# Plankton active response to turbulence enables efficient transport

Michelle Dibenedetto, Rémi Monthiller, Christophe Eloy, Lauren Mullineaux

► To cite this version:

Michelle Dibenedetto, Rémi Monthiller, Christophe Eloy, Lauren Mullineaux. Plankton active response to turbulence enables efficient transport. Journal of Experimental Biology, 2025, 228 (24), pp.jeb251123. 10.1242/jeb.251123. hal-05478798

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# Plankton active response to turbulence enables efficient transport

Michelle H. DiBenedetto<fnref n="1" />,*, Rémi Monthiller<fnref n="2" />, Christophe Eloy<fnref n="2" /> and Lauren S. Mullineaux<fnref n="3" />

**Summary:** Plankton actively tilt against flow rotation, exploiting turbulent flow structures to enhance their upward transport beyond their swimming capabilities alone.

## ABSTRACT

Although plankton typically have slow swim speeds relative to ocean flows, they can potentially enhance their transport by exploiting certain flow features. For example, a theorized “surfing” strategy describes how plankton can preferentially sample upwelling areas of the flow by simply sensing and actively reorienting in response to local velocity gradients. In this study, we present the first experimental evidence that real plankton may be able to surf turbulence. We study the bottom-heavy, planktonic larval snail *Crepidula fornicata* as our model organism. By observing these plankton in a jet-stirred turbulence tank, we show that they indeed have complex responses to velocity gradients. In particular, we find that they actively rotate to oppose the local vorticity, which contrasts with the typical passive, gyrotactic response. We compare our observations to those of simulated surfing plankton to demonstrate the applicability of the surfing theory to our data, where we see good agreement. Finally, we observe that the real plankton can preferentially sample upwelling areas of the flow in some cases, enhancing their transport relative to their swimming speed alone, similar to the proposed surfing theory.

**Keywords:** Turbulence, Plankton, Hydrodynamics

## INTRODUCTION

The plankton in the ocean consist of a wide variety of organisms from bacteria to jellyfish. And while plankton, by definition, are species that predominantly drift along with the currents, this does not mean that assuming plankton “go with the flow” is an accurate representation of their behavior in a turbulent ocean (Wheeler et al., 2019). Plankton respond to the flow around them both actively and passively in ways which can alter their net transport. Ecologically, understanding planktonic vertical transport is important because it modulates the plankton’s position

<footnote n="1">¹Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA</footnote>

<footnote n="2">²Aix Marseille Univ, CNRS, Centrale Med, IRPHE, Marseille, France</footnote>

<footnote n="3">³Department of Biology, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA</footnote>

<footnote>*Author for correspondence (mdiben@princeton.edu)</footnote>
in the water column, controlling their exposure to currents, nutrients, and sunlight. Water column position is in part mediated by plankton swimming; however plankton swimming speeds are often much slower than the mean currents or characteristic fluctuating velocities near the ocean surface (Franks et al., 2022). And yet, their net speed may in fact differ due to their interactions with turbulence (e.g. see (Fuchs et al., 2013; Michalec et al., 2015; DiBenedetto et al., 2022)). Of interest is a recently theorized "surfing strategy" which describes how swimming plankton could effectively double their net speed in turbulence compared to still water by preferentially sampling the fluid velocity fluctuations (Monthiller et al., 2022). However, it is still unclear whether real plankton exhibit this behavior.

