2.1 Morphodynamic Model Description
 HSTAR solves the two-dimensional, depth-averaged shallow water equations written in conservative form. Model equations are solved on a structured grid (resolution Δx, Δy) within which each grid cell is defined as either active river bed or floodplain (including vegetated islands). The conservation of mass and momentum equations solved by the model are expressed as:
(1) (2a) (2b)
where h is flow depth; t is time; qx and qy are unit discharge in the x and y directions; g is acceleration due to gravity; ρ is fluid density; z is bed elevation; τxx, τyy, τxy, and τyx are turbulent stresses; τbx and τby are bed shear stresses in the x and y directions; and FSx and FSy are stresses resulting from the effects of secondary flow. Turbulent stresses are treated using the Boussinesq approximation combined with a zero-order eddy viscosity model [Begnudelli et al., 2010]:
(3a) (3b) (3c) (4)
where u=qx/h and v=qy/h are the depth-averaged velocities in the x and y directions, υt is the turbulent viscosity, and U* is the shear velocity, which is determined from the local bed shear stress. Bed shear stresses are modeled using a quadratic friction law:
where C is the Chezy friction coefficient, which can be treated as a constant or determined using the Colebrook-White equation:
where ks is a roughness length scale, which includes the effects of both grain and form roughness. In the current application, floodplain cells are assigned a constant Chezy roughness (Cf = 10 m0.5s−1), which is representative of forested surfaces [Straatsma and Baptist, 2008]; hence, variations in vegetation structure (and roughness) with surface age are neglected.
 The stress terms in the momentum equations (2a) and (2b) resulting from secondary currents are determined as a function of the intensity of spiral motion of the flow from the following expressions:
(7a) (7b) (8a) (8b) (8c) (9) (10)
where |q| is the unit discharge magnitude, κ is the von Karman constant, and Φ is the spiral motion intensity of the flow, which is related to the radius of curvature (see below). This approach to modeling the stresses associated with secondary flow is adopted within the commercial code Delft3D [Lesser et al., 2004; Kleinhans et al., 2008; Deltares, 2010]. It is less sophisticated than a full treatment of the dispersive stresses, which is sometimes implemented when simulating flow in high curvature bends in laboratory-scale channels [e.g., Duan, 2004; Begnudelli et al., 2010]. However, a full treatment of the dispersive stresses is not necessary where curvature is low [Duan, 2004], such as in large anabranching rivers where secondary circulation may be weak or absent [Parsons et al., 2007; Sandbach et al., 2012].
 The spiral motion intensity of the flow is derived by solving an advection-diffusion transport equation of the form [de Vriend, 1981; Wu and Wang, 2004]:
where R is the local streamline radius of curvature (determined from the depth-averaged velocity field), DH is the horizontal diffusivity, and L is an adaption length scale, which is given by:
 The final term on the left-hand side of (11) accounts for the adjustment of Φ toward the local equilibrium value |q|/R.
 Two sediment size fractions are represented by the model: one sand fraction and one silt fraction. The proportion of sediment in each size class is stored in a series of vertical layers within each grid cell. Vertical layer thickness is set at 1 m for the simulations reported herein, as in previous morphodynamic simulations of the river Rhine [e.g., Sloff and Mosselman, 2012]. The initial sand fraction is set at 100% for channel cells and 30% for floodplain cells (i.e., floodplain cells are initially composed of 70% silt). Over the course of simulations, the sand:silt ratio for individual cell layers evolves due to erosion and deposition processes. For simplicity, a constant bed porosity is assumed, independent of the silt/sand ratio in each layer. Sediment transport calculations differ between active bed cells and vegetated floodplain cells. For active river bed cells, total sand transport capacity (qT), which includes sand moved as bed load and suspended load, is determined using a form of the relation of Engelund and Hansen :
where |U| is the depth-averaged velocity magnitude, ψ is the sediment relative density, D is the representative (median) sand diameter, χ is a parameter that accounts for the effect of the local bed slope, and which is defined below (see equation (21)), and C0 is a roughness coefficient determined from equation (6) with ks = 3D90 = 6D (see section 4 for further discussion of this choice of transport formula). The Engelund-Hansen transport law has been used widely in morphodynamic modeling of fluvial, estuarine, and coastal settings [Marciano et al., 2005; van der Wegen et al., 2008; Kleinhans et al., 2008]. The total sand transport capacity in each grid cell is separated into two components to reflect the local availability of sand-sized bed material. These components are the actual sand transport rate (qS) and an excess transport capacity (qE):
where p is the proportion of sand in the bed and (1 − p) is the proportion of silt.
