Emergence of phytoplankton patchiness at smallscales in mild turbulenceRebekka E. Breiera, Cristian C. Lalescub, Devin Waasa, Michael Wilczekb, and Marco G. Mazzaa,c,d,1

aDepartment of Dynamics of Complex Fluids, Max Planck Institute for Dynamics and Self-Organization, 37077 Gottingen, Germany; bMax Planck Institutefor Dynamics and Self-Organization, 37077 Gottingen, Germany; cInterdisciplinary Centre for Mathematical Modelling, Loughborough University,Loughborough, Leicestershire LE11 3TU, United Kingdom; and dDepartment of Mathematical Sciences, Loughborough University, Loughborough,Leicestershire LE11 3TU, United Kingdom

Edited by David R. Nelson, Harvard University, Cambridge, MA, and approved October 3, 2018 (received for review May 21, 2018)

Phytoplankton often encounter turbulence in their habitat. Asmost toxic phytoplankton species are motile, resolving the inter-play of motility and turbulence has fundamental repercussions onour understanding of their own ecology and of the entire ecosys-tems they inhabit. The spatial distribution of motile phytoplank-ton cells exhibits patchiness at distances of decimeter to millime-ter scales for numerous species with different motility strategies.The explanation of this general phenomenon remains challeng-ing. Furthermore, hydrodynamic cell–cell interactions, which growmore relevant as the density in the patches increases, have beenso far ignored. Here, we combine particle simulations and con-tinuum theory to study the emergence of patchiness in motilemicroorganisms in three dimensions. By addressing the combinedeffects of motility, cell–cell interaction, and turbulent flow condi-tions, we uncover a general mechanism: The coupling of cell–cellinteractions to the turbulent dynamics favors the formation ofdense patches. Identification of the important length and timescales, independent from the motility mode, allows us to eluci-date a general physical mechanism underpinning the emergenceof patchiness. Our results shed light on the dynamical character-istics necessary for the formation of patchiness and complementcurrent efforts to unravel planktonic ecological interactions.

phytoplankton | patchiness | turbulence

The spatial distribution of plankton is key to understandingmyriad aspects of marine ecology: from the flux of nutri-

ents in the water column, or predation strategies (1, 2), to theglobal repercussions of harmful algal blooms (3). It is one of theoldest observations in biological oceanography that phytoplank-ton exhibit a heterogeneous spatial distribution (patchiness) atlength scales of kilometers (4, 5). Recent technological advanceshave ushered in measurements at small scales (decimeters to mil-limeters) that revealed patchiness also at these smaller scales(6). These small distances form the natural arena where bio-logical and physical processes such as nutrient uptake, growth,and community composition take place (2, 7–10). Due to variousfactors such as wind or tidal currents, the local, small-scale envi-ronment of phytoplankton is nearly invariably turbulent (11), andturbulence has a profound impact on the biology and ecology ofplankton (12). For example, turbulent diffusion reduces the sizeof the concentration boundary layer of depleted nutrients andthus facilitates the nutrient uptake of the microorganism (13–15).

Numerous phytoplankton species are motile, and the vastmajority of harmful algal blooms are caused by motile phyto-plankton (3, 11). Thus, a comprehension of the consequences ofactive motion in a turbulent environment can clarify importantaspects of the phytoplankton ecology, such as division rates andnutrient retrieval. Recent work has established that gyrotacticplankton swimming in mildly turbulent environments form densepatches (16–19). However, patchiness is exhibited by a myriadof motile species with different motility modes, such as photo-taxis, chemotaxis, pattern swimming, and autoregulated aggrega-tion (3, 20–22). Furthermore, flow visualization techniques havedemonstrated that motile plankton create microflows which in

turn modify the fluid environment (2) and will introduce effectivehydrodynamic interactions among individual cells at the scale ofabout 100 µm. Recurrent algal blooms produce the initial con-ditions for the subsequent emergence of patchiness (7). In thiscontext, a basic yet unresolved question is to determine what fea-tures of marine turbulence conspire with planktonic motility toengender patchiness.

