To resolve how phytoplankton chains migrate through the DNS of turbulence, we initialized our computational domain with 100,000 Lagrangian swimmers after the turbulent flow had reached a statistical steady state. The initial positions of swimmers were randomly (Poisson) distributed in space, and their initial swimming directions, p, were randomly distributed on a unit sphere. At each time step of the simulation, the fluid flow properties (u, ω, and E) at the location of each swimmer were calculated using trilinear interpolation, and the swimmer orientation and position were integrated using a second-order Adams-Bashforth scheme. All of our analyses were performed after swimmers had reached a statistically steady distribution, which, in our simulations, corresponds to one to two integral time scales, which is the turnover time of the largest eddies in the flow (equivalent to approximately 40 Kolmogorov time scales). We verified the convergence of our analyses using separate tests that integrated the trajectories of 300,000 swimmers until they reached a statistical steady-state distribution.

Our simulations modeled cells as point-like particles, which previous studies have found to be a good approximation provided that the particles are smaller than approximately 2ηK or 3ηK (39, 40). The Kolmogorov scale for the largest turbulent dissipation rate considered here (ε = 10−6 W kg−1) is ηK = 1 mm, suggesting that the point approximation is valid because the longest chains are typically less than few hundred micrometers in length (3, 6, 8, 14, 16, 17, 19). We also note that Eqs. 1 and 2 do not include the effect of phytoplankton inertia, owing to their small size and density close to that of the water in which they live (33). Furthermore, our model also assumes that phytoplankton chains are rigid and do not bend in flow. While previous studies have directly measured the stiffness of nonmotile diatom chains (41), we are not aware of any studies that directly measure the stiffness of motile phytoplankton chains or how bending might affect their motility.

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