ANOMALOUS DIFFUSION OF ISOLATED FLOW SINGULARITIES AND OF FRACTAL OR SPIRAL STRUCTURES

Structures with isolated singularities and structures with nonisolated singularities (fractal structures) arise in many different physical s ituations such as in vertical and turbulent flows, chaotic mixing, clo uds, shear layers, lightning, snowflakes, and flames, to name but a fe w. Whereas the mathematical description of the instantaneous geometry of many fractal structures has been widely explored, less attention ha s been devoted to the evolution of such structures under the action of physical fluxes. Do space-filling properties, for example, affect the way in which physical fluxes act on the structure? Here we consider t he effects of molecular diffusion on scalar fields with singularities that are in the field itself rather than in the held's environment. Th ese singularities may be simple power-law singularities in which case the stronger the singularity the faster its early decay by diffusive a ttrition; or may be fractal or isolated spiral singularities, in which case early diffusive decay is accelerated by the space-filling proper ties of fractal or spiral structures, and the diffusive length scale i s a simple function of the geometry and structure of the held.