# Continuum models of particle entrainment and deposition in snow drift and avalanche dynamics

## Abstract and Figures

Particle entrainment from, and deposition to, the snow cover are important processes in avalanche dynamics that have been neglected in most models because very little experimental information is available to guide modelers. For powder-snow avalanches, we present a two-layer model that expresses the dynamics in terms of physically well-defined sub-processes. A priori estimates of the model parameters can be based on the analogy with snow drift; detailed studies of the sub-processes will provide more precise values. We formulate hypotheses concerning the nature of different candidate entrainment mechanisms in dense-snow avalanches and their possible dependence on the flow regime. A series of experiments on a chute and in existing avalanche test sites are proposed for studying the basic processes and calibrating future models. Particle-dynamics simulations are expected to become instrumental in the development of continuum entrainment models.

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... The purpose of this paper is to present AVAL-1D, an avalanche dynamics program that we have developed at the SLF. We do not want to stress the details of the actual avalanche dynamics models -which have been presented elsewhere (Issler et al., 1998), (Issler, 1998), (Bartelt et al., 1999 ) rather the software system as a whole. The problem we want to address directly is the difficulty of introducing numerical models in practice. ...

... This approach is well suited for coupling to a dense-flow avalanche model. For more details about the model and the validation see Issler et al. (1998), Issler (1998) and Förster (2000). ...

Avalanche hazard maps are prepared by engineers and land use planners. These experts rely both on practical experience and calculation models to predict avalanche runout distances and flow velocities. Both dense flow and powder snow avalanche dynamics models are using sophisticated numerical schemes to track the motion of avalanches from initiation to runout. Since avalanche experts are seldom numerical specialists, the software must be stable and user-friendly and model limitations must be clearly communicated. In this paper we present a new avalanche dynamics program called AVAL-1D that meets these requirements. This program is intended to be used by avalanche practitioners to supplement the well-tested Voellmy-Salm model. In the following we discuss the program structure and present an example calculation with the new program.

... During the second stage, the detached part of the bed material moves between the overlaying fluid and the stationary bed material (see t=t c ¼ 1:5). Indeed, this flowing part of the bed material is entrained using two mechanisms: a) ploughing and b) basal erosion (Gauer and Issler, 2004;Issler et al., 2000) (see the inset of Fig. 10 at t=t c ¼ 1:5). The bed material, which is ploughed and pushed forward, forms the front of entraining flow. ...

... Turbidity currents in oceans or lakes, which can lead to substantial erosion and transport of sediment, are typical examples of such flows in geophysical situations (see Parker et al., 1986;Middleton, 1993;Dade and Huppert, 1994). Another well-known example are powder snow avalanches on sloping terrain which pose a serious threat in mountain areas (Issler et al., 1998). Clearly, the ability to predict the relevant features of such flows, such as the speed or the final runout length, is of great importance in practice. ...

High-resolution simulations are presented of particle-driven gravity currents in the lock-exchange configuration. The study concentrates on dilute flows with small density differences between particle-laden and clear fluid. Moreover, particles are considered which have negligible inertia, and which are much smaller than the smallest length scales of the buoyancy-induced fluid motion. For the mathematical description of the particulate phase a Eulerian approach is employed with a transport equation for the local particle-number density. The governing equations are integrated numerically with a high-order mixed spectral/spectral-element technique. In the analysis of the results, special emphasis is placed on the sedimentation of particles and the influence of particle settling on the flow dynamics. Time-dependent sedimentation profiles at the channel floor are presented which agree closely with available experimental data. A detailed study is conducted of the balance between the various components of the energy budget of the flow, i.e. the potential and kinetic energy, and the dissipative losses. Furthermore, the simulation results, along with a modified Shields criterion, are used to show that resuspension of sediment back into the particle-driven current is unlikely to occur in the cases considered. Two-dimensional (2D) and three-dimensional (3D) computations are compared which reveal that, for the present configuration, a 2D model can predict reliably the flow development at early times. However, concerning the long-time evolution of the flow, more substantial differences exist between 2D and 3D simulations.

This paper is a first step towards a synoptic analysis of the experimental information on snow avalanche flow provided by field observations and dedicated experiments over the past 60 years. Both full-size tests in instrumented avalanche tracks and laboratory experiments with snow or substitute materials are used to extract information on two major questions: (i) Which flow regimes are possible in avalanches and under which conditions do they occur? (ii) By which mechanisms and at which rate do avalanches entrain snow from the snow cover? The major types of sensors used in avalanche experiments are briefly discussed, and it is seen that a large variety of sensors and experimental techniques—including laboratory experiments—have to be combined in order to obtain definitive answers to the open questions.

Avalanches, landslides and debris flows are devastatingly powerful natural phenomena that are far too little understood. These granular matters are mixtures of solid particles and of an interstitial fluid and are easily modelled on the microscopic level by the laws of mechanics. On mesoscopic and macroscopic levels, the different scales of the influence of the particles, the fluid and their interaction lead to various models of avalanching flows. In this survey, we consider several models of granular materials characterised by height only or by height and momentum, discuss the existence of similarity solutions, existence of arbitrary solutions and particle segregation. The main part concerns the Savage-Hutter equations for dense flow avalanches.

