bar becomes the same as that of the adjacent floodplain. Submergence then becomes very rare, occurring only during large flood events. Those processes may play out over years or decades.
 Simulating the slow process of bank accretion requires enormous calculation times. It is furthermore very difficult to specify the process because it depends on several local effects (e.g., climate and the type of vegetation). Therefore, the resultant effect of bank accretion on river planform change is considered in the numerical model only through an effective time scale for bare sediment to become vegetated. Here we define the inner bank of the meandering channel in terms of the line along which the height of the point bar becomes identical to that of the adjacent floodplain, which is considered dry land in our computation. Considering this, the process of land accretion is treated as follows in the calculation:
where h(i,j) is the water depth in the calculation grid corresponding to the i, j grid indices, Δt is the time step in the calculation, and Tdry(i,j) is the sum of the time of consecutive steps over which h = 0 at each i, j grid. In this model, grid points for which Tdry becomes greater than some specified time TLand are considered to have accreted to the floodplain and are thus removed from computational domain (Figure 9). Each grid point is then shifted according to a smoothing method [Crosato, 2007] as follows:
where and are the distances of shift of each bank in the direction as a result of accretion of channel to the floodplain. The parameter TLand thus quantifies the total consecutive time necessary for a dry portion of the channel to be incorporated into the floodplain. Here TLand is a specified parameter for each numerical run.
 Also the return time of flooding is important factor that limits bank accretion at the time scale TLand. If the return time of flooding is short enough, vegetation can be flushed out before it has time to stabilize bare sediment. Hence, in this study, the step hydrograph is set keeping the ratio between the time scale TLand and the return time of flooding.
2.5 Natural Cutoff Model
 In the course of the evolution of channel planform, two cross sections separated by a finite centerline arc length may grow so close to each other that their banks touch (Figure 10). When this occurs in the model, it is assumed that a neck cutoff occurs, resulting in the formation of a new, straighter channel segment. Neck cutoffs are known to occur naturally and are an important feature of meander morphology, giving rise to cusps in the bank line and other notable features which are otherwise difficult to explain (e.g., Figure 1).
 In the model present here, channel cutoff is modeled as an instantaneous change in planform which occurs at any time step during which two migrating banks meet. Cutoff formation is implemented by the following procedures, as shown in Figure 10. First, the model domain is searched for any crossing or meeting point of riverbanks. If a point of intersection is found, the part of the bank that is not part of that intersection is realigned (typically the outer bank in two bends) into a single bank, leaving a cutoff in the form of an oxbow lake. In order to ensure that the new bank is smooth, cubic spline interpolation is used to determine the new bank shape so as to match the original outer banks. After this path is determined, the coordinate system is remeshed, a new channel centerline is computed, and the bed elevation is interpolated onto the new grid coordinates. After this process is complete, the model recomputes hydraulic conditions using the new geometry, and channel evolution is allowed to continue.
3 Numerical Experiments With the Meandering Model
 As an initial step in the testing and verification of the numerical model developed in this study, the approach was applied to a simple, hypothetical channel. Shimizu et al.  conducted meandering experiments using flume with various meandering shapes. The flow and bed evolution model, also used in this study, was used to investigate those experiments and it is confirmed that the model can be reproduced stable bed forms of various meandering shape (e.g., different wavelength and different width-depth ratio) channels with fixed wall [Shimizu et al., 1995, 1996]. Hence, one of those meandering shapes is used for one of the numerical tests in this study. Bed evolution and bank shift were predicted for two cases of constant discharge: one corresponding to a “large flood” chosen so that notable bank erosion occurs, i.e., “bankfull,” and the other for a lower discharge corresponding to a “low flow” at which the nondimensional shear stress is set to be smaller than the critical shear stress of initial river shape. Next, meander evolution is investigated for unsteady flow which has a step hydrograph including the two types of discharges that are used in the numerical studies of steady flow. Furthermore, features of meandering development are investigated focusing on the relationship between the parameter TLand and the return time of flooding Treturn.
