An inverse cascade model for self-organized complexity and natural hazards

Yakovlev, G.; Newman, William I.; Turcotte, Donald L.; Gabrielov, Andrei

doi:10.1111/j.1365-246x.2005.02717.x

Cited by 27 publications

(19 citation statements)

References 36 publications

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“…Drainage network growth has been shown to be analogous to invasion‐percolation models on the basis of similar fractal dimensions of the drainage network [ Stark , 1991; Yakovlev et al , 2005]. Moreover, Stark [1994] studied the relative effects of three processes and elements involved in the retreat of a drainage front: weathering and weakening of the substratum, seepage erosion and initial rock strength.…”

Section: Discussionmentioning

confidence: 99%

Tectonic interpretation of transient stage erosion rates at different spatial scales in an uplifting block

et al. 2009

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[1] We explore the extent to which it is possible to convert erosion rate data into uplift rate or erosion laws, using a landscape evolution model. Transient stages of topography and erosion rates of a block uplifting at a constant rate are investigated at different spatial scales, for a constant climate, and for various erosion laws and initial topographies. We identify three main model types for the evolution of the mountain-scale mean erosion rate: ''linear''-type, ''sigmoid''-type and ''exponential''-type. Linear-type models are obtained for topographies without drainage system reorganization, in which river incision rates never exceed the uplift rate and stepped river terraces converge upstream. In sigmoid-type and exponential-type models (typically detachment-limited or transportlimited models with a significant transport threshold), drainage growth lasts a long time, and correspond to more than linear transport laws in water discharge and slope. In exponential-type models, the mean erosion rate passes through a maximum that is higher than the rock uplift rate. This happens when the time taken to connect the drainage network exceeds half the total response time to reach dynamic equilibrium. River incision rates can be much greater than the uplift rate in both cases. In the exponential-type model, river terraces converge downstream. Observations of a mountain in the Gobi-Altay range in Mongolia support the exponential-type model. This suggests that the erosion of this mountain is either detachment-limited or transport-limited with a significant transport threshold. This study shows that drainage growth could explain differences in erosion rate measurements on different spatial scales in a catchment.

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Section: Discussionmentioning

confidence: 99%

Tectonic interpretation of transient stage erosion rates at different spatial scales in an uplifting block

et al. 2009

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“…For large and medium-sized LS, LS frequency distributions are generally negative power-law (scale-invariant) functions of LS area (Table 1), even though results have been presented (i) as cumulative and non-cumulative frequency distributions, (ii) for LS scar areas, total LS areas, or LS volumes, (iii) for relative complete inventories of LS caused by one triggering event or for historical LS inventories, which are the sum of many single events and which are assumed to be incomplete as morphological characteristics of old shallow LS can be obliterated by erosion and human activities, and (iv) for inventories of LS triggered by rainfall, earthquakes, or rapid snowmelt. Therefore these power-law distributions of LS observed under various conditions have been associated with the concept of self-organized criticality (SOC; [26][27][28][29][30]). According to its definition, selforganized critical behaviour is characterized by steady inputs, and outputs that are a series of 'events' satisfying power-law frequency-size statistics [9].…”

Section: Introductionmentioning

confidence: 99%

Characteristics of the size distribution of recent and historical landslides in a populated hilly region

Eeckhaut

Poesen

Govers

et al. 2007

Earth and Planetary Science Letters

183

155

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Despite the availability of studies on the frequency density of landslide areas in mountainous regions, frequency-area distributions of historical landslide inventories in populated hilly regions are absent. This study revealed that the frequency-area distribution derived from a detailed landslide inventory of the Flemish Ardennes (Belgium) is significantly different from distributions usually obtained in mountainous areas where landslides are triggered by large-scale natural causal factors such as rainfall, earthquakes or rapid snowmelt. Instead, the landslide inventory consists of the superposition of two populations, i.e. (i) small (b 1-2 · 10 − 2 km 2 ), shallow complex earth slides that are at most 30 yr old, and (ii) large (N 1-2 · 10 − 2 km 2 ), deep-seated landslides that are older than 100 yr. Both subpopulations are best represented by a negative power-law relation with exponents of −0.58 and −2.31 respectively. This study focused on the negative power-law relation obtained for recent, small landslides, and contributes to the understanding of frequency distributions of landslide areas by presenting a conceptual model explaining this negative power-law relation for small landslides in populated hilly regions. According to the model hilly regions can be relatively stable under the present-day environmental conditions, and landslides are mainly triggered by human activities that have only a local impact on slope stability. Therefore, landslides caused by anthropogenic triggers are limited in size, and the number of landslides decreases with landslide area.The frequency density of landslide areas for old landslides is similar to those obtained for historical inventories compiled in mountainous areas, as apart from the negative power-law relation with exponent −2.31 for large landslides, a positive power-law relation followed by a rollover is observed for smaller landslides. However, when analysing the old landslides together with the more recent ones, the present-day higher temporal frequency of small landslides compared to large landslides, obscures the positive power-law relation and rollover.

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“…An explanation for this robust behavior is given in terms of an inverse cascade of metastable clusters (Gabrielov, et al, 1999;Turcotte, et al, 1999;Yakovlev, et al, 2005). A metastable cluster is the region over which an avalanche spreads once triggered.…”

Section: Introductionmentioning

confidence: 99%

Recurrence and interoccurrence behavior of self-organized complex phenomena

Abaimov

Turcotte

Shcherbakov

et al. 2007

Nonlin. Processes Geophys.

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Abstract. The sandpile, forest-fire and slider-block models are said to exhibit self-organized criticality. Associated natural phenomena include landslides, wildfires, and earthquakes. In all cases the frequency-size distributions are well approximated by power laws (fractals). Another important aspect of both the models and natural phenomena is the statistics of interval times. These statistics are particularly important for earthquakes. For earthquakes it is important to make a distinction between interoccurrence and recurrence times. Interoccurrence times are the interval times between earthquakes on all faults in a region whereas recurrence times are interval times between earthquakes on a single fault or fault segment. In many, but not all cases, interoccurrence time statistics are exponential (Poissonian) and the events occur randomly. However, the distribution of recurrence times are often Weibull to a good approximation. In this paper we study the interval statistics of slip events using a slider-block model. The behavior of this model is sensitive to the stiffness α of the system, α=k C /k L where k C is the spring constant of the connector springs and k L is the spring constant of the loader plate springs. For a soft system (small α) there are no system-wide events and interoccurrence time statistics of the larger events are Poissonian. For a stiff system (large α), system-wide events dominate the energy dissipation and the statistics of the recurrence times between these system-wide events satisfy the Weibull distribution to a good approximation. We argue that this applicability of the Weibull distribution is due to the power-law (scale invariant) behavior of the hazard function, i.e. the probability that the next event will occur at a time t 0 after the last event has a power-law dependence on t 0 . The Weibull distribution is the only distribution that has a scale invariant hazard function. We further show that the onset of system-wide events is a well defined critical point. We find that the number of system-wide events N SWE Correspondence to: S. G. Abaimov (abaimov@geology.ucdavis.edu) satisfies the scaling relation N SWE ∝ (α−α C ) δ where α C is the critical value of the stiffness. The system-wide events represent a new phase for the slider-block system.

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An inverse cascade model for self-organized complexity and natural hazards

Cited by 27 publications

References 36 publications

Tectonic interpretation of transient stage erosion rates at different spatial scales in an uplifting block

Tectonic interpretation of transient stage erosion rates at different spatial scales in an uplifting block

Characteristics of the size distribution of recent and historical landslides in a populated hilly region

Recurrence and interoccurrence behavior of self-organized complex phenomena

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