 The sand transport direction deviates from the mean flow direction (defined by qx, qy) due to the effects of secondary circulation [Stuiksma et al., 1985; Kleinhans et al., 2008]:
(15) (16) (17)
where R* is the effective radius of curvature of the flow, which differs from the local streamline curvature (R) and is determined from the spiral motion intensity (Φ). The parameter ϵ varies between 0 and 1, depending upon the mode of sand transport (i.e., bed load versus suspension). To determine the value of ϵ, the sand load is separated into bed load and suspended load fractions (ξ and 1 − ξ, respectively) using the approach of van Rijn . The bed load fraction moves in the direction of the near bed flow (ϵ = 1). The net direction of transport for the suspended sand fraction (1 − ξ) lies between the direction of the mean flow (ϵ = 0) and that of the near-bed flow (ϵ = 1) and is determined by integrating functions for the primary flow, secondary flow, and sand concentration profile [van Rijn, 1984; Kalkwijk and Booij, 1986]. Sand transport rates in the x and y directions (qSx and qSy, respectively) are then determined from:
where q*Sx and q*Sy are additional sand fluxes resulting from the effect of gravity acting on sand in motion on a sloping bed [Ikeda, 1982; Deltares, 2010] given by:
where ∂z/∂n is the bed slope normal to the sediment transport direction and the parameter λ is defined using the expression of Talmon et al. :
where τ* is the dimensionless shear stress and k is a constant that takes a value in the range 1–2 [Talmon et al., 1995]. Sediment transport rates are also adjusted to account for the effect of the local bed slope parallel to the direction of sediment transport (∂z/∂s), via the parameter χ in equation (13):
 As in a number of morphodynamic models, bed slope effects are applied only to the bed load fraction of the sand load, because it is this component that is in contact with the bed [Olesen and Tjerry, 2002; Lesser et al., 2004].
 Transport of silt is modeled using a two-dimensional advection-diffusion equation:
where φ is the depth-averaged sediment concentration and DR, ER, and BR are source terms representing rates of deposition, erosion from the bed, and sediment supply from banks, respectively. The deposition rate is modeled using the Krone equation:
where wS is the particle-settling velocity, τb is the bed shear stress, and τC is the critical shear stress (above which no deposition occurs).
 The rate of silt erosion from the bed in river cells is determined as:
where qEx and qEy are the components of the excess transport capacity in the x and y directions (determined from equation (14b)). This approach is based on the assumption that the rate of erosion in active bed cells is set by the amount of material that must be entrained to satisfy the total sand transport capacity (defined by equation (13)) and that the fraction of silt stored in the bed is released into suspension when this erosion occurs. For the simulations reported herein, the silt content of river bed cells is very low except in slack water zones located in the lee of the bar head. The bank sediment supply term (BR) in equation (22) is determined from the silt-sized sediment fluxes delivered by lateral bank erosion, summed over all floodplain cells bordering the channel cell under consideration:
where (1 − p) is the silt fraction in the floodplain cells, and the bank erosion fluxes are calculated as:
where E is a dimensionless bank erodibility constant, qTx and qTy are the total sand transport capacities in the x and y directions, and Sx and Sy are the bank slopes.