3.1 Initial Channel Conditions
 Figure 11 shows the original channel and grid which is used for all of the calculations. The bed slope is 1/250 and the width is 0.2 m. (As noted above, the model is here implemented on a trial experimental scale.) A single sine-generated bend is set at the inlet of the flume. This bend corresponds to a down valley wavelength of 2.0 m and an amplitude of 0.5 m. Straight intervals are set upstream and downstream of the sine curve, with an upstream length of 1.25 m and a downstream length of 20 m. The bed material is assumed to be 0.9 mm uniform sand, which is within the range of applicability of the bed load transport relation of Ashida and Michiue . The initial bank shape of the channel is vertical, and the bank height is set to 0.03 m along the channel. As described in the model above, the bank is assumed to have two layers. The thicknesses of which are set as described in Figure 6: A lower layer composed of noncohesive material has a thickness of 0.02 m, and an upper layer composed of cohesive material has a thickness of Hc = 0.01 m. Furthermore, the characteristic size of slump blocks is set equal with the cohesive layer thickness, so that Dchunk = 0.01 m. The number of computational grid points is 195 in the streamwise direction and 11 in the cross-stream direction. The angle of repose of the sediment is set to 26.5°, so thattan θc = 0.5. Parameters characterizing slump blocks are specified as Rwc = 0.5 and α=0.00001 s2/cm2 based on the experimental work of Nishimori and Sekine . At any given time, the flow depth at the upstream end is computed from the specified flow discharge and the bed elevation at the two cross sections farthest upstream. The upstream sediment feed rate is then set to the equilibrium value associated with this flow. As a result, the sediment feed rate changes as the channel evolves.
3.2 Calculation Results for Steady Flow
 Calculations were performed for three cases, i.e., Cases 1–3 of Table 1. Two of these correspond to constant discharge, and one corresponds to a simple hydrograph corresponding to two cycled discharges. It is seen in Table 1 and Figure 12 that the flow conditions used here correspond to laboratory rather than field scale.
Table 1. Discharges and Froude Numbers for the Three Cases Investigated
Bank Height (m)
 The two cases of constant discharge are Case 1 and Case 2. In both cases, the flow is Froude subcritical. Figure 13 shows channel planform and depth contours at five times during Case 1, which corresponds to the higher discharge. Figure 14 shows the corresponding contours of constant bed elevation. The times shown are 120 s, 3600 s, 10,800 s, 18,000 s, and 25,200 s. The channel width expands from the initial value by bank erosion as shown in Figures 13a and 13b. After this initial expansion, the channel width stays nearly constant from upstream to downstream as it evolves, because slump block roughly protects both banks from bank erosion. A point bar is created at the inner bank of the bend constituting the initial perturbation, as shown in Figures 14a and 14b. This promotes bank attack, so that channel meandering commencing from the initial bend develops both bars and a meandering planform. As illustrated in Figures 13 and 14, the bends propagate downstream but maintain a rather modest amplitude. Note that cutoffs do not develop in the simulation.
 In Case 2, i.e., that of lower constant discharge, the time development of the channel is documented in Figures 15 (depth contours) and 16 (bed elevation contours). Again, the times shown are 120 s, 3600 s, 10,800 s, 18,000 s, and 25,200 s. Similarly to Case 1, a point bar grew along the inner bank of the perturbation bend. However, unlike Case 1, channel meandering did not develop, because in Case 2, the shear stress in most area of the channel was nearly the same value as critical shear stress. Instead, the initial bed perturbation was gradually obliterated and a pattern of alternate bars was created from upstream to downstream as documented in Figures 15 and 16.
 It was seen from Figures 13-16 that in neither case of constant discharge did a complex pattern similar to the Omolon River of Figure 1 evolve. In Case 1, a meandering planform developed, but never attained an amplitude sufficient to result in cutoffs. In Case 2, a meandering channel did not develop. The results of Case 1 and Case 2 serve to motivate the case of cyclically varying discharge considered in Case 3.
3.3 Calculation Results for Cyclically Varying Hydrograph
 While neither Case 1 nor Case 2 resulted in the development of high-amplitude meandering, they serve to illustrate the roles of different discharges in planform evolution. The morphology of a river channel is, however, ultimately the result of the supplied flow and sediment integrated over a relatively long period of time, during which the flow can be expected to vary from low flow to flood conditions, and then back to low flow. Thus, when considering long-term river evolution, the roles of various discharges and their temporal sequence must be considered. Strictly speaking, a morphodynamic model of planform evolution should account for a repeated full hydrograph. This, however, remains impractical, as such long-term simulations (over a hundred years or more) would require an unacceptably long calculation time. With this in mind, in Case 3, a full hydrograph is replaced with a simple, repeated two-step hydrograph, with a high discharge corresponding to relatively large, rare flood events that have direct impact on bed deformation and bank erosion and a lower discharge that plays little role in bank erosion or bed deformation, but allows point bar stabilization and land accretion. The large discharges are directly simulated by the computational approach, whereas the effect of the low discharge can be modeled parametrically in so far as it does not have direct impacts on bed and bank erosion. Thus, in the model, the duration of low flow can be cut dramatically, with the time over which land accretion occurs correspondingly adjusted to reflect the compression of the duration of low flow. In this way, the model still includes the processes of vegetation growth and fine sedimentation on bar tops, but does so in a manner that greatly reduces computational time.
