Numerical simulation of river meandering with self-evolving banks

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: EARTH SURFACE, VOL. 118, , doi: /jgrf.20150, 2013 Numerical simulation of river meandering with self-evolving banks Kazutake Asahi, 1 Yasuyuki Shimizu, 2 Jonathan Nelson, 3 and Gary Parker 4 Received 2 February 2013; revised 17 September 2013; accepted 17 September 2013; published 18 October [1] 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. Citation: Asahi, K., Y. Shimizu, J. Nelson, and G. Parker (2013), Numerical simulation of river meandering with selfevolving banks, J. Geophys. Res. Earth Surf., 118, , doi: /jgrf Introduction [2] 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. Additional supporting information may be found in the online version of this article. 1 Planning Department, River Center of Hokkaido, Sapporo, Japan. 2 Laboratory of Hydraulic Research, Graduate School of Engineering, Hokkaido University, Sapporo, Japan. 3 Geomorphology and Sediment Transport Laboratory, U.S. Geological Survey, Golden, Colorado, USA. 4 Department of Civil and Environmental Engineering and Department of Geology, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA. Corresponding author: K. Asahi, Planning Department, River Center of Hokkaido, Ito 110 Bldg., N-7 W-4, Kita-ku, Sapporo, Hokkaido , Japan. (kazutake.asahi@gmail.com) American Geophysical Union. All Rights Reserved /13/ /jgrf 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. [3] 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 [1974], 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 2208

2 ASAHI ET AL.: RIVER MEANDERING SIMULATION Figure 1. Aerial photo of the Omolon River in northeastern Siberia (Google Earth). This image shows both the complex shape of the river and the history of planform evolution. The flow is from left to right. 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 [2011] 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. [4] Several numerical models of meandering have used the formulation for bank migration of Hasegawa [1978, 1984] and Ikeda et al. [1981] 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. [5] 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. Figure 2. Definition diagram for the plane coordinate system used in the model: x and y are the axes of an orthogonal Cartesian coordinate system; ξ and η are the axes of a nondimensional generalized orthogonal coordinate system, and the tildes denote dimensioned versions of ξ and η. 2209

3 Figure 3. Diagram indicating the relationships between depth-averaged velocity and near-bed velocity in Cartesian (x,y) and streamwise-normal (s,n) coordinates. [6] 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. [2008] 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 [1999] 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. [2005] and Gurnell et al. [2006]. Perruca et al. [2007] 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. [7] 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. [8] The goal of this approach is the modeling of the longterm 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 [9] 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: u ξ t t h J þ ξ ξ t þ u ξ h J þ ξ t þ u ξ u ξ ξ þ ð η t þ u η Þ u ξ η þα 1 ξ t þ u ξ u ξ þ α 2 ξ t þ u ξ þ ðη η t þ u η Þ h ¼ 0 (1) J u η þ η t þ u η ð Þu ξ þα 3 ðη t þ u η Þu η D ξ H H ¼ g ξ 2 x þ ξ2 y ξ þ ξ xη x þ ξ y η y η C rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du ξ 2 η hj y u ξ ξ y u η þ ð ηx u ξ þ ξ x u η Þ 2 Figure 4. Definition diagram for the cross-sectional coordinate system model used in (a) dividing the section into central, left bank, and right bank regions and (b) defining parameters for the bank erosion model. (2) 2210

