Phototactic clustering of swimming microorganisms in a turbulent velocity field

We study the distribution of swimming microorganisms advected by a two-dimensional smooth turbulent flow and attracted towards a light source through phototaxis. It is shown that particles aggregate along a dynamical attractor with fractal measure whose dimension depends on the strength of the phototaxis. Using an effective diffusion approximation for the flow, we derive an analytic expression for the increase in light exposure over the aggregate and by extension an accurate prediction for the fractal dimension based on the properties of the advection and the statistics of the attracting field.