Snow erosion and entrainment processes in avalanches are classified according to their mechanisms, the flow regimes in which they occur, and their spatial position within the avalanche. Simple, but process-specific, models are proposed for erosion by impacts, abrasion, plowing and blasting. On the basis of order-of-magnitude estimates, the first three mechanisms are clearly expected to be important. The fourth mechanism stipulates that the compaction of the snow cover ahead of the avalanche leads to the flow of escaping air just in front of the avalanche that may disrupt the snow cover and support formation of a saltation layer. The effects of this hypothetical mechanism resemble those of the plowing mechanism. All mechanisms depend strongly on the snow properties, but, with plausible parameter values, erosion rates at or above the experimentally found rates are obtained. The entrainment rate of an avalanche is most often limited by the shear stress needed to accelerate the eroded snow to avalanche speed.

In this paper we summarise a survey report on computational models for snow avalanche motion that was developed within the frame-work of the EU research project SAME (Snow Avalanche Modelling, Mapping and Warning in Europe). An examination of existing models shows that: (1) there is not - and probably never will be - a single model that adequately describes all avalanche types; (2) in order to account for the extraordinary variability of avalanche motion in response to initial and boundary conditions, flow-regime transitions and the snow mass balance should be properly described in future models; (3) calibration and validation of these models will require a comprehensive measurement programme; (4) determination of realistic initial conditions is a serious problem. We suggest that using simple models to scan the relevant parameter space with more advanced models for detailed simulations of selected scenarios could improve this situation. Finally, we discuss the needs for, and benefits of, a co-ordinated programme of avalanche research. The main features of the SAME proposal for an extensive joint experimental programme are described. We suggest that international collaboration could produce high-quality models covering all essential practical needs. Increased interdisciplinary collaboration would be advantageous for model development and facilitate incorporation of other scientific disciplines.

Following Norem’s description of powder-snow avalanche formation and structure, we propose a mathematical model that consists of a suspension layer and a so-called saltation layer. The latter is only a few meters deep and is modelled by depth-averaged mass and momentum balances. In the suspension layer, the mass and momentum balance equations for the mixture are supplemented by the snow mass balance and the transport equations for turbulent kinetic energy and dissipation. Mass and momentum exchange between the two layers is determined by particle settling, turbulent diffusion against the concentration gradient and aerodynamic shear forces. The net erosion or deposition rate is a function of the kinetic energy of the impacting particles. The saltation layer reacts on the suspension layer in that saltating particles extract momentum from the air flow. The preliminary estimates of the model parameters can be refined by means of saltation-trajectory simulations. Three-dimensional simulations with a simplified model have clearly shown the importance of snow erosion and deposition in practical applications. This approach is well suited for coupling to a dense-flow avalanche model.

A small avalanche path near the Bridger Bowl ski area in southwestern Montana has been instrumented to measure density, velocity and dynamic friction in a flowing avalanche. These measurements, made by an array of sensors mounted in the avalanche path, have been carried out for several dry-snow avalanches. Measurements of density were made using a capacitance probe that measures the dielectric constant of any material that passes in front of it. Through a calibration procedure, the dielectric constant of a given type of snow can be related to the density of that snow. Optical sensors were used to measure light reflected from the avalanche as it passed by the sensors. Signals from adjacent optical sensors were cross-correlated to determine velocity. Density and velocity measurements were made at several heights in the avalanche, with particular attention directed near the running surface. Results indicate that avalanche deformation is concentrated near the running surface where the snow density is found to be largest. Upward from the surface, the velocity gradient falls off greatly while the density also declines.
Finally, the dynamic-friction coefficient at the base of the avalanche was found by measuring shear and normal forces on a roughened 23 cm × 28 cm aluminum plate mounted parallel and flush with the avalanche running surface. The ratio of the shear force to normal force on the plate provides a measure of the dynamic-friction coefficient at the base of the avalanche.

The monitoring system of Mount Pizzac has been created in order to study the avalanche dynamics and the effect of its impact on the structures. In its extreme dimensions the monitored avalanche gets off at 2200 meters and stops at 1745 meters, thus following a trajectory of 836 meters with an average gradient of 31°. The necessary structures for monitoring have been installed in the track of the gully between the above limit of the flowing zone and the central track of the accumulation zone, for a whole length of 418 meters. They allow to observe the development of the event, thus recording continuously: pressures, speed and geometric variations of the body of the avalanche. In particular six steel poles, each one fitted out with n. 8 pressure measuring devices (each one with an area of 7850 mm2) allow to determine the profile of the pressures continually with a resolution of 50 cm up to a maximum height of 5 m. Moreover, five of the six poles placed along the flowing zone of the avalanche are fitted out with measuring devices able to check the flow height, allowing the recreation both in time and space. A small-sized wedge-shaped obstacle (area 1 m2) allows to estimate the influence of the form and area over the power of the avalanche impact. Furthermore, knowing the time when the avalanche flow passes by means of sections placed at known distances, it is possible to determine the average speed of the front in 14 tracks of the flow line. The monitoring system has three cameras which permit to record the event automatically 15 events have been recorded since 1993, when the monitoring system was first installed. The avalanches which occurred most frequently were wet snow flowing avalanches in springtime and dry snow flowing avalanches in wintertime, with average volumes of 2000 m3. Particularly for average speeds of the avalanche front ranging from 2.5 to 23 m/s, the pressures have recorded variable readings going from 5 to 175 kPa.

