An energy-consistent depth-averaged Euler system: derivation and properties

In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by aminimal energy constraint instead of an asymptotic expansion. The model slightly differs from thewell-known Green-Naghdi model and is confronted with stationary andanalytical solutions of the Euler system corresponding to rotationalflows. At the end of the paper, we givetime-dependent analytical solutions for the Euler system that are alsoanalytical solutions for the proposed model but that are not solutionsof the Green-Naghdi model. We also give and compare analytical solutions of thetwo non-hydrostatic shallow water models

A weakly non-hydrostatic shallow model for dry granular flows

A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the $\mu(I)$ rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. A simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure, and it is based on a hydrostatic reconstruction. Two key points are: (i) the friction force has to be taken into account before solving the non-hydrostatic pressure. Otherwise, the incompressibility condition is not ensured; (ii) both the hydrostatic and the non-hydrostatic pressure are taken into account when dealing with the friction force. The model and numerical scheme are then validated based on several numerical tests, including laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models. An important result is that the simulated mass profiles up to the deposit and the front velocity are greatly improved. As expected, the influence of the non-hydrostatic pressure is shown to be larger for small values of the slope

Frictional velocity-weakening in landslides on Earth and on other planetary bodies

One of the ultimate goals in landslide hazard assessment is to predict maximum landslide extension and velocity. Despite much work, the physical processes governing energy dissipation during these natural granular flows remain uncertain. Field observations show that large landslides travel over unexpectedly long distances, suggesting low dissipation. Numerical simulations of landslides require a small friction coefficient to reproduce the extension of their deposits. Here, based on analytical and numerical solutions for granular flows constrained by remote-sensing observations, we develop a consistent method to estimate the effective friction coefficient of landslides. This method uses a constant basal friction coefficient that reproduces the first-order landslide properties. We show that friction decreases with increasing volume or, more fundamentally, with increasing sliding velocity. Inspired by frictional weakening mechanisms thought to operate during earthquakes, we propose an empirical velocity-weakening friction law under a unifying phenomenological framework applicable to small and large landslides observed on Earth and beyond

An analytic approach for the evolution of the static/flowing interface in viscoplastic granular flows

International audienceObserved avalanche flows of dense granular material have the property to present two possible behaviours: static (solid) or flowing (fluid). In such situation, an important challenge is to describe mathematically the evolution of the physical interface between the two phases. In this work we derive analytically a set of equations that is able to manage the dynamics of such interface, in the thin-layer regime where the flow is supposed to be thin compared to its downslope extension. It is obtained via an asymptotics starting from an incompressible viscoplastic model with Drucker-Prager yield stress, in which we have to make several assumptions. Additionally to the classical ones that are that the curvature of the topography, the width of the layer, and the viscosity are small, we assume that the internal friction angle is close to the slope angle (meaning that the friction and gravity forces compensate at leading order), the velocity is small (which is possible because of the previous assumption), and the pressure is convex with respect to the normal variable. This last assumption is for the stability of the double layer static/flowing configuration. A new higher-order non-hydrostatic nonlinear coupling term in the pressure allows us to close the asymptotic system. The resulting model takes the form of a formally overdetermined initial-boundary problem in the variable normal to the topography, set in the flowing region only. The extra boundary condition gives the information on how to evolve the static/flowing interface, and comes out from the continuity of the velocity and shear stress across it. The model handles arbitrary velocity profiles, and is therefore more general than depth-averaged models

A two-layer shallow flow model with two axes of integration, well-balanced discretization and application to submarine avalanches

We propose a two-layer model with two different axes of integration and a well-balanced finite volume method. The purpose is to study submarine avalanches and generated tsunamis by a depth-averaged model with different averaged directions for the fluid and the granular layers. Two-layer shallow depth-averaged models usually consider either Cartesian or local coordinates for both layers. However, the motion characteristics of the granular layer and the water wave are different: the granular flow velocity is mainly oriented downslope while water motion related to tsunami wave propagation is mostly horizontal. As a result, the shallow approximation and depth-averaging have to be imposed (i) in the direction normal to the topography for the granular flow and (ii) in the vertical direction for the water layer. To deal with this problem, we define a reference plane related to topography variations and use the associated local coordinates to derive the granular layer equations whereas Cartesian coordinates are used for the fluid layer. Depthaveraging is done orthogonally to that reference plane for the granular layer equations and in the vertical direction for the fluid layer equations. Then, a finite volume method is defined based on an extension of the hydrostatic reconstruction. The proposed method is exactly well-balanced for two kind of stationary solutions: the classical one, when both water and granular masses are at rest; the second one, when only the granular mass is at rest. Several tests are presented to get insight into the sensitivity of the granular flow, deposit and generated water waves to the choice of the coordinate systems. Our results show that even for moderate slopes (up to 30◦), strong relative errors on the avalanche dynamics and deposit (up to 60%) and on the generated water waves (up to 120%) are made when using Cartesian coordinates for both layers instead of an appropriate local coordinate system as proposed here.Ministerio de Economía y Competitividad (MINECO). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Agence Nationale de la Recherche. FranceEuropean Research Council (ERC

Viscoplastic modeling of granular column collapse with pressure-dependent rheology

