Geometric diagnostics of complex patterns: Spiral defect chaos

Abstract

Motivated by the observation of spiral patterns in a wide range of physical, chemical, and biological systems, we present an automated approach that aims at characterizing quantitatively spiral-like elements in complex stripelike patterns. The approach provides the location of the spiral tip and the size of the spiral arms in terms of their arc length and their winding number. In addition, it yields the number of pattern components ͑Betti number of order 1͒, as well as their size and certain aspects of their shape. We apply the method to spiral defect chaos in thermally driven RayleighBénard convection and find that the arc length of spirals decreases monotonically with decreasing Prandtl number of the fluid and increasing heating. By contrast, the winding number of the spirals is nonmonotonic 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 ͑Prտ 1͒. 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. Many systems in nature exhibit complex patterns that may be stationary or time dependent, possibly in a chaotic fashion. To understand these patterns and any transition they might undergo, it is important to have quantitative measures that characterize the relevant properties of the patterns and their time dependence. Due to the multitude of different types of patterns, it is not to be expected that a single measure would be sufficient to capture the qualitatively different aspects of the various patterns. Thus, while quite a few different measures have been developed over the years, so far no convincing approach is available that extracts the characteristic features of patterns dominated by spiral-like pattern components. This paper presents an automated method that allows one to determine quantitatively features like the length of a spiral and its winding number. The approach is, however, not limited to proper spirals and yields additional insightful measures.

Hexagons and spiral defect chaos in non-Boussinesq convection at low Prandtl numbers

“…To characterize the components better and to distinguish cellular and roll-like structures we introduce the 'compactness' C of components [26],…”

“…To identify spiral components in the pattern directly we also measure the winding number of the components [26]. It is defined via the angle θ by which the (spiral) arm of a pattern component is rotated from its tip to its end at the vertex at which it merges with the rest of the component.…”

“…It is defined via the angle θ by which the (spiral) arm of a pattern component is rotated from its tip to its end at the vertex at which it merges with the rest of the component. In cases in which a component has no vertices we split it into two arms at the location of minimal curvature [26]. The winding number is then defined as |W| = θ/2π.…”

Homology and symmetry breaking in Rayleigh-Bénard convection: Experiments and simulations

“…In planform, patterns of spiral defect chaos, which are observed just above convective onset in low Prandtl number (∼ 1) fluids, are composed of convection rolls deformed into numerous rotating spirals and riddled with dislocations, disclinations and grain boundaries. Spiral defect chaos has been quantitatively described by a wide variety of approaches, including structure factors, correlation lengths and times as well as wavenumber, spectral and spiral number distributions ( [8]; see also [9] and references therein).…”

Superlattice Patterns in the Complex Ginzburg–Landau Equation with Multiresonant Forcing

“…However a more detailed quantitative characterization of the transients that captures also the competition between multi-mode structures like super-squares and anti-squares, which have the same number of participating modes, is still an open problem (cf. [55]). The long-time scaling of the ordering process of such complex structures and a comparison with the ordering in stripe [56,57] or hexagon patterns [58] may also be interesting to study.…”