Extended scale invariance in the vortices of freely evolving two-dimensional turbulence

We report the existence of a self-similar scaling in the vortices of freely evolving two-dimensional turbulence: The vortex number density $n(A,t)$ compensated by the mean vortex intensity $\overline{{\omega }_{\text{v}}^2}(A,t)$, where $A$ is vortex area and $t$ is time, follows the self-similar form $n(A,t)/\overline{{\omega }_{\text{v}}^2}(A,t)\sim t^{-2/3}{A}^{-1}$. This extended scale-invariant behavior holds for different initial conditions, despite very different scaling of $n(A,t)$ and $\overline{{\omega }_{\text{v}}^2}(A,t)$ taken separately, and ensures that the number of turnaround times ${\left[\overline{{\omega }_{\text{v}}^2}(A,t)\right]}^{-1/2}$ taken to cross the characteristic intervortex distance is independent of scale.

Figure 1

(a) Coherent vortices on a ${4096}^2$ subdomain from simulation mGauss-CA at $\tau =3667$ as selected by ${\omega }_{\text{thr}}=0.125{\omega }_{\text{rms}}$ and (b) $F\left(A\right)$ as defined in Eq. (9) for two thresholds at the indicated times, with best fit line for ${\omega }_{\text{thr}}=0.125{\omega }_{\text{rms}}$ and $A^{-1}$ scaling for comparison.

Figure 2

(a) mGauss-CA and (c) Gauss-PS intensities $\overline{{\omega }_{\text{v}}^2}$ and normalized number densities $N_{\text{v}}^{-1}n(A,t)$, where $N_{\text{v}}\left(t\right)$ is the total number of vortices, for ${\omega }_{\text{thr}}=0.125{\omega }_{\text{rms}}$ (mGauss-CA) and ${\omega }_{\text{thr}}={\omega }_{\text{rms}}$ (Gauss-PS). (b) mGauss-CA and (d) Gauss-PS vortex peak vorticities $\overline{{\omega }^{\text{ext}}}$ and compensated number densities $N_{\text{v}}^{-1}n\left(A\right)/\overline{{\omega }_{\text{v}}^2}$, with best fit lines $A^{-1.02}$ and $A^{-0.98}$; here $\tilde{r}=1.02\pm 0.04$ (mGauss-CA) and $\tilde{r}=0.98\pm 0.03$ (Gauss-PS), where the error is the standard deviation. Cyan and pink symbols are instantaneous and open black symbols are time-averaged values.

Figure 3

(a) Tophat-CA and (c) Tophat-PS intensities $\overline{{\omega }_{\text{v}}^2}(A,t)$ and normalized number densities $N_{\text{v}}^{-1}n(A,t)$, where $N_{\text{v}}\left(t\right)$ is the total number of vortices, for ${\omega }_{\text{thr}}={\omega }_{\text{rms}}$. (b) Tophat-CA and (d) Tophat-PS compensated number densities $N_{\text{v}}^{-1}n(A,t)/\overline{{\omega }_{\text{v}}^2}(A,t)$ and vortex peak vorticities $\overline{{\omega }^{\text{ext}}}$. Cyan and pink symbols are instantaneous values, while open black symbols are time-averaged values. The error bars on the scaling exponents $r$ and $\tilde{r}$ can be found in Table 3.

Figure 4

Fit range used to compute the best-collapse temporal scaling exponent $\tilde{\alpha }$ for simulations mGauss-CA (left) and Gauss-PS (right), illustrated for extractions with ${\omega }_{\text{thr}}=0.125{\omega }_{\text{rms}}$ (mGauss-CA) and ${\omega }_{\text{thr}}={\omega }_{\text{rms}}$ (Gauss-PS), $e=0.85$, and ${\omega }_{\text{thr}}^{\text{ext}}=2{\omega }_{\text{rms}}$. Points between the vertical dash-dotted lines are used in computing the values $\tilde{\alpha }=0.64\pm 0.07$ (mGauss-CA, left) and $\tilde{\alpha }=0.65\pm 0.06$ (Gauss-PS, right), which best collapse the number densities over these ranges.

Figure 5

(a) Tophat-PS number density $n(A,t)$ and vortex translational speed $U_{\text{v}}$ as functions of $A$ at three times. Also shown are (b) $N_{\text{v}}$ (dotted line), $Z_{\text{v}}$ (dash-dotted line), and $E_{\text{v}}$ (solid line) for the $A^{-1}$ range. The scaling exponents are $-0.65\pm 0.01,-0.34\pm 0.01$, and $-0.03\pm 0.01$. (c) End points of the corresponding comoving interval $[\mu A_0\left(\tau \right),A_0\left(\tau \right)]$ and area $A_{\text{max}}$ of the largest vortex. The scaling exponent for $A_{\text{max}}$ is $0.36\pm 0.03$.

Figure 6

Plot of $N_{\text{v}}$ (dotted line), $Z_{\text{v}}$ (dash-dotted line), and $E_{\text{v}}$ (solid line) for (a) a comoving interval $[\mu A_{\text{typ}}\left(\tau \right),A_{\text{typ}}\left(\tau \right)]$ and (b) a comoving interval $[A_{\text{typ}}\left(\tau \right),A_{\text{max}}\left(\tau \right)]$. The measured scaling exponents are (a) $-0.67\pm 0.01,-0.35\pm 0.03$, and $-0.07\pm 0.05$ and (b) $-0.83\pm 0.02,-0.47\pm 0.01$, and $-0.09\pm 0.01$. (c) Interval end points $\mu A_{\text{typ}},A_{\text{typ}}$, and $A_{\text{max}}$, with $A_{\text{p}}$ for reference and a fit line for comparison. The scaling exponent for $A_{\text{typ}}$ is $0.329\pm 0.005$.