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.

Figure 1

Distribution of 500 000 phototactic particles after relaxation of transients. Top: $\chi =0.15$. Bottom: $\chi =0.05$. Light distribution is shown in yellow (shaded light gray region).

Figure 2

Top: Gain in light exposure. Squares—numerical results, solid line—theoretical approximation, dashed line—first order theoretical approximation. Bottom: Information dimension. Squares—values calculated from distribution by weighted box count [16], dashed line—results from numerically calculated Lyapunov exponents, solid line—theoretical model.

Figure 3

Radial plot of probability density function (solid line) and numerical results (points) for average particle distribution, $\chi =0.05$. Inset: Scatter plot of predicted density (${\overline{\rho }}_T$) to numerically observed density (${\overline{\rho }}_N$) for range of $\chi$ from 0.025 to 0.15. Points lying on the diagonal represent exact match between effective diffusion model and numerics.

Figure 4

Lyapunov exponents and dissipation rate. Symbols represent numerical values found by the Benettin method, line values calculated from the theoretical approximation defined by Eqs. (11, 12). Squares and solid line represent ${\lambda }_1$; crosses and heavy dashed line represent $\alpha$; asterisks and fine dashed line represent ${\lambda }_2$.