Defect-Mediated Turbulence in Systems with Local Deterministic Chaos

Defect-mediated turbulence is shown to exist in media where the underlying local dynamics is deterministically chaotic. While many of the characteristics of defect-mediated turbulence, such as the exponential decay of correlations and a squared Poissonian distribution for the number of defects, are identical to those seen in oscillatory media, the fluctuations in the number of defects differ significantly. The power spectra suggest the existence of underlying correlations that lead to a different and nonuniversal scaling structure in chaotic media.

Figure 1

Left: projection of the chaotic attractor in the $(c_1,c_2)$ plane of the homogeneous WR model for ${\kappa }_1=31.2,{\kappa }_{-1}=0.2,{\kappa }_2=1.45,{\kappa }_{-2}=0.072,{\kappa }_3=10.8,{\kappa }_{-3}=0.12,{\kappa }_4=1.02,{\kappa }_{-4}=0.01,{\kappa }_5=16.5,{\kappa }_{-5}=0.5$. Snapshots of the inhomogeneous $\mathbf{c}$ field in the defect turbulent state closely resemble this attractor. Center: phase field for $L=128$ [16]. Right: normalized correlation function in the defect-mediated turbulent state for $L=128$ and different values of ${\kappa }_{-2}$.

Figure 2

Left: $n(t)$ for the parameters given in Fig. 1 and $L=128$. Right: normalized histogram $h(n)$. The solid curve is the corresponding squared Poissonian distribution.

Figure 3

Left: $c(n)$ in the WR system for the parameters given in Fig. 1 and $L=128$. The solid line corresponds to a constant creation rate. Right: $a(n)$ in the WR system for the same parameters. The solid line corresponds to an increase in the annihilation rate proportional to $n^2$.

Figure 4

Left: power spectrum of $n(t)$ for different parameters in WR. For clarity, the curve for ${\kappa }_{-2}=0.075$ has been shifted down by one decade. The thick lines are to guide the eye. The thick solid line decays with $\gamma =1.43$ and the thick dotted line with $\gamma =1.60$. Center: power spectrum of $n(t)$ for different parameters in the CGLE and $L=128$. The thick line is to guide the eye and decays with $\gamma =1.9$. Right: normalized pair correlation function $h(|\mathbf{\Delta }\mathbf{r}|/r_0)$ with $r_0=\sqrt{L^2/⟨n⟩}$.