Small-scale convective turbulence constrains microbial patchiness

1 Microbes play a primary role in aquatic ecosystems and biogeochemical cycles. 2 Patchiness is a critical component of these activities, inﬂuencing biological productiv- 3 ity, nutrient cycling and dynamics across trophic levels. Incorporating spatial dynamics 4 into microbial models is a long-standing challenge, particularly where small-scale tur- 5 bulence is involved. Here, we combine a realistic simulation of turbulence with an 6 individual-based microbial model to test the key hypothesis that the coupling of motil- 7 ity and turbulence drives intense microscale patchiness. We ﬁnd that such patchiness is 8 depth-structured and requires high motility: Near the ﬂuid surface, strong convective 9 turbulence overpowers motility, homogenising motile and non-motile microbes equally. 10 In deeper, thermocline-like conditions, highly motile microbes are up to 1 . 6 -fold more 11 patch-concentrated than non-motile microbes. Our results demonstrate that the del- 12 icate balance of turbulence and motility that triggers micro-scale patchiness is not a ubiquitous consequence of motility, and that the intensity of such patchiness in real- world conditions is modest.

Secondly, only the most agile motile microbes, sustaining both high swim speeds and fast 136 reorientation timescales, achieved a significant difference in patchiness from their non-motile slow to reorient and more vulnerable to "disorientation" due to viscous torque. We note that 152 swim speed did not appear to impact the distribution of microbe orientations. 153 Microbe orientation alone is only part of the story underlying the patchiness trends  In the Deep region (Fig. 6e,f), the polar angle is approximately horizontal but again exhibits 177 little variation between simulations, while the magnitude varies by a margin (∼ 13 mm s −1 ) 178 significantly greater than the largest difference in microbe swim speeds between simulations 179 (0.49 mm s −1 ). Overall, with increasing depth, effective velocity slows and becomes increas-180 ingly horizontal, with microbes in the 'Deep' region moving near-horizontally. Microscale microbe patchiness may have far-reaching implications both for the microbes 183 themselves and for the wider ecosystem, but it is first and foremost essential to accurately 184 understand the prevalence and intensity of such patchiness in real-world conditions. By 185 combining state-of-the-art individual-based modelling tools with a realistic 3D model of tur-186 bulence, we have shown that patchiness driven by the interaction of turbulence and gyrotactic 187 motility is likely to be modest in intensity, and limited in practise to particularly agile motile 188 microbes and to deeper regions of the mixed layer where convective turbulence is less intense 189 than at the surface.

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Although our DNS simulates only a relatively small physical volume of water, physical 191 scaling arguments (see SI Section 2) establish limits on the differences in turbulent fluid 192 motion between our DNS and in larger real-world flows. Turbulent velocity fluctuations in the ocean mixed layer are between 0.93 − 7.88-fold stronger than in our DNS, while we 194 could not find sufficient data to compare our DNS to mixed layer turbulence in lakes. These fold stronger than in our simulation (see again SI Section 2), and patch enhancement may 220 be especially difficult to achieve. It must also be stressed that the ocean mixed layer is 221 not constantly overturning, but also undergoes periods of minimal or negative heat loss 222 through the surface, when our turbulence regime is not applicable. Due to computational 223 constraints, we have not modelled the effects of other sources of turbulence, such as wind or 224 waves. We predict that, since turbulence from these sources would also be strongest near the 225 fluid surface and decline with depth, our results would not qualitatively change with their 226 inclusion; additional turbulence near the fluid surface would continue to disperse microbe 227 patches, while at greater depth, highly motile microbes may begin to form patches through 228 coupling with weaker turbulence. for microbe patches to affect reproduction or nutrient distributions, for example, the lifetime 277 of a patch cannot be smaller than the timescale of reproduction or of resource consumption 9 .

