# THIS IS THE BRANCH THAT WILL BE PUBLICALLY RELEASE
 -> this contains the full version without force computation and resulting outputs

# FORCES ARE REMOVED FROM THIS Version
 -> This is done, but some checks are still undergoing

# TEST ON COMPILATION AND RUNNING IS UNDERGOING
 -> download, compilation and running have been tested on GNU/Linux (Fedora x86_64) and on macOS (Apple Silicon, arm64) with no issue

# VALIDATION IS PENDING BEFORE RELEASE
  -> as the four examples run with no issue, cross-validation should be easy.

  Once done, the documentation needs to be adjusted, we need to define the licence for the public branch and add it to the repository, we also need to adapt the header for each field,

  Finaly, we need to adjust the Python library that load/read the data and display the outputs.









# **SHALTOP**

 **Copyright 2022, Francois BOUCHUT, Anne MANGENEY, Marc PERUZZETTO, Antoine LUCAS,  Jacques SAINTE-MARIE**







Shallow depth-averaged model for granular flows and landslides over a complex topography including different friction laws

- Version 2003, Francois Bouchut

- Updated version 2003-2006 Francois Bouchut, Anne Mangeney

- Updated Version 2007-2014 François Bouchut, Anne Mangeney, Antoine Lucas

- Updated Version 2017-2018 Marc Peruzzetto, François Bouchut, Jacques Sainte-Marie, Anne Mangeney

- Updated Version 2020 Marc Peruzzetto, François Bouchut, Anne Mangeney

This code results from a long term collaboration between:

- Departement de Mathematiques et Applications
DMA - UMR 8553
Ecole Normale Superieure & CNRS
45, rue d'Ulm
75230 Paris cedex 05, France

- Laboratoire d'Analyse et de Mathématiques Appliquées
LAMA - UMR 8050
Université Gustave Eiffel
5, boulevard Descartes
Cité Descartes - Champs-sur-Marne
77454 Marne-la-Vallée cedex 2 - France 

- Institut de Physique du Globe de Paris
1 place Jussieu
75005 Paris 

- Laboratoire Jacques-Louis Lions
4 place Jussieu, Paris 5ème
Secrétariat : 3ème étage - 1516-3-07 

- INRIA
2 rue Simone Iff
75012 Paris

When using this model refer to :

* *Bouchut, F., Mangeney-Castelnau A., Perthame, B., and Vilotte, J. P., 2003, A new model of Saint-Venant and Savage-Hutter type for gravity driven shallow water flows, C. R. Acad. Sci. Paris, Ser. I 336, 531-536*

* *Bouchut F., M. Westdickenberg. 2004, Gravity driven shallow water models for arbitrary topography. Comm. in Math. Sci., 2(3):359--389*

* *Mangeney-Castelnau A. , F. Bouchut, J.-P. Vilotte, E. Lajeunesse, A. Aubertin, M. Pirulli, 2005, On the use of Saint-Venant equations for simulating the spreading of a granular mass, J. Geophysical Research 110(B9):B09103.*
 
* *Mangeney A. , F. Bouchut, N. Thomas, J.-P. Vilotte, M.-O. Bristeau, 2007, Numerical modeling of self-channeling granular flows and of their levee/channel deposits, to appear in  J. of Geophysical Research - Earth Surface.*

For comparison of the simulation with SHALTOP and data from real landslides refer to :

* *Lucas, A., A. Mangeney, Bouchut F., Bristeau, M.O., Mege, D., 2007. Benchmark Excercices for Granular Flows.Proceedings of the 2007 International Forum on Landslide Disaster Management, Hong-Kong, Vol.2, ISBN: 9627619302, arXiv:2106.13572, hal:hal-03256825.*

* *Lucas, A., Mangeney, A., and Ampuero, J. P., 2014. Frictional weakening in landslides on Earth and on other planetary bodies, Nature Communications, 5(3417)*

* *Favreau, P., Mangeney, A., Lucas, A., Crosta, G., and Bouchut, F., 2010. Numerical modeling of landquakes, Geophys. Res. Lett., 37, L15305.*

* *Hibert, C., Mangeney, A., Grandjean, G., and Shapiro, N., 2011. Slopes instabilities in the Dolomieu crater, la Réunion island : from the seismic signal to the rockfalls characteristics, J. Geophys. Res. - Earth Surface, 116, F04032.*

