Nonequilibrium Ribbon Model of Twisted Scroll Waves

We formulate a reduced model to analyze the motion of the core of a twisted scroll wave. The model is first shown to provide a simple description of the onset and nonlinear evolution of the helical state appearing in the sproing bifurcation of scroll waves. It then serves to examine the experimentally studied case of a medium with spatially varying excitability. The model shows the role of sproing in this more complex setting and highlights the differences between the convective and absolute sproing instabilities.

Figure 1

(a) Dispersion relations obtained from a direct linear stability analysis of a twisted scroll wave of Eq. (8) [8], with $\alpha =0.8$, $ϵ=0.025$, and $\beta =0.01$, and (b) from Eq. (9), with $a=0.2+0.2i$, $d=3.5+i$, and $b=2+i$, chosen to give a semiquantitative overall agreement between the two sets of curves. Above a critical twist, $\mathrm{Re}(\Omega )$ is positive for $k$ around a nonzero $k_c$, and the straight scroll becomes unstable to a finite wave number Hopf instability. Since the instability occurs at finite wavelength, a better fit at $k=0$ (using the exact value of $a$ [8]) typically deteriorates the overall fit.

Figure 2

Distributions of twist along a straight filament in simulations of RD equation (8) (solid line) and given by Eq. (10) (dashed line). (a) Jump of excitability obtained by taking $\beta (z)={\beta }_b+({\beta }_t-{\beta }_b)\Theta (z-L/2)$ in Eq. (8) or in the model by taking the allied spiral frequency jump ${\omega }_0={\omega }_b+({\omega }_t-{\omega }_b)\Theta (z-L/2)$. (b) Linear gradient of excitability: $\beta (z)={\beta }_b+({\beta }_t-{\beta }_b)z/L$ in Eq. (8) or ${\omega }_0={\omega }_b+({\omega }_t-{\omega }_b)z/L$ for the model. The parameters are the same as in Fig. 1, with ${\beta }_b=0.01$, ${\beta }_t=0.03$, ${\omega }_b=1.80$, ${\omega }_t=1.696$, and coefficients $D=0.578$, $c=0.720$, at the bottom, and $D=0.614$, and $c=0.856$, at the top, with equivalent expressions. This gives a theoretical prediction of ${\tau }_w^{\max }\simeq (\Delta {\omega }_0/c{)}^{1/2}=0.35$ in (a). Here and in the following, we plot twist in absolute values.

Figure 3

(a) Center filament maximal radius vs ${\beta }_t$. The straight scroll becomes unstable at ${\beta }_t^c\simeq 0.03939$, resulting in a small hysteresis. (b) A 3D view of the solution for ${\beta }_t=0.0399$.

Figure 4

(a) Instability onset vs domain size $L$, for domains with a jump in excitability. For fixed $L$, ${\beta }_t$ is increased until the system becomes unstable. The corresponding critical twist (solid circles) is plotted together with the fit (dashed line): ${\tau }_{\max }^c=0.344+31.3/L^2$. For comparison, the onset of sproing for periodic BC (${\tau }_w^c\simeq 0.29$) is also shown (dotted line), for $\beta =0.03$, that corresponds to the critical value of ${\beta }_t$ in a large system. (b) Solutions for ${\beta }_b=0.01$ and ${\beta }_t=0.031$. (c) Model critical twist obtained by solving the eigenvalue problem vs $L$ for the distribution of twist shown in Fig. 2a (diamonds), a constant twist in half of the system (squares), and vs $L/2$ for a constant twist in the whole system (circles). For comparison, we also show the onset of convective (dotted line), absolute instability (long-dashed line), and the onset obtained with the double root criterion (short-dashed line). (d) Critical modes obtained from the eigenvalue problem, using the distribution of twist, for $L=150$.