Persistence Problem in Two-Dimensional Fluid Turbulence

We present a natural framework for studying the persistence problem in two-dimensional fluid turbulence by using the Okubo-Weiss parameter $\Lambda$ to distinguish between vortical and extensional regions. We then use a direct numerical simulation of the two-dimensional, incompressible Navier-Stokes equation with Ekman friction to study probability distribution functions (PDFs) of the persistence times of vortical and extensional regions by employing both Eulerian and Lagrangian measurements. We find that, in the Eulerian case, the persistence-time PDFs have exponential tails; by contrast, this PDF for Lagrangian particles, in vortical regions, has a power-law tail with an exponent $\theta =2.9\pm 0.2$.

Figure 1

(a) A pseudocolor plot of the stream function $\psi$, at a representative time in the statistically steady state, with a representative Lagrangian-particle track (blue squares) superimposed on it from our run R2. The symbol ○ indicates the beginning of the trajectory and the $\times$ sign marks its end. For an animated version, see the movie file at 16. (b) log-log (base 10) plot of the energy spectrum for our run R4 (line with dots); the black line with a slope $-3.6$ is shown for reference.

Figure 2

(a) Representative semilog plots (base 10) of the cumulative persistence-time PDFs $Q_E^{+}$ (red crosses), $Q_E^{-}$ (black open circles), and $Q_L^{-}$ (magenta full circles); the subscripts $E$ and $L$ signify Eulerian and Lagrangian frames, respectively; and the superscripts $+$ or $-$ distinguish PDFs from vortical points from those from extensional ones. The dashed lines indicate exponential fits to these cumulative PDFs. These plots use data from our run R4. (b) A representative log-log plot (base 10) of the cumulative PDF $Q_L^{+}(\tau )$ (●) versus $\tau$ for our run R4; the full red line, with a slope equal to $-2$, is drawn for reference. The vertical arrows indicate the time scales (from left to right) $T_{\eta }$, $T_{\mathrm{inj}}$, and $T_{\mathrm{cf}}$, respectively. The inset shows the local slope $\chi =d{log}_{10}{Q}_L^{+}(\tau )/d{log}_{10}(\tau )$ versus $\tau$; the horizontal line is drawn at the mean $⟨\chi ⟩\simeq -1.93$ and the black dashed lines, drawn at $⟨\chi ⟩\pm {\sigma }_{\chi }$, where ${\sigma }_{\chi }\simeq 0.2$ is the standard deviation of $\chi$. (c) Plots of the autocorrelation function of the Okubo-Weiss parameter $C_{\Lambda }(t/T_{\eta })$ (see text) versus $t/T_{\eta }$ in Eulerian (blue open circles) and Lagrangian (red full circles) frames for our run $\mathtt{R}\mathtt{1}$. In the inset we compare the data points for these plots with their fits (full lines) to the form $G\exp [-(t/T_{\Lambda }{)}^2]$.