Gyrotaxis in a Steady Vortical Flow

We show that gyrotactic motility within a steady vortical flow leads to tightly clustered aggregations of microorganisms. Two dimensionless numbers, characterizing the relative swimming speed and stability against overturning by vorticity, govern the coupling between motility and flow. Exploration of parameter space reveals a striking array of patchiness regimes. Aggregations are found to form within a few overturning time scales, suggesting that vortical flows might be capable of efficiently separating species with different motility characteristics.

Figure 1

(a) Gyrotactic microorganisms, such as the toxic marine phytoplankton Heterosigma akashiwo shown here (diameter $\approx 14\mu \mathrm{m}$), swim in a direction, $\theta$, set by a balance of torques. The torque due to flow ($T_F$) tends to rotate the cell, whereas the torque due to cell asymmetry ($T_A$)—for example bottom heaviness—tends to restore the cell to its preferential orientation, $\mathbf{k}$. $V_C$ is the swimming speed. (b) Three cells, with different $\Psi$ and $\Phi$, initialized at the same location and orientation ($x=z=-\pi /2$; $\theta =\pi /4$; white arrow) in a TGV flow follow very different trajectories. $(\Psi ,\Phi )=(0.1,20)$ [light gray (green)], $(1,0.2)$ [medium gray (red)], $(100,0.5)$ [dark gray (blue)]. The TGV velocity and vorticity fields are shown by arrows and by shading, respectively. The domain is doubly periodic. (c) The most intense cell accumulation occurs when cells converge to equilibrium points, where total cell velocity ${\mathit{V}}_T=(dx/dt,dz/dt)=(0,0)$. Shown here is the “equilibrium double” regime ($\Psi =1.1$, $\Phi =0.25$). White crosses are numerical predictions of the equilibria, small pink circles are analytical results. Arrows and shading show ${\mathit{V}}_T$ and $|V_T|$, respectively, assuming cell orientation is static. Large red circles denote regions where vorticity can overturn cells ($\omega \Psi >1$).

Figure 2

(a) Parameter space of gyrotactic swimming in TGV flow, showing different patchiness regimes. Each square represents one of 900 simulations. In each simulation the trajectories of 400 randomly initialized cells were integrated until $t=2000$. Ten distinct patterns emerge (b)–(k), not including the cases in which accumulation does not occur (m) or has not converged (l). For the “equilibrium” regimes (b),(c), all cells reside at the equilibrium points. The symbols in (a) correspond to $(\Psi ,\Phi )$ values that are analyzed in Fig. 4b to investigate the role of cell elongation.

Figure 3

(a) The degree of patchiness at time $t=10$, quantified by the accumulation index $N$. $N>0$ indicates aggregation. (b) The time, $\tau$, required for cells to reach a time-invariant spatial pattern. (c) The normalized vertical migration rate $W$.

Figure 4

(a) Gyrotactic cells form aggregations for almost any vortex orientation. Plotted here is the accumulation index $N$, at $t=10$ and $\Phi =\Psi =1$, as a function of the polar and azimuthal angles, $\eta$ and $\beta$. (b) Cell elongation (aspect ratio $\gamma >1$) produces an increase in patchiness in most regimes, compared to the case of spherical cells ($\gamma =1$). Shown is the change in the accumulation index, $N_E-N$, due to elongation, at $t=2000$. Symbols correspond to $\Phi ,\Psi$ values and in Fig. 2a. Representative cell distributions can be found in Fig. S1 in 13.