* *Moretti, L., Mangeney, A., Walter, F., Capdeville, Y., Bodin, T., Stutzmann, E., and Le Friant, A., 2020. Constraining landslide characteristics with Bayesian inversion of field and seismic data, Geophys. J. Int., 221(2), 1341-1348.* 

* *Moretti, L., Allstadt, K., Mangeney, A., Capdeville, Y., Stutzmann, E., and Bouchut, F., 2015. Numerical modeling of the Mount Meager landslide constrained by its force history derived from seismic data, J. Geophys. Res. - Solid Earth, 120(4), 2579-2599.*

* *Levy, C., Mangeney, A., Bonilla, F., Hibert, C., Calder, E., and Smith, P. J., 2015. Friction weakening in granular flows deduced from seismic records at the Soufrière Hills Volcano, Montserrat, J. Geophys. Res. - Solid Earth, 120(11), 7536-7557.*

* *Moretti, L., Mangeney, A., Capdeville, Y., Stutzmann, E., Christian Huggel, C., Schneider, D., and Bouchut, F., 2012. Numerical modeling of the Mount Steller landslide flow history and of the generated long period seismic waves, Geophys. Res. Lett., 39, L16402.*



## NUMERICAL METHOD

The basic features of the numerical method are summarized in 

* *Mangeney-Castelnau et al., 2005* and in the appendix of *Mangeney et al., 2007.*

For more details about the hydrostatic reconstruction method, refer to

* *Audusse E. , F. Bouchut, M.-O. Bristeau, R. Klein, B. Perthame, 2004, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comp. 25, 2050-2065*

or

* *Bouchut F. , Nonlinear stability of finite volume methods for  hyperbolic conservation laws, and well-balanced schemes for sources, 2004, Frontiers in Mathematics series, Birkhauser, ISBN 3-7643-6665-6*



## DESCRIPTION OF MODEL

Gravitational flow model over a topography with small slope variation (small curvature) and friction. The equations are expressed in a horizontal/vertical reference frame and the shallow approximation is imposed in the direction normal to the slope. The equations and their derivation are described in

* *Bouchut F. , M. Westdickenberg, 2004, Gravity driven shallow water models for arbitrary topography, Comm. in Math. Sci. 2 , 359-389, in particular (2.54)-(2.55).*, *Mangeney-Castelnau et al., 2005* and in the appendix of *Mangeney et al., 2007.*, and *Peruzzetto et al., 2020.*

 x,y : horizontal coordinates

 functions of x,y :

 z : topography altitude above (x,y)

 p,pt : gradient in x,y of z (constant in the case of constant slope)

 ota = 1/sqrt(1+p^2+pt^2) = cosinus of the slope angle. The 3d unit vector normal to the topography is [-p\*ota, -pt\*ota, ota]

 thet = 1./sqrt(1.+p^2+pt^2) same as ota, but used differently at the discrete level, as volume changes between horizontal coordinates and curvilinear coordinates

 rho : thickness of the layer in direction normal to topography

 u, ut : velocity components. In the horizontal/vertical 3d frame, the material velocity has the 3 components
 [ ota\*u, ota\*ut, ota*(p\*u+pt\*ut) ], which is tangent to topography.

 For example in the call fricloc=fric(...), the second argument
 of "fric" is the norm of the velocity vector (see its computation).

 In the case of constant slope, the resolved system is equivalent to
 the usual Saint Venant system written in the variables tangent
 and normal to the slope.

 In the case of topography z(x) (ie pt=0), we have ota=dx/dX
 with X the curvilinear coordinate following the topography.
 Therefore, in the frame attached to the local slope,
 the velocity has components (u, ota*ut), and the system is
 equivalent to the Savage Hutter system in curvlinear variable,
 with the curvature correction described in

* *Bouchut F. , A.Mangeney-Castelnau, B. Perthame, J.-P. Vilotte, 2003, A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows, C.R. Acad. Sci. Paris, serie I 336, 531-536.*

 Coefficient k_act/pass is not considered in this version

 Note: velocity values become irrelevant when rho is small

 To use this code for solving the usual Saint Venant system
 corresponding to assuming small topography slope,
 take initp=0, initthet=0, initota=0, and initial data
 thet0(x,y)=1., ota0(x,y)=1., p0(x,y)=0., pt0(x,y)=0.
