Simulation tools are important to investigate and predict mobility and the destructive potential of gravitational mass flows (e.g., snow avalanches). AvaFrame – the open avalanche framework – offers well-established computational modeling approaches, tools for data handling and analysis, and ready-to-use modules for evaluation and testing. This paper presents the theoretical background, derivation, and model verification for one of AvaFrame's core modules, the thickness-integrated computational model for avalanches with flow or mixed form of movement, named

Simulation tools for gravitational mass flows – with a focus on snow avalanches in this article – are in great demand for operational engineering applications and scientific model development and are gaining increasing attention in academic education. Each of these application requires different outputs. For operational engineering applications, the runout outcome for different scenarios is usually of highest interest. Scientific applications aim at better understanding the processes and will require outputs such as flow variable evolution. Existing tools for simulating snow avalanches cover a wide range of numerical implementations and vary from proprietary (e.g.,

At their core, avalanche simulation tools are based on a large variety of flow models, differing in their basic assumptions (what physical processes they include, degree of complexity), mathematical derivation, and/or their numerical implementation. These range from Eulerian methods

The resulting equations are solved using a mixed particle–grid method, in which mass is tracked using particles. Pressure gradients are computed using a smooth particle hydrodynamics (SPH) method adapted to steep terrain and flow thickness is computed on the grid

Verifying and validating the methods applied in our implementation is a challenging but crucial step, as it is for all simulation tools. Verification is done by comparing the numerical model results to an analytical solution (e.g.,

The flow variable tests

The article is structured as follows. Section

Besides the in-depth description within this article,

In this section, the mathematical model and associated equations used to simulate DFA processes are presented. The derivation is based on the thickness integration of the three-dimensional Navier–Stokes equations, using a Lagrangian approach with a terrain-following coordinate system. The equations are simplified using the assumption of shallow flow on a mildly curved topography, meaning flow thickness is considerably smaller than the length and width of the avalanche, and it is considerably smaller than the topography curvature radius.

We consider snow as the material; however, the choice of material does not influence the derivation in the first place. We assume constant density

In order to solve the previously described equations, a local coordinate system is introduced. The avalanche flows on a surface

A control volume

The normal vector

Example of a small Lagrangian volume considered in the equations and corresponding notations. The gray surface (denoted

The following volume (indicated by the superscript

The NCS has some interesting properties that will be useful for projecting and solving the equations. Because of the orthogonality of this NCS, we have

To complete the conservation Eqs. (

traction-free top surface –

impenetrable bottom surface without detachment –

bottom friction law –

To close the momentum equation, a constitutive equation describing the basal shear stress tensor

The model verification tests (Sect.

With Mohr–Coulomb friction an avalanche starts to flow if the slope inclination exceeds the friction angle

Different friction models accounting for the influence of flow velocity, flow thickness, etc., have been proposed. Three friction models are available in the

Taking advantage of the NCS and using the boundary conditions, it is possible to split the surface forces into bottom, lateral, and free surface forces and perform further simplifications:

Using the definitions of average values given in Sect.

We can project the momentum equation (Eq.

Using Eq. (

The expression of the thickness-integrated pressure is used to derive the pressure gradient

Using the derived expression of the thickness-integrated pressure (Eq.

After replacing the velocity derivative component in the normal direction with the expression developed in Eq. (

The curvature acceleration is in the normal direction to the tangent plane in order to keep the flow on the surface.

In the previous section, the equation of motion was derived using a Lagrangian approach. In order to solve this set of equations numerically, we employ a mix of particle and grid approaches. We discretize the material into numerical particles and solve the equation of motion, with the total avalanche mass being the sum of the mass associated with each particle. The grid is used to compute several parameters that are required for the computations, e.g., surface-normal vectors and flow thickness. Combining both approaches allows us to best exploit the advantages of each. The particle approach is used to track the mass, compute the curvature terms and the gradient of the flow thickness, and update the particle positions. The grid is used to handle the topography information and compute the flow thickness and artificial viscosity. We found this to help with numerical stability, and it is more efficient as it decreases the required number of particles. A theoretical convergence criterion is described in the last section.