 In Case 3 then, the two-step hydrograph of Figure 12 was used to simulate channel evolution. The time scale of land accretion TLand was set to 10 s. High flow was continued for 800 s and low flow for 400 s. Thus, the time scale Treturn between high flows was 400 s. The flow and bed deformation features occurring during high flows are essentially the same as those seen in constant discharge Case 1 as described in section 3.2 above. The land accretion along inner banks and channel narrowing occurring at low flows, however, modify conditions at high flow in such a way as to facilitate the eventual evolution of sinuosity that is high enough to cause meander bend cutoff.
 Evolution of the entire reach of 23.25 m of down valley length is documented in Figures 17 (planform, contours of constant depth) and 18 (planform, contours of constant bed elevation) and Animation S1 (planform, bed elevation contour, and velocity vector) in the supporting information. The times shown are 120 s (high flow), 3600 s (high flow), 10,800 s (low flow), 18,000 s (high flow), and 25,200 s (low flow). As can be seen from the hydrograph in the figures, the high and low flows are not consecutive. The figures illustrate the evolution of a rich and complex pattern of meandering, displaying both streamwise and temporal width variation and the effects of multiple cutoffs on meander planform. The late-stage patterns show remarkable similarities with the planform of the Omolon River shown in Figure 1.
 Figures 19 and 20 illustrate the evolution of planform and bed morphology over a time span of 3960 s. More specifically, snapshots are shown for the following times: 720 s (high flow), 1320 s (next low flow), 2280 s (next high flow), 2880 s (next low flow), and 3960 s (next high flow). Although the down valley length of the computational domain was 23.25 m (Figures 17 and 18), only the upstream portion with a length of 4.0 m is shown for clarity.
 Water depth and bed elevation contours are shown with velocity vector fields during the initial stages (first 3960 s) of channel evolution Figures 19 and 20. During the first episode of high discharge (stage a), the bed along outer bend is scoured, so that the depth in this region increases. On the other hand, the bed height along the inner bank increases due to sediment deposition, and the depth there becomes shallower (Figures 19a and 20a). During the lower discharge episode immediately following the first high discharge, the channel width becomes narrower due to shifting of the bank line of the inner bend as land accretes. The scoured bed region near the outer bend remains relatively unchanged (Figures 19b and 20b).
 In stage c, corresponding to the second high discharge, the river width expands by bank erosion at the outer bank so that the bank line of outer bend erodes and the point bar grows due to sediment deposition along the inner bend. As in the case of the first high discharge, scour along the outer bend reduces the bed height there. In stage d, corresponding to the second low discharge, the land accretion process once again shifts the inner bank line in the direction of the channel centerline.
 In stage e, corresponding to the third high discharge, bank erosion again shifts the outer bank line away from the earlier centerline. As a result, the bend corresponding to the initial perturbation migrates downstream, increasing the amplitude of both meander planform and width variation.
 Figures 21 and 22 document the evolution of water depth and bed elevation in Case 3 over a much later period in the evolution of the channel, i.e., 18,960 s (low flow), 19,920 s (next high flow), 20,520 s (next low flow), and 21,480 s (next high flow). More specifically, they document the channel pattern immediately before and after a cutoff. The reach shown in Figures 21 and 22 is only a part of the 23.25 m computational reach (Figures 17 and 18). Just before the cutoff, the outer bank line is shifted by bank erosion during high discharge, and the inner bank line is shifted by land accretion process during low discharge (Figures 21a–21c and 22a–22c). However, during the time periods c to d, the bank lines cross and the channel shape is changed drastically by the formation of a cutoff. During this process, the channel width maintains a nearly constant value from upstream to downstream throughout the reach.
3.4 Numerical Tests Focusing on TLand and Treturn
 The results of Case 3 indicate that discharge variation plays an important role in defining meander planform shape. As demonstrated in this section, the effect of discharge variation is primarily felt through the ratio TLand/Treturn. The amount of time required for vegetation to become established on newly deposited sediment is a function of vegetation type, hydrologic and climactic regime, the character of the deposited sediment, and other factors, all of which are folded into the single parameter TLand. If TLand is small compared to the return time Treturn for the next flood, vegetation can quickly act to narrow the channel. In doing so, it strengthens a single-thread configuration and forces erosion on the opposite bank during subsequent floods. If, on the other hand, TLand is approximately equal to Treturn, vegetation is relatively ineffective at stabilizing deposited sediment, narrowing the channel, and forcing erosion at the opposite bank.
