Critical exponents for boundary avalanches in two-dimensional Abelian sandpile

Equivalence between the Abelian sandpile model and the q→0 limit of the Potts model

Abelian avalanches and Tutte polynomials

<strong class="journal-contentHeaderColor">Abstract.</strong> Accurately predicting the location, timing and size of natural snow avalanches is crucial for local and regional decision-makers, but remains one of the major challenges in avalanche forecasting. So far, forecasts are generally made by human experts, interpreting a variety of data, and drawing on their knowledge and experience. Using avalanche data from the Swiss Alps and one-dimensional physics-based snowpack simulations, we developed a model predicting the probability of dry-snow avalanches occurring in the vicinity of automated weather stations based on the output of a recently developed instability model. This new avalanche day predictor was compared to benchmark models related to the amount of new snow. Evaluation on an independent data set demonstrated the importance of snow stratigraphy for natural avalanche release, as the avalanche day predictor outperformed the benchmark model based on the three-day sum of new snow (F1 scores: 0.71 and 0.65, respectively). The averaged predictions of both models resulted in the best performance (F1 score: 0.75). In a second step, we derived functions describing the probability for certain avalanche size classes. Using the 24-hour new snow height as proxy of avalanche failure depth yielded the best estimator of typical (median) observed avalanche size, while the depth of the deepest weak layer, detected using the instability model, provided the better indicator regarding the largest observed avalanche size. Validation of the avalanche size estimator on an independent data set of avalanche observations confirmed these findings. Furthermore, comparing the predictions of the avalanche day predictors and avalanche size estimators with a 21-year data set of re-analysed regional avalanche danger levels showed increasing probabilities for natural avalanches and increasing avalanche size with increasing danger level. We conclude that these models may be valuable tools to support forecasting the occurrence of natural dry-snow avalanches.

We present an experimental study of the influence of quenched disorder on the distribution of flux avalanches in type-II superconductors. In the presence of much quenched disorder, the avalanche sizes are power-law distributed and show finite-size scaling, as expected from self-organized criticality (SOC). Furthermore, the shape of the avalanches is observed to be fractal. In the absence of quenched disorder, a preferred size of avalanches is observed and avalanches are smooth. These observations indicate that a certain minimum amount of disorder is necessary for SOC behavior. We relate these findings to the appearance or nonappearance of SOC in other experimental systems, particularly piles of sand.

We characterize the distributions of size and duration of avalanches propagating in complex networks. By an avalanche we mean the sequence of events initiated by the externally stimulated excitation of a network node, which may, with some probability, then stimulate subsequent excitations of the nodes to which it is connected, resulting in a cascade of excitations. This type of process is relevant to a wide variety of situations, including neuroscience, cascading failures on electrical power grids, and epidemiology. We find that the statistics of avalanches can be characterized in terms of the largest eigenvalue and corresponding eigenvector of an appropriate adjacency matrix that encodes the structure of the network. By using mean-field analyses, previous studies of avalanches in networks have not considered the effect of network structure on the distribution of size and duration of avalanches. Our results apply to individual networks (rather than network ensembles) and provide expressions for the distributions of size and duration of avalanches starting at particular nodes in the network. These findings might find application in the analysis of branching processes in networks, such as cascading power grid failures and critical brain dynamics. In particular, our results show that some experimental signatures of critical brain dynamics (i.e., power-law distributions of size and duration of neuronal avalanches) are robust to complex underlying network topologies.

We investigate numerically temporal correlations in a one-dimensional critical-slope sandpile model with rules that on average conserve the number of particles. Our work is motivated by the existence of two well-separated time scales in self-organized sandpile models, one related to the spreading of avalanches and the other imposed by the external driving. We assume that avalanches are instantaneous events on the time scale imposed by the external deposition and study the autocorrelation function of the series of successive avalanche amplitudes. We find that the autocorrelation function has a log-normal form and for large system sizes tends to a constant, implying that the temporal correlations become stronger in the limit of large system size. We independently test this result by calculating the power spectrum of the series of successive avalanche lifetimes and sizes. For large system sizes L there is a frequency regime where the power spectrum tends to a 1/f type of noise, in agreement with the tendency of the autocorrelation function to approach a constant in large systems. @S1063-651X~96!07212-1#

Self-organized criticality within fractional Lorenz scheme

Effects of the underlying topology on perturbation spreading in the Axelrod model for cultural dissemination

Self-organized criticality and stock market dynamics: an empirical study

It is impractical to create gridded numerical models of coastal circulation with sufficient resolution around small topographic features, such as submarine canyons, and still have the alongshore boundaries placed beyond the decay distance of coastal trapped waves. Two solutions to this problem are to make the alongshore boundaries either open or periodic. Numerical simulations were performed with upwelling and downwelling winds to compare the effects of these different choices for boundary conditions. Several open boundary formulations were tried and three are discussed in detail. The offshore boundary was specified as “no gradient” for all variables with no serious effect. The “modified” Orlanski radiation condition is used for all variables at the alongshore boundaries, except the vertically integrated flow that has the strongest effect on the model solution. An alongshore pressure gradient, opposing the wind, develops in the model if the modified Orlanski radiation condition is applied to the barotropic flow, causing slower currents near the surface and deep undercurrents away from the shelf. The other cases, which combined either a radiation or a relaxation boundary condition with a local solution of the barotropic equations on the boundary, were at least initially similar to the periodic case but with slower alongshore flow. The initial impact of these differences on the circulation within the canyon was small. The models with the open boundaries were more stable (did not develop strong flow meanders) than the cases with periodic conditions as initial transients are not trapped, and amplified, within the domain. Thus, open cases, especially with the upwelling winds, could run for extended times.

