Multifractal clustering of passive tracers on a surface flow

– We study the anomalous scaling of the mass density measure of Lagrangian tracers in a compressible flow realized on the free surface on top of a three dimensional flow. The full two dimensional probability distribution of local stretching rates is measured. The intermittency exponents which quantify the fluctuations of the mass measure of tracers at small scales are calculated from the large deviation form of stretching rate fluctuations. The results indicate the existence of a critical exponent nc ≃ 0.86 above which exponents saturate, in agreement with what has been predicted by an analytically solvable model. Direct evaluation of the multifractal dimensions by reconstructing the coarse-grained particle density supports the results for low order moments. Advection of homogeneous distribution of passive particles in a compressible flow generically results in clusters of particles’ concentration [1–3]. The effects of compressibility in three-dimensional flows are typically relevant only at Mach numbers near or larger than 1 when density inhomogeneity reacts back on the fluid velocity. However there are simple physical systems in which the phenomenology of passive particle advection in compressible flows is applicable. Effective flows of inertial particles in incompressible flows at small Stokes numbers [4–6] and flows of surface suspensions [7–9] are among such instances. Consider the distribution of particles advected by a velocity field u(x, t) assumed homogeneous and stationary in time. The evolution of particle density ρ(x, t) in the limit of vanishing diffusivity is given by ∂ρ ∂t +∇ · (ρu) = 0 (1) If ∇ · u 6= 0 the flow is compressible and particle trajectories asymptotically converge to a non-uniform distribution. Compressibility can be characterized by the dimensionless ratio C = 〈(∂iui) 〉 〈(∂iuj) 〉 which assumes values in 0 ≤ C ≤ 1, the limiting values corresponding to an incompressible and a potential flow respectively. The particle density depends on the particular realization of the velocity field and for a stationary random flow the density is expected to converge to a statistically stationary state ρ∗.