It has been well-established that small changes to planktonic behavior can have large-scale effects on their transport. One well-studied passive behavior in the plankton is bottom-heaviness, i.e., gyrotaxis in a viscous fluid (Kessler, 1985). Gyrotactic swimmers have been of interest due to the array of behaviors they exhibit in different flows. Gyrotactic trapping was described by (Durham et al., 2009) where gyrotactic swimmers can form horizontal thin layers under steady horizontal shear flow. In isotropic turbulence, gyrotactic swimmers can “unmix” into patches (Durham et al., 2013), and can preferentially sample different parts of the turbulence depending on their shape and relative swimming speed. *Preferential sampling* in this case refers to when plankton oversample certain turbulence statistics, on average, relative to a fluid tracer which faithfully follows the flow. For example, in homogeneous turbulence with no mean flow, a tracer would also experience no mean flow. However, a plankter (or particle) which preferentially samples the turbulent fluctuations could experience a net flow. While previous work has shown that spherical gyrotactic swimmers will preferentially sample downwelling areas of the flow and thereby reduce their upward transport (Zhan et al., 2014; Gustavsson et al., 2016), elongated, non-spherical swimmers can preferentially sample upwelling areas of the flow to enhance their upward transport (Borgnino et al., 2018; Lovecchio et al., 2019). These studies show how even simple behavioral models, such as a swimmer with a constant swim speed and a passive gyrotactic torque, can have meaningfully altered transport when studied in a turbulent flow. Many small, motile phytoplankton with small Reynolds numbers ($Re \ll 1$) have been shown to appropriately behave as simple gyrotactic swimmers (Durham et al., 2009, 2013; Zeng et al., 2022), however it is not clear that these dynamics apply to other types of plankton with more complex behavior.

Zooplankton are typically larger than phytoplankton and have more diverse morphologies and swimming behaviors. Therefore, phytoplankton models of behavior cannot directly be mapped onto larger zooplankton which have intermediate Reynolds numbers. While many planktonic larvae are also bottom-heavy and thus have passive gravitational stability (Strathmann and Grünbaum, 2006; Clay and Grünbaum, 2010; Wheeler et al., 2016), it is not clear that gyrotaxis adequately describes all of their orientational behavior, even though the gyrotactic model has been previously applied to some planktonic larvae (Fuchs et al., 2013, 2015a,b). The deviation from a passive model of orientation, i.e. gyrotaxis, is in part due to the fact that many plankton can also actively control their orientation. For example, many larvae exhibit helical swimming behaviors with varied angular velocities (e.g. (Chan and Grünbaum, 2010; Bearon, 2013; Maciejewski et al., 2019; Krishnamurthy et al., 2020)), or more complex swimming patterns that cannot be categorized as helical swimming (DiBenedetto et al., 2021). However, studies have yet to fully elucidate how both the active and passive orientational behavior by larvae modulate their transport in turbulence.
This study builds on previous work we conducted studying the swimming response of zooplankton to turbulence, in which we observed *Crepidula fornicata* larvae swimming in a jet-stirred turbulence tank (DiBenedetto et al., 2022). Overall, we found that these plankton do increase their upward swimming speed in response to turbulence, and that their response is best correlated to the local fluid acceleration. However, we also found that the plankton modulated their response based on the average background level of turbulence, suggesting that they are able to integrate flow information over time to influence their behavior. In contrast, in this study, we focus on orientational behavioral responses to turbulence, studying how the plankton both actively and passively respond to the flow around them. Because we cannot explicitly distinguish between these two behaviors in our data, we compare our results to simulations of plankton with two different behavioral models in order to interpret the observations. We examine both the models of passive gyrotaxis and active surfing: the former of which theoretically results in a slowdown of upward swimming (spherical) plankton in turbulence, and the latter of which results in a speed up of the same plankton in turbulence. Overall, we find that the behavior of the real plankton in this study tends to agree more with that of surfing than gyrotactic plankton.

In the remainder of this paper we first describe the methodology, including both the laboratory experiments and the simulations. Next, we review the gyrotaxis and surfing models of behavior. Finally, we report and discuss the results of the study.

# METHODS AND MATERIALS

## Laboratory experiments

We use the veliger larval stage of the slipper snail *Crepidula fornicata* as our model organism. These plankton are approximately spherical with a heavy shell. They have a ciliated velum divided into two velar lobes which they use to swim that sits above their shell in their upright posture, as drawn in Fig. 1A. Their cilia create a current used for both feeding and propulsion. In general, their swimming speed smoothly varies, and they have a strong preference for swimming up due to their bottom-heaviness (Chan et al., 2013; DiBenedetto et al., 2021). Their bias for swimming up is also related to the fact that they are negatively buoyant, so they need to create upward propulsion in order not to sink. The diameter of the larvae range from about 0.3 to 1 mm and they can swim at speeds of up to 1 to 4 mm s⁻¹ in still water (Chan et al., 2013) which puts them in an intermediate Reynolds number regime defined as