Atomistic simulations of binary amorphous systems with over 4 million atoms are performed. Systems of two interatomic potentials of the Lennard-Jones type, LJ12-6 and LJ9-6, are simulated. The athermal quasistatic shearing protocol is adopted, where the shear strain is applied in a stepwise fashion with each step followed by energy minimization. For each avalanche event, the shear stress drop (Δσ), the hydrostatic pressure drop (Δσ_{h}), and the potential energy drop (ΔE) are computed. It is found that, with the avalanche size increasing, the three become proportional to each other asymptotically. The probability distributions of avalanche sizes are obtained and values of scaling exponents fitted. In particular, the distributions follow a power law, P(ΔU)∼ΔU^{-τ}, where ΔU is a measure of avalanche sizes defined based on shear stress drops. The exponent τ is 1.25±0.1 for the LJ12-6 systems, and 1.15±0.1 for the LJ9-6 systems. The value of τ for the LJ12-6 systems is consistent with that from an earlier atomistic simulation study by Robbins et al. [Phys. Rev. Lett. 109, 105703 (2012)]PRLTAO0031-900710.1103/PhysRevLett.109.105703, but the fitted values of other scaling exponents differ, which may be because the shearing protocol used here differs from that in their study.

We consider the non Abelian sandpile model introduced by Y.-C. Zhang on a\ntwo-dimensional square lattice. The static and dynamical properties of the\nmodel are investigated and compared to the Abelian sandpile model of Bak, Tang\nand Wiesenfeld. A detailed analysis which takes the finite size effects into\naccount yields that the exponents of the avalanche probability distribution are\nthe same as in the Abelian model.\n

Abstract. Consistent estimates of avalanche size are crucial for communicating not only among avalanche practitioners but also between avalanche forecasters and the public, for instance in public avalanche forecasts. Moreover, applications such as risk management and numerical avalanche simulations rely on accurately mapped outlines of past avalanche events. Since there is not a widely applicable and objective way to measure avalanche size or to determine the outlines of an avalanche, we need to rely on human estimations. Therefore, knowing about the reliability of avalanche size estimates and avalanche outlines is essential as errors will impact applications relying on this kind of data. In the first of three user studies, we investigate the reliability in avalanche size estimates by comparing estimates for 10 avalanches made by 170 avalanche professionals working in Europe or North America. In the other two studies, both completed as pilot studies, we explore reliability in the mappings of six avalanches from oblique photographs from 10 participants and the mappings of avalanches visible on 2.9 km2 of remotely sensed imagery in four different spatial resolutions from 5 participants. We observed an average agreement of 66 % in the most frequently given avalanche size, while agreement with the avalanche size considered “correct” was 74 %. Moreover, European avalanche practitioners rated avalanches significantly larger for 8 out of 10 avalanches, compared to North Americans. Assuming that participants are equally competent in the estimation of avalanche size, we calculated a score describing the factor required to obtain the observed agreement rate between any two size estimates. This factor was 0.72 in our dataset. It can be regarded as the certainty related to a size estimate by an individual and thus provides an indication of the reliability of a label. For the outlines mapped from oblique photographs, we noted a mean overlapping proportion of 52 % for any two avalanche mappings and 60 % compared to a reference mapping. The outlines mapped from remotely sensed imagery had a mean overlapping proportion of 46 % (image resolution of 2 m) to 68 % (25 cm) between any two mappings and 64 % (2 m) to 80 % (25 cm) when compared to the reference. The presented findings demonstrate that the reliability of size estimates and of mapped avalanche outlines is limited. As these data are often used as reference data or even ground truth to validate further applications, the identified limitations and uncertainties may influence results and should be considered.

Exploring influence of sectional flexural yielding on experimental pile response analysis and applicability of distributed plastic hinge model in inelastic numerical simulation for laterally loaded piles

The avalanches in the Abelian sandpile model have a compact structure and consist of shells. The different shells consist solely of sites that have toppled a fixed number of times. For the outermost shell the number is one and increases successively by one from an outer shell to the next inner shell. In this paper, we determine the size-distribution exponents of various categories of sites: sites that have toppled at least once, at least twice, at least thrice, only once and only twice. A particular subclass of avalanches is identified which have no shell structure and consist solely of sites which have toppled only once. The size distribution exponent of this type of clusters is determined.