Pattern synthesis from singular solutions in the Debye limit: helical waves and twisted toroidal scroll structures

Pattern formation in physiochemical systems through the interaction of spatial diffusion and the local properties of the chemical reactions involved has been the subject of active research in recent years (Gmitro and Scriven 1966, Nicolis and Prigogine 1977, Winfree 1980). While the role of diffusion in such a synthesis is open to debate (Thones 1973), the reaction-diffusion (R-D) hypothesis has been successful in generating a variety of testable global and local geometrical concepts in relation to possible patterns (Auchmuty and Nicolis 1976). The object of this investigation is to show that there are several features in common between the geometrical properties of the concentration contours implied by the Debye limit (Abramowitz and Stegun 1965, p 366) of the singular solutions of the linear R-D equations and experimentally observed patterns in Belousov-Zhabotinskitype (Winfree 1980, p 300) reactions. This then suggests that the class of nonlinear models that may be relevant to observed patterns is the one that stabilises the above-mentioned concentration contours. In our opinion, linear models, despite their well-documented defects (Tyson 1976), might contain valid information on geometrical aspects of concentration contours that could be usefully incorporated into nonlinear generalisations. The emphasis here is not on amplitudes of concentrations or the facility of superposition intrinsic to linear models, but on the structure of functionals that emerge which are candidates for describing experimentally observed concentration contours. DeSimone, Beil and Scriven (1973) (referred to hereinafter as DBS) have provided such a beginning by constructing a two-dimensional model in which the singular solutions have been taken as elementary pattern functions. They were able to obtain Archimedean spiral patterns in the asymptotic limit of large distances. Admittedly, singularity of the solution is a defect of the model. But the intriguing result is that the singular solution considered in the Debye limit leads to the well-known involute of a circle for concentration contours (see § 2). This encourages one to apply such