$$ Re_p = V_{swim} d / \nu \approx O(1) $$ where $V_{swim}$ is their swim speed, $d$ is their diameter and $\nu \approx 10^{-6} \text{m}^2/\text{s}$ is the kinematic viscosity of sea water. This preference for swimming up, active orientational behavior, spherical shape, and intermediate Reynolds number is what makes them an appropriate model organism for this study. We expect them to be able to respond to the flow, but to not vary their swimming speed too much, when e.g., compared with copepods which exhibit highly unsteady and irregular jump responses in turbulence (Kiørboe and Visser, 1999; Michalec et al., 2015).
We conducted controlled laboratory experiments to observe the effects of turbulence on plankton swimming behaviors. Two sets of larvae were used in the experiments: an “early stage” and a “late stage”. The former were 2 days old, with an average diameter of 488 micron, and the latter were 12 days old, with an average diameter of 758 micron. The larvae were put in a 30.5 cm cubic jet-stirred turbulence tank designed to create low Reynolds number isotropic turbulence. This data comes from a previously published study (DiBenedetto et al., 2022), where more details can be found on the plankton collection and rearing, the turbulence tank, and the experimental procedure. To summarize, experiments were run at randomized turbulence levels: “low,” “medium,” and “high”. Average flow statistics are reported for each case in Table 1. One common way to characterize turbulence intensity is via the dissipation rate $\varepsilon$. The levels in our experiments range from values on the order of $10^{-3}$ to $10^{-2} \text{ cm}^{-2} \text{ s}^{-3}$. This range corresponds to observations in the upper 20 m of the continental shelf, estuaries, and tidal channels (Fuchs and Gerbi, 2016), and the upper 10 m of the open ocean (Franks et al., 2022). We used particle image velocimetry (PIV) to measure the flow field in the center plane of the tank and used particle tracking to measure the velocity of the plankton that intersected the 2D plane. All measurements were taken in the $x-z$ plane, corresponding to $u$ and $w$ velocities, and therefore only the out-of-plane vorticity component $\omega_y$ was measured. The image field-of-view was approximately 5.4 cm×5.4 cm with 377 pixels $\text{cm}^{-1}$ resolution. PIV processing was done with LaVision software and the resultant PIV vectors were resolved to 32×32 pixel boxes with 25% overlap.

## Laboratory data analysis

In still water, the plankton velocity is due to the effects of swimming and gravitational sinking which depend on the forces from propulsion, drag, buoyancy, and gravity. This means that the larval relative velocity $\mathbf{u}_r$ can be approximated as the sum of a swimming velocity $\mathbf{u}_s$ and terminal settling velocity $\mathbf{u}_g$, as depicted in Fig. 1B. We label the swimming speed (the magnitude of $\mathbf{u}_s$) as $V_{swim}$. We emphasize that there is inherent uncertainty in estimating $\mathbf{u}_g$ which is a function of the mass, volume, and drag coefficient of the larvae. The mass and volume has variability across individuals, and the drag coefficient depends on velar posture which is variable and depends on the larval swimming behavior. Therefore, we do not attempt to quantify $\mathbf{u}_g$, and instead we analyze $\mathbf{u}_r$ as a proxy for swimming. Note that the horizontal component of $\mathbf{u}_r$ is entirely due to swimming. In turbulence, the larval velocity is additionally affected by the background flow. In this case, we measure the total larval velocity $\mathbf{u}_p$ which is the sum of $\mathbf{u}_r$ and the fluid velocity $\mathbf{u}_f$, which represents the flow field that the plankton are experiencing (see Fig. 1C). By interpolating the PIV field around the plankton onto the centroids of the plankton ($x_p, z_p$), we were able to estimate $\mathbf{u}_f$. We interpolate the fluid velocity and spatial gradients using a least-squares optimization of a 2D, second-order Taylor series function of the PIV data in an annulus greater than 0.27 cm and less than 0.53 cm away from the larval centroids (100 and 200 pixels). A sensitivity analysis showed that our results were not very sensitive to this choice. We calculate the plankton relative velocity $\mathbf{u}_r$ by taking the total plankton velocity $\mathbf{u}_p$ measured via particle tracking and subtracting the interpolated fluid velocity $\mathbf{u}_f$:
$$ \mathbf{u}_r = \mathbf{u}_p - \mathbf{u}_f(x_p, z_p). \qquad (1) $$

Note that we define the vector $\mathbf{u}_r = (u_r, w_r)$ in 2D where $u_r$ is the horizontal component in the x-direction and $w_r$ is the vertical component in the z-direction.

