Caustic formation in turbulent aerosols at weak particle inertia

Caustic singularities of the spatial distribution of particles in turbulent aerosols increase collision rates and accelerate coagulation. The dominant mechanisms of caustic formation depend sensitively on the dynamical time scales of the particle motion and of the turbulent flow. For strongly inertial particles in rapidly fluctuating flows, caustic formation is described by a Kramers escape of the particle-velocity gradients. Weakly inertial particles in persistent flows have recently been studied in two spatial dimensions, where caustics are induced by rare excursions of the fluid strain, while vorticity remains small. Here we show that in three dimensions, caustics form by a related, yet different mechanism. As in two dimensions, caustics are induced by a large, strain-dominated excursion of the fluid-velocity gradients. In three dimensions, however, the excursion occurs in the $Q$-$R$-plane, where it must exceed a characteristic threshold line. The most likely way to reach this threshold is by an ``optimal fluctuation'' that propagates along the Vieillefosse line and is unique up to similarity transformations. We explicitly determine the probability and shape of the optimal fluctuation, and find that it is dominant in numerical simulations even for moderate particle inertia.