Research Fellows Directory
Professor Jonathan Dawes
University of Bath
Well-known natural examples of `pattern formation' include the coloured markings on tropical fish, leopards, tigers and zebras, cloud streets and `mackerel skies', fingerprints, and visual hallucinations. Although the physical, chemical or biological details vary from one situation to another, there are mathematical models that explain the surprising similarities between these different cases.
Physical laboratory experiments are also a rich source of examples: a convenient simplified model for various geophysical and astrophysical processes is to suppose that a layer of fluid is placed between two horizontal plates. Convection currents are then driven by heating the lower plate. Usually the convection currents result in a regular pattern of `rolls' rotating in alternate senses which fill the fluid domain. It turns out, though, that in some cases the system forms localised convection currents. While this doesn't happen often for animal markings (imagine a zebra with only one stripe!), it happens naturally in some fluid flows, and in other areas of physics, including nonlinear optics and experiments with granular materials such as sand. The spontaneous appearance of localised activity is an intriguing phenomenon: even though energy is supplied to the system in a uniform manner, the response of the system is to become active in only a single region.
The ways in which localised states appear and disappear in these experiments are surprisingly similar; many of these similarities can be explained by the general mathematical theory. Turning to a future challenge, one very interesting, and experimentally motivated, extension of the mathematical theory would be to cope with localised states that are not steady convection rolls, but where the localised activity is itself a patch of complicated `turbulent' fluid flow. Such localised turbulence spots appear to organise the transition to turbulence that is a general feature of many vitally important fluid flows.