Here, we combine molecular dynamics simulations and a con-tinuous description of the incompressible Navier–Stokes flow todetermine a general mechanism for patchiness in motile phy-toplankton cells swimming in a turbulent environment. Unlikeprevious investigations of gyrotactic plankton, we include a min-imal model of cell–cell interactions among the swimmers. Ourwork explores the fundamental spatial and temporal scales asso-ciated to the reorientation of the swim direction when cellsencounter each other and the scales associated to the mild tur-bulent motion. We identify dimensionless numbers that comparethe characteristic time scales associated to the small-scale motionof the fluid with the cell–cell interactions and that compare thecharacteristic length scales associated to the turbulent motionwith the active motion of the cells.

Associating time and length scales to turbulent dynamics is afamiliar operation from classical studies of turbulence, but therole of these scales in the context of phytoplankton with hydro-dynamic coupling is not known. We consider two complementarymethods to generate a turbulent flow: (i) a simplified but realisticrepresentation of turbulent flow via random Fourier modes (23,24), the so-called kinematic simulation method first proposedby Kraichnan (25), and (ii) direct numerical simulations (DNS)of the Navier–Stokes equations with a pseudospectral method.

Significance

Recent observations found that swimming phytoplanktonspecies have a patchy spatial distribution down to the mil-limeter scale in oceans and lakes. This is rather surprisingbecause it goes against the intuitive expectation that theturbulent flow mixes the microorganisms and spreads themrather homogeneously. Here, we identify the relevant scalesthat rationalize the observation of patchiness. We find that amild turbulent field can lead to the formation of small densepatches of motile cells when the typical scales of the tur-bulence roughly match the scales of the interaction amongmicroorganisms, which is mediated by hydrodynamics.

Author contributions: M.G.M. designed research; R.E.B., C.C.L., D.W., M.W., and M.G.M.performed research; R.E.B. and C.C.L. analyzed data; and R.E.B., C.C.L., M.W., and M.G.M.wrote the paper.y

The authors declare no conflict of interest.y

This article is a PNAS Direct Submission.y

Published under the PNAS license.y1 To whom correspondence should be addressed. Email: marco.mazza@ds.mpg.de.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1808711115/-/DCSupplemental.y

Published online November 8, 2018.

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Our investigations are carried out in a parameter range closeto the ecologically relevant Taylor-based Reynolds number (26)Rλ≈ 20, which corresponds to realistic conditions found, e.g., atthe pycnocline (7).

We find that the correlation length scale associated to the vor-ticity of the flow constitutes a characteristic scale, and when itmatches the length scale associated to the cell–cell interactions,strong patchiness emerges. Experimental evidence at such a levelof precision is scarce. However, the current evidence points tothe fact that minuscule changes in the cellular morphology candramatically affect the hydrodynamic cell–cell interactions (27)and how different organisms respond to turbulent microflows(28), thus giving further motivation for our work.

ResultsModeling Phytoplankton Dynamics in Mildly Turbulent Environ-ments. Consider a motile microorganism, such as phytoplank-ton, swimming in a turbulent aqueous milieu. A fundamentallength scale characterizing the scale where the energy associ-ated with turbulent motion dissipates is the Kolmogorov lengthscale ηK = (ν3/ε)1/4, where ν is the kinematic viscosity of thefluid (∼10−6 m2/s for sea water), and ε is the energy dissipa-tion rate. For realistic values of dissipation rate (ε≈ 10−10−10−5 m2/s3) (29, 30), the Kolmogorov length scale ηK is ofthe order of hundreds of micrometers. Within dense patches,phytoplankton cells are separated by distances comparable orsmaller than ηK (16). Thus, for typical phytoplankton cell sizes(1−1,000 µm) (31) cell–cell interactions should be directlyaddressed.

