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 1), as well as their size and certain aspects of their shape. We apply the method to spiral defect chaos in thermally driven Rayleigh-Bénard convection in the Boussinesq as well as the non-Boussinesq regime.
The diagnostics program, which is written in MATLAB, takes as input tiff-files and is therefore also directly applicable to experimental results. It generates a periodic extension of the mid-level contours and analyzes the resulting closed contour lines. The code is freely available from H. Riecke.
A typical snap-shot of spiral defect chaos and an example of a pattern
element is shown below. In this case `white' is inside the closed contour.
The arrow points to the location of the pattern element in the full
pattern. The figure of the pattern element shows the raw
closed contour (green open circles) as well as a slightly smoothed
closed contour (reddish open circles). Local maxima of the closed contour are
marked by red dots; blue dots mark local minima and green solid
squares denote inflection points.
The joint probability distribution for the compactness of the closed contours and their length
shows that for small Prandtl numbers a large number of compact structures arise (marked by
the arrow in the left figure below), while for large Prandtl
numbers target-like patterns appear, which are characterized by contours with lengths that are multiples
of the smallest `bubbles' (marked by an arrow in the right figure below).