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Since we do not currently have the tools to reliably determine patch lifetimes for compari-279 son between simulations, we are not able to report patch lifetimes for the results from this 280 study. Modelling or measuring in-situ the longevity of microbe patches, and their longevity's 281 dependence on conditions such as turbulence, will be critical to future research in this space.  The flow targeted in this paper is convection in the top-layer of a two-layer stratified fluid 291 due to surface cooling. These flows are of direct relevance to the oceanic mixed layer and 292 convective mixing in lakes 38-40 . As we aim to simulate the behaviour of microbes, it is 293 essential that all dynamic scales of the turbulence are resolved. This implies (1) that a 294 highly accurate code for direct numerical simulation needs to be employed; and (2) that 295 the problem should be scaled down to a lab-scale, since it is impossible to resolve all the 296 turbulence at real-world oceanic or lacustrine scales.

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The domain is 0.6×0.6×0.3 m (length × width × height), and the fluid inside the domain 298 comprises two layers of thickness h 0 = 0.15 m of which the top layer has a density that is 299 75% lower than that of the bottom layer, thus creating a stable stratification. The density 300 jump used here is much stronger than in lakes or the ocean, but is chosen to limit turbulent of the cooling of the water surface due to long wave radiation during night time, winter or 306 autumn cooling, or during a cold-air outbreak. As a result, the fluid near the surface cools 307 and begins to descend, forming a convective mixed layer above the density interface.

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In order to be able to resolve all the turbulent scales of motion, a kinematic viscosity 309 ν = 5 × 10 −6 m 2 s −1 was chosen, which is slightly higher than that of water. A thermal 310 diffusivity was set to 4 × 10 −6 m 2 s −1 . Setting the buoyancy flux B = 5 × 10 −4 m 2 s −3 implies 311 a characteristic velocity scale of the mixed layer w * = (Bh 0 ) 1/3 = 0.042m s −154,55 (see  plementary Methods). This implies that the initial bulk Richardson number and Reynolds 313 number are: The computational grid is 720 × 720 × 360, which corresponds to 186,624,000 cubic mixed layer is formed. The microbe simulations commence after these initial transients, and 323 will thus only use the data for 30 ≤ t ≤ 90 s.

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The velocity and density fields are obtained using the direct numerical simulation code 325 SPARKLE, which employs a symmetry-preserving fourth-order-accurate finite volume dis-326 cretization scheme, preserving mass, momentum and energy 56,57 . SPARKLE solves the 327 Navier-Stokes equations in 3D in the Boussinesq approximation: where space is denoted as x = (x, y, z) and fluid velocity by p is kinematic pressure, ν is kinematic viscosity, b = b(x, t) is buoyancy, κ is thermal dif- where p is a unit vector describing the swimming direction, ω = ∇ × u is the fluid vorticity  We compute motile and non-motile particle trajectories using the OceanParcels Lagrangian where X is the microbe's position, v swim is the swimming velocity of the microbe and once 371 again p is its swimming direction and u is the fluid velocity. This approach has previously 372 been shown to accurately capture the trajectories of passive and active swimmers in a turbu-    To quantify patchiness in our simulations, we first performed the 3D Voronoi tessellation 396 described above. Then, following the approach outlined in 32 , we defined patches to consist

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of the fraction f of microbes with the largest local concentration (i.e. smallest Voronoi 398 polyhedron volume), and we used the concentrations within these patches to calculate, at 399 every second, the 'patch concentration enhancement factor' Q: where C is the median concentration among motile microbes inside patches, C P is the median   Each uniquely-coloured sequence of dots represents a single microbe's trajectory. Owing to the periodic boundaries in the longitudinal and latitudinal directions, trajectories may appear discontinuous when a microbe moves through such a boundary (e.g. green trajectory). Microbes actively traverse the mixed layer, mostly due to rapid advection in upwelling or downwelling regions of fluid, but also through individual locomotion in less fast-moving regions of the fluid. In particular, long sojourns are noticeable at greater depths where turbulent fluid motion is less intense.