International audienceA mechanical and numerical model of dry granular flows is proposed that quantitatively reproduce laboratory experiments of granular column collapse over inclined planes. The rheological parameters are directly derived from the experiments.The so-called \mu(I) rheology is reformulated in the framework of Drucker-Prager plasticity with the yield stress and viscosity \eta(||D||,p) depending on both the pressure p and the norm of the strain rate tensor ||D||. The granular domain, velocities, stress deviator and pressure fields are calculated using a finite element method based on an iterative decomposition-coordination formulation coupled with the augmented Lagrangian method. 2-D simulations using this model well reproduce the dynamics and deposits of collapsing granular columns. The flow is essentially located in a surface layer behind the front, whereas it is distributed over the whole depth near the front where basal sliding occurs. The computed runout distances and slopes of the deposits agree very well with the values found in the experiments. Using an easily calculated order of magnitude approximation of the mean viscosity during the flow (\eta = 1 Pa s here), we show that a Drucker-Prager rheology with a constant viscosity gives results very similar to the \mu(I) rheology and agrees with experimental height profiles, while significantly reducing the computational cost. Within the range of viscosities 0.1 < \eta < 1 Pa s, the dynamics and deposits are very similar. The observed slumping behavior therefore appears to be mainly due to the flow/no-flow criterion and to the associated strain-independent part of the "flowing constitutive relation" (i.e. related to plastic effects). However, the results are very different when an unrealistically large value of viscosity (10 Pa s) is used

Slope instabilities in Dolomieu crater, Réunion Island: From seismic signals to rockfall characteristics

International audienceThe seismic signals of hundreds of rockfalls within Dolomieu crater, Piton de la Fournaise volcano, Réunion Island, have been analyzed to investigate a possible link between physical rockfall-generating processes and associated seismic signal features. Moreover, indirect observation of rockfalls via the seismic signals they generate can provide useful data for studying volcanoes and the temporal variations of their structure. An increase in the number of rockfall events and their volumes might be an indicator of structural weakness and deformation of the volcano associated with potential eruptive activity. The study focuses on a 10 month period following the 6 April 2007 crater floor collapse within Dolomieu crater, from May 2007 to February 2008. For granular flows a scaling law is revealed between seismic energy and signal duration. A semiempirical approach based on both analytical analysis and numerical simulation of these flows shows that a similar scaling law exists between the difference of potential energy computed for an event and its propagation times and also emphasizes the effect of local topography on this scaling law. Simulated and observed data were compared to evaluate the proportion of potential energy dissipated in the form of seismic waves and confirm a direct link between the seismic energy and potential energy of a given granular flow. The mean ratio of seismic to potential energy is of the order of 10−4, comparable to the range of values observed in previous studies. A simple method based on these ratios is proposed to estimate the volumes of rockfalls from their seismic signal. Observed seismic energy and the frequency of rockfalls decreased at the beginning of the studied period and reached a stable level in July, thus suggesting a postcollapse relaxation time of Dolomieu crater structure of about 2 months from seismic signal analysis, which is confirmed by deformation data. The total rockfall volume over the study period is estimated to be 1.85 Mm3

Influence of the scar geometry on landslide dynamics and deposits: Application to Martian landslides

International audienceLandslides dynamics prediction remains difficult in spite of a considerable number of studies. The runout distance is widely used in analysis of landslide dynamics and in the calibration of the rheological parameters involved in numerical modeling. However, the unknown impact of the significant uncertainty in the shape of the initial released mass on the runout distance and on the overall shape of the deposit raises questions about the relevance of these approaches. The impact of the initial scar geometry on flow and distribution of the deposits is studied here using satellite data and numerical modeling of theoretical landslides, and Martian landslides informed by geomorphological analysis, by varying the initial scar geometry from spoon‐shaped to steep wall geometry. Our results show that the runout distance is a very robust parameter that is only slightly affected by the change in the geometry of the initial scar. On the contrary, the lateral extent of the deposit is shown to be controlled by the scar geometry, providing unique insights into the initial landsliding conditions on Mars and makes it possible to accurately recover the volume initially involved, an essential ingredient for volume balance calculation. A feedback analysis of Valles Marineris landslides can be drawn, showing good agreement between numerical results and geomorphological analysis; the geometry of the initial scar inferred from numerical modeling is strongly correlated with the regional tectonic history in Valles Marineris area

Mixed layer in a stably stratified fluid

International audienceWe present a numerical study of the generation and evolution of a mixed layer in a stably stratified layer of Boussinesq fluid. We use an external forcing in the equation of motion to model the experimental situation where the mechanical energy input is due to an oscillating grid. The results of 2D and 3D numerical simulations indicate that the basic mechanism for the entrainment is the advection of the temperature field. This advection tends to produce horizontally thin regions of small temperature vertical gradients (jets) where the hydrodynamics forces are nearly zero. At the bottom of these structures, the buoyancy brakes the vertical motions. The jets are also characterized by the presence of very short horizontal scales where the thermal diffusion time turn out to be comparable with the dynamics time. As a result, the temperature field is well mixed in a few dynamics times. This process stops when the mechanical energy injected becomes comparable with the energy dissipated by viscosity