Influence of the medium's dimensionality on defect-mediated turbulence

Spatiotemporal chaos in oscillatory and excitable media is often characterized by the presence of phase singularities called defects. Understanding such defect-mediated turbulence and its dependence on the dimensionality of a given system is an important challenge in nonlinear dynamics. This is especially true in the context of ventricular fibrillation in the heart, where the importance of the thickness of the ventricular wall is contentious. Here, we study defect-mediated turbulence arising in two different regimes in a conceptual model of excitable media and investigate how the statistical character of the turbulence changes if the thickness of the medium is changed from (quasi-) two- dimensional to three dimensional. We find that the thickness of the medium does not have a significant influence in, far from onset, fully developed turbulence while there is a clear transition if the system is close to a spiral instability. We provide clear evidence that the observed transition and change in the mechanism that drives the turbulent behavior is purely a consequence of the dimensionality of the medium. Using filament tracking, we further show that the statistical properties in the three-dimensional medium are different from those in turbulent regimes arising from filament instabilities like the negative line tension instability. Simulations also show that the presence of this unique three-dimensional turbulent dynamics is not model specific.

Figure 1

2D bifurcation diagram of the modified Bär model for $D_v=D_u=1$ and $a=0.84$. The Hopf and transcritical bifurcation lines were computed analytically, the saddle-loop line was computed numerically, whereas all the other lines have been estimated from simulations of 1D (backfiring) and 2D (spiral breakup) systems. The dashed line with dots highlights the main set of simulations discussed in this paper.

Figure 2

(a) Snapshots of typical $u$-field patterns in 2D simulations (showing a subsystem with $L=102$ for better readability) for varying control parameter $e$ along the dashed line shown in Fig. 1. The vortices are shown as red and black circles. (b) Two views of a single snapshot of 3D filament turbulence in the modified Bär model for $e=11.0$: with a surface field showing that the filaments are an extension of surface defects and uniquely labeled filaments as indicated by the different colors. All filaments are tracked in time during the simulation allowing us to record different types of events such as the generation of new filaments.

Figure 3

Scaled average vortex and surface defect densities $\tilde{n}=\langle n\rangle \times A_{\perp }/A$ with $A_{\perp }={102}^2$ as a function of the system's size at $e=11.0$ (left panel) and as a function of the control parameter $e$ for $L=204$ in 2D and $L_z=76.4$ in 3D (right panel). 2D (3D) system size is characterized by $L$ ($L_z$). In all cases, $A$ is the total surface of the medium $A=2L^2+4L{L}_z$ and $L=L_z$ for the 3D-cube cases and $L_z=0$ for the 2D cases. $A_{\perp }$ is simply a convenient scaling factor that plays no fundamental role. 2D systems are all $L$ squares with periodic (P) or no-flux (NF) boundary conditions, whereas there are two sets of 3D systems: $L$ cubes, and slabs of fixed cross section $A_{\perp }$ and varying thickness $L_z$. Note that the slabs interpolate between the 2D system with no-flux boundary conditions and the 3D cube.

Figure 4

Scaled average defect rate density ${\tilde{r}}_n={\langle r\rangle }_n\times A_{\perp }/A$ for the same systems and cases as in Fig. 3.

Figure 5

Surface turbulence statistics with (a) varying control parameter $e$ for a set of 2D systems ($L=204.6$ with periodic boundary conditions) and thick 3D slabs [cross section $A_{\perp }={102}^2$ and thickness $L_z=76.4$, corresponding to the dashed line in (b)] and (b) varying thickness $L_z$ of the medium at $e=11.0$. All 3D simulations were performed with no-flux boundary conditions. The rightmost dashed line in (a) indicates the spiral breakup (bu). Notice how the ratio and constant component seem to collapse on the left side of the backfiring instability (bf) for $e<9.0$. The observed deviations from ${\gamma }_n^{\text{3D}}=1$ are likely accounted for by the finite size effect in the 3D simulation since 2D systems with the same cross section surface showed qualitatively the same deviations. Consequently, the no-flux boundary conditions have a considerable influence on ${\gamma }_n$ for the thin, and relatively small, 3D systems. The sequence of 2D systems (down triangles), which corresponds to system sizes $L=102$, $153.2$, $204.4$, $306.8$, and $409.2$, respectively, from top to bottom in (b), reconciles the value observed at $e=11.0$ in (a).

Figure 6

(a) Decomposition of the surface defect creation rate based on the filament analysis far from onset at $e=6.5$ and (b) close to onset $e=11.0$ with (c) and (d) the corresponding annihilation rate decompositions. The creation and annihilation rate fits are shown as solid lines and are superimposed on both graphs to facilitate comparison. The annihilation rates in both cases seem to follow a linear plus quadratic relation.

Figure 7

Surface creation rate decomposition in two paradigmatic models and regimes. (a) Amplitude turbulence in the CGLE ($b=1.0$ and $c=-1.5$ corresponding to the model presented in [16]) where all components have a significant constant contribution to the overall rate. (b) Filament turbulence in the Barkley model after onset of the NLT instability ($\varepsilon =0.02$, $a=1.1$, $b=0.19$, $D_u=1$, and $D_v=0$ corresponding to the model presented in [15]) where only filament splitting contributes significantly to the generation of new surface defect pairs, at a rate directly proportional to the number of existing surface defect pairs.

Figure 8

Surface turbulence statistics in the CGLE of Ref. [16] with varying control parameter $c$ at fixed $b=5.0$ for a set of 2D systems ($L=512$ with periodic boundary conditions) and 3D cubes ($L=L_z=256$ with no-flux boundary conditions). The dotted vertical line shows the onset of amplitude turbulence (AT) in 2D and the dashed line marks the absolute Eckhaus instability (EI).