Anomalous velocity fluctuation in one-dimensional defect turbulence

In this paper various eccentric hole dynamics are presented in defect turbulence of the one-dimensional complex Ginzburg-Landau equation. Each hole shows coherent particlelike motion with nonconstant velocity. On the other hand, successive hole velocities without discriminating each hole exhibit anomalous intermittent motions being subject to multi-time-scale non-Gaussian statistics. An alternate non-Markov stochastic differential equation is proposed, by which all these observed statistical properties can be described successfully.

Figure 1

Spatiotemporal profiles of amplitude $\left|A\right|$ and phase arg $\left(A\right)$ in the whole system. In the amplitude profile, black lines indicate trajectories of locally depressed amplitude. In the phase profile, slits correspond to phase discontinuities.

Figure 2

Snapshots of (a) a defect and (b) a hole. Black lines and white circles are amplitude and phase, respectively. The defect has a phase singularity at its core, whereas the hole does not have the same feature.

Figure 3

Sample hole trajectories with their velocity diagrams. The hole trajectories in (a1) and (b1) were obtained by the smoothing with the weighting function. The hole velocities in (a2) and (b2) were evaluated by the local regression at each point in (a1) and (b2), respectively.

Figure 4

Hole velocity fluctuation obtained from the successive hole velocities without discriminating each hole in the whole space at each time.

Figure 5

Probability distribution for the hole velocity fluctuation. The black circles are estimated from the hole velocity fluctuation with the help of the Rosenblatt-Parzen density estimator. The solid line denotes the generalized Cauchy distribution with the parameters $(a,b)=(0.709,2.179)$ estimated by the LSM, of which mean-square error (MSE) is $5.12\times {10}^{-3}$. The dashed line denotes the Gaussian distribution with the mean and the variance estimated from the hole velocity fluctuation.

Figure 6

Autocorrelation coefficients and function for the hole velocity fluctuation. The black circles denote the autocorrelation coefficients for the hole velocity fluctuation estimated from the simulated time series. The solid line denotes the ACF of the NSGCP in Eq. (14) with the estimated parameters $\gamma =0.429$, $D_a=0.064$, $D_m=0.128$, $K=0.112$, and $\eta =8.4\times {10}^4$. The MSE is $4.64\times {10}^{-5}$.

Figure 7

MSD for the hole velocity fluctuation. White circles denote the MSD obtained by the time averaging in Eq. (3). Solid lines approximate the obtained MSD with power-law exponent as $t^{\alpha }$: $\alpha =1.53$ for $t\in [{10}^{-3},{10}^{-2}]$, $\alpha =1.96$ for $t\in [{10}^{-2},{10}^1]$, and $\alpha =1.05$ for $t\in [{10}^1,{10}^3]$. The MSE for the time scales are as follows: $1.69\times {10}^{-4}$ for $t\in [{10}^{-3},{10}^{-2}]$, $6.71\times {10}^{-5}$ for $t\in [{10}^{-2},{10}^1]$, and $2.00\times {10}^{-2}$ for $t\in [{10}^1,{10}^3]$.

Figure 8

Sample path of the NSGCP generated from the recurrence formula in Eq. (15) with estimated parameters $\gamma =0.429$, $D_a=0.064$, $D_m=0.128$, $K=0.112$, and $\eta =8.4\times {10}^4$.

Figure 9

Theoretical MSD calculated with the representation in Eq. (16). The diffusive behavior changes from superdiffusion with $\alpha =1.74$ to nearly normal diffusion $\alpha =1.01$. The MSE are estimated as $9.32\times {10}^{-4}$ $(t<{10}^1)$ and $1.81\times {10}^{-4}$ $(t<{10}^1)$.