Jc. Vassilicos, ANOMALOUS DIFFUSION OF ISOLATED FLOW SINGULARITIES AND OF FRACTAL OR SPIRAL STRUCTURES, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 52(6), 1995, pp. 5753-5756
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.