Because of their sizes, the motion of microorganisms inaquatic environments is dominated by viscous forces, which aretwo to five orders of magnitude larger than the inertial forces(32). This means that the microorganisms’ motion and the small-scale perturbations to the fluid produced by their motility aredescribed by the Stokes equation ∇p−µ∇2u = f , where p isthe hydrostatic pressure, u is the fluid velocity, µ is the fluidviscosity, and f is a body force. The fact that this equation islinear and not explicitly dependent on time allows a great sim-plification with respect to the full Navier–Stokes equations, andwe can write the general solution (in unbounded domains) asu =

∫O(r− r′) · fdr′, where O(r) = 1

8πµr(I + r⊗r

r2) is the Oseen

tensor. Because flagellated microorganisms move autonomously,the hydrodynamic perturbations produced by their active motioncan be well described as the sum of two equal and oppo-site forces f =±f0e separated by a distance δ, i.e., a forcedipole. The fluid velocity field generated by a force dipole readsudip = f0δ

8πµr2[3(r · e)2− 1]r, where r and e are the unit vectors

representing the direction of the position and orientation ofthe dipolar swimmer, respectively. We note that the expres-sion for udip has a symmetry: It remains invariant under theexchange e→−e. Hence, hydrodynamically mediated cell–cellinteractions among flagellated swimmers generate a flow fieldwhich to leading order exhibits up–down symmetry (33, 34).Hydrodynamic interactions between motile cells are in princi-ple long ranged. However, the intrinsic biological stochasticityof the cells’ motility together with the strongly fluctuating con-ditions of turbulent microflows effectively produce short-rangecell–cell interactions (34). The range ε of this cell–cell interac-tion mediated by hydrodynamics has been estimated to be of thesame order of magnitude as the cell body length for Chlamy-domonas (34, 35). Beyond this length scale ε the stochastic natureof the flow prevails. Short- to medium-range cell–cell interac-tions with up–down symmetry are the key ingredient for ourmodeling.

We model the cells as particles with swimming speed v0 andorientation e. The cells’ reorientation is subject to two torques:(i) 1

2(ω× e) induced by the fluid vorticity ω=∇× u, where u is

the fluid velocity, and (ii) a torque representing the short-rangecell–cell interactions with up–down symmetry. The interactionis restricted to particles within the range ε and takes place ona time scale 1/γ. Rotational diffusion is introduced through astochastic noise ξ. For the sake of comparison we also con-sider select simulations with extended particles to examine theimpact of repulsive contact forces. See Materials and Methods formore details.

To model the turbulent flow, we combine results from twocomplementary techniques to generate a turbulent flow. In thefirst approach, we use kinematic simulations; that is, the tur-bulent velocity field is modeled by a sum of unsteady randomFourier modes which obey a prescribed energy spectrum. Werefer to this model as the “Kraichnan fluid” after its first pro-ponent (25). In the second approach, we perform DNS of theincompressible Navier–Stokes equations with a pseudospectralmethod to model the turbulent flow directly (more details in SIAppendix). (We note that the Kolmogorov scale as a measurefor the small-scale turbulent motions is meaningful only in thecontext of 3D Navier–Stokes flow. In that sense, comparisonsto real-world data are more sensible for this setting than forsimplified 2D flows.)

The orientational dynamics can be characterized by twodimensionless numbers. The first one is the rotational Pecletnumber P = γ/Dr , which describes the ratio between thestrength of cell–cell interaction γ and the rotational diffusionDr due to stochastic noise. The second dimensionless num-ber serves to compare the strength of the cell–cell interactionwith the turbulent field. The typical vorticity of the turbulentfield is quantified by the Kolmogorov shear rate ωK =

√〈ω2〉=√

2∫E(k)k2dk , where E(k) is the energy spectrum and k the

wavenumber. We define the vortical Stokes number as Sω =τturb/τint = 2γ/ωK , where τturb and τint are the characteristic timesof the turbulent field and of the cell–cell interaction.

Phase Diagram of Patchiness. In our simulations, we observephytoplankton patchiness as a consequence of the cell motil-ity and the cell–cell interaction for a wide range of parame-ters. This can be reasoned as follows: The dynamics describedby the equations of motion of the swimming cells (Eq. 2 inMaterials and Methods) produce a compressing phase-space vol-ume Γ≡∇r · r +∇e · e =−4γ(eav · e), where eav is the averageorientation around a reference cell. Because cells swimmingin the same patch will necessarily have a similar orienta-tion, the average 〈Γ〉< 0. Thus, the coupling of active motionand turbulence potentially produces regions of increased localdensity.