4 Figure 5. Illustration of cohesive and noncohesive layers in the Vermillion River, a natural meandering river in Minnesota, USA. u η u þ ξ t t þ u ξ η ξ þ ð η t þ u η Þ u η η þ α 4 ξ t þ u ξ u ξ þα 5 ξ t þ u ξ u η þ ðη t þ u η Þu ξ þ α6 ðη t þ u η Þu η D η H H ¼ g η x ξ x þ η y ξ y ξ þ η2 x þ η2 y η C rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du η 2 η hj y u ξ ξ y u η þ ð ηx u ξ þ ξ x u η Þ 2 (3) D ξ, D η are terms describing the diffusion of momentum in the ξ and η directions, respectively. The parameters α 1 α 6 are given as follows: 2 x α 1 ¼ ξ x ξ 2 þ ξ 2 y y ξ 2 2 x α 2 ¼ ξ x ξ η þ ξ 2 y y ξ η 2 x α 4 ¼ η x ξ 2 þ η 2 y y ξ 2 2 x α 5 ¼ η x ξ η þ η 2 y y ξ η 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); ξ t, ξ x, ξ y,andη t, η x, η y 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 = ξ x η y ξ y η x ; u ξ, u η 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/s 2 ); and C d is a bed friction coefficient given by following equation: C d = gn 2 m /h 1/3.Heren m is Manning s roughness coefficient given by n m ¼ d 1=6 p = 6:8 ffiffiffi g [Kishi and Kuroki, 1973], where d is the diameter of the bed material. In equations (2) and (3), 2 x α 3 ¼ ξ x η 2 þ ξ 2 y y η 2 2 x α 6 ¼ η x η 2 þ η 2 y y η 2 (4) [10] Here we consider the relation between the dimensioned and dimensionless coordinate system. Let e ξ andeηdenote dimensioned versions of ξ and η parameters. These have the same direction axes as ξ and η (Figure 2b). The parameters ξ r and η r are defined using local grid width as follows (Figure 2b): ξ r ¼ ξ ξ e ¼ Δξ Δξ e η r ¼ η eη ¼ Δη Δ eη (5) Figure 6. Riverbank is treated as a two-layer system with a cohesive layer above a noncohesive layer. Protection of the noncohesive layer from erosion due to slump block armoring is illustrated. 2211

5 ASAHI ET AL.: RIVER MEANDERING SIMULATION Figure 7. Photographs of the Nakashibetsu River, Hokkaido, Japan, illustrating the evolution of the inner bank [Yasuda and Watanabe, 2008]. [11] Using these factors, the physical parameters in equations (1) (3) can be transformed from the nondimensional general coordinates system (ξ, η) into the dimensioned coordie ηe) as follows: nate system (ξ; 1 u ¼ uξ ξr ξ 1 u ¼ uη ηr η Dξ ¼ (7) where νt is a coefficient of eddy viscosity given as follows: κ νt ¼ u h 6 u¼ (6) where u ξ ; u η are depth-averaged flow velocity components in the ξe and ηe directions, respectively. Also, the parameters Dξ and Dη can be described as follows: u ξ u ξ vt ξ 2r vt η2r þ ξ η ξ η η η 2 u 2 u η vt ξ r vt ηr þ D ¼ ξ η ξ η [12] 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, nm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðu2 þ v2 Þ h1=6 1 ηx uξ þ ξ x uη J (10) 2.2. Bed Evolution Model [13] 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: (8) (9) v¼ 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 ξ zb 1 qb qb η þ þ ¼0 t J 1 λ ξ J η J and u ¼ 1 η uξ ξ y uη J y (11) where zb is riverbed elevation, λ is porosity of the bed material (=0.4 here), and qξb ; qηb are bed load sediment volume 2212

