Geometric Diagnostics of Complex Patterns: Spiral Defect Chaos

Motivated by the observation of spiral patterns in a wide range of physical, chemical, and biological systems we present an approach that aims at characterizing quantitatively spiral-like elements in complex stripe-like patterns. The approach provides the location of the spiral tip and the size of the spiral arms in terms of their arclength and their winding number. In addition, it yields the number of pattern components (Betti number), as well as their size and certain aspects of their shape. We apply the method to spiral defect chaos in thermally driven Rayleigh-B\'enard convection and find that the arclength of spirals decreases monotonically with decreasing Prandtl number of the fluid and increasing heating. By contrast, the winding number of the spirals is non-monotonic in the heating. The distribution function for the number of spirals is significantly narrower than a Poisson distribution. The distribution function for the winding number shows approximately an exponential decay. It depends only weakly on the heating, but strongly on the Prandtl number. Large spirals arise only for larger Prandtl numbers ($Pr \gtrsim 1$). In this regime the joint distribution for the spiral length and the winding number exhibits a three-peak structure indicating the dominance of Archimedean spirals of opposite sign and relatively straight sections. For small Prandtl numbers the distribution function reveals a large number of small compact pattern components.