Topography information is usually provided in a raster format which corresponds to a regular rectilinear mesh on the global horizontal

Discretizing the material into particles (particle quantities are denoted by the subscript

By assuming that the Lagrangian control volume

To assess the flow thickness gradient, we employ an SPH method (smoothed particles hydrodynamics method

In theory, an SPH method does not require any mesh to compute the gradient. However, applying this method requires finding neighbor particles. This process can be sped up with the help of an underlying grid; different neighbor search methods are presented in

The SPH method is used to express a quantity (the flow thickness in our case) and its gradient at any particle location as a weighted sum of its neighbors' properties. The principle of the method is described well in

This method is usually either used in a 3D space, in which particles move freely and where the weighting factor for the summation is the volume of the particle, or on a 2D horizontal plane, where the weighting factor for the summation is the area of the particle and the gradient is 2D. Here we want to compute the gradient of the flow thickness on a 2D surface (the topography) embedded in 3D space. The method used is analogous to the SPH gradient computation on the 2D horizontal plane but the gradient is 3D and tangent to the surface (co-linear to the local tangent plane). The theoretical derivation in Appendix

The particle flow thickness is computed with the help of the grid. The mass of the particles is interpolated onto the grid using a bilinear interpolation method (described in Sect.

We do not compute the flow thickness directly from the particle properties (mass and position) using an SPH method because it induced instabilities. Indeed, the cases where too few neighbors are found lead to very small flow thickness, which becomes an issue for flow-thickness-dependent friction laws. Note that using such an SPH method would lead to a full particle method. But since the flow thickness is only used in some cases for the friction force computation, using the previously described grid method should not affect the computation significantly.

The Coulomb friction force term in Eq. (

The momentum equation is solved numerically in time using an Euler time scheme. The time derivative of any quantity

We are looking for a criterion that relates the properties of the spatial and temporal discretization to ensure convergence of the numerical solution. Simply decreasing the time step and increasing the spatial resolution, by decreasing the grid cell size and kernel radius and increasing the number of particles, does not ensure convergence.

In this section, the numerical implementation and algorithm of the

To start a simulation with

Then the material is discretized into particles. The field of normal vectors to the surface is computed from the input DEM and the different grid fields are initialized. The details of the initialization process are given in the initialization section (

Because the lateral shear force term was removed when deriving the model equations (because of its relative smallness,

In this expression, let

Potential solutions could be taking the physical shear force into account, using for example the

The last term of the particle momentum equation (Eq.

Note that the curvature acceleration term is needed to compute the bottom pressure (Eq.

Adding the driving forces is done after adding the artificial viscosity as described in Fig.

The friction force related to the bottom shear force needs to be taken into account in the momentum equation and the velocity needs to be updated accordingly. Friction force acts against motion, hence it only affects the magnitude of the velocity and cannot be a driving force

The last term in Eq. (

Dense-flow avalanche solver (

In this section, the numerical implementation of the mathematical model is tested. We present different tests where, for specific conditions, a (semi-)analytical solution exists. The tests described here are implemented in the

In the second test, the energy line test, we investigate global variables, such as mass-averaged position and kinetic energy, that are derived from the DFA simulations. This test is based on energy conservation considerations for simplified topographies. This allows us to verify the accuracy of the DFA simulations in terms of mass-averaged runout. All the tests presented and used in what follows are implemented and available in AvaFrame (both data and helper functions). All results and figures can be reproduced using the code available on the AvaFrame GitHub repository (

Before performing the abovementioned similarity solution and dam break tests, it is necessary to describe the quantities that are compared and the measures that are used to assess the convergence, accuracy or precision of the numerical model. Both the flow thickness (

The uniform norm (

The Euclidean norm (

The similarity solution is one of the few cases where a semi-analytic solution is available for solving the thickness-integrated equations. This makes it very useful for validating the implementation of dense-flow avalanche numerical methods (here

This test is implemented in the

Flow variable test setups.

The dam break test is the second test for which an analytical solution of the thickness-integrated equations is known. In this test, we also consider an avalanche governed by a dry-friction law (Coulomb material), released from rest on an inclined plane (see Fig.

