Power-law distributions of particle concentration in free-surface flows

Particles floating on the surface of a turbulent incompressible fluid accumulate along string-like structures, while leaving large regions of the flow domain empty. This is reflected experimentally by a very peaked probability distribution function of $c_r$, the coarse-grained particle concentration at scale $r$, around $c_r=0$, with a power-law decay over two decades of $c_r$, $\mathrm{\Pi }(c_r)\propto c_r^{-{\beta }_r}$. The positive exponent ${\beta }_r$ decreases with scale in the inertial range, and stays approximately constant in the dissipative range, thus indicating a qualitative difference between the dissipative and the inertial ranges of scales, also visible in the first moment of $c_r$.