Critical transitions in colliding cascades

We consider here the interaction of direct and inverse cascades in a hierarchical nonlinear system that is continuously loaded by external forces. The load is applied to the largest element and is transferred down the hierarchy to consecutively smaller elements, thereby forming a direct cascade. The elements of the system fail (i. e., break down) under the load. The smallest elements fail first. The failures gradually expand up the hierarchy to the larger elements, thus forming an inverse cascade. Eventually the failures heal, ensuring that the system will function indefinitely. The direct and inverse cascades collide and interact. Loading triggers the failures, while failures release and redistribute the load. Notwithstanding its relative simplicity, this model reproduces the major dynamical features observed in seismicity, including the seismic cycle, intermittence of seismic regime, power-law energy distribution, clustering in space and time, long-range correlations, and a set of seismicity patterns premonitory to a strong earthquake. In this context, the hierarchical structure of the model crudely imitates a system of tectonic blocks spread by a network of faults (note that the behavior of such a network is different from that of a single fault). Loading mimics the impact of tectonic forces, and failures simulate earthquakes. The model exhibits three basic types of premonitory pattern reflecting seismic activity, clustering of earthquakes in space and time, and the range of correlation between the earthquakes. The colliding-cascade model seemingly exhibits regularities that are common in a wide class of complex hierarchical systems, not necessarily Earth specific.

Exactly soluble hierarchical clustering model: inverse cascades, self-similarity, and scaling.

We show how clustering as a general hierarchical dynamical process proceeds via a sequence of inverse cascades to produce self-similar scaling, as an intermediate asymptotic, which then truncates at the largest spatial scales. We show how this model can provide a general explanation for the behavior of several models that has been described as "self-organized critical," including forest-fire, sandpile, and slider-block models.

Geometric incompatibility in a fault system.

Interdependence between geometry of a fault system, its kinematics, and seismicity is investigated. Quantitative measure is introduced for inconsistency between a fixed configuration of faults and the slip rates on each fault. This measure, named geometric incompatibility (G), depicts summarily the instability near the fault junctions: their divergence or convergence ("unlocking" or "locking up") and accumulation of stress and deformations. Accordingly, the changes in G are connected with dynamics of seismicity. Apart from geometric incompatibility, we consider deviation K from well-known Saint Venant condition of kinematic compatibility. This deviation depicts summarily unaccounted stress and strain accumulation in the region and/or internal inconsistencies in a reconstruction of block- and fault system (its geometry and movements). The estimates of G and K provide a useful tool for bringing together the data on different types of movement in a fault system. An analog of Stokes formula is found that allows determination of the total values of G and K in a region from the data on its boundary. The phenomenon of geometric incompatibility implies that nucleation of strong earthquakes is to large extent controlled by processes near fault junctions. The junctions that have been locked up may act as transient asperities, and unlocked junctions may act as transient weakest links. Tentative estimates of K and G are made for each end of the Big Bend of the San Andreas fault system in Southern California. Recent strong earthquakes Landers (1992, M = 7.3) and Northridge (1994, M = 6.7) both reduced K but had opposite impact on G: Landers unlocked the area, whereas Northridge locked it up again.