6 the onset of sediment motion, and u * c is the corresponding critical bed shear velocity. The Shields number is given as τ ¼ u2 s g gd (15) [14] Both the critical Shields number and the critical shear velocity are computed using the formula of Iwagaki [1956]. [15] The near-bed velocity in the direction of the vertically averaged flow is given as follows [Engelund, 1974]: u s b ¼ βv (16) β ¼ 31 ð σþð3 σþ (17) Figure 8. Diagram illustrating the submodel describing land accretion associated with sediment deposition on point bars and vegetal encroachment. 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: q ξ b ¼ ξ rq b q η b ¼ η rq b " e u ξ b γ z b V b ξ e þ cosθ z # b eη eη u b γ z b V b eη þ cosθ z b ξ e (12) (13) whereu b ξ; u b η are near-bed velocities in the ξ and η directions; V b is the p resultant velocity of the near-bed velocities, i.e., V b ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u b2 þ v 2 b, γ is a correction coefficient to account for the bed slope (described below); and θ is the intersection angle between the ξ and η axes. In addition, q b is the magnitude of the volume bed load sediment transport rate per unit width. Here q b is given by the relation of Ashida and Michiue [1972] which was developed for, and is applicable to, the transport of sand and fine gravel: q q ffiffiffiffiffiffiffiffiffiffiffi b ¼ 17τ 3=2 s g gd 3 1 τ *c 1 u *c τ u (14) where s g 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 where u s b is the near-bed velocity in streamwise direction (Figure 3); V is the resultant p velocity of the depth averaged velocities, i.e., V ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 þ v 2 ; 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, u n b is estimated as follows (Figure 3): u n b ¼ us b N h (18) r s where N * is a coefficient of the strength of the secondary flow taking the constant value N * = 7.0 [Engelund, 1974] and r s is the radius of curvature of the streamline in question. Here r s is given as follows: 1 ¼ 1 v r s V 3 u2 ξ x ξ þ η v v x þ uv ξ η y ξ þ η v y η # u uv ξ x ξ þ η u x v 2 u ξ η y ξ þ η u y η [16] In addition, u b ξ; u b η can be obtained as follows: u b ξ¼ 1 ξ x cosθ s þ ξ y sinθ s u s ξ b þ ξ x sinθ s þ ξ y cosθ s u n b r u b η ¼ 1 h η η x cosθ s þ η y sinθ s u s b þ η x sinθ s þ η y cosθ s r u n b i (19) (20) (21) where cos θ s = u/v and sin θ s = v/v, respectively. [17] According to Hasegawa [1984], the parameter γ in (12) and (13) is given as rffiffiffiffiffiffiffiffiffiffiffiffiffi τ γ ¼ *c (22) μ s μ k τ where μ s is the static coefficient of Coulomb friction (=1.0) and μ k 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. 2213

7 Figure 9. Illustration of the computational model for the redistribution of calculation area and grid points subsequent to land accretion Bank Erosion Model [18] 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 θ Bc to be the angle of incipient collapse of the bank, z BL and z BR can be written as follows: z BL ¼ z 0 BL þ eη eη0 L tanθbc (23) z BR ¼ z 0 BR eη eη0 R tanθbc (24) where z BL and z BR are the local elevations of the left and right banks, respectively, z 0 BL and z0 BR are the corresponding bed elevations at the base of the corresponding bank, and eη 0 L, eη0 R, eη 00 L,eη00 R are the values of the transverse coordinate at the bottom and top of both banks (Figure 4). [19] 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: ( eη 0 L ¼ 1 z 0 BL þ 1 e!) q ξ b t tanθ Bc t 1 λ e 1 q η bs (25) ξ B L leftbank leftbank 8 0 eη 0 R ¼ 1 z 0 BR þ 1 e 19 < q ξ b t tanθ Bc t 1 λ e þ 1 q η bs A : ξ B R rightbank ; (26) rightbank whereb L ¼ eη 00 L eη0 L andbr ¼ eη 0 R eη00 R denote the widths of left and right bank regions, respectively. The parameters Figure 10. Illustration of how the model treats cutoffs. 2214

8 Figure 11. Initial bed and planform used in the numerical simulations. The total down valley length of the reach modeled is m. Δeη 0 L and Δeη0 R are calculated along each riverbank at each time step. The calculation grid is then shifted using those values. [20] 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. [2011]. 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 eη direction is found by considering the armoring effect of the slump blocks [Parker et al., 2011]. Thus, whereq bseη is the transverse transport rate of noncohesive material in the presence of slump block armoring and q η bs is the corresponding value in the absence of armoring, then q η bs ¼ Kqη b (27) where 0 K 1 is an armoring coefficient given as 8 1 A chunk ; 0 < A chunk >< < 1 D chunk B D chunk B K ¼ A >: chunk 0; > 1 D chunk B (28) where D chunk is characteristic size of slump block, and B * is the length B L or B R (Figure 4). In addition, A chunk is the volume per unit area of slump block material, here estimated as follows [Parker et al., 2011]: da chunk dt ¼ q chunk A chunk T chunk (29) where q chunk is the volume rate of production per unit streamwise distance of slump block material and T chunk is the characteristic lifetime of a chunk before it breaks down. Here q chunk is given as q chunk ¼ c H c (30) where c denotes the transverse speed of bank migration and H c denotes the thickness of the cohesive layer (Figure 6). The characteristic lifetime of a slump block T chunk is estimated herein as T chunk ¼ D chunk 2 E s (31) where E s is an entrainment rate of cohesive material (loss of slump block volume per unit surface area). Nishimori and Sekine [2009] proposed the following equation using Table 1. Discharges and Froude Numbers for the Three Cases Investigated Case Stage Q (m 3 /s) Duration (s) W 0 (m) Slope Bank Height (m) H c (m) d (mm) D 0 (m) V 0 (m/s) T dry (s) Fr τ * Case Constant Case Constant Case 3 high low

