## 1. Introduction

[2] Ripples and dunes are alluvial river bed forms on the smallest scales of fluvial morphodynamics. For flows of increasing strength, a typical sequence of bed forms occurs: lower flat bed → ripples → dunes → upper flat bed → antidunes → pools and chutes [*Simons et al*., 1961; *Guy et al*., 1966; *Simons and Richardson*, 1966]. Features up to and including dunes are generally termed lower-flow-regime bed forms, characterized by a low bed-material transport rate and a relatively high flow resistance.

[3] Early studies of river bed forms were based on results from flume experiments. Those studies attempted to use mean flow and sediment properties to indicate what bed forms are to be expected under which conditions, and to derive parameterized prediction formulae for bed form geometries, called roughness predictors. *Kennedy* [1963] is considered to be the first to analyze the formation and the geometry of wave-like phenomena on mobile sand beds. His model was based on a two-dimensional potential flow over an erodible bed. To produce an unstable wave, he related the local sediment transport rate to the local fluid velocity, with a lag distance between sediment transport and flow velocity. *Engelund* [1970] investigated the stability of a sand bed by a two-dimensional mathematical model based on the vorticity transport equation. The model takes account of the internal friction and describes the nonuniform distribution of the suspended sediment. The inclusion of the fluid friction and a model of the sediment transport mechanism leads to bed forms rather different from those obtained by potential flow analysis. *Richards* [1980] added viscous effects to the flow model, including a one-dimensional turbulence model for flow over a hydrodynamically rough bed, to study the formation of ripples and dunes. His results showed that ripple formation is independent of the flow depth. *Sumer and Bakioglu* [1984] extended this work to hydrodynamically smooth flows to analyze ripple formation.

[4] Both two-dimensional and three-dimensional dunes are observed in nature and laboratory experiments [*Venditti*, 2007]. Two-dimensional ripples and dunes are fairly regular in their spacing, height, and length. Their crest lines are straight or weakly sinuous and are oriented perpendicular to the flow. Contrastingly, 3-D features are more irregular in spacing, height, and length, with highly sinuous or discontinuous crest lines [*Ashley*, 1990]. *Southard and Boguchwal* [1990] provide a plotting methodology and extensive bed form phase diagrams, showing the occurrence of ripples, dunes, antidunes, or plane bed for different sediment and flow properties. The formation and development of these bed forms are associated with local hydrodynamic and sediment transport characteristics, as well as with the flow-induced forces on the bed, which in turn are influenced by the bed forms. Few attempts have been made so far to study the generation, migration, splitting, merging, and superimposition of dunes under constant or variable discharges [*Wilbers*, 2004; *Venditti et al*., 2005]. These phenomena are still not fully understood and difficult to study in the field or even in the laboratory.

[5] Recently, significant progress has been made in understanding bed form dynamics, thanks to significant advances in the ability to monitor flow and bed form morphology in laboratory and field [*Venditti et al*., 2005; *Tuijnder et al*., 2009], as well as in its numerical modeling [*Giri and Shimizu*, 2006; *Shimizu et al*., 2009; *Niemann et al*., 2011]. Nowadays, numerical modeling captures not only the characteristics of the mean flow field but also those of turbulence, including coherent flow structures above nonflat beds [*Kraft et al*., 2011; *Nabi et al*., 2012b]. These advances of numerical modeling bring us into a position to make radical progress in quantifying, modeling, and understanding the dynamics and kinematics of alluvial bed forms [*Best*, 2005]. Thanks to the increased computer power and novel numerical techniques, detailed descriptions of turbulent flow and sediment motion can be used for process-based simulation of ripples and dunes. The local flow field is determined from well-established high-resolution hydrodynamic modeling concepts like direct numerical simulation (DNS), large eddy simulation (LES), and unsteady Reynolds-averaged Navier-Stokes turbulence closure (URANS). The description of the local and instantaneous sediment motion incorporated in these models is equally important but much less well established (see the companion paper *Nabi et al*. [2013]).

[6] Several researchers have applied numerical methods to simulate the flow over fixed ripples, in order to understand the effects of bed forms on the flow field and the implications for the sediment transport. *Zedler and Street* [2001], for instance, focused on the initial entrainment and transport of suspended sediment in flows over fixed ripples. A well-resolved large eddy simulation (LES) was employed to examine in some detail the role and effect of coherent structures that occur near the bed.

[7] None of the existing numerical models is capable of simulating the generation and migration of dunes in an entirely physics-based way. Yet, numerical models were used to address these issues. *Fredsøe* [1982] proposed a model in which the dune height was determined by assuming the dune to move as a migrating front. The length of the dune was determined using a semiempirical flow description. *Tjerry and Fredsøe* [2005] refined the Fredsøe model by describing the flow with a numerical flow model based on a two-equation turbulence closure. They were able to explain how the streamline curvature above the mildly sloping upstream part of the dune influences the dune length. The dune height, however, was not explained by this model; hence, the earlier results from *Fredsøe* [1982] were employed.