In our experiments, the relative velocities of the larvae were on average larger in the vertical direction than in the horizontal direction due to their preference for swimming up. We report the average vertical relative velocity for each larval stage and each turbulence level in Table 1. Note that there is some response to the turbulence level: in our earlier study, we found that the plankton tend to increase their upward swim speed under increasing turbulence intensity and increasing local fluid acceleration (DiBenedetto et al., 2022).

Whereas in our previous study we focused on plankton vertical swimming behavior, in this study we instead focus on the orientational behavior of the plankton relative to the local flow gradients. Because the plankton are approximately spherical, measuring their orientation in the experimental images was not possible. As a proxy for orientation, we can consider the horizontal relative velocity $u_r$; the larvae are typically observed in their upright position, and therefore their orientation should correlate to their horizontal swimming velocity which is estimated via their horizontal relative velocity. We chose to avoid using a model to infer plankton kinematics, which e.g. could be done by either using a Maxey Riley model to estimate the forces and torques on the plankter (e.g. (Fuchs et al., 2015a,b)), or by estimating swimming by subtracting an assumed sinking speed from the measured relative velocity. We chose this approach to avoid uncertainty in estimating plankton drag coefficients and mass distribution, which would be needed to estimate the forces and the sinking speed of the larvae. The drag coefficient of these plankton is not constant because they manipulate their velum to adjust their swimming velocity (Chan et al., 2013). Finally, to further constrain our analysis, we only consider the relative horizontal velocity $u_r$ when the larvae are swimming up, to avoid circumstances when the plankton are either upside down or sinking.

## Numerical simulations

To model a homogeneous isotropic turbulent flow, we use the pseudo-spectral, open-source solver Snoopy (Lesur and Longaretti, 2005, 2007). The Navier-Stokes equations are solved for an incompressible fluid with kinematic viscosity $\nu = 0.004$ (arbitrary unit), in a tri-periodic domain of size $l = 1$ (arbitrary unit) with resolution of $n \times n \times n$ with $n = 128$. We used an external forcing delta-correlated in time and localized in spectral space ($3/2 < |k|l < 5/2$, with $k$ the wavevector) to make the flows statistically steady. The simulations used in this study come from (Monthiller, 2022), where more details can be found. We chose a simulation with $Re_\lambda = 11$ because although this Reynolds number is somewhat lower than that in the experiments, (Monthiller, 2022) showed their results to be qualitatively similar over a wide range of Reynolds numbers.
We used the root mean square velocity $u_{rms}$ and the integral length scale $L$, computed as follows

$$L = \frac{\pi}{2u_{rms}} \int \frac{E(k)}{k} dk, \qquad (2)$$

where $E$ is the spectral energy density, to characterize the simulated turbulent flows. The parameters of the simulations performed with *Snoopy* are summarized in Table 2. The Lagrangian equations of plankter motion are integrated using the open-source code, SHELDO N (Monthiller et al., 2022).

## Conceptual plankton model

We use an idealized framework to describe plankton swimming in turbulence to compare to and contextualize our laboratory observations. In this framework, we assume plankton are smaller than or on the same order as the Kolmogorov lengthscale $\eta$. This means that the flow gradients are uniform across the plankter and that the flow they experience can be described by the velocity at their centroid $\mathbf{x}_p(t)$. We further assume the plankton do not affect the flow around them, and that they are advected by the fluid velocity $\mathbf{u}_f$ plus a constant swimming speed $V_{swim}$. Their swimming direction is denoted by the unit vector $\mathbf{p}$:

$$\frac{dx_p}{dt} = \mathbf{u}_f(x_p, z_p) + V_{swim} p(t). \qquad (3)$$

Although we have assumed $V_{swim}$ is constant in this case in order to focus on orientational dynamics, in general both the plankton we consider in this study and others can vary their swim speed (e.g. (Chan et al., 2013; DiBenedetto et al., 2021; Visser and Kiørboe, 2006; True et al., 2020)). This description of plankton swimming allows us to directly compare two models of plankton orientation: *passive* gyrotaxis and *active* surfing. In reality, we expect real plankton to have both a passive and active orientational response, and we cannot precisely separate the two responses in our experimental observations.

