Twisted scroll wave

Fig. 6. Twisted scroll wave in the FitzHugh-Nagumo model with stepwise refractoriness (from [62] reprinted with permission from A. M. Pertsov).

Three-Dimensional Systems. Twisted Scroll Waves... 

Winfree discovered scroll waves [60] and classified them [61]. The simplest cylindric scroll wave can be produced by a parallel shift of a plane spiral wave in the direction perpendicular to its plane. A shift in the phase of rotation between cross-sections perpendicular to the axis (filament) of the scroll creates a twisted scroll wave which is unstable in a uniform medium, and evolves back to a cylindric scroll. 

Pertsov et al. [63] created twisted scroll waves in a 3-D BZ-RDS with a gradient of the velocity of the spiral rotation which was due to a temperature gradient along the rotation axis. In their experiments, the increase in temperature by 1°C resulted in a decrease of the rotation period from 10 to 9.5 min. The wavelength of a twisted scroll decreases with increasing twist. 

Fig. 7. Twisted scroll wave giving birth to a train of quasiplanar waves in the Barkley model with stepwise excitability (from [67] courtesy of S. C. Mtiller).

That is, the twist rate of a scroll ribbon is the sum of the torsion of the filament and the twist rate of the scroll wave with respect to the Frenet frame. It follows that the total twist of a scroll ribbon is... 

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... 

Twisted scrolls were also found by Yamaguchi and Muller [64] in thin tubes with longitudinal gradients of concentrations of the BZ initial reagents. They also observed the transition from a twisted scroll to a set of plane traveling waves. 

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.)...

Equation (3)]. The period of plane waves at this shorter wavelength. A, can be read from the dependence of ID wave propagation speed or period on the interval of time or distance separating consecutive waves (called the dispersion curve). The argument is that this must be the rotation period of the scroll if the filament s twist becomes stationary then its rotation period must be the same as in remote regions where we see only a periodic train of plane waves [42]. 

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