Scroll filament twisted

Fig. 6. The curvature-torsion plane for helical scroll filaments. Helices evolve along semicircles, Equation (33), according to a law (34) for the evolution of torsion with time. Stationary helices exist along the nullcline (dashed curve) given by a(/c, r) = 0. (a) The leading order terms in a(/c, r). Equation (35), predict a parabolic nullcline. Straight, twisted filaments (k = 0, lu = r < 0) are stable if o (0, r) > 0 and unstable if a(0, r) < 0. (b) By including higher order terms in o (/c, r), e.g. Equation (38), we change the shape of the nullcline so that straight filaments with small twist are stable whereas ones with large twist are unstable.

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]. 

For some small positive number s, define v(s) to be the family of vectors v(s) = V(s), where the tail of v(s) lies on the filament R(s) at the coordinate point s. Then the surface 7v(s), 0<7< l,0[Pg.97]

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... 

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. 

Consider a helical filament supporting a scroll ribbon that is untwisted with respect to the Frenet frame. Let = 7t/2 so that V = B and the scroll ribbon is identical to the Frenet ribbon. In particular, the twist of the scroll ribbon is identical to the torsion of the filament. The ribbon is wrapped flat... 

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... 

We attach a scroll ribbon to this filament such that (j)g = 0, i.e. w = t. o this case, the twist rate of the scroll is the same as the twist rate of the Frenet frame, namely... 

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... 

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. 

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