We study the dynamics of scroll vortices in excitable reaction-diffusion systems analytically and numerically. We demonstrate that intrinsic three-dimensional instability of a straight scroll leads to the formation of helicoidal structures. This behavior originates from the competition between the scroll curvature and unstable core dynamics. We show that the obtained instability persists even beyond the meander core instability of two-dimensional spiral wave. [5], and many others. These seemingly unrelated phenomena share a common feature: they often allow for a description within the framework of two-component reaction-diffusion type systems, for which spiral solutions are generic.
PACSThe three-dimensional analog of a spiral wave is a scroll vortex, which can be represented by translating the spiral wave along the third direction. Thus, the point singularity of two-dimensional spiral wave (tip) develops into a line singularity (vortex filament). The dynamics of the vortex filaments in reaction diffusion systems has attracted a great deal of attention [6,7] in connection with sudden heart fibrillation, where it is believed that scroll and ring vortices play a crucial role [8,9]. Extensive numerical simulations [6][7][8] and recent experiments on gel-immobilized BZ reaction [10,11] show that in a wide range of parameters the scrolls are unstable and may assume helicoidal or even more complicated dynamic configurations. However, the theoretical analysis performed first by J. Keener, predicts an ultimate collapse of vortex rings and stability of straight vortex filaments [12]. This conclusion was drawn on the basis of multiscale analysis, which shows that a vortex ring is similar to an elastic line with a positive line tension. The persistence of nontrivial vortex configurations and turbulence in reaction-diffusion systems was attributed to a negative line tension of the filament [6].In this Letter we demonstrate, on the basis of numerical and analytical calculations, that the formation of helicoidal vortices can be related to the intrinsic threedimensional instability of a straight scroll, caused by a nontrivial response of the filament core to a bending of the filament. We show that the limit of this instability, resulting in the formation of spontaneous helicoidal vortices, goes beyond the corresponding two-dimensional core meander instability.The dynamics of a scroll vortex can be consistently described by the two-component reaction-diffusion system