Turbulent Fluid Acceleration Generates Clusters of Gyrotactic Microorganisms

The motility of microorganisms is often biased by gradients in physical and chemical properties of their environment, with myriad implications on their ecology. Here we show that fluid acceleration reorients gyrotactic plankton, triggering small-scale clustering. We experimentally demonstrate this phenomenon by studying the distribution of the phytoplankton Chlamydomonas augustae within a rotating tank and find it to be in good agreement with a new, generalized model of gyrotaxis. When this model is implemented in a direct numerical simulation of turbulent flow, we find that fluid acceleration generates multifractal plankton clustering, with faster and more stable cells producing stronger clustering. By producing accumulations in high-vorticity regions, this process is fundamentally different from clustering by gravitational acceleration, expanding the range of mechanisms by which turbulent flows can impact the spatial distribution of active suspensions.

Figure 1

Spatial distribution of gyrotactic swimmers in a rotating cylindrical vessel (radius 2 cm, volume 50 ml, rotation rate 5 Hz), obtained for (a) cells killed with 8% v/v ethanol, (b) swimming cells, and (c) simulated cells. The white dashed line denotes the axis of rotation and time is measured since the onset of the cylinder’s rotation. (a),(b) A culture of C. augustae ($\sim {10}^5\mathrm{cells}/\mathrm{ml}$) illuminated with a green laser (50 mW) sheet. Brightness increases with cell concentration. (c) ${10}^4$ synthetic swimmers, whose positions were obtained by integrating Eqs. (1) and (2) with flow velocity $\mathit{u}=(-\mathrm{\Omega }y,\mathrm{\Omega }x,0)$, with $\mathrm{\Omega }=10\pi \mathrm{rad}{\mathrm{s}}^{-1}$ and cell parameters $v_C=100\mu \text{m}{\mathrm{s}}^{-1}$, $B=v_o/g=5\mathrm{s}$, and rotational diffusivity $D_r=0.067{\mathrm{rad}}^2{\mathrm{s}}^{-1}$, closely approximating previous estimates for C. augustae [18].

Figure 2

Slice of a 3D turbulent flow, at ${\mathrm{R}\mathrm{e}}_{\lambda }=62$, showing cell clustering (black dots) in high vorticity regions when the stabilizing torque aligns them with the local fluid acceleration [$\mathit{A}=-\mathit{a}$ in Eq. (1)], ${\mathrm{\Psi }}_a=1.5$ and $\mathrm{\Phi }=1$. Shading shows the magnitude of the fluid vorticity relative to the Eulerian average.

Figure 3

Correlation dimension $D_2$ versus stability number ${\mathrm{\Psi }}_g$ for increasing ${\mathrm{R}\mathrm{e}}_{\lambda }$ (and ratio $\alpha =g/a_{\mathrm{rms}}$) at fixed dimensionless swimming speed $\mathrm{\Phi }=1/3$. Semifilled symbols refer to the complete model with $\mathit{A}=\mathit{g}-\mathit{a}$ in Eq. (1) with ${\mathrm{R}\mathrm{e}}_{\lambda }=20$ ($\alpha =0.34$) (orange diamonds), ${\mathrm{R}\mathrm{e}}_{\lambda }=36$ ($\alpha =0.50$) (blue squares), and ${\mathrm{R}\mathrm{e}}_{\lambda }=62$ ($\alpha =0.84$) (red circles). Open (red) circles denote the case where cell orientation is determined by gravity only, $\mathit{A}=\mathit{g}$ at ${\mathrm{R}\mathrm{e}}_{\lambda }=62$. Inset: the generalized dimension $D_q$ as a function of $q$, at ${\mathrm{R}\mathrm{e}}_{\lambda }=62$ when only gravitational ($\mathit{A}=\mathit{g}$, open circles) or fluid acceleration ($\mathit{A}=-\mathit{a}$, filled circles) is considered.

Figure 4

Results of simulations for the model with fluid acceleration only, $\mathit{A}=-\mathit{a}$. (a) Correlation dimension and (b) square vorticity averaged over particle positions $⟨{\omega }^2⟩$ and normalized by the Eulerian value $⟨{\omega }^2{⟩}_E$ as a function of the stability number ${\mathrm{\Psi }}_a$ for different values of ${\mathrm{R}\mathrm{e}}_{\lambda }$ and nondimensional swimming speeds $\mathrm{\Phi }$. Inset of panel (a): the codimension $3-D_2$ as a function of $\mathrm{\Phi }$ at different ${\mathrm{R}\mathrm{e}}_{\lambda }$ and ${\mathrm{\Psi }}_a$ in log-log plot. The straight line shows the theoretical prediction $3-D_2\propto {\mathrm{\Phi }}^2$.