Scroll filament rings

Figure 6.11 Simulation of a scroll ring. (Reprinted from Winfree, A. T. Strogatz, S. H. 1983. Singular Filaments Organize Chemical Waves in Three Dimensions. II. Twisted Waves, Physica 9D, 65-80, with kind permission from Elsevier Science, The Netherlands.)...

It was appreciated two decades ago when scroll rings were first measured and timed in Belousov-Zhabotinsky media [24] that they contract and vanish, possibly faster the greater the curvature it was supposed that they would collapse in time 0(diameter / >) [43, p. 255]. But it was not until one decade ago that Yakushevitch [68] and Panfilov and Pertsov [69] noticed and confirmed numerically (in the case of equal diffusion of all reactants, and radius of curvature/wavelength large and slowly varying or constant along the filament) that the reaction-diffusion equation prescribes such motion strictly in proportion to curvature, with coefficient equal to the diffusion coefficient ... 

Fig. 1. Scroll wave filaments (dashed curves) move slowly through space as the scroll rotates, (a) An elongated spiral becomes symmetric, and (b) an elongated ring becomes circular and then disappears (after Winfree [10]). (c) A scroll ring shrinks and disappears, and (d) a figure-eight ring splits into two circular rings which then shrink and disappear (after Welsh [17]).

For example, a planar scroll ring offixed radius a has R(s) = (acos(s/a), a sin(s/a), 0). Clearly, k= fa, and r = 0. The Frenet ribbon has zero twist. As long as the filament is a planar closed curve without self intersection, the scroll ribbon must have an integral value of total twist. 

The simplest example of a scroll wave filament is a planar scroll ring. An initially planar filament remains planar for all time if Rt B is independent of s, i.e., if K.S = Ws = 0, see Equation (16c). Thus, in general, an untwisted or uniformly twisted circular filament is the only filament that will remain... 

If the filament is a perfect circle then the radius of the circle satisfies the differential equation dr/dt — -D/r, with solution = ro(f) - 2Dt, where tq is the initial radius of the ring. In other words, a circular scroll ring should collapse and disappear in the finite time T = Tq/ID, and a plot of as a function of time should be a straight line with slope -2D. 

To confirm this equation, Panfilov et al. [35] made measurements on initially noncircular scroll rings in BZ reagent. Their results are reproduced in Figure 5, where the area contained inside a closed planar filament (A0 = 27t) is plotted as a function of time. The plot shows a straight line with slope that corresponds to a diffusion coefficient of 0.12 mm /min. 

Fig. 5. The area inside the filament of a scroll ring decreases linearly with time with slope —2ttD, ZJ = 0.12 mm /min (from Panfilov et al. [35]). A is measured in mm and t in min.

The study of twisted circular scroll rings is easier than that of helical filaments. For a circular scroll ring of radius a t), curvature is la t), torsion is zero, and the uniform twist rate is locked in since the filament is closed, hence w = (f)s must be an integer multiple of k. The dynamics of such a scroll ring are governed by... 

Compared to scroll rings and helices, the geometry of even the simplest knotted filament (the trefoil) is quite complex. Knotted scroll waves are unknown (or unrecognized) in the BZ reaction, and there have been few thorough studies of knotted scroll wave solutions to reaction-diffusion equations modeling excitable media [26,28]. The analytical theory [32] of invariant knotted solutions to the filament equations (15) is not only difficult but also (probably) inapplicable to the invariant knots that have been computed numerically, because the latter are compact structures whose dynamics seem to be dominated by interactions between the closely spaced segments of the knotted filament. 

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