For a more detailed picture of the mechanism we map outthe nonequilibrium phase diagram of cell patchiness, identify-ing the relevant length and time scales. To quantify the degreeof patchiness in the spatial distribution of swimming cells wecalculate the patchiness factor Q , which is a normalized localdensity based on the Voronoi tessellation of the 3D domain(16) (more details in Materials and Methods). Fig. 1A showsthe nonequilibrium phase diagram obtained from our moleculardynamics simulation of swimming cells immersed in a Kraichnanfluid. At fixed Peclet number, as the vortical Stokes numberSω increases to values larger than unity (indicating a dominantreorientational dynamics of the cells), we find that the patchi-ness factor Q exhibits a sharp increase. At Sω ≈ 10 a well-definedmaximum occurs. This maximum is robust upon variation overa wide range of Peclet numbers and upon replacing point-likewith extended particles. At large values of Sω the patchinessalways saturates to a finite value.

The relevant length scales associated to phytoplankton patch-iness warrant the investigation of the impact of the cells’ finitesize—an aspect so far largely neglected by models of motile

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A B

Fig. 1. (A) Nonequilibrium phase diagram of point-like swimming cells immersed in a Kraichnan turbulent flow field. As the reorientation of the swimdirection due to cell–cell interactions increases its strength with respect to the turbulent vorticity (i.e., increasing vortical Stokes number Sω), small-scalepatchiness emerges. We calculate the dependence of the patchiness factor Q on Sω and the Peclet number P for N = 27,000 cells at a number densityρε3 = 4× 10−3. For P > 1, motile cells swimming in a turbulent field and interacting with each other exhibit a maximum in the patchiness, as Sω increases.When we consider the ratio ζ of characteristic turbulent to active length scales, we observe emergence of strong patchiness at ζ≈ 1. The lines indicate theloci P = 1 (dotted) and P = Sω (solid) which serve as boundaries for the region where emergence of patchiness is expected (main text). (B) The maximum inpatchiness does not depend on the microscopic details of the interaction, as the emergence of patchiness is robust upon replacing point-like with extendedparticles (both cases for P→∞); nor does the maximum in patchiness depend on the details of the flow field.

particles. The finite extension of the cells induces correlationsin their spatial distribution, and these correlations will affectthe characteristics of the patches. To study whether our resultsare robust upon inclusion of more realistic spatial correlations,we perform simulations of extended particles immersed in aKraichnan fluid. Fig. 1B (circles) shows that patchiness emergesalso for extended cells for Sω & 10 and exhibits the same qualita-tive dependence on Sω as for point-like cells. Due to the spatialextension of the cells, however, the patchiness factor Q reacheslower values than for point-like cells because the local densitycannot grow arbitrarily large.

We can derive some limits on the location of the maximum forpatchiness in the following way. First, for the cell–cell interactionto overcome the rotational diffusion, the Peclet number P mustbe larger than unity (this condition corresponds to the regionabove the dotted line in Fig. 1A). Second, to have an effect on thedynamics, the vorticity associated to the turbulent flow should bestronger than the rotational diffusion, which is quantified by thecondition Sω < P (corresponding to the region above the solidline in Fig. 1A). This characterization of patchiness in terms oftime scales can be complemented by considering the relevantlength scales. To this end, we define a dimensionless numberζ ≡Lturb/Lint that compares the length scale associated to theturbulent vortical motion Lturb = τturburms with the length scaleLint = τintv0 associated to the cell–cell interactions. The regionof enhanced patchiness is delimited by the two loci describedabove and as visible in Fig. 1A. In the current setting, we find thatthe condition ζ ≈ 1 marks the emergence of strong patchinessfor our system of motile cells for both point-like particles andextended ones (Fig. 1). This fact points to a general mechanismunderpinning the emergence of patchiness. The region wherecell–cell interactions considerably change cells’ swim directionshas characteristic size Lint. When Lint roughly matches the sizeLturb, where turbulent vortical motion acts most strongly, densepatches emerge.

