Emergence of phytoplankton patchiness at small scales in mild turbulence

Abstract

Phytoplankton often encounter turbulence in their habitat. As most toxic phytoplankton species are motile, resolving the interplay of motility and turbulence has fundamental repercussions on our understanding of their own ecology and of the entire ecosystems they inhabit. The spatial distribution of motile phytoplankton cells exhibits patchiness at distances of decimeter to millimeter scales for numerous species with different motility strategies. The explanation of this general phenomenon remains challenging. Furthermore, hydrodynamic cell-cell interactions, which grow more relevant as the density in the patches increases, have been so far ignored. Here, we combine particle simulations and continuum theory to study the emergence of patchiness in motile microorganisms in three dimensions. By addressing the combined effects of motility, cell-cell interaction, and turbulent flow conditions, we uncover a general mechanism: The coupling of cell-cell interactions to the turbulent dynamics favors the formation of dense patches. Identification of the important length and time scales, independent from the motility mode, allows us to elucidate a general physical mechanism underpinning the emergence of patchiness. Our results shed light on the dynamical characteristics necessary for the formation of patchiness and complement current efforts to unravel planktonic ecological interactions.

Keywords: patchiness; phytoplankton; turbulence.

Fig. 1.

(A) Nonequilibrium phase diagram of point-like swimming cells immersed in a Kraichnan turbulent flow field. As the reorientation of the swim direction due to cell–cell interactions increases its strength with respect to the turbulent vorticity (i.e., increasing vortical Stokes number ${\mathcal{S}}_{\omega }$), small-scale patchiness emerges. We calculate the dependence of the patchiness factor $Q$ on ${\mathcal{S}}_{\omega }$ and the Péclet number $\mathcal{P}$ for $N=\mathrm{27,000}$ cells at a number density $\rho {ϵ}^3=4\times 10^{-3}$. For $\mathcal{P}>1$, motile cells swimming in a turbulent field and interacting with each other exhibit a maximum in the patchiness, as ${\mathcal{S}}_{\omega }$ increases. When we consider the ratio $\zeta$ of characteristic turbulent to active length scales, we observe emergence of strong patchiness at $\zeta \approx 1$. The lines indicate the loci $\mathcal{P}=1$ (dotted) and $\mathcal{P}={\mathcal{S}}_{\omega }$ (solid) which serve as boundaries for the region where emergence of patchiness is expected (main text). (B) The maximum in patchiness does not depend on the microscopic details of the interaction, as the emergence of patchiness is robust upon replacing point-like with extended particles (both cases for $\mathcal{P}\to \infty$); nor does the maximum in patchiness depend on the details of the flow field.

Fig. 2.

(A) Typical steady-state configuration of the motile cells in the Navier–Stokes flow field (at ${\mathcal{S}}_{\omega }=86.52$). Cells are shown as arrows whose color indicates the normalized local density (obtained from a Voronoi tessellation) and whose orientation indicates the local alignment. The background (red) color represents the value of the normalized turbulent vorticity. The elongated, dark red regions correspond to the vortex cores moving through the system. (B) Close-up of the system where the clusters are discernible.

Fig. 3.

Dependence of the maximum patchiness factor $Q$ on the ratio of correlation length scale of the vorticity and cell–cell interaction range $L_{\omega }/ϵ$. In all these calculations of the Kraichnan flow, we set $k_{max}/k_{min}=63$ and $\mathcal{P}\to \infty$. The patchiness increases as $L_{\omega }/ϵ$ increases. This behavior is robust upon variation of the average filling fraction (solid symbols, $\rho {ϵ}^3=0.41$; open symbols: $\rho {ϵ}^3=0.041$). Furthermore, this conclusion is not altered upon variation of the ratio of length scales—either by changing the cell–cell interaction range $ϵ$ (blue stars) or by changing $L_{\omega }/ϵ$ via a shift of the minimum wavenumber of the turbulent flow (circles). Keeping a fixed (yellow) or variable (red) turbulent kinetic energy also shows the same qualitative result.