 Net changes in bed elevation for active river bed cells are determined from the Exner mass balance relation:
where δ is the bed porosity. In this expression, M is a morphological scaling factor that is used to accelerate the rate at which the bed evolves, effectively decoupling the hydrodynamic and morphodynamic time steps used in the model. This is a common modeling strategy [e.g., Lesser et al., 2004; Crosato et al., 2012], justified by the fact that the bed morphology evolves much more slowly than the hydrodynamic variables; hence, vertical changes in bed elevation during individual time steps remain very small. For most simulations reported herein, M is set at a value of 200. In effect, this means that a time step of ~3 s in the hydrodynamic model is equivalent to ~10 min of real time in the context of sediment transport and channel change. This approach is discussed below in the context of model sensitivity to M and choice of sediment transport relation. The third and fourth terms inside the square brackets in (27) represent the sand fluxes delivered by lateral bank erosion, summed over all floodplain cells bordering the channel cell under consideration. Equivalent bank-derived silt fluxes are not included in (27) because this material is assumed to enter suspension (via the BR term in equation (22)). In addition to the changes in bed elevation calculated using equation (27), the sand fraction (p) in each grid cell is updated based on the net changes in the depth of sand (the first four terms inside the square brackets of (27)) and net changes in the depth of silt (fifth and sixth terms). Sand transport rate and associated mass balance calculations are implemented here using a capacity-based approach, which assumes that the transport rate is in local equilibrium with the flow. This approach remains valid while the sand transport adaption length is less than several multiples of the grid resolution [Begnudelli et al., 2010]. This criterion was evaluated using the adaption length definition of Begnudelli et al.  and was satisfied for all simulations reported herein.
 Floodplain cells are treated differently to river bed cells in that they cannot be eroded vertically, unless they experience velocities in excess of a critical value (Vcr), when they are reactivated by vertical scour (as opposed to lateral bank erosion). Consequently, floodplain reworking occurs predominantly by lateral erosion, parameterized by equations (26a) and (26b) above. To prevent diffusion of topography and maintain distinct channel bank lines, bank erosion does not lead to a reduction in bank height when floodplain cells are eroded. Instead, the volume of material removed from the floodplain cell by bank erosion is recorded, and the floodplain cell is converted to a channel cell at the level of the channel bed in the bank adjacent cell once sufficient material has been removed to lower the floodplain to that level. Despite the simplicity of this bank erosion scheme, it has proven capable of simulating the development of high sinuosity meanders [Nicholas, 2013a]. Silt deposition is an important component of floodplain construction. Active channel cells are converted to floodplain cells when the maximum depth of inundation experienced over a specified time period (Tveg) does not exceed a given threshold depth (hcr). Low Tveg and high hcr values promote rapid vegetation colonization. This representation of floodplain development and bank erosion aims to capture the first-order controls on channel evolution and is simple in order to avoid over-parameterization of processes.
 Model equations are solved by explicit time integration using a finite volume scheme in which all variables are stored at the cell centers. The solution of fluid mass and momentum equations utilizes a higher-order Godunov scheme. Such schemes are commonly used to solve the shallow water equations in a range of applications including the simulation of within-channel flows, floodplain inundation, and dam-break floods [Fraccarollo and Toro, 1995; Mingham and Causon, 1998; Liang et al., 2008]. Mass and momentum fluxes are computed using the Hartex-Lax-Van Leer approximate Riemann solver [Harten et al., 1983]. Second-order accuracy in time and space is achieved using a predictor-corrector scheme and the montone upwind scheme for conservation laws approach to variable reconstruction [van Leer, 1979]. Spurious oscillations in the solution domain are prevented using the double minmod limiter, which has been recommended because of its neutral dissipation properties [Sanders and Bradford, 2006]. The model hydrodynamic time step (Δt) is defined to satisfy the Courant-Friedrichs-Lewy stability criterion. Validation of the hydrodynamic model in a large sand-bed river (the Rio Paraná, Argentina) is described elsewhere [Nicholas et al., 2012]. This study focuses on the simulation of river morphodynamics. The model outlined above is parallelized using a combination of openMP and MPI, and all simulations reported herein took approximately 2 weeks to complete using 12 cores on a SGI Altix ICE 8200 system with 2.8 GHz Intel Westmere dual hexcore nodes. The main flow and/or sediment transport equations used in HSTAR are similar to those used in a number of other two-dimensional morphodynamic models [e.g., Enggrob and Tjerry, 1999; Lesser et al., 2004; Jang and Shimizu, 2005; Kleinhans et al., 2008; Crosato and Saleh, 2010; Wang et al., 2010]. The key differences between HSTAR and these alternative models are in the simple two-fraction sediment transport approach adopted here, the treatments of bank erosion and bar conversion to floodplain by vegetation, and the numerics of the hydrodynamic finite volume scheme.