## Gyrotaxis

Gyrotactic swimmers passively reorient based on the local fluid vorticity and their gyrotactic reorientation timescale $B$ (Kessler, 1985). This reorientation timescale characterizes the restoring torque due to plankton bottom-heaviness in a viscous fluid. The torque is in part controlled by how offset the plankton's center of mass is from their center of buoyancy. We write the equation of motion for their orientation following (Pedley and Kessler, 1987):

$$\frac{d\mathbf{p}}{dt} = \frac{1}{2B}(\mathbf{n}_g - (\mathbf{n}_g \cdot \mathbf{p})\mathbf{p}) + \frac{1}{2}\omega(x_p, z_p) \times \mathbf{p}, \qquad (4)$$
where $\mathbf{n}_g$ is a unit vector which denotes their stable upright orientation due to gravity, and $\omega$ is the fluid vorticity evaluated at the plankter's centroid. This model is appropriate for spherical plankton, which is a good approximation of the plankton considered in our experiments. If the plankton were non-spherical, there would be an additional term that involves the aspect ratio of the plankton and the rate of strain tensor of the flow (Jeffery, 1922; Zhan et al., 2014; Gustavsson et al., 2016), which is most important for elongated plankton with high aspect ratio. Eqn 4 gives a fixed-point orientation determined by the vorticity and reorientation timescale: in two dimensions, we can define an orientation angle $\theta$ that is the angle between $\mathbf{p}$ and the vertical direction. In this case, $\theta$ is then given by the equation $\sin\theta = B\omega$ for $B\omega < 1$. (In three dimensions, the equation is more complicated; e.g. see (Eloy, 2024).) We see clearly from this expression that stronger vorticity results in a larger rotation.

Under this passive behavioral model, we expect vorticity to be correlated with the orientation of the plankton. We expect plankton to tilt with the vorticity, as is depicted in Fig. 1E.

## Surfing

Surfing describes an active orientational control strategy by plankton in response to the local flow gradients. This is a strategy that has only been described theoretically and was derived in order to optimize plankton navigation with minimal sensing of the flow (Monthiller et al., 2022). Under the assumption that the plankton can reorient instantaneously, we can write an equation for their orientation directly as follows:

$$ \mathbf{p}(t) = \left[ \exp(\tau \nabla \mathbf{u}_f) \right]^T \cdot \mathbf{n}_s \quad (5) $$

where $\mathbf{n}_s$ is a unit vector in the target surfing direction, $\tau$ is a free parameter which we refer to as the surfing time horizon, $\exp(\cdot)$ is the matrix exponential and $[\cdot]^T$ is the transpose. Using simulations, we have previously shown that there exists an optimal $\tau$ for a given flow and $V_{swim}$ value (Monthiller et al., 2022). Initial results from simulations of surfers in turbulence over a range of Reynolds numbers and $V_{swim}$ values indicate that the optimal $\tau$ is on the order of the Kolmogorov timescale $\tau_\eta$ and that increasing swimming velocity tends to reduce the optimal $\tau$ (Monthiller, 2022). In the surfer simulations used in this study, we found the optimal $\tau = 2\tau_\eta$.

Under this active behavioral model, we expect vorticity to be anti-correlated with the orientation of the plankton. We expect plankton to tilt into the vorticity, as is depicted in Fig. 1F.
# RESULTS

## Plankton observed to actively oppose fluid vorticity

The main result from our experimental observations indicates a strong anti-correlation between the larval horizontal relative velocity and the fluid vorticity, consistent with surfing behavior. We observe this effect in both the early- and late-stage larvae and in all levels of turbulence, as seen in Fig. 2 where horizontal plankton relative velocity $u_r$ was binned and plotted against vorticity $\omega_y$ for each data set. In this coordinate system, which is the same as that depicted in Fig. 3, negative vorticity