# Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers
## 8 Conclusions  
In this study the dune evolution model of [PERSON] et al. (2009) [21] was extended with an alternative sediment transport model using sediment pick-up, deposition, and step length models instead of the usual equilibrium transport formula. The model was devised in such a way that similar dune behaviour is obtained as with the more complex k-\(\varepsilon\) model of [PERSON] et al. (2009) [14] for a flood wave corresponding to flume conditions.  
Step lengths have been allowed to become larger than with the step length model of [PERSON] et al. (2009) [14]. In addition, higher step lengths than the maximum observed in the experiments of [PERSON] and [PERSON] (1980) [23] have been adopted such that effects of suspended load are implicitly taken into account. We would like to note that this implicit formulation is a first step to account for suspended sediment transport processes in a strongly schematized way, which needs to be extended and improved in further studies. It was shown that in this way, the model is able to predict a transition to upper-stage plane bed and clearly shows hysteresis effects due to the lag between bedform dimensions and discharge. While the simulated associated moments in time and the maximum dune height are close to the results of [PERSON] et al. (2009) [14], the exact results in terms of water depth still showed differences of about 25%.  
A scenario corresponding to the 1998 flood in the river Waal was modelled as well. While the qualitative behaviour of the dunes was represented well, the dune height was overestimated. This signifies the need for further model improvement/calibration. For the river scenario, the step length is significantly higher than the values used for the flume conditions. It was shown that the Waal model is also computationally fast. By slightly increasing the discharge of the original river scenario, a transition to upper-stage plane bed was also modelled. Although in 1995, for much higher discharges, no transition to upper-stage plane bed has occurred ([PERSON] and [PERSON] (2003) [51]), this does show the sensitivity of the dune dynamics in this transitional regime and also indicates the need for a well-validated dune evolution model for extreme discharges.  
It should be noted that when a transition to upper-stage plane bed indeed occurs during extreme floods (i.e., during dike design conditions), this has the potential to greatly reduce the design water depth because the hydraulic roughness of the main channel would be significantly lower. However, in a bifurcation river system such as the river Waal, balancing effects distribute this lowering of the water levels over both branches, if it would occur in one branch only [4]. For the model results presented here, the difference in water levels in one branch is approximately 0.5 m. Although in reality this effect would also be dampened due to the presence of floodplains, which are not accounted for here, this is still considered as a significant effect.  
Conceptualization, O.J.M.v.D., S.J.M.H.H. and J.S.R.; methodology, O.J.M.v.D., S.J.M.H.H. and J.S.R.; software, O.J.M.v.D.; resources, S.J.M.H.H.; writing--original draft preparation, O.J.M.v.D., S.J.M.H.H. and J.S.R.; writing--review and editing, O.J.M.v.D., S.J.M.H.H. and J.S.R.; visualization, O.J.M.v.D.; supervision, S.J.M.H.H. and J.S.R.; project administration, S.J.M.H.H.; funding acquisition, S.J.M.H.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by NWO-AES, grant number 10483.  
Not applicable.  
This study was carried out as part of the project 'BedFormFlood', supported by the Technology Foundation STW, the applied science division of now, and the technology programme of the Ministry of Economic Affairs. The authors thank [PERSON] and [PERSON] for their help in editing the manuscript.  
The authors declare no conflict of interest.