  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.
