Clustering of plankton plays a vital role in several biological activities, including feeding, predation, and mating. Gyrotaxis is one of the mechanisms that induces clustering. A recent study

Plankton are known to form small-scale clusters in a turbulent environment

Gyrotactic plankton can form different kinds of clusters depending on the flow characteristics. For instance, plankton accumulate in the center or the wall regions in downward or upward pipe flow, respectively

Previous studies have suggested that gyrotaxis originates from asymmetric body structures, such as nonuniform mass distribution (bottom-heaviness)

In this study, we aim to analyze the clustering of planktonic swimmers under the influence of fluid inertial torque. We model plankton as point-like spherical microswimmers undergoing gravity sedimentation. We use direct numerical simulations of swimmers in homogeneous isotropic turbulence to analyze their clustering characteristic. In Sect.

In the present study, we consider a spherical swimmer undergoing gravitational sedimentation, as shown in Fig.

A sketch of a settling swimmer.

The dynamics of the swimmer is governed by the following expressions:

Using the characteristic scales for velocity and time of the flow,

In turbulence, we can take the respective turbulence Kolmogorov velocity and timescales

The typical value of

The motion of swimmers in homogeneous isotropic turbulence is simulated by Eulerian–Lagrangian direct simulations. The flow field is resolved in the Eulerian frame, while the motions of individual swimmers are solved along the Lagrangian trajectories using local flow information at swimmers' positions. The incompressible turbulent flow is directly simulated by solving the Navier–Stokes equations:

Swimmers are initialized with random positions and orientations after turbulence is fully developed. When solving the trajectories of swimmer, fluid velocity and its gradients at Eulerian grid points are interpolated by a second-order Lagrangian method at the positions of swimmers. Equations (

The instantaneous location and orientation of swimmers are depicted in Fig.

Instantaneous spatial distribution of swimmers in homogeneous isotropic turbulence. Black dots and tiny arrows represent the position and swimming direction of each swimmer, respectively. Background contour represents the vertical fluid velocity

The clustering of swimmers is quantified by a three-dimensional Voronoï tessellation

We use the MATLAB toolbox “voronoi.m” and “convhull.m” functions to compute the vertices of Voronoï polyhedrons and calculate their volumes. Figure

To show how clustering depends on the settling and swimming speeds, we depict the variance of Voronoï volumes for different

The clustering of spherical gyrotactic swimmers in turbulence has been shown to be associated with the preferential sampling of downwelling regions

Mean vertical fluid velocity at swimmers' positions,

Comparing Figs.

Voronoï analysis allows us to track the Voronoï volume of each swimmer. Based on the values of volumes, we can distinguish whether each swimmer is inside a cluster (with a small Voronoï volume) or located away from other swimmers (with a large Voronoï volume). Figure

Joint probability distribution function (PDF) of the vertical fluid velocity

A settling spherical squirmer experiences a fluid inertial torque that causes it to swim against gravity, acting as an effective gyrotactic torque

Using direct numerical simulation, we investigated the clustering of swimmers under fluid inertial torque. We quantified the clustering using Voronoï analysis. When swimmers are not settling, the fluid inertial torque vanishes and the swimmers are randomly distributed, resulting from a random swimming direction, with no clustering observed. Settling swimmers experience a fluid inertial torque and behave similarly to gyrotactic swimmers. We observed that swimmers form more intense clustering when

We also examined how the clustering of spherical swimmers is related to their preferential sampling of downwelling regions. We found that when swimmers are not settling, their dynamics remains isotropic and no preferential sampling is observed in the gravity direction. However, the fluid inertial torque and the settling speed break this symmetry and drive settling swimmers to sample downwelling regions. The sampling is more pronounced with larger

The fluid inertial torque on settling swimmers can cause the formation of small-scale clusters, highlighting the importance of fluid inertial effects on the dynamics of plankton. However, most earlier studies did not consider gravitational sedimentation, leading to the neglect of fluid inertial torque. This results in an underestimate of the intensity of gyrotaxis, as the total gyrotactic torque is contributed by both fluid inertial torque and bottom-heaviness. In addition, the fluid inertial torque is proportional to the swimming and settling speeds, making the reorientation time a dependent parameter. Therefore, planktonic swimmers have the potential to tune their reorientation behavior and, thus, control clustering intensity by adjusting their swimming speed, which further impacts their mating, predation, and feeding.

Finally, it is necessary to clarify the assumptions of our model. First, we considered only spherical swimmers. Nonspherical plankton, such as elongated ones, probably experience a fluid inertial torque stemming from both their nonspherical shape

Raw data from the simulation are available upon request from the corresponding author.

JQ, EC, and LZ designed the project; JQ and LZ performed the research; JQ and ZC developed the numerical tools; JQ analyzed data; and JQ, EC, and LZ wrote the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This article is part of the special issue “Turbulence and plankton”. It is a result of the EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022.

This research has been supported by the National Natural Science Foundation of China (grant nos. 92252104, 12388101, and 92252204) and the China Postdoctoral Science Foundation (grant no. 2022M721849).

This paper was edited by François G. Schmitt and reviewed by four anonymous referees.