The dam break assumes an invariance in the

DFA simulations are computed using the

Parameter variation used to study convergence of the DFA simulation solution for both similarity solution and dam break test ( the parameters used for the figures presented in this article are given in bold).

Comparison of the analytical (dashed lines) and numerical (solid lines) solution for the similarity solution test case at

For both of the tests, the numerical schemes to apply friction and the method used to compute the SPH gradient are crucial for obtaining a proper starting and stopping behavior of the flow. Some intermediate developments showed that adding the friction with methods differing from as the one presented here and computing the SPH gradient without taking the slope inclination into account leads to unsatisfactory results. This is why the friction force is added as described in Sect.

Figure

Results from both the similarity solution tests and the dam break test validate the convergence criterion from

Comparison of the analytical and numerical solution for the dam break test case at

The energy line test compares the results of the

In this test, we use the

Example of trajectory where the steepest descent path hypothesis fails. The mass point is traveling from

Energy line test for an inclined plane smoothly transitioning to a horizontal plane with particles following Coulomb friction and not subject to pressure forces. All conditions for the energy line test to be applicable are satisfied, and the geometrical solution can be used as reference to compute the numerical error (here less than

The first three are related to the analytical runout point defined by the intersection between the

The horizontal distance between the analytical runout point and the end of the path profile defines the

The vertical distance between the analytical runout point and the end of the path profile defines the

The runout angle difference between the

The root-mean-square error (RMSE) between the

It is essential to state where the assumptions of this test hold. One of the important hypotheses for the energy solution is that the inclination of the material point trajectory is equal to the slope angle of the surface, i.e., where

The

Energy line test for an inclined plane smoothly transitioning to a horizontal plane with particles following Coulomb friction, not subject to pressure forces but taking curvature acceleration into account. The geometrical solution and the numerical solution match on the inclined plane and horizontal parts and differ on the curved part. This shows the effect of curvature on the runout (decrease of runout because of the added friction due to curvature). Panel

Finally, the effects of the pressure force can be studied. For example, with this test it can be shown that adding the pressure forces does not influence the simulation runout (not shown here). This can be explained by the fact that pressure forces do not dissipate any energy and hence should not affect the energy balance. However, pressure forces lead to particle trajectories that do not necessarily follow the steepest direction, which means that the fundamental hypothesis illustrated in Fig.

The energy line test previously described is also used to test whether the numerical method implemented in

The presented

The default setup of the module targets very large to extremely large avalanches of catastrophic intensity, corresponding to avalanches of size 4–5 of the European Avalanche Warning Service (EAWS) size classification or a 150-year return period in the case of Austrian hazard mapping

To verify the numerical implementation of

Note that the computational efficiency of the

Current and future potential improvements to

The conservation of energy for a material point (block model) flowing downslope from point O to point B reads as follows (assuming only Coulomb friction):

Considering a system of material points flowing down a slope with Coulomb friction, we can sum the previous equation (Eq.

Panel

The SPH method used in shallow-water equations is in most applications applied on a horizontal surface. The theoretical development on a horizontal plane is described in Appendix

Let us start with the computation of the gradient of a scalar function

We now want to express a function

Tangent plane and local coordinate system used to apply the SPH method

The vector

The AvaFrame software is publicly available at

The supplement related to this article is available online at:

AW, FO, and MT are the principal programmers and jointly developed the dense-flow avalanche numerical model code. MT prepared the article and performed the model simulations to produce the results and figures of this article with contributions from AW. FO and JTF jointly supervised the project and contributed to the discussion and paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to acknowledge Peter Sampl, who developed the theory and source code for SamosAT which represented the starting point of the

AvaFrame is supported by the Austrian Federal Ministry for Agriculture, Forestry, Regions and Water Management through cooperation between the Austrian Avalanche and Torrent Control (WLV) and the Austrian Research Centre for Forests (BFW). Additional support has been provided by the international cooperation project “AvaRange – Particle Tracking in Snow Avalanches” supported by the German Research Foundation (DFG, project no. 421446512) and the Austrian Science Fund (FWF, project no. I 4274-N29).

This paper was edited by Ludovic Räss and reviewed by Dieter Issler and Martin Mergili.