Theory of the one-dimensional forest-fire model

Age distribution of trees in stationary forest system

A statistical theory for the age distribution of spatially dominant trees in a stationary forest system is developed. The result depends whether or not mortality is spatially correlated, as well as whether or not the stand boundaries are pre-determined. In the case of spatially non-correlated mortality, the tree age distribution is an exponential with survival rate as the base. In the case of spatially correlated mortality within a stand with pre-determined boundaries, the age distribution within the stand is an exponential with natural base. For a small stand, the median life span of the stand is inversely proportional to the number of trees (n); the median life span in relation to stand closure time is inversely proportional to nln(n). For a large stand, the stand life does not extend to the closure time. The behaviour of a forest system without fixed stand boundaries depends on the dimensionality of the system. In the case of a one-dimensional system, the longevity distribution is exponential, most of the trees however having the same longevity. Consequently, the probability density of tree age is constant. However, the probability mass of size of catastrophe destroying a particular tree is evenly distributed. This is due to trees being rapidly born on empty areas in the beginning of the life cycle, and clusters rapidly growing into larger ones close to the end of tree life.

Dynamics of the one-dimensional self-organized forest-fire model

We examine the dynamical evolution of the one-dimensional self-organized forest-fire model (FFM), when the system is far from its statistically steady state. In particular, we investigate situations in which conditions change on a time scale that is faster than, or of the order of the typical time needed for relaxation. An analytical approach is introduced based on a hierarchy of first-order nonlinear differential equations. This hierarchy can be closed at any level, yielding a sequence of successively more accurate descriptions of the dynamics. It is found that our approximate description can yield a faithful description of the FFM dynamics, even when a low order truncation is used. Employing both full simulations of the FFM and our approximate descriptions, we examine the time scales and cluster-size-dependent dynamics of relaxation to the statistical equilibrium. As an example of changing external conditions in a natural forest, the effects of a time-dependent lightning frequency are considered.

Coexistence of self-organized criticality and intermittent turbulence in the solar corona

An extended data set of extreme ultraviolet images of the solar corona provided by the SOHO spacecraft is analyzed using statistical methods common to studies of self-organized criticality (SOC) and intermittent turbulence (IT). The data exhibit simultaneous hallmarks of both regimes: namely, power-law avalanche statistics as well as multiscaling of structure functions for spatial activity. This implies that both SOC and IT may be manifestations of a single complex dynamical process entangling avalanches of magnetic energy dissipation with turbulent particle flows.

Solitons in the one-dimensional forest fire model

Fires in the one-dimensional Bak-Chen-Tang forest fire model propagate as solitons, resembling shocks in Burgers turbulence. The branching of solitons, creating new fires, is balanced by the pairwise annihilation of oppositely moving solitons. Two distinct, diverging length scales appear in the limit where the growth rate of trees, p, vanishes. The width of the solitons, w, diverges as a power law, 1/p, while the average distance between solitons diverges much faster as d approximately exp(pi2/12p).

Scale-dependent dimension in the forest fire model

The forest fire model is a reaction-diffusion model where energy, in the form of trees, is injected uniformly, and burned (dissipated) locally. We show that the spatial distribution of fires forms a geometric structure where the fractal dimension varies continuously with the length scale. In the three-dimensional model, the dimensions vary from zero to three, proportional with ln(l), as the length scale increases from l approximately 1 to a correlation length l=xi. Beyond the correlation length, which diverges with the growth rate p as xi approximately p(-2/3), the distribution becomes homogeneous. We suggest that this picture applies to the "intermediate range" of turbulence where it provides a natural interpretation of the extended scaling that has been observed at small length scales. Unexpectedly, it might also be applicable to the spatial distribution of luminous matter in the universe. In the two-dimensional version, the dimension increases to D=1 at a length scale l approximately 1/p, where there is a crossover to homogeneity, i.e., a jump from D=1 to D=2.

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.