Possible reasons for the scale invariance of avalanche starting zone sizes J. Faillettaz1 & F. Louchet2
(1) Sols, Solides, Structure, ENSHMG, B.P. 75, 38402 St Martin d’Hères (France):
[email protected] (2) LTPCM/ENSEEG, BP 75, 38402 St Martin d’Hères cedex (France)
[email protected]
,1752'8&7,21
4- CELLULAR AUTOMATON independence of H and L ↓ problem of L distributions treated in 2 dimensions ↓ specific 2-d simulation ↓ physical mechanisms responsible for avalanche triggerings
What is scale invariance?
f( x)=
bf(x)
(Power law) Local rules used in the cellular automaton: Red: shear failure. Stars: "tensile" failure Grey: load redistribution.
Observed or most geophysical phenomena landslides, rockfalls, earthquakes, etc
local rules for: i) shear failure of a cell:
grid of square cells cell states defined by ξ ∝ τa / τr (local applied shear stress divided by local shear resistance) 0 small H values are more frequently found than large ones => the H distribution has a negative slope - but does not necessarily mean that it obeys a power law. Cumulative frequency distributions of crown crack heights H for La Plagne and Tignes artificial triggerings
=> The question of the origin of the scale invariance of H is thus still open.
3-2- Crown crack lengths statistics:
Similar exponents: b = 2.4 for L, (and 1.2 for L2). Equivalent to a Non Cumulative (NC) L2 exponent: b≈2.2
Cumulative (C) frequency distributions of crown cracklengths L and starting zone area L2 for La Plagne and Tignes artificial triggerings
b >> b =1 (simulations in the literature) intermediate between: - landslides (2.3 to 3.3) - rockfalls (1.75)
exponents from field data and the automaton, not very different from those for landslides (2.8) or rockfalls (1.75) ↓ Our model suggests that:
But slopes disagree with field data
=> statistical distributions of crown crack depths H and crown crack lengths L should not be correlated - Data are in close agreement with this prediction:
- Very similar exponents (between -2.5 and -2.6) => some kind of "universality"
Main difference with sand pile simulations (or forest fire or slider block models): tensile threshold ↓ Our simulation deals with cohesive materials
(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 geophysical failures
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, (2002). 3- B. D. Malamud & D. L. Turcotte, Self Organized Criticality applied to natural hazards. Natural Hazards 20, 93-116 (1999).
