Scroll rings

A. V. Panfilov, R. R. Aliev and A. V. Mushinsky, An integral invariant for scroll rings in a reaction-diffusion system . Physica, D36, 181 (1989). 

S. Alonso, F. Sagues, and A. S. Mikhailov. Periodic forcing of scroll rings and control of Winfree turbulence in excitable media. Chaos, 16 1-11, 2006. 

T. Bansagi and O. Steinbock. Nucleation and Collapse of Scroll Rings in Excitable Media. Preprint, 2006. 

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.)...

Moscovich, T., Hughes, J. Navigating documents with the virtual scroll ring. In Proc. HIST 2004, pp. 57-60 (2004)... 

Krinsky [40, p. 13] remarks that organizing centers other than the vortex ring discovered in 1973 [41] (there called a scroll ring ) have little interest... 

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... 

Fig. 4. The lifetime of a scroll ring increases in proportion to the square of its initial radius (from Agladze etal. [15]). T is measured in min and rl in mm. ...

In the special case where the diffusion matrix is proportional to D times the identity matrix, b2 = D and c = C3 = 0. The equation R( N = Dk was first derived for circular scroll rings with equal diffusion coefficients by Panfilov et al. [33]. Henze etal. [24, p. 703] reported the results of numerical simulations of circular scroll rings using the Oregonator model for BZ reagent, with equal diffusion coefficients, finding no vertical drift and Rt N = 0.93Dn, close to the theoretical predictions. 

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. 

Panfilov and Rudenko [34] published results of numerical experiments using Pushchino kinetics, with Dj = 0, that lend credence to equations (21). (By convention, when Di D2, Dy refers to the excitation variable and D2 to the recovery variable.) They found R< N and R( B to be linear functions of K even up to very high curvatures, 0 < k < 3/Aq (where Aq is the wavelength of the underlying spiral wave). They also found that scroll rings may expand (62 < 0) as well as shrink (62 > 0) when diffusion coefficients are not equal. 

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. 

Because of the symmetry of the problem, the curvature k of the meridian coincides with one of the two principal curvatures of the scroll ring surface. Using the standard methods of differential geometry it is possible to show that the second principal curvature is k2 — sin a. Therefore, the mean curvature of the surface element located at distance p from the ring s centre is... 

The motion of the curve representing the meridian of the scroll ring surface obeys the general kinematical equation (9). However, in this equation the velocities V and G are given now by (80) and (81) where H and n are determined by (82) and (84). Substituting these dependencies into (9), we obtain after some transformations the equation... 

Note that the difference between the original (G) and the modified (G) tangential velocities is small because we consider the scroll rings of large radius R. 

Hence, effectively we have a small-amplitude periodic modulation of the excitability of the medium. This modulation is resonant because its frequency coincides with the rotation frequency of the spiral wave. Now we can apply the results of our analysis of the resonance of spiral waves that was performed in Section 5. We know that under the conditions of the complete resonance a spiral wave drifts along a straight line. In the case of the scroll ring the spiral is the meridian cross-section of its surface. The drift of the spiral seen in this cross-section means that generally the scroll ring shrinks or expands while moving along the symmetry axis. 

Omitting the details of calculations, we give the final results (derived in a different but equivalent form in [25]). The scroll ring expands (or contracts if this quantity is negative) with the velocity. 

It was mentioned at the beginning of this section that, when the diffusion constants of the activator and the inhibitor are equal, 7 = 0 and 7 = D. As we see from (95), the resonant contribution Vs to the drift velocity of the scroll ring vanishes in this case and we obtain... 

Hence, in the activator-inhibitor systems with equal diffusion constant of both species the scroll rings always collapse and do not drift along the symmetry axis. This result has been derived directly from the reaction-diffusion equations in [52]. 

The resonant interactions in the scroll ring force its expansion. As the difference in the diffusion constants of the activator and the inhibitor is decreased, the coefficients 7 and 7 grow and the rate of collapse of the scroll ring decreases. It vanishes when the condition... 

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