Kinematic simulations provide a synthetic flow field with real-istic spatiotemporal correlations. However, they lack a numberof hallmark features of real hydrodynamic turbulence such as aself-consistent energy spectrum with realistic large- and small-scale cutoffs and non-Gaussian velocity and vorticity fluctua-tions. Although the degree of non-Gaussianity is low for themild turbulence considered here, departures could have a sensi-

tive impact on the clustering behavior. We therefore turn to fullDNS calculations (SI Appendix) to further corroborate our find-ings and compare them with the results of the Kraichnan fluid.Fig. 1B (squares) shows the patchiness factor Q for the samerange of Sω as for the Kraichnan fluid (P→∞). The emergenceof clusters occurs again for ζ ≈ 1. It is remarkable to note thata maximum in Q occurs for intermediate values of Sω close tothe maximum found in the Kraichnan fluid. The fully nonlinearNavier–Stokes flow and the Kraichnan flow therefore generateconsistent results.

This coincidence corroborates our argument above about themechanism of patch formation: The cell–cell interactions lead toa patchy distribution of swimming cells provided that the scalesassociated to turbulent motion and cell–cell interactions match.The height and precise position of the maximum in Q dependon the details of the fluid flow. We find that both the Taylor-based Reynolds number and the ratio of cell–cell interactionlength scale to Kolmogorov length scale influence the height ofthe maximum in Q (details in SI Appendix).

Fig. 2 shows a typical snapshot of the system. The visualizationof vorticity fields of the Navier–Stokes flow exhibits the fila-mentary vortex structures, which are characteristic of turbulentflows. The position and orientation of the phytoplankton cellsare indicated with arrows. It appears that the locations of thephytoplankton patches are subtly influenced by the joint effectof active motion and flow vorticity. While there are some eventsof considerable reorientation of the cells within the vortex cores,in a more typical situation cells are mildly deflected by the localvorticity.

Finally, to bolster our discussion of the influence of the cor-relation length of the turbulent field compared with the inter-action range ε, we perform simulations of the Kraichnan flowat fixed Rλ and with varying correlation length scale of the tur-bulent field. We introduce the correlation length scale of thevorticity

Lω ≡∫ ∞0

fω(λ)dλ, [1]

where fω(λ)≡〈ωx(r +λex, t)ωx(r, t)〉/〈ω2x〉, ωx =ω · ex, and ex

is one of the unit vectors specifying the Cartesian coordi-nate axes.

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Fig. 2. (A) Typical steady-state configuration of the motile cells in the Navier–Stokes flow field (at Sω = 86.52). Cells are shown as arrows whose colorindicates 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 shows the dependence of the maximum in patchinessfactor Q on the ratio of Lω to the cell–cell interaction lengthscale ε for different cell densities and also physical parameterscharacterizing the turbulent flow (i.e., the wavenumbers delim-iting the inertial regime and turbulent kinetic energies). As Lωincreases, the patchiness in the spatial distribution of motile cellsincreases monotonically, indicating a stronger influence on thesystem of the turbulent vorticity that favors cluster formation.This conclusion is robust upon variation of the physical param-eters characterizing the system, while the slope of the curvedepends on the precise features of the microenvironment.

DiscussionNoninertial particles in an ergodic incompressible flow cancluster only if they are motile because they can cross thestreamlines of the flow. Passively advected particles which ini-tially are homogeneously distributed will be advected followingvolume-conserving dynamics. We find that small-scale patchinessemerges when hydrodynamic cell–cell interactions couple withthe vorticity in a turbulent flow field.

The following physical picture emerges from our results. Whenthe reorientational strength of the cell–cell interaction is strongerthan the rotational diffusion of the fluid, the cell–cell interactionleads to (some degree of) alignment of neighboring cells. Theturbulent fluctuations may then statistically bring two motile cellscloser together. From this point on the cells will experience asimilar physical environment and will be very likely to remain atclose distance.

When a balance is struck between the length scales wherereorientational interactions and turbulence act, marked by ζ ≈ 1,patchiness is likely to emerge. Thus, the concerted action ofmotility, cell–cell interaction, and turbulent microflows approx-imately produces an absorbing state, that is, a dynamical con-figuration that remains statistically preserved in time. This ismathematically represented by a negative, average phase-spacevolume compression factor, 〈Γ〉< 0.