 Here we explore these effects by varying TLand/Treturn. The base case we use to do this is Case 3, for which TLand/Treturn = 0.025. Table 2 outlines the conditions for this case, and also Cases 3-1, 3-2, 3-3, and 3-4. These four extra cases are identical to Case 3, except that TLand/Treturn takes the respective values 0.125, 0.25, 0.5, and 1. The results of the calculations are compared in Figure 23. The results shown there are for the same length of reach and at the same time. The smallest value of TLand/Treturn corresponds to a narrow, intensely sinuous channel which has already developed cutoffs (see Figure 22). As TLand/Treturn increases, the channel becomes wider, less sinuous, and less prone to cutoffs. In Case 3-4, for which TLand/Treturn = 1, the sinuosity is barely above unity, and the presence of central bars suggests incipient braiding.
Table 2. Relationship Between the Parameters TLand and the Return Time of Flood Treturn
 The shape of natural river meanders is produced by the interrelationship of several natural phenomena. This study suggests that discharge variation plays an important role in determining the shape and behavior of realistic meandering river reaches. During relatively high discharges, bank erosion causes river width to expand by shifting the outer bank line away from the channel centerline. This process is driven by bank erosion, which is in turn driven by bed erosion near the outer bank. During lower flows, the process of land accretion along the inner bank narrows the channel. These two processes evolve to be in overall balance, such that the channel maintains a roughly constant width. An explanation of the salient observed features of natural meandering rivers also requires the inclusion of a channel cutoff model, as this process drastically changes river planform shape. Calculations of channel evolution considering all these phenomena can be carried out for relatively long periods of time, provided that some parametric compression of the duration of low flow discharge is employed. The calculation results suggest that the details of river channel evolution depend on the magnitudes and durations of the high and low flow discharges. In particular, the characteristic time for the process of the land accretion is a key factor governing the shift of the inner bank line in response to vegetal encroachment.
 The effect of varying the ratio TLand/Treturn is studied in Figure 23. This figure documents a clear tendency for sinuosity to decrease, and for cutoff to become less frequent, as TLand/Treturn increases. Evidently, vegetation must stabilize bare sediment rather quickly in order for high-amplitude sinuosity and relatively frequent cutoffs to develop.
 Our model represents the first implementation of the framework of Parker et al.  for migration of meandering rivers that self-select width. In nearly all previous models, river width has been held constant, or at most only bank erosion has been included. Here we can study both widening and narrowing as the river erodes or deposits sediment on each bank. Application of our model specifically shows that for the same bank erosion law, a shorter time for vegetation to take hold leads to a more sinuous, narrower channel, whereas a time that is so long that it is of the order of the time between floods leads to a less sinuous, nearly braided channel.
 The calculations presented here have been specifically performed at laboratory scale, i.e., a reach length of 23.25 m, an initial channel width of 0.2 m, and a maximum time duration of 25,200 s in order to allow a first test of the model. The model is thus most directly applicable to self-formed meandering or sinuous channels at the experimental scale, such as the experiments of Dulal and Shimizu , where the banks gain strength due to cohesive slump blocks, and the experiments of Tal and Paola  and Braudrick et al. , where the banks gained strength from alfalfa sprouts. The extension of the model to field-scale meandering stream with a vegetated floodplain represents a future challenge. In order to do this, it will be necessary to define the accretion time scale in accordance with observed time scales of vegetation growth, sediment deposition, slump block formation and decay, and interaction between the two processes. Most importantly, it will be necessary to delineate a relation for slump block lifetime Tchunk that is applicable at field scale, rather than the experimental-scale relation of Nishimori and Sekine  used here. Extensive field work will be necessary to do this; a start has been made by Micheli and Kirchner  for slump blocks and Yasuda and Watanabe  for vegetal encroachment on point bars.
 These comments notwithstanding, the results shown here confirm that the prediction of meander evolution over relatively long time scales can be described using a relatively simple long-term discharge variation along with conventional formulations for in-channel flow and morphodynamics and appropriate submodels for bank erosion and accretion. They show a remarkable correspondence with the field results of Pizzuto : Smaller floods favor channel narrowing via inner bank deposition, and larger floods favor channel widening via outer bank erosion. Within the appropriate range, both factors act together to allow the maintenance of a coherent channel as meander amplitude increases.
 This paper describes a numerical model for understanding and predicting the evolution of rich and complex meandering planforms. The numerical model considers bank erosion moderated by slump block armoring, land accretion due to vegetal encroachment, and the formation of channel cutoffs.
