Abstract. This study describes an investigation of channelbed entrainment of sediment by debris flows. An entrainment model, developed using field data from debris flows at the Illgraben catchment, Switzerland, was incorporated into the existing RAMMS debris-flow model, which solves the 2-D shallow-water equations for granular flows. In the entrainment model, an empirical relationship between maximum shear stress and measured erosion is used to determine the maximum potential erosion depth. Additionally, the average rate of erosion, measured at the same field site, is used to constrain the erosion rate. The model predicts plausible erosion values in comparison with field data from highly erosive debris flow events at the Spreitgraben torrent channel, Switzerland in 2010, without any adjustment to the coefficients in the entrainment model. We find that by including bulking due to entrainment (e.g., by channel erosion) in runout models a more realistic flow pattern is produced than in simulations where entrainment is not included. In detail, simulations without entrainment show more lateral outflow from the channel where it has not been observed in the field. Therefore the entrainment model may be especially useful for practical applications such as hazard analysis and mapping, as well as scientific case studies of erosive debris flows.

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In the first part of this article, we extend the formal upscaling of a diffusion–precipitation model through a two-scale asymptotic expansion in a level set framework to three dimensions. We obtain upscaled partial differential equations, more precisely, a non-linear diffusion equation with effective coefficients coupled to a level set equation. As a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one strong solution is shown by applying Schauder's fixed point theorem. Additionally, non-negativity, uniqueness, and global existence or existence up to possible closure of some pores, i.e. up to the limit of degenerating coefficients, is guaranteed.

Density gradient theory (DGT) allows fast and accurate determination of surface tension and density profile through a phase interface. Several algorithms have been developed to apply this theory in practical calculations. While the conventional algorithm requires a reference substance of the system, a modified "stabilized density gradient theory" (SDGT) algorithm is introduced in our work to solve DGT equations for multiphase pure and mixed systems. This algorithm makes it possible to calculate interfacial properties accurately at any domain size larger than the interface thickness without choosing a reference substance or assuming the functional form of the density profile. As part of DGT inputs, the perturbed chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) was employed for the first time with the SDGT algorithm. PC-SAFT has excellent performance in predicting liquid phase properties as well as phase behaviors. The SDGT algorithm with the PC-SAFT EoS was tested and compared with experimental data for several systems. Numerical stability analyses were also included in each calculation to verify the reliability of this approach for future applications. List of symbols Symbol Units DescriptionA 0 J Homogeneous Helmholtz free energy A id 0 J Ideal gas contribution to A 0 A hs 0 J Hard sphere contribution to A 0 A hc 0 J Hard chain contribution to A 0 A disp 0

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a b s t r a c tThis is the first in a series of papers on implementing a discontinuous Galerkin (DG) method as an open source MATLAB/GNU Octave toolbox. The intention of this ongoing project is to provide a rapid prototyping package for application development using DG methods. The implementation relies on fully vectorized matrix/vector operations and is carefully documented; in addition, a direct mapping between discretization terms and code routines is maintained throughout. The present work focuses on a two-dimensional time-dependent diffusion equation with space/time-varying coefficients. The spatial discretization is based on the local discontinuous Galerkin formulation. Approximations of orders zero through four based on orthogonal polynomials have been implemented; more spaces of arbitrary type and order can be easily accommodated by the code structure.

Abstract. Debris-flow volumes can increase due to the incorporation of sediment into the flow as a consequence of channel-bed erosion along the flow path. This study describes a sensitivity analysis of the recently introduced RAMMS (Rapid Mass Movements) debris-flow entrainment model, which is intended to help solve problems related to predicting the runout of debris flows. The entrainment algorithm predicts the depth and rate of erosion as a function of basal shear stress based on an analysis of erosion measurements at the Illgraben catchment, Switzerland (Frank et al., 2015). Starting with a landslide-type initiation in the RAMMS model, the volume of entrained sediment was calculated for recent well-documented debris-flow events at the Bondasca and the Meretschibach catchments, Switzerland. The sensitivity to the initial landslide volume was investigated by systematically varying the initial landslide volume and comparing the resulting debris-flow volume with estimates from the field sites. In both cases, the friction coefficients in the RAMMS runout model were calibrated using the model, whereby the entrainment module was (1) inactivated to find plausible values for general flow properties by adjusting both coefficients (ξ and µ) and then (2) activated to further refine coefficient µ, which controls erosion (patterns). The results indicate that the model predicts plausible erosion volumes in comparison with field data. By including bulking due to entrainment in runout models, more realistic runout patterns are predicted in comparison to starting the model with the entire debris-flow volume (initial landslide plus entrained sediment). In particular, lateral bank overflow -not observed during these events -is prevented when using the sediment entrainment model, even in very steep (≈ 60-65 %) and narrow (4-6 m) torrent channels. Predicted sediment entrainment volumes are sensitive to the initial landslide volume, suggesting that the model may be useful for both reconstruction of historical events and the modeling of scenarios as part of a hazard analysis.

This is the second in a series of papers on implementing a discontinuous Galerkin (DG) method as an open source MATLAB / GNU Octave toolbox. The intention of this ongoing project is to offer a rapid prototyping package for application development using DG methods. The implementation relies on fully vectorized matrix / vector operations and is comprehensively documented. Particular attention was paid to maintaining a direct mapping between discretization terms and code routines as well as to supporting the full code functionality in GNU Octave. The present work focuses on a two-dimensional time-dependent linear advection equation with space / time-varying coefficients, and provides a general order implementation of several slope limiting schemes for the DG method.