Phototactic Clustering of Swimming Micro-organisms in a Turbulent Velocity Field

We study the distribution of swimming micro-organisms advected by a model turbulent flow and attracted towards a localised 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 phototactic gain (increase in light exposure over the aggregate) and by extension an accurate prediction for the fractal dimension based on the properties of the advection dynamics and the statistics of the attracting field. This shows that the fractal characteristics of the aggregate are determined by the non-dimensional ratio of the kinetic energy of swimming to that of the turbulent flow.