The results of Fig. 1, together with Fig. 3, also explain themaximum in clustering observed upon increasing Sω . In the limitof large Sω and for Sω > P , the vorticity cannot effectively per-turb the orientations, and the spatial distribution of cells remainshomogeneous. For very small values of Sω , instead, the turbulentvorticity is much stronger than the cell–cell alignment mecha-nism, and in this case the turbulent field will decorrelate theorientation of closely neighboring particles and eventually tearthem apart. For intermediate values of Sω , the interplay between

cell–cell alignment together with the motility of the cells andthe effect of the turbulent vorticity leads to patchiness. Largerregions of influence lead to larger amounts of patchiness.

Let us now put our findings in the context of experimentalobservations. Recent technological advances have expanded therange of length scales we can access and consequently also ourview of the problem of plankton patchiness. In situ fluorome-try (36, 37), rapid freezing (38, 39), and pneumatically operatedsampling (40, 41) allow one to reach submillimeter scales (6) inmeasurements of planktonic distributions.

Experiments using high-resolution gradient multisamplers cangive access to spatial resolutions of the order of decimeterswhich correspond to the scale of patchiness predicted by ourresults. Measurements at the pycnocline in the Kattegat foundan increase of a factor of 20 with respect to the surface layersand a variation of a factor of 3–5 within 30 cm (7) in Gyrodinium

Fig. 3. Dependence of the maximum patchiness factor Q on the ratio ofcorrelation length scale of the vorticity and cell–cell interaction range Lω/ε.In all these calculations of the Kraichnan flow, we set kmax/kmin = 63 andP→∞. The patchiness increases as Lω/ε increases. This behavior is robustupon variation of the average filling fraction (solid symbols, ρε3 = 0.41;open symbols: ρε3 = 0.041). Furthermore, this conclusion is not alteredupon variation of the ratio of length scales—either by changing the cell–cell interaction range ε (blue stars) or by changing Lω/ε via a shift ofthe minimum wavenumber of the turbulent flow (circles). Keeping a fixed(yellow) or variable (red) turbulent kinetic energy also shows the samequalitative result.

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aureolum. Dinobryon species concentrations were observed tovary by up to a factor of 10 in a river estuary within a meter scale(42). Variations in concentration of factors up to 2 or 3 below themeter scale were observed for different species of dinoflagellatesin the Aarhus Bay (8). These numbers are in agreement with thepredictions of our theory (Fig. 1B).

A combination of traditional turbulence sensors [e.g., shearprobes and fast response thermistors (43, 44)] and a backscatterfluorometer measured microstructure in the velocity and in theintensity of chlorophyll concentrations at the centimeter scale(45), so-called biophysical microstructures. As high-frequencyacoustic scattering techniques grow in precision (46, 47), addi-tional tools can be deployed to map marine and freshwaterenvironments to increasing accuracy, which may facilitate futureexperimental confirmations of our findings.

Our results ignore the presence of stratification. While this isan important feature of real-world settings, we do not expect thatour conclusions will change qualitatively, as Q will result in aweighted average of the individual strata.

ConclusionAiming to identify a minimal model for motile phytoplankton,we investigated the collective behavior of a large number of bothpoint-like and extended, motile cells immersed in a mildly turbu-lent flow field. We quantify the influence of cell–cell interactionsvia a dimensionless number, the vortical Stokes number Sω , thatmeasures the ratio of time scales associated to vorticity andcell–cell interaction. Strong patchiness emerges for Sω & 10. Thispoints to a balance of time scales underpinning the dynamics ofthe motile cells.

We explore the microhydrodynamics characterized by aTaylor-based Reynolds number of the order of Rλ≈ 20 whichmatches the typical conditions found, e.g., in the pycnocline(48). We identify the relevant dimensionless number ζ whichcompares the length scales associated to cell–cell interactionsand to turbulent dynamics. Patchiness is likely to emerge whenζ ≈ 1, indicating the coupling of turbulent structures with thecell–cell spatial correlations. Our results indicate that the cou-pling of cell–cell interactions with mildly turbulent flows con-trols the emergence of a patchy spatial distribution of motilephytoplankton.

We confirmed that both models (Kraichnan flow and Navier–Stokes flow) show a near quantitative agreement. We find, how-ever, an influence of the kinematic details, such as the ratio ηK/ε,on the magnitude of the maximum patchiness (more details in SIAppendix). This means that the occurrence of small-scale patchesstrongly depends on the specific turbulent field which acts onthe cells.