2.2 Numerical Simulations
 Numerical simulations were conducted to study the evolution of large anabranching sand-bed rivers over periods of several hundred years. All simulations used the same initial conditions: a straight channel, 2.4 km (60 grid cells) wide and 12 m deep, having a constant slope (S) with small (±0.1 m) white noise elevation perturbations. Trial simulations confirmed that qualitatively, styles of river evolution are independent of initial channel width, although narrower initial widths promoted greater initial channel widening, as might be expected. Boundary conditions consisted of a series of inflow hydrographs and a perturbation to the inlet bathymetry (both described below). Initial channel depth was set so that peak flows remain in-bank. However, vegetated islands that form during simulations as the channel widens are inundated during peak flows. All simulations were carried out using a model domain 50 km long by 16 km wide. In the default model setup, this was composed of 625 × 400 cells, each measuring 80 m long by 40 m wide. This cell aspect ratio (Δx:Δy) provides the optimal balance between model efficiency and resolution, because it enables the use of larger model time steps (controlled predominantly by Δx), while maintaining the capability to resolve finer bars and channels in the cross-stream direction. Additional model runs were conducted for a subset of simulations using different cell sizes (including 60 m by 30 m and 60 m by 60 m) to assess model sensitivity to grid resolution (see section 3.2).
 Inflow conditions consisted of a series of sinewave hydrographs with a minimum discharge of 10,000 m3 s−1 and peak discharges that varied from 15,000 to 30,000 m3 s−1 between individual events. The magnitude of this discharge is not unreasonable given that the world's largest rivers are usually associated with a mean annual discharge greater than 10,000 m3 s−1 [Latrubesse, 2008] although the largest flows can peak at around 100,000 m3 s−1. Moreover, it is recognized that hydrograph shapes for natural rivers are more complex than the sinewave hydrograph used here and may vary substantially between rivers. However, the aim of this study is to investigate generic characteristics of river behavior for relatively simple boundary conditions, rather than to simulate specific hydrologic regimes. A sequence of floods with varying peak discharge was generated and applied in all simulations. In model simulations that used the default setup, each hydrograph lasted 3.65 days, which, for the default value of 200 used for M, is equivalent to 2 years of morphodynamic time (note that all times reported hereafter represent morphodynamic times scaled in this way). Trial simulations were conducted to establish that the short hydrograph duration did not lead to substantial attenuation of peak discharge along the model domain. The effective morphodynamic duration (e.g., 2 years) of individual events simulated here is likely more appropriate in large sand-bed rivers characterized by an annual hydrograph than in smaller channels that experience more frequent, short-duration events. Sand supply rates at the inlet to the model domain are assumed to be at capacity. Silt concentrations at the inlet (φIN) are held constant throughout simulations (i.e., they do not vary over the course of the hydrograph).
