1. independence of H and L  problem of L distributions treated in 2 dimensions  specific 2-d simulation  physical mechanisms responsible for avalanche triggerings grid of square cells cell states defined by    a /  r = α (local applied shear stress divided by local shear resistance) 0 <  < 4   load  (snowing) or shear resistance  (evolution) Possible reasons for the scale invariance of avalanche starting zone sizes J. Faillettaz 1 & F. Louchet 2 (1) Sols, Solides, Structure, ENSHMG, B.P. 75, 38402 St Martin d’Hères (France): jerome.faillettaz@inpg.fr (2) LTPCM/ENSEEG, BP 75, 38402 St Martin d’Hères cedex (France) francois.louchet@ltpcm.inpg.fr 1. INTRODUCTION What is scale invariance? f( x)= b f(x) (Power law) Observed or most geophysical phenomena landslides, rockfalls, earthquakes, etc Concept of SOC introduced by Bak, Tang and Wiesenfeld (1) on the famous sand pile model Bak's cellular automaton => scale invariance b  1 in cumulative plots Snow Scale invariance of snow avalanches evidenced 2 years ago (2) Present work - confirms scale invariance for starting zone sizes of slab avalanches - "universal" exponents! - modeling carried out using an original 2-threshold cellular automaton - results compared to other geological failures 2- FIELD DATA RETRIEVAL AND TREATMENT Databases: - La Plagne: 4500 events 3450 events (1998-2002) - Tignes: 1445 events (1999-2002) 3- RESULTS - Shear stress in the basal plane increases with snow depth h - Tensile stress independent of snow depth - statistical distributions of crown crack depths T and crown crack lengths L should not be correlated 3-1- Slab thickness statistics: Cumulative frequency distributions of crown crack heights H for La Plagne and Tignes artificial triggerings - Very similar exponents (between -2.5 and -2.6) => some kind of "universality" - Available snow depth varies along the season => small T values are more frequently found than large ones => the T distribution has a negative slope but does not necessarily mean that it obeys a power law. => The question of the origin of the scale invariance of T is thus still open. 3-2- Crown crack lengths statistics : Cumulative (C) frequency distributions of crown cracklengths L and starting zone area L 2 for La Plagne and Tignes artificial triggerings Similar exponents: b = 2.4 for L, (and 1.2 for L 2 ). Equivalent to a Non Cumulative (NC) L 2 exponent: b  2.2 b >> b =1 (simulations in the literature) intermediate between: - landslides (2.3 to 3.3) - rockfalls (1.75) Local rules used in the cellular automaton: Red: shear failure. Stars: "tensile" failure Grey: load redistribution. 5 RESULTS AND DISCUSSION local rules for: i) shear failure of a cell:   given threshold. ii) tensile failure of the links between two cells   given threshold Load increased by  steps randomly distributed on the grid until avalanche release occurs Non cumulative surface (L 2 ) distribution obtained from the automaton (load increments and tensile thresholds between 0 and 4). A slope of - 2 is shown for comparison. 5- CONCLUSIONS 1- Scale invariance confirmed for crown crack lengths, heights and surfaces of slab avalanche starting zones. 2- Field data exponents seem to be "universal" 3- Field data exponents "comparable" to landslides or rockfalls 4- Specific 2 parameters cellular automaton takes into account: - shear basal failures - tensile crown ruptures 5- Exponents very sensitive to initial conditions: - Exponents comparable to field data obtained only for random and comparable shear and tensile thresholds 6- Agreement with field data much better than for Bak's sand pile model or forest fire models.  cohesive character of the material 7- The robustness and universality of this scaling law suggests that it may be used for a statistical prediction of large events based on recordings of much more frequent small events. References 1- P. Bak, C. Tang and K. Wiesenfeld, Self Organised Criticality. Phys Rev. A 38, 364-374 (1988). 2- F. Louchet, J. Faillettaz, D. Daudon, N. Bédouin, E. Collet, J. Lhuissier and A-M. Portal 2001, XXVI General Assembly of the European Geophysical Society, Nice (F), march 25-30 2001, Natural Hazards and Earth System Sciences, 2, nb 3-4, 157-161 (2002). 3- B. D. Malamud & D. L. Turcotte, Self Organized Criticality applied to natural hazards. Natural Hazards 20, 93-116 (1999). J. Faillettaz, F. Louchet and J.R. Grasso. Scaling laws for isolated snow fracture. Submitted to Geology (2004) J. Faillettaz, F. Louchet and J.R. Grasso. A two threshold model for scaling laws of non-interacting snow avalanches. Submitted to Physical Review Letter (2004) Slope  2  Good agreement with field data (NC slopes  2.2)  Quite different from simulations in the literature (b  1) Load increment discretisation reflects basal plane heterogeneity   equivalent to shear resistance scatter  4- CELLULAR AUTOMATON Spatial variability α = 0.5 WHY? Main difference with sand pile simulations (or forest fire or slider block models): tensile threshold  Our simulation deals with cohesive materials exponents from field data and the automaton, not very different from those for landslides (2.8) or rockfalls (1.75)  Our model suggests that: (i) cohesion is essential (ii) shear and tensile resistances should be scattered but with similar magnitudes (iii) this model might apply to a wider range of gravitational failures -b