Active Caustics

TL;DR

The paper demonstrates that caustics—singular concentration events previously associated with inertial particles—can arise in inertialess, self-propelled swimmers due to a formal correspondence with inertial-particle dynamics in a vortical background. By combining a singular perturbation analysis near a point vortex, two minimal active-dimer models (Hookean and preferred-length), and direct numerical simulations in 2D turbulence, the authors show caustic formation at intermediate activity levels, with a characteristic inner-region scaling $r\sim t^{2/3}$ and density peaks that drive burst-like encounters. The results reveal a universal mechanism for multivalued velocity fields and enhanced near-neighbor collisions in active suspensions, with potential ecological implications for marine microorganisms and reproduction. This framework provides a minimal, general route to extreme local concentrations in macroscopically dilute active matter and offers a basis for exploring caustics in three-dimensional and more complex active flows.

Abstract

Inertial particles (IPs) in vortical fluid flow cluster strongly, forming singular structures termed caustics for their resemblance to focal surfaces in optics. Here we show that such extreme aggregation onto low-dimensional submanifolds can arise without mechanical inertia for self-propelled particles (SPPs), through a formal correspondence between the dynamics of IPs and SPPs in a generic background flow. We establish that a singular perturbation underlies caustics formation by SPPs around a single vortex, and numerical studies of SPPs in two-dimensional Navier-Stokes turbulence reveal intense caustics in straining regions of the flow, peaking at intermediate levels of self-propulsion. Our work offers a route to singularly high local concentrations in a macroscopically dilute suspension of zero-Reynolds-number swimmers. Caustics generate burst-like encounters through large relative velocities between neighboring swimmers, with potentially significant implications for communication and sexual reproduction. An intriguing open direction is whether the active turbulence of a suspension of swimming microbes could serve to generate caustics in its own concentration

Figures (9)

• Figure 1: Active dimer model for self-propelled particles in ambient flow (a) We depict typical values of the ratio of flow time scale $1/\norm{\nabla \mathbf{U}}$ to swimming time scale $1/\beta$, and Reynolds number $Re$ for a marine bacterium Vibrio alginolyticusdoi:10.1073/pnas.1602307113, various dinoflagellates, ciliates Lauga2019, invertebrate larvae FUCHS2016109 and copepods FUCHS2016109. For $\norm{\nabla \mathbf{U}}$ we substitute the Kolmogorov shear rate for the range of previously measured energy dissipation rates in the upper mixed layer of the ocean $10^{-8} - 10^{-6}$ m$^{2}$s$^{-3}$StockerARFM2012 [see Appendix \ref{['plankton']}]. (b) Schematic of an active dimer of extension $\mathbf{w}$ in a flow $\mathbf{U}$. (c) Caustics based on the inner solution are marked by the intersection of representative trajectories (blue circles) of particles starting at closely separated initial radial distances, with $\alpha = 0.1$. A continuous variation in $\tilde{r}_0$ would give a continuous curve. The inset (Photo credit: R. Chajwa) is an image of optical caustics with a similar cusp on the surface of coffee in a mug.
• Figure 2: Comparison between asymptotic analysis and full numerical solution:(a,b) Comparing trajectories obtained analytically from the inner solution in \ref{['eqn:5.13b']} (dashed lines), and from evolving the full system following \ref{['eqn:rLeqns']} (curves). Both the radial velocity in (a) and angular momentum in (b) follow the analytical solution till particles reach $O(1)$ distance from the vortex origin, where the inner solution is no longer valid. (c,d) The outer solution in black dotted lines following \ref{['eqn:osol']} also matches well with the numerical solution of the full system at late times. At such times, particles are far away from the vortex origin (c), and their angular momentum saturates to the background fluid angular velocity (d).
• Figure 3: Centrifugation and caustics of a Hookean dimers in point vortex: (a) Time frames showing the positions of the particles (blue dots) around a point vortex at the origin for IP and noiseless AOUP, by numerically solving the Maxey-Riley equation, and \ref{['eqn:dimerX']} & \ref{['eqn:dimerw']}, respectively [see Appendix \ref{['videos']} Video 1]. $N=10^{5}$ particles were initialised with uniformly random initial positions and orientations/velocities. (b) An initially homogeneous number-density (grey circles) peaks near a critical radial distance (green) in the steady state, compared to unsteady density of IP (purple) at a representative $t \gg 1$. (c) Trajectory (rays) of particles averaged over all initial orientations. The envelope of rays, for particles starting at various initial $r$, gives rise to caustics (red circles). The green curve shows $R = t^{\nu}$, with $\nu =2/3$. In (b), (c) & (d), $\alpha=1$. (d) The crossing time of adjacent rays separated by $\Delta r = 0.001$ starting at various radial distances $r$, averaged over uniformly random initial orientations, diverges at a finite critical radius $r_c$ for an active particle (red) and for an inertial particle (blue) averaged over uniformly random initial velocity of unit magnitude. (e) For various $\alpha$, plotting $r_c$ demarcates the region of caustics, which is the radial distance below which adjacent rays cross each other in a finite time.
• Figure 4: (a) Caustics times for particles starting from different initial radii as calculated from equations \ref{['nd_prtclEvol']} (triangles) and \ref{['nd_ZEvol']} (squares). The two definitions of caustics give the same answer. (b) Average final radius for a range of initial radius, for different activity levels $\alpha$, of particles starting near a point vortex.
• Figure 5: Centrifugation and caustics of active preferred-length dimers in a point vortex: (a) & (b) show time frames of particle positions around a point vortex for inertial particles and preferred-length dimers (or ABP without noise) respectively, shown for $\tilde{\beta} = 0.5$ and $\alpha=1.0$ in \ref{['eqn:dimerX5']} - \ref{['eqn:dimerX6']}. (c) radial number-density of active particles (green) compared with inertial particles (purple) shown for time $10\tau$, and the number-density at $t=0$ (grey). Arrows schematically depict the radial drift of the unsteady state. (d) Intersections of adjacent trajectories marking the caustics curve (red circles) obtained for $\tilde{\beta} = 0.5$, $\alpha = 1.0$ and $\Delta r = 0.01$.