 In natural rivers and laboratory channels with low Froude numbers, meander bends and alluvial bars propagate predominantly in a downstream direction [Seminara, 1998, 2006; van Dijk et al, 2012]. Consequently, in numerical models, unless such bed disturbances are introduced at the model domain inlet, the channel morphology downstream of the inlet is likely to evolve toward a relatively static configuration [Defina, 2003; Federici and Seminara, 2003]. A perturbation was therefore introduced at the upstream domain boundary to mimic an effect similar to the migration of lateral bars or a meandering thalweg through the inlet and to encourage the development of a non-symmetrical channel. This was achieved by representing the inlet cross section as a laterally inclined plane with a maximum amplitude (ZIN) that tipped back and forth over time period (TIN). Appropriate values for these parameters can be estimated from the amplitude, wavelength, and migration rate of bed disturbances (bars and thalwegs) in large rivers. For example, in the case of the Rio Paraná, Argentina [Ramonell et al., 2002], typical thalweg wavelengths (approximately 10 km) and migration rates (100 m yr−1) imply TIN = 100 years. In contrast, typical compound bar lengths (approximately 2.5 km) and migration rates (125 m yr−1) imply TIN = 20 years. In the case of the Jamuna, Coleman  reports sand wave (unit bar) lengths (approximately 1 km) and migration rates (50–200 m d−1) that imply TIN = 5–20 days. Consequently, appropriate values for TIN vary over a wide range (e.g., from <1 to 100 years). In contrast, all these features have typical amplitudes on the order of 10 m. In the default model setup, values of TIN = 40 years and ZIN = 10 m were used. The sensitivity of the model results to the amplitude and period of the inlet oscillation is examined in section 3.2 below.
2.3 Metrics of Channel Morphology
 Model results are described and evaluated using a set of simple quantitative metrics. The number of individual channels (separated by dry bars/islands) was determined for each cross section in the model domain (e.g., each row of grid cells). The average value of this metric for all sections in the model domain is termed the braid intensity (or braiding index) and varies with flow stage (see below). The total planform area of wet cells divided by the domain length is termed the water surface width. Simulated channel depths are examined by considering the mean depth, maximum depth, and 99th percentile of the depth distribution, for any given channel digital elevation model (DEM) and discharge. The full depth frequency distribution is also considered when comparing simulated channel morphology with bathymetric data from natural rivers. Bar and island shapes are quantified in terms of the lengths of their major and minor axes (see Figure 2d). These lengths were determined for simulated channels by identifying each individual bar or island as a group of adjacent dry pixels that is entirely surrounded by water. The major axis was determined as the longest straight line that could be drawn on the bar/island. Note that this implies nothing about the orientation of the major axis, and it was not assumed that the major axis was aligned with the grid axes. Typically, the major axis is aligned with the mean flow on either side of the bar. The minor axis was then determined as the longest line perpendicular to the major axis with start and end points on the bar/island. This definition allows the minor axis to cross a topographic low that contains water (e.g., a slackwater zone between the limbs that typically extend downstream of the bar head) and is similar to that adopted by other researchers who have quantified midchannel bar shape in large rivers [Orfeo and Stevaux, 2002; Sambrook Smith et al., 2005; Kelly, 2006]. Bar shape (denoted by Ω) is quantified here in terms of the ratio of major to minor axes lengths. The terms bar length and width are used interchangeably with major and minor axis lengths, respectively. In order to compare bar lengths for rivers of different sizes with model output, bar lengths are made dimensionless by dividing the major axis length by the mean width of all individual channels (<W>) within a reach. Data required for model evaluation were generated by digitizing bars and islands in several large sand-bed rivers. The rivers chosen for this analysis were the middle Paraná (mean annual discharge ~17,000 m3 s−1), the Japurá (mean annual discharge ~14,000 m3 s−1), and the Jamuna (mean annual discharge ~21,000 m3 s−1). These rivers were selected because of their contrasting planform morphology (see Figure 1). Further details on their geomorphic characteristics are summarized elsewhere [Latrubesse, 2008]. Data for the Paraná were obtained from satellite images in which sandbars were visible (i.e., at low water). At moderate to high stages in this river, sandbars are typically submerged, but vegetated islands remain predominantly dry; hence, island shape changes little until overbank flows occur. In contrast, bar and island dimensions in the Jamuna depend strongly on flow stage [cf. Ashworth and Lewin, 2012, Figures 12b and 12c]. Consequently, distributions of bar and island shape were derived separately for low (February), intermediate (November), and high (September) stage conditions for the Jamuna. In the case of the Japurá River, only island shape is considered because few distinct sandbars are visible in satellite images (see Figure 1b).