Stefan Wirtz Vom Fachbereich VI (Geographie/Geowissenschaften ...

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Experimentelle Rinnenerosionsforschung vs. Modellkonzepte – Quantifizierung der hydraulischen und erosiven Wirksamkeit von Rinnen Using one of the Navier-Stokes equations, an incompressible fluid can be completely described; thus reducing hydrodynamic questions to a mathematical problem. But this problem consists of a system of second order nonlinear partial differential equations which require the most powerful computers to numerically solve even the easiest cases. For the general, 3-dimensional case, existence- uniqueness- and regularity statements are not yet proven. Indeed, the Clay Mathematics Institute (CMI) included this in their “Millennium Prize Problems” which represent the most important open problems in mathematics, and has offered a prize of US$ 1,000,000 for a solution or a counter example (Constantin, 2001, Fefferman, 2006, Seiler, 2002, Schneider, 2008, Temam, 2000, Wiegner, 1999). The inconsistencies between the experimental results and the process based model assumptions can be the consequence of several reasons, such as uncertainties in measurements on the one hand or, on the other hand, inconsistent and incomplete process representations within the models. Uncertainty in the measurement of soil erosion has been a strong point of discussion (Stroosnijder, 2005), which is certainly still not fully resolved. The experimental setup applied in this study aimed to minimize systematic errors and their propagation. The design of the inlet, the flume for runoff measurement and the monitoring of flow and sediment transport reduced disturbance to a minimum (<strong>Wirtz</strong> et al., 2010, <strong>Wirtz</strong> et al., 2012). The results of measurements were also within the range measured in other experiments (e.g. Knapen et al., 2007). This, in combination with qualitative process observations made during the experiments, allows us to draw the conclusion that the source of errors is found in the model concepts. In models, different parameters play an important role. It is to distinguish between the hydraulic parameter flow shear stress τ and the soil parameter critical shear stress τ c or τ cr . Shear stress exerted by flow must exceed the critical shear stress to cause erosion. In critical shear stress calculations, soil parameters like dry bulk density, grain size distribution and organic content are used, in shear stress calculation hydraulic parameters, roughness, flow velocity and fluid density are variables. Different research groups deleted or added different factors to adapt the equation to their research topic. As a consequence, the equations for calculating transport and detachment capacity, critical shear stress and shear stress have different forms in different publications. We assumed that shear stress or critical shear stress values are always defined in the unit [Pa = kg m -1 s -2 ]. Note that in each paper where the units of the variables were not always given, we assumed [Pa] when no other unit was defined by the author and calculated the unknown units of the used parameters considering this default. For a better comparison with the original 159
Experimentelle Rinnenerosionsforschung vs. Modellkonzepte – Quantifizierung der hydraulischen und erosiven Wirksamkeit von Rinnen literature, we did not homogenize the labelling of the different parameters, but used the ones given by the authors. That means different parameters can be labelled equally. As it can be seen in Table 5, for the calculation of shear stress exerted by flowing water, many different parameters have been taken into account. Even the physical definition of the parameter “shear stress” seems to be unclear. Some factors have been developed from empirical studies (equation 10, equation 11). In most cases, the theoretical basis of the equations is not clear. Equation 12 is derived from Landau and Lifchitz (1971). The critical shear stress is the force needed to detach a soil particle, so it corresponds to a soil parameter and therefore input for calculation should also be depending on soil characteristics. In the WEPP model (Table 6, equation 26-28), the critical shear stress is calculated using only soil parameters like texture, organic matter content and dry bulk density. However in some equations the so called “critical” shear stress consists of hydraulic parameters like water depth, water width or fluid density (Table 6, equations 17-20). In equation 21 hydraulic and soil parameters are used equally. In equation 22, 23 and 24 the empirical nature of the development is clear. According to equation 23 and 24 the critical shear stress is only dependent on a constant value and the relationship between given particle size and the subsurface d 50 . In equation 26, the typical parameters for calculating shear stress of flowing water are used to define critical shear stress. This method uses additionally the Manning friction factor, which is a descriptor of the soils surface roughness. Thus, the use of the term “critical shear stress” seems to be very unclear. The different equations for detachment and transport capacity were developed from data sets created from controlled laboratory experiments, field observations or field experiments. In equations 29-32 (Table 7), neither critical shear stress nor shear stress is used for calculation of the transport capacity. In equation 33, shear stress is used to calculate transport capacity, in equation 34 transport rate is calculated using shear stress and critical shear stress and in equation 35 the detachment capacity is calculated using critical shear stress and shear stress (see Table 7). Equation 33 is a modification of the Yalin (1963) equation from 1963 (Foster et al., 1995). In equation 34 and 35 it is clear to see that shear stress and critical shear stress are opponents, the important parameter is the difference between these two variables. If critical shear stress is higher than shear stress, no erosion can occur. A summary of these equations is given by Reid and Dunne (1996), on the EPA-homepage (2009) and in Hessel and Jetten (2007).It has been shown that the physical definition of the parameters is not clear and their mostly empirical foundation does not fit the high temporal and spatial variability of processes within a rill. Different processes dominate at different intensities and this fact causes high 160
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Stefan Wirtz Vom Fachbereich VI (Ge
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Danksagung Danksagung Die vorliegen
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Experimentelle Rinnenerosionsforsch
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Dipl. Geogr. Stefan Wirtz; Physisch
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