9 Figure 12. Case 3. Patterns of discharge and Froude number variation associated with the cycled hydrograph of experimental results for the E s (cm/s) in terms of shear stress, moisture content, and water temperature: E s ¼ α u 3 R wc 2:5 (32) where R wc is the moisture content of the slump block and α (s 2 /cm 2 ) is a dimensioned coefficient varying with water temperature T ( C) Land Accretion Model [21] Figure 7 shows the time evolution of the inner bank of the Nakashibetsu River in Hokkaido, Japan, from November 2003 to August 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 As the bar became higher, it spent less time submerged, resulting in the vegetal encroachment seen in the photo from August 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 Figure 13. Calculation results for Case 1 illustrating channel planform and contours of constant depth at five times: 120 s, 3600 s, 10,800 s, 18,000 s, and 25,200 s. The entire m valley length of the modeled reach of Figure 11 is shown. 2216

10 Figure 14. Calculation results for Case 1 illustrating channel planform and contours of constant bed elevation at five times: 120 s, 3600 s, 10,800 s, 18,000 s, and 25,200 s. The entire m valley length of the modeled reach of Figure 11 is shown. 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. [22] 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: 8 hi; ð jþ 0 T dry ði; jþ ¼ T dry ði; jþþ Δt >< >: hi; ð jþ > 0 T dry ði; jþ ¼ 0 (33) 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 T dry (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 T dry becomes greater than some specified time T Land 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: Δeη ði; nyþ leftbank ¼ Δeη ði; 1Þ rightbank ¼ Δeη ð i 1; nyþþ Δeη ð i; nyþþ Δeη ð i þ 1; ny Þ 3 Δeη ði 1; 1Þþ Δeη ði; 1Þþ Δeη ð i þ 1; 1 Þ 3 (34) (35) where Δeη leftbank and Δeη rightbank are the distances of shift of each bank in the eη direction as a result of accretion of channel to the floodplain. The parameter T Land thus quantifies the total consecutive time necessary for a dry portion of the channel to be incorporated into the floodplain. Here T Land is a specified parameter for each numerical run. [23] Also the return time of flooding is important factor that limits bank accretion at the time scale T Land. 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 T Land and the return time of flooding Natural Cutoff Model [24] 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 2217

11 Figure 15. Calculation results for Case 2 illustrating channel planform and contours of constant depth at five times: 120 s, 3600 s, 10,800 s, 18,000 s, and 25,200 s. The entire m valley length of the modeled reach of Figure 11 is shown. 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). [25] 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 [26] 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. [1995] 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 T Land and the return time of flooding T return Initial Channel Conditions [27] 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- 2218

12 Figure 16. Calculation results for Case 2 illustrating channel planform and contours of constant bed elevation at five times: 120 s, 3600 s, 10,800 s, 18,000 s, and 25,200 s. The entire m valley length of the modeled reach of Figure 11 is shown. 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 [1972]. 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 H c = 0.01 m. Furthermore, the characteristic size of slump blocks is set equal with the cohesive layer thickness, so that D chunk = 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 θ c = 0.5. Parameters characterizing slump blocks are specified as R wc =0.5andα= s 2 /cm 2 basedonthe experimental work of Nishimori and Sekine [2009]. 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 Calculation Results for Steady Flow [28] 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. [29] 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. [30] 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 2219

