Buckling of Scroll Waves

A scroll wave in a sufficiently thin layer of an excitable medium with negative filament tension can be stable nevertheless due to filament rigidity. Above a certain critical thickness of the medium, such a scroll wave will have a tendency to deform into a buckled, precessing state. Experimentally this will be seen as meandering of the spiral wave on the surface, the amplitude of which grows with the thickness of the layer, until a breakup to scroll wave turbulence happens. We present a simplified theory for this phenomenon and illustrate it with numerical examples.

Figure 1

Buckled scroll and filament, with the tip path on the top of the box. Barkley model with $a=1.1$, $b=0.19$, $c=0.02$, $D_v=0.10$, box size $20\times 20\times 6.9$ [19].

Figure 2

(a) Bifurcation diagram ($a=1.1$, $b=0.19$, $c=0.02$, $D_v=0.1$): the amplitude (upper panel) and precessing frequency (lower panel) of the straight and buckled scrolls. (b) The corresponding translational branch: real part (upper panel) and imaginary part (lower panel). For the meandering mode, $\mathrm{Re}(\lambda )<-0.24$ [19].

Figure 3

(a) Bifurcation diagram (buckling amplitude) and (b) translational branch (real part), for $a=1.1$, $b=0.19$, $c=0.02$, $D_v=0$. (c, d) Same, for $a=1.1$, $b=0.19$, $c=0.02$, $D_v=0.25$. For the meandering modes, $\mathrm{Re}(\lambda )<-0.098$ and $-0.32$, respectively [19].

Figure 4

Development of (a) autowave turbulence ($a=1.1$, $b=0.19$, $c=0.02$, $D_v=0$) and (b) wrinkled scroll as restabilized solution after 3D meandering bifurcation ($a=0.66$, $b=0.01$, $c=0.025$, $D_v=0$, which corresponds to the leftmost point of Fig. 10(a) in Ref. 18). Wave fronts are cut out by clipping planes halfway through the volume, to reveal the filaments. Curves on the right are real parts of rotational, translational and meandering eigenvalue branches of ${\widehat{\mathbf{L}}}_{k_z}$ from Eq. (2).