Helical deformation of the filament of a scroll wave

In three-dimensional excitable systems scroll waves may lose their originally straight shape through different instabilities. In experiments, the formation of zigzag-shaped or helical filaments is often observed. Such a deformation may be due to either a twist-induced instability or a 3D meandering instability. We performed a systematic study using a Belousov-Zhabotinsky (BZ) system with a vertical gradient of excitability and determined the necessary twist for the onset of undulation in the filament. Thus, we demonstrate that in the case of a BZ system with a gradient parallel to the filament, the deformation of the filament is induced by a twist-induced instability.

Figure 1

Height dependence of the duration of one revolution of the scroll wave in a closed reactor. The revolutions are shorter at the top of the reactor ($z$ $=$ 0 mm) than they are at the bottom ($z$ $=$ 25.0 mm) due to the presence of a vertical gradient of excitability. The red open circles stand for the rigidly rotating scroll of Fig. 2, and the black squares for a meandering scroll wave (using 160 mM H${}_2$SO${}_4$). In these filaments the twist extents from the top to $z\sim$ 11 mm.

Figure 2

Temporal evolution of an initially straight, rigidly rotating scroll wave (top) and of its filament (bottom). With time, the scroll wave became twisted and the filament adopted a zigzag shape. (a), (g) 60 min; (b), (h) 180 min; (c), (i) 220 min; (d), (j) 270 min; (e), (k) 330 min; and (f), (l) 360 min after initiation. Sample dimensions $\sim$13 mm $\times$ 13 mm $\times$ 24 mm.

Figure 3

Evolution of the twist $w$ for a rigidly rotating scroll wave (dotted red line and circles; using 210 mM H${}_2$SO${}_4$) and a meandering scroll wave (solid black line and squares; using 180 mM H${}_2$SO${}_4$).

Figure 4

Temporal evolution of the filament of a meandering scroll wave (i.e., 180 mM H${}_2$SO${}_4$). The time points are indicated below the respective filament.

Figure 5

Relative length of the filament. Slightly bent filaments straighten due to positive line tension until their length attains a minimum value. The undulation of the filaments begins a $t\ge 200$ min and $t\ge 250$ min for rigidly rotating (dotted red line and circles) and meandering scroll waves (solid black line and squares), respectively. As the filament becomes undulated its relative length $L/L_z$ increases.

Figure 6

Evolution of the wavelength $\lambda$ of the undulation of the filament for a meandering scroll wave (180 mM H${}_2$SO${}_4$).

Figure 7

Perspective views of the undulated filament of a meandering scroll wave (i.e., 180 mM H${}_2$SO${}_4$). The filament is shown from different perspectives from 0 to 180${}^{\circ }$ in intervals of 10${}^{\circ }$, at $t=248$ min. The zigzag can be seen from all perspectives; therefore, the twisted filament is helical.

Figure 8

Twist $w$ for BZ media of different excitabilities (H${}_2$SO${}_4$ concentrations). (Black) dots mark filaments, where undulation set in, whereas (red) squares stand for filaments that remained straight. The two horizontal lines indicate the domain of the critical twist ${\omega }_{\mathrm{crit}}$.