 The evolution of river planform shape under conditions of both constant discharge and a simplified hydrograph are investigated by applying the model over time periods sufficiently long to describe asymptotic behavior. The case of high constant discharge yields a channel that develops well-defined downstream migrating meanders of moderate amplitude, which do not develop to a cutoff. The case of low constant discharge yields a channel that evolves to be nearly straight, but contains migrating alternate bars. The case of a simplified hydrograph using alternating high and low discharges from the previous two runs yields a complex, ever-evolving planform characterized by repeated bend extension and cutoff. This planform, although obtained from a numerical simulation at laboratory scale, shows remarkable similarity with that of field-scale meandering rivers.
 It was possible to verify through numerical runs that the relationship between the time scale of transition from channel to exposed land TLand in areas of net sediment deposition and the return time of flooding Treturn plays an important role in defining meander planform shape. This relationship can be expressed in terms of the ratio TLand/Treturn. The smaller the value of this parameter, the higher is the amplitude of meandering that developed, and the more frequent was the occurrence of cutoffs. A meandering configuration did not develop when the value is asymptotic to 1.
 The main achievement of this model is the ability to capture the coevolution of channel sinuosity and width. The results further suggest that discharge variation plays an important role in determining natural river planform shape. Bank erosion is favored by higher flows, whereas land accretion is favored by lower flows. The interaction of the two allows evolution of high-amplitude meander bends that cut off, which in turn leads to realistic, complex meander planform morphology. This result is in general agreement with the field observations of Pizzuto  on the meandering Powder River, which tends to widen due to bank erosion during relatively extreme floods and narrow due to bank accretion in response to relatively moderate flood flows. Extension of the numerical model from laboratory to field scale will be made possible through the determination of the relevant field-scale input parameters. This represents a challenge for future research.
 The symbols used for the flow model and the bed deformation model:
bed friction coefficient [-];
diameter of bed material [L];
acceleration due to gravity [L/T2];
water surface elevation [L];
water depth [L]
Manning's roughness coefficient [T/L1/3];
coefficient of the strength of the secondary flow [-];
total volume bed load sediment transport rate per unit width [L2/T];
volume bed load sediment transport rate in the ξ and η directions, respectively [L/T];
volume bed load sediment transport rate in the and directions, respectively [L2/T];
radius of curvature of the streamline [-];
submerged specific weight of a sediment grain in water (=1.65) [-];
depth-averaged flow velocity components in x and y directions, respectively [L/T];
depth-averaged flow velocity components in ξ and η directions, respectively [1/T];
depth-averaged flow velocity components in and directions, respectively [L/T];
flow velocities near the bed in s and n directions, respectively [L/T];
flow velocities near the bed in ξ and η directions, respectively [L/T];
shear velocity [L/T];
critical shear velocity [L/T];
resultant velocity of the depth averaged velocities [L/T];
resultant velocity near bed [L/T];
bed elevation [L];
velocity factor (=V/u*) [-];
correction coefficient to account for the bed slope [-];
Von Karman constant (=0.4) [-];
porosity of bed material [-];
dynamic coefficient of Coulomb friction (=0.45) [-];
static coefficient of Coulomb friction (=1.0) [-];
eddy viscosity coefficient [L2/T];
intersection angle between ξ and η directions [rad];
intersection angle between x and s directions [rad];
nondimensional shear stress [-];
critical nondimensional shear stress [-].
 The symbols used for the description of bank erosion model:
volume per unit area of slump block [L2];
bank width of left and right bank regions, respectively [L];
transverse speed of bank migration [L/T];
characteristic size for slump block [L];
erosion speed of slump block [L/T];
thickness of cohesive layer [L];
parameter of the armoring effect of slump block [-];
transverse sediment transport rate at bank region considered slump block effect [L2/T];
volume rate of production per unit streamwise distance of slump block [L2/T];
moisture content of slump block [-];
characteristic time for slump block to decay [T];
local elevations of the left and right banks, respectively [L];
local elevations of left and right banks, respectively [L];
parameter of cohesive sediment defined by water temperature [T2/L2];
axis value of left and right bank edges position, respectively [L];
porosity of bed material [-];
angle of incipient collapse of the bank [-].
 The symbols used for the description of land accretion model:
characteristic time for slump block to decay [T];
characteristic time for slump block to decay [T].
 The coordinate axes and the coordinate transformation coefficients:
axes in orthogonal coordinate system [L];
dimensionless generalized axes in moving boundary-fitted coordinate system [-];
dimension generalized axes in moving boundary-fitted coordinate system [L];
grid width in the and directions, respectively [L];
axes in the streamwise direction and cross-sectional direction [L];
Jacobian determinant of the coordinate transformation [1/L2];