Earlier works on numerical modelling are analysed. Anderson and Haff (1991) proposed a model using the “splash” function which was defined for cohesionless sand. The Uematsu and others (1989, 1991) and Liston and others (1993,1994) approaches are based on fluid-mechanics conservation laws where the snow is transported and diffused by the air flow. These models consider the saltation layer as a boundary condition.
For the flow, and for the suspension, we adopt the same model as that of Uematsu and Liston. For mass exchange between the flow and snow surface, we have developed an erosion–deposition model where mass exchange is defined in relation to flow turbulence, threshold-friction velocity and snow concentration. Our snow-erosion model was calibrated using Takeuchi's(1980) field measurements. The deposition model was tested by comparing numerical results with wind-tunnel ones, for sawdust-accumulation windward and leeward of a solid snow fence with a bottom gap. The numerical results obtained are close to the experimental results. The main results of the various sensitivity experiments are: the leeward accumulation is very sensitive to the ratio ( u* / u*t ) (it appears for ( u* / u*t ) close to 1 and disappears for ( u* / u*t ) > 1.2), the global accumulation produced by the fence increases as ( u* / u*t ) decreases and the back reaction of particles on turbulence extends slightly the windward accumulation.

This paper describes a frequency modulated, continuous wave (FMCW) microwave radar system used for different types of investigations in snow and avalanche research. Different semi-empirical equations describing transmission and backscatter of electromagnetic energy in snow are compared and applied to analyse the frequency domain spectra of the backscattered radiation. The FMCW scatterometers are either buried in the ground looking upward into the snow cover or are towed on skis looking downward into the snow. The backscatter of electromagnetic radiation from avalanche snow moving perpendicular to the radar beam is analysed to estimate the height of dense flow in the avalanche. The geometrical layering, density, water equivalence, settlement, total snow height, percolation of water through the snow cover and moisture content of the snow are determined from the backscatter of the stratigraphy of a static snow pack.

The interaction between a turbulent wind and the motion of uniform saltating grains of sand or soil, so massive as to fail to enter into suspension, is examined on the basis of two complementary hypotheses. The first asserts that the effect of the moving grains on the fluid outside the region to which saltation is confined is similar to that of solid roughness of height comparable with the depth of the saltation layer. The second requires the concentration of particles engaging in the saltation to adjust itself so that the shear stress exerted by the wind on the ground—different from that acting on the fluid outside the saltation layer by an amount accountable to the change in horizontal momentum suffered by the particles in their passage through the fluid—is just sufficient to maintain the sand-strewn surface in a mobile state.
Existing experimental data on the wind profiles outside the saltation region and the horizontal flux of particles through it are shown to be consistent with these hypotheses.
The second hypothesis implies a self-balancing mechanism for controlling the concentration of saltating particles. For if the concentration is too low the shear stress at the surface rises above the value required merely to secure mobility and more particles are encouraged to leave the surface; conversely, too large a concentration depresses the surface stress, and the consequent loss of surface mobility inhibits saltation and reduces th concentration of particles until equilibrium is restored.

A description of an installation is given that allows to observe and to measure artificially released masses of snow sliding over an inclined plane and especially to measure the impact pressure on an obstacle. The installation consists of an acceleration section 20 meters long and of a measuring section 35 meters long on which obstacles can be placed. The forces are measured with strain gauges and for measuring the velocity of the moving snow masses light barriers are arranged along the acceleration section. The impact pressure upon a yielding obstacle is calculated assuming an unidimensional elastic continuum.

In this paper the powder snow avalanche is considered as a two-phase flow (air and snow particles). The equations governing this flow are the fluid mechanics conservation laws. The mass and the momentum conservation are considered for each phase. The interaction between the two phases takes into account the drag force between the particle and the air. Owing to high turbulence in the powder flow, a closure model was used based on a modified k - model in order to take into account the reduction of turbulence energy by the particles. The dense avalanche is modeled using the shallow water equations. The formation and the development of the powder avalanche is modeled using a mass and momentum exchanges between the powder flow and the dense flow. The flow area is digitized horizontally and vertically using a finite elements mesh. The numerical scheme is obtained by integrating the equations on each cell. The model thus built was calibrated using laboratory measurements of density current carried out in a flume. The model was successfully applied to reproduce many avalanches observed in France. At the end of this paper, an application of this model to an engineering case study is presented. It concerns the Uzengili path where an avalanche occurred in 1993. In this paper we use the integrated dense/powder avalanche model to define the effect of a powder avalanche flow in this path. Different simulations allow display of maps of the exposed zones for different available snow depths in the starting zone. The results were mapped in terms of dynamic pressure field and recommendations are proposed to the local authorities.