This finding can complement discussions on the influence ofphysical forces on marine ecology. Although for smaller phyto-plankton species it is conceivable that the local environment isa linear shear field, the microflows change rapidly on the scaleof Lω . Our results show that Lω has considerable impact on theamount of patchiness. Because the correlation length scale of thevorticity may substantially vary in realistic situations, our resultcan inform future hydrographic and biophysical measurementsat small scales.

By addressing the combined effect of turbulent flow and cell–cell interactions, our work paves the way to unravel the complexinterplay of physical and biological interactions determiningphytoplankton’s microenvironment.

Materials and MethodsMolecular Dynamics Simulations. Our system is composed of N identical,finite-size particles which are self-propelled and have an up–down symmet-ric interaction with each other. These self-propelled particles inhabit a cubicbox with side length L and periodic boundary conditions. We integrate theequations of motion for particle i at position ri and with orientation ei ,

ri = v0ei + u(ri , t)−1

γF

N∑j=1

∂UWCA(rij)

∂ri[2a]

ei =

(−γ

∂U

∂ei+

1

2(ω(ri , t)× ei) + ξi(t)

)⊥

, [2b]

where the subscript ⊥ refers to the part of the torque which is perpen-dicular to ei , with e2

i = 1 at all times. The orientation vector indicates thedirection of self-propulsion with speed v0 and can be changed by three dif-ferent mechanisms: First, each particle interacts with its ni neighbors withinan interaction range ε, which we take as our unit of length. This inter-action is given by the Lebwohl–Lasher potential U =− 1

2

∑Ni

∑j∈ni

(ei · ej)2

(49) together with the relaxation constant γ. Second, the vorticity ω=∇×u of the turbulent field changes the particle’s orientation (16); third, a noiseξi(t) [uniformly distributed on the surface of a sphere (50)] models rota-tional diffusion and is delta correlated in time and space:

⟨ξiα(t)ξiβ (t′)

⟩=

2Drδ(t− t′)δαβ . The finite size of the particles is modeled as hard coresthrough the Weeks–Chandler–Anderson (WCA) potential (51) UWCA(rij) =

4εF

[(σ/rij

)12−(σ/rij

)6]+ εF , for rij ≤

6√2σ, and 0 otherwise, where σ

denotes the particle diameter, εF is the strength of the potential, andrij ≡ |ri − rj|.

All simulations are performed in a cubic simulation domain of sidelength ranging L = (124− 374)ε. The dimensionless number density is setto ρε3 = N(ε/L)3 = 0.0041, except where stated otherwise (see SI Appendixfor discussion on influence of ρε3). The number of particles varies between8,000≤N≤ 216,000, depending on the choice of the cell–cell interactionrange ε.

We perform two sets of simulations: In the first one we use εF = 0, (i.e.,point particles). In the second set, the particle diameter is chosen to be halfof the cell–cell interaction range (σ= 0.0168) which leads to a packing frac-tion of φ= N π

6

(σL

)3≈ 0.027%. For extended particles the relaxation timeand the strength of the WCA force are kept fixed (γF = 1, εF = 10−3). Dif-ferent values of P and Sω are achieved by changing the relaxation constantγ and the rotational diffusion constant Dr , if not stated otherwise.

Analysis of Patchiness. To quantify the degree of clustering in a given con-figuration, we use the patchiness factor defined in ref. 16. To derive thisfactor, the Voronoi tessellation of a given configuration is calculated, whichattaches a volume vi to every particle. The local (number) density C is thendefined as the inverse of this volume. We use the fraction f of the mostdense particles to calculate a mean density C(f) = 〈vi〉−1

i∈f . This mean den-sity is compared with a random distribution of cells (Crandom(f)) and theoverall number density ρ= N/V as Q(f) = ρ−1[C(f)− Crandom(f)]. We takethe absolute value of Q which differs from Q only for roughly homoge-neously distributed particles (small values). All data shown in this paper arecalculated with f = 5%.

ACKNOWLEDGMENTS. The authors thank Jens Elgeti, Stephan Herming-haus, and Anupam Sengupta for useful discussions. The authors gratefullyacknowledge support of the Max Planck Society.

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