Emergence of Local Ordering and Mesoscale Giant Number Fluctuations in Active Turbulence

Abstract

We study spatiotemporal chaos in two-dimensional dense active suspensions using a generalized hydrodynamic model. Increasing activity induces a structural transition marked by the formation of intense vortices and giant number fluctuations at the mesoscale. The flow self-organizes into locally polar-ordered regions coexisting with chaotic domains, producing a bimodal velocity distribution and enhanced correlations. This mixed-state morphology underlies the universal statistical behavior observed beyond a critical activity threshold. Reducing the instability timescale yields similar transitions, showing that both activity and instability act as control parameters for pattern formation. An energy-based order parameter derived from the system's budget quantifies and unifies these structural transitions across the phase space of activity and instability timescales.

Figures (11)

• Figure 1: The vorticity field ($\omega$) is shown for two different activity levels: (a I) $\alpha = -1$, and (a II) $\alpha = -9$ at $t =85\tau_{\Gamma}$ with $\tau_{\Gamma}=0.4$. The scale bar in both panels corresponds to $25\Lambda$. Panels (a III) and (a IV) present zoomed-in sections of the respective vorticity fields, with magnified regions chosen to be of the same order as the scale bar. The vorticity snapshot at higher activity displays spatial inhomogeneities, characterized by intense vortex regions coexisting with areas of negligible vorticity. The color bar is normalized by the maximum vorticity $\omega_0$ of the respective $\alpha$. (b I) and (b II) display the velocity streamlines superimposed on the velocity magnitude within a subdomain of size $25\Lambda$, corresponding to the regions marked by the red and green boxes in (a II). (c) Standard deviation $\Delta n$ of the number of vortex centers within subregions is plotted against the corresponding mean $\bar{ n}$ as the subregion size $\ell$ increases. The scaling behavior $\Delta n \sim \bar{n}^\delta$ reveals the presence of giant number fluctuations for $\alpha < \alpha_c$. Inset: the scaling exponent $\delta$ is plotted as a function of the activity parameter $\alpha$, showing that fluctuations become increasingly pronounced (larger $\delta$) with higher activity.
• Figure 2: (a) The normalized velocity correlation length $\zeta/\Lambda$ versus $\alpha$. Inset: The spatial velocity correlation function, defined as $C_v(r) = \langle \mathbf{u}(\mathbf{r}_0) \cdot \mathbf{u}(\mathbf{r}_0 + \mathbf{r}) \rangle$ is shown as a function of distance $r$ for increasing values of $|\alpha|$. $\zeta$ is extracted by fitting the decay of $C_v(r) \sim e^{-r/\zeta}$. For $\alpha > \alpha_c$, $\zeta/\Lambda \sim 1$, whereas it increases significantly for $\alpha < \alpha_c$, indicating the emergence of long-range velocity correlations in the high-activity regime. (b) Probability distribution function of the normalized velocity component $u_x/\sqrt{|\alpha|/\beta}$ for different activity values $\alpha$. A transition from a unimodal to a bimodal distribution is observed as $\alpha$ decreases beyond the critical point $\alpha_c$. (c) The order parameter $\phi = E_{\rm active} - E_{\Lambda} - E_{\rm kin}$ is plotted as a function of $\alpha$ for three values of $\tau_\Gamma$ obtained by varying $\Gamma_2$. The baseline case with $\tau_{\Gamma} = 0.4$ ($\Gamma_2 = 9.0\times10^{-5}$, as specified in the parameters) is shown by circles, while increased and decreased $\tau_{\Gamma}$ are represented by triangles and squares, respectively. For intermediate and larger $\Gamma_2$, $\phi$ remains positive below the corresponding critical activity $\alpha_c$ and becomes negative above it, marking the transition. In contrast, for the smallest $\Gamma_2$, $\phi$ stays negative across all $\alpha$, and no transition point $\alpha_c$ is observed.
• Figure 3: The energy spectra by varying the instability growth timescale $\tau_{\Gamma}$ by varying $\Gamma_0$ for fixed $\alpha = -4$, $\Gamma_2=9\times10^{-5}$, $\beta=0.5$ and $\lambda_0=3.5$. Inset: The corresponding energy-based order parameter $\phi$ is plotted as a function of $\tau_{\Gamma}$, showing a sign reversal that is consistent with a transition.
• Figure S1: (a) $u_{rms}$ increases with activity, (increasing $-\alpha$). (b)Energy spectra for different $\alpha$, showing power scaling for $\alpha < \alpha_c$.
• Figure S2: (a) Velocity streamlines overlaid on the velocity magnitude are shown for $\alpha = -1$ in a $25\Lambda \times 25\Lambda$ domain, highlighting flow structures predominantly governed by the length scale $\mathcal{O}(\Lambda)$. The color bar is normalized by $u_{\rm rms}$. (b) The time-averaged number of identified vortex centers as a function of $\alpha$, showing a decrease in the number of distinct vorticity centers with increasing activity. The shaded region represents the fluctuations (standard deviation) over time.