# Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers
## 1 Introduction  
Hydraulic roughness values play an important role in correctly predicting water levels in rivers [1, 2, 3], which is critical for flood management purposes. This especially holds for bifurcated rivers [4], due to feedback mechanisms between the downstream branches. River dunes increase the hydraulic roughness significantly: their shapes cause form drag. Because of their significant impact on hydraulic roughness, water level forecasts during a high river water discharge depend on accurate predictions of the evolution of river dune dimensions. One aspect of this is the correct prediction of a transition to upper-stage plane bed conditions. The general motivation of this paper is to contribute to the development of a relatively simple and physics-based dune evolution model that is able to capture proper dune dimensions and dynamic behaviour under transitional conditions. In effect, the goal is to model river dune behaviour under varying discharges between the lower-stage plane bed to the upper-stage plane bed. Such a model should work under flume and field conditions, and should predict transitions at the appropriate discharges. In the past, many approaches have been used to model dune dimensions, varying from empirical equilibrium dune height predictors (e.g., [4, 5, 6, 7]) to different forms of stability analyses (e.g., [8, 9, 10, 11]). Recently, more advanced models have been developed that calculate the turbulent flow field over bedforms, in some cases in combination with morphological computations (e.g., [12, 13, 14, 15, 16, 17, 18, 19, 20]). These models are valuable to study detailed hydrodynamicprocesses, but are computationally intensive, which makes them unsuitable for the prediction of bedform evolution and roughness over full-scale river reaches during a longer period (e.g., that of a flood wave). To enable efficient predictions of dune dimensions over the time-scale of a flood wave, [PERSON] et al. (2009) [21] have developed a model in which the flow and sediment transport at the flow separation zone is parameterized instead of using full hydrodynamic equations. This model is able to predict the evolution of dunes from small initial disturbances up to equilibrium dimensions with limited computational time and good accuracy. In addition, this model has been coupled with an existing large-scale depth-averaged hydraulic model to form a 'dynamic roughness model' [22]. The coupled model clearly shows the expected hysteresis effects in dune roughness and water levels and different behaviour of sharp-peaked versus broad-peaked flood waves within the dune regime [22]. As [PERSON] and [PERSON] (1980) [23] have argued, a lag distance between flow properties (and thereby bed shear stress) and sediment transport is the principal cause of bed instability and thereby regime transitions. One of the contributing factors is the step length of sediment particles, which is the distance travelled from dislodgement to rest according to [PERSON] (1950) [24]. This creates a phase-lag effect which is not taken into account in equilibrium transport formulas, such as that of [PERSON] and [PERSON] (1948) [25]. The latter formula was used in the original model of [PERSON] et al. (2009) [21], which made it impossible to model a transition to upper-stage plane bed. The pick-up and deposition model, as proposed by [PERSON] and [PERSON] (1980) [23], inherently allows a phase-lag effect over distance. The deposition of sediment away from the pick-up point is determined by using a probability distribution function that relies on the mean step length. This pick-up and deposition model has already been used in the dune evolution model of [PERSON] et al. (2009) [14], with good results regarding prediction of generated dunes and upper-stage plane bed in flumes. However, this model is far too complicated to allow for simulations of river dune dynamics, that is, at full river scale during a longer period (e.g., a flood wave and afterwards, typically a month). Therefore, the [PERSON] et al.