Abstract
We study a predator–prey model of plankton dynamics in the two and three-dimensional wakes of turbulent flows behind a cylinder, focusing on the impact of the three-dimensional character of the carrying velocity field on population variance spectra and spatial distributions. By means of direct numerical simulations, we find that the qualitative behavior of the biological dynamics is mostly independent of the space dimensionality, which suggests that only the relation between the typical flow and biological timescales is crucial to observe persistent blooms. Similarly, in both cases, we find that the spectral properties of the planktonic populations are essentially indistinguishable from those of an inert tracer. The main difference arising from the comparison of the two- and three-dimensional configurations concerns the local spatial distribution of plankton density fields. Indeed, the three-dimensional turbulent dynamics tend to destroy the localized coherent structures characterizing the two-dimensional flow, in which the planktonic species are mostly concentrated, thus reducing the phytoplankton biomass in the system.
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This manuscript has associated data in a data repository. [Authors’ comment: The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.]
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This work was granted access to the HPC resources [MESU] of the HPCaVe centre at UPMC-Sorbonne University.
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Impact of the Schmidt number in the 2D case
Impact of the Schmidt number in the 2D case
In order to assess the possible effect of the Schmidt number, \(Sc = \nu /D\), we report here on a comparison between the results from 2D simulations performed at \(Re = 2000\), and (A) \(Sc = 1\), or (B) \(Sc = 100\), focusing on scalar fluctuation spectra. To be consistent, in both these simulations the maximum resolution was chosen to be \(N=2^{12}\), in order to resolve all the scales down to the Batchelor one, \(\ell _B\). As shown in Fig. 7, for both (A) and (B), the wavenumber phytoplankton variance spectrum is compatible with the scaling \(\sim k^{-1}\), but in case (A) (i.e., for \(Sc=1\)) this behavior is detectable only over a decade, for wavenumbers close to, or smaller than, \(k_d\), before a rapid decay of the spectrum at larger wavenumbers. In case (B) (i.e., for \(Sc=100\)) the power-law range extends over about two decades, indicating higher variance at large k, i.e., small scales, than in (A). This is clearly an effect due to the value of Sc. Indeed, when \(Sc >1\), scalar fluctuations are not dissipated at the viscous dissipative scale but are transferred to smaller ones and finally dissipated at scale \(\ell _B = \ell _{\nu } Sc^{-1/2}<\ell _\nu \). Considering that for simulation (B) \(\ell _B \sim 0.1\ell _{\nu }\), this simple estimate explains the additional decade of scaling range present at \(Sc=100\).
The impact of the Schmidt number can be appraised also in physical space, by comparing visualizations of the phytoplankton density field in the two cases (Fig. 8). For this purpose, one can consider the estimate for the width of a plankton filament in a purely elongational flow. According to the model originally developed in [37], such width is \(\ell _f \sim \sqrt{D/s}\), where D is the diffusivity and s the strain rate. Note that this quantity does not depend on any biological parameters. In spite of the additional complexity of our flow and biological model, simulations (A) and (B) display a clear qualitative agreement with this estimate of \(\ell _f\). The reduction of Sc, corresponding to the increase of D, makes filaments considerably thicker, as it is apparent from the comparison of Figs. 8a and 8b.
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Jaccod, A., Berti, S., Calzavarini, E. et al. Three-dimensional turbulence effects on plankton dynamics behind an obstacle. Eur. Phys. J. Plus 137, 184 (2022). https://doi.org/10.1140/epjp/s13360-022-02396-1
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DOI: https://doi.org/10.1140/epjp/s13360-022-02396-1