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, St Martin d’Hères (France): (2) LTPCM/ENSEEG, BP 75, St Martin d’Hères cedex (France) 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 ( ) - Tignes: 1445 events ( ) 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 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, (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 , Natural Hazards and Earth System Sciences, 2, nb 3-4, (2002). 3- B. D. Malamud & D. L. Turcotte, Self Organized Criticality applied to natural hazards. Natural Hazards 20, (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