# Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers
## 4 New Step Length Model  
Because using the [PERSON] et al. (2009) [14] step length model leads to results closest to the results of the [PERSON] et al. (2009) [14] dune evolution model in flumes, it was taken as the basis for further extensions to enable river dune simulations. To compensate for not modelling suspended transport explicitly, and since \(u\)-\(/w_{s}>1\) for most of the scenario (it varied between 0.9 and 1.8), the parameter \(\alpha\) needs to become much higher with the new step length model to implicitly take into account the effects of suspended transport. Suspended sediment load differs from bed load in the sense that turbulent vortices have the potential to move the sediment to higher parts in the water column. Due to the vertical mixing and settling process of suspended sediment, the suspended sediment load does not respond to variations in bed-shear stress (and thereby sediment pick-up) immediately, but with a spatial (and/or time) lag. To some extent, this is similar as bed-load, which also experiences a (smaller) spatial lag due to the step length of sediment grains. The turbulent mixing capacity \(\varepsilon_{s}\) is related to eddy viscosity \(\
u_{t}\), which for wall boundary layer flows is related to the friction velocity \(u\)- and the water depth \(h\) as follows (see e.g., [PERSON] (1993) [43]):  
\[\varepsilon_{s}\,\sim\,\
u_{t}\,\sim\,u_{s}h \tag{16}\]  
The suspension mixing height above the bed is controlled by the turbulent mixing capacity \(\varepsilon_{s}\), and the settling by gravity (settling velocity \(w_{s}\)) and scales with \(\varepsilon_{s}/w_{s}=u\)-\(h/w_{s}\) ([PERSON], 1937) [44]. [PERSON] and [PERSON] (1985) [45] derived a depth-averaged model for suspended sediment based on the usual advection-diffusion equation. They show that the vertical processes of turbulent diffusion and settling of suspended sediment can be  
Figure 4: Dune crest and trough position (black lines) and water depth (blue line) over time with the step length model of (**a**) [PERSON] and [PERSON] (1992) [42] and (**b**) [PERSON] et al. (2009) [14]. translated into a relaxation or adjustment process of suspended sediment in flow direction, involving an adjustment length (or spatial lag) of suspended sediment. A generalized expression is as follows:  
\[\Lambda_{sj}=\text{ function }\left(\frac{u_{*}}{w_{s}}\right).h \tag{17}\]  
The equation shows that the spatial lag directly scales with the water depth \(h\). The process of turbulent diffusion is represented by the friction velocity \(u\)., the settling process by the settling velocity \(w_{s}\). In addition, [PERSON] et al. (2011) [PERSON] et al. (2011) [PERSON] et al. (2011) derived a similar relation for the relaxation length of suspended sediment. There is a difference in how the spatial lag of bed-load and suspended load relate to the water motion: where the step length of bed load depends on (bed) friction velocity, the relaxation length of suspended sediment scales with friction velocity and the water depth (see Equation (17)). Therefore, as a first step, it is assumed in the present study that the step length of particles \(\alpha\) should also scale with the water depth \(h\) in order to simulate the influence of suspended sediment. A new model for non-dimensional step length \(\alpha\) is proposed by the following equation, which depends on both non-dimensional grain shear stress and water depth. \[\alpha\left(\theta^{\prime},h\right)=\alpha_{g}\left(\theta^{\prime}\right) \frac{h}{h_{ref}} \tag{18}\]  
Here \(h_{ref}\) is a reference water depth equal to the water depth at the start of the transitional regime of the case used to tune the step length model, scenario A4 of [PERSON] et al. (2009) [PERSON] et al. (2009), of which the results can been seen in Figure 5. The value of the non-dimensional grain shear stress-dependent step length \(\alpha_{g}\) follows from a modified version of the [PERSON] et al. (2009) [PERSON] et al.