[8] *Giri and Shimizu* [2006] proposed a (vertically) two-dimensional morphodynamic model that successfully reproduces fluid and bed form dynamics in a coupled manner under steady flows. They described turbulence with a nonlinear model. A nonequilibrium sediment transport approach, treating pickup and deposition of sediment empirically, was used along with an assumed mean sediment particle step length. Despite its attractive features, the shape of the dunes is strongly dependent on the definition of the particle step length, which is unknown. Hence, this model cannot always predict the shape of the dunes correctly. Furthermore, this model uses a URANS turbulence closure, which removes the turbulent fluctuations. Hence, it removes the shear stress fluctuations and their effects on the sediment pickup. The stress fluctuations are an important element in the formation of ripples on the stoss sides of the dunes, which cannot be simulated by RANS or URANS approaches (resulting in smooth dunes).

[9] *Paarlberg et al*. [2009] developed a two-dimensional vertical model assuming hydrostatic pressure conditions. The sediment transport was computed using a Meyer-Peter-Müller type of equation, including gravitational bed slope effects and a critical bed shear stress. The flow model was simplified by parameterization of the flow in the recirculation zone and by considering the separation streamline as an artificial bed. This model successfully simulated the bed form evolution from a flat bed, with initial perturbations. However, in this model, dunes keep merging until one dune covers the full domain, which is unrealistic. Moreover, bed load sediment transport is evaluated using the turbulence-averaged bed shear stress as flow parameter, which is not accurate in case of nonuniform flow with developing boundary layers associated with significant spatial variations in turbulence structures [*Nelson et al*., 1995].

[10] An interesting study on numerical modeling of dune dynamics was presented by *Niemann et al*. [2011]. Their hydrodynamic model is based on a *k* − *ω* turbulence closure and the transport model is based on a conventional bed-load transport formula (Meyer-Peter and Müller), but including a slope effect and an avalanche model to stabilize the slope of the lee face. The model is found to be capable of predicting the dune evolution from an initial perturbation. The model uses a filter to smoothen the bed form. *Niemann et al*. [2011] argue that the filter locally “rearranges” the sediment on the bed to some extent and that its effect can thus be interpreted as “artificial” erosion or deposition.

[11] *Kraft et al*. [2011] simulated the sediment transport in a turbulent channel flow over the sediment bed with a ripple structure by means of a large eddy simulation. The distribution of the suspended sediment concentration is calculated with the convection-diffusion equation. The rate of sedimentation depends on the concentration near the bed and the settling velocity of the sediment. The migration and deformation of the interface between the sediment bed and the fluid flow is captured by the level-set method. A global effect of these local processes is the migration of two-dimensional ripples. However, the migration of the ripples is relatively small and the bed starts its motion from prior initialized ripples. The migration of the ripples is not sufficiently significant for practical applications.

[12] The detailed modeling approaches, mentioned above, are all two-dimensional, but the nature of flow over three-dimensional dunes is very different from that in two dimensions, to the extent that the application of 2-D models to field situations requires careful attention [*Best*, 2005]. Field observations suggest that 3-D models are necessary to describe natural bed forms.

[13] It is against this background that we developed a high-resolution 3-D numerical model for morphodynamic processes on small temporal and spatial scales, based on large eddy simulation, particle-based transport of sediment, and adaptive grid refinement and immersed-boundary techniques for mobile sediment beds. The flow and sediment transport submodels are presented in two companion papers by *Nabi et al*. [2012b, 2013]. The flow model (part I) simulates the detailed hydrodynamics by large eddy simulation on a multilevel (i.e., locally refined) Cartesian grid. In the sediment transport model (part II), the sediment grains are considered as spherical particles moving with the fluid (in a Lagrangian framework). A discrete-element model is developed for sediment pickup from the bed, movement of particles along the bed and in the water column, and deposition on the bed. In the present paper (part III), we focus on the formation and development of ripples and dunes, presenting the first fully process-based 3-D model for the simulation of bed form evolution and migration. This model gives a better insight into the generation and migration of ripples and dunes and clarifies the effect of coherent turbulence structures on the development of the bed features. The novelty of the model is also its physics-based prediction of the bed forms for relatively long temporal scales (experimental scale) within reasonable computational times. We verify the model on four flume experiments by *Bakker et al*. [1986] and one flume experiment by *Crosato et al*. [2011]. Subsequently, we use the settings for the experiment by *Crosato et al*. [2011] to investigate the effect of sediment grain size on ripples and dunes. Finally, we extend the verification to pronounced 3-D morphologies by simulating the experiments by *Khosronejad et al*. [2012] on the development of local scour at a pier.