13 Figure 17. Calculation results for Case 3 showing channel migration, contours of constant depth, and velocity vectors. 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 illustrated in the hydrograph plot. The entire m down valley length of Figure 11 is shown. 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. [31] It was seen from Figures 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 Calculation Results for Cyclically Varying Hydrograph [32] 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 longterm 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 2220

14 Figure 18. Calculation results for Case 3 showing channel migration, contours of constant bed elevation, and velocity vectors. 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 illustrated in the hydrograph plot. The entire m down valley length of Figure 11 is shown. 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. [33] In Case 3 then, the two-step hydrograph of Figure 12 was used to simulate channel evolution. The time scale of land accretion T Land was set to 10 s. High flow was continued for 800 s and low flow for 400 s. Thus, the time scale T return 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. [34] Evolution of the entire reach of 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. [35] 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 m (Figures 17 and 18), only the upstream portion with a length of 4.0 m is shown for clarity. [36] Water depth and bed elevation contours are shown with velocity vector fields during the initial stages (first 2221

15 Figure 19. Calculation results for Case 3 showing channel migration, contours of constant depth, and velocity vectors. The times shown are 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). The figure shows only a part of calculation domain (0 4m) of Figure 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). [37] 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. [38] Instagee,correspondingtothethirdhighdischarge, 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. [39] 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 2222

16 Figure 20. Calculation results for Case 3 showing channel migration, contours of constant bed elevation, and velocity vectors. The times shown are 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). The figure shows only a part of the calculation domain (0 4 m) of Figure ,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 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 Numerical Tests Focusing on T Land and T return [40] 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 T Land /T return. 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 T Land.IfT Land is small compared to the return time T return 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, T Land is approximately equal to T return, vegetation is relatively ineffective at stabilizing deposited sediment, narrowing the channel, and forcing erosion at the opposite bank. [41] Here we explore these effects by varying T Land /T return. The base case we use to do this is Case 3, for which 2223

17 Figure 21. Calculation results for Case 3 showing channel migration, contours of constant depth, and velocity vectors. The times shown are 18,960 s (low flow), 19,920 s (next high flow), 20,520 s (next low flow), and 21,480 s (next high flow). The figure shows only a part of calculation domain (10 22 m from upstream) of Figure 17. T Land /T return = 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 T Land /T return 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 T Land /T return corresponds to a narrow, intensely sinuous channel which has already developed cutoffs (see Figure 22). As T Land /T return increases, the channel becomes wider, less sinuous, and less prone to cutoffs. In Case 3-4, for which T Land /T return =1, the sinuosity is barely above unity, and the presence of central bars suggests incipient braiding. 4. Discussion [42] 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 2224

18 Figure 22. Calculation results for Case 3 showing channel migration, contours of constant bed elevation, and velocity vectors. The times shown are 18,960 s (low flow), 19,920 s (next high flow), 20,520 s (next low flow), and 21,480 s (next high flow). The figure shows only a part of calculation domain (10 22 m from upstream) of Figure 18. 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. [43] The effect of varying the ratio T Land /T return is studied in Figure 23. This figure documents a clear tendency for sinuosity to decrease, and for cutoff to become less frequent, as T Land /T return increases. Evidently, vegetation must stabilize bare sediment rather quickly in order for high-amplitude sinuosity and relatively frequent cutoffs to develop. 2225

19 Table 2. Relationship Between the Parameters T Land and the Return Time of Flood T return Case Stage Q (m 3 /s) Duration (s) T return (s) T Land (s) T Land /T return Case 3 high low Case 3-1 high low Case 3-2 high low Case 3-3 high low Case 3-4 high low [44] Our model represents the first implementation of the framework of Parker et al. [2011] 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. [45] The calculations presented here have been specifically performed at laboratory scale, i.e., a reach length of 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 [2008], where the banks gain strength due to cohesive slump blocks, and the experiments of Tal and Paola [2007] and Braudrick et al. [2009], where the banks gained strength from alfalfa sprouts. The extension of the model to field-scale meandering stream Figure 23. Comparison of channel migration and contour of bed elevation under different conditions depending on the relationship between the parameters T Land and T return. All results are for the same run time of 25,200 s. Run conditions are shown in Table 2. The entire m down valley length of Figure 19 is shown. 2226