# Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers
## 7 Discussion
This phase lead arises from the phenomenon that the pressure gradient of the front of the flood wave is higher than in the tail of the flood wave, as described by [PERSON] (1915) [54] and later studied by various authors (e.g., [55; 56; 57]). The spatial phase lag of the sediment transport relative to the local bed shear stress is the crucial element of the model for representing the growth and washing out of bedforms. This sediment transport phase lag is not an explicit part of our model, but it is brought into the model through the step-length of grains. This stepping of grains leads to the situation that the local sediment transport along the dune is coupled to the bed shear stress (and grain pick-up) some distance upstream. The original [PERSON] step length model in principle is a bed load formulation, as it explicitly models grains to make a step at a distance of some grains (i.e., rolling or sliding a small distance). To incorporate the suspension effect, we are stretching the [PERSON] step length model towards much larger steps. For  
Figure 22: Dune crest and trough position for the first 45 min of scenario A4 of [PERSON] et al. (2009) [14]) with (**a**) \(\alpha_{max}=300\) and (**b**) \(\alpha_{max}=350\). suspended sediment, this phase-lag is caused by the relatively slow process of upward turbulent diffusion and downward settling of suspended sediment particles. The length scale of this suspension adjustment process is a function of the suspension numbers _u-/w\({}_{s}\)_ and increases with the water depth \(h\) (see Equation (17)). Now, by incorporation of the depth in the step length formulation, we allow for much larger steps. Such larger steps can be thought of as overflowing the flow separation zone in a dune, so that the sediment accumulates at the stoss side of the next dune. If this step becomes very large, suspended sediment will be able to pass even more than one dune crest. It could be interesting to use a separate suspended transport model similar to the bed load models used in this study, with the lower step lengths used for bed load and the higher step lengths used for suspended load. With such a dual method, it is important to know what the appropriate step lengths are for each of the transport modes. This could be done by either implementing more physics-based predictors of the step length for bed load and suspended load, or by calibrating the step length models using dune data sets from the field, such as that of [PERSON] (2004) [50], as a benchmark. Another interesting exercise would be to test the Froude number effect in this model with laboratory and field experiments (see e.g., [PERSON] et al. (2014) [28]). In general, a well-calibrated version of the dune evolution model could then be applied within a model of a river system (replacing a part of the hydraulic roughness model) to better determine if and under which conditions a transition to upper-stage plane bed may occur. As far as we know for the Dutch rivers, washing out of dunes with a high discharge has never occurred, and it is unknown what will happen with the extreme (design) discharges that may occur in the future. In current practice, roughness values are calibrated by matching observed and modelled water levels [58; 59]. This means that there is significant uncertainty in roughness for extreme discharges that have not been observed yet. Using a dynamic roughness model which includes the dune dynamics may lead to more physically correct results. The river Waal case in this study for the 1998 flood has shown that a relatively small increase in river discharge can lead to the occurrence of upper-stage plane beds together with a considerable drop in water depth in a relatively short time. This shows that the occurrence of upper-stage plane bed can be an unexpected and rapid occurrence.