Symmetries in geology and geophysics

Symmetries have played an important role in a variety of problems in geology and geophysics. A large fraction of studies in mineralogy are devoted to the symmetry properties of crystals. In this paper, however, the emphasis will be on scale-invariant (fractal) symmetries. The earth’s topography is an example of both statistically self-similar and self-affine fractals. Landforms are also associated with drainage networks, which are statistical fractal trees. A universal feature of drainage networks and other growth networks is side branching. Deterministic space-filling networks with side-branching symmetries are illustrated. It is shown that naturally occurring drainage networks have symmetries similar to diffusion-limited aggregation clusters.

Origin and evolution of circular waves and spirals in Dictyostelium discoideum territories.

Randomly distributed Dictyostelium discoideum cells form cooperative territories by signaling to each other with cAMP. Cells initiate the process by sending out pulsatile signals, which propagate as waves. With time, circular and spiral patterns form. We show that by adding spatial and temporal noise to the levels of an important regulator of external cAMP levels, the cAMP phosphodiesterase inhibitor, we can explain the natural progression of the system from randomly firing cells to circular waves whose symmetries break to form double- and single- or multi-armed spirals. When phosphodiesterase inhibitor is increased with time, mimicking experimental data, the wavelength of the spirals shortens, and a proportion of them evolve into pairs of connected spirals. We compare these results to recent experiments, finding that the temporal and spatial correspondence between experiment and model is very close.

Scaling in geology: landforms and earthquakes.

Landforms and earthquakes appear to be extremely complex; yet, there is order in the complexity. Both satisfy fractal statistics in a variety of ways. A basic question is whether the fractal behavior is due to scale invariance or is the signature of a broadly applicable class of physical processes. Both landscape evolution and regional seismicity appear to be examples of self-organized critical phenomena. A variety of statistical models have been proposed to model landforms, including diffusion-limited aggregation, self-avoiding percolation, and cellular automata. Many authors have studied the behavior of multiple slider-block models, both in terms of the rupture of a fault to generate an earthquake and in terms of the interactions between faults associated with regional seismicity. The slider-block models exhibit a remarkably rich spectrum of behavior; two slider blocks can exhibit low-order chaotic behavior. Large numbers of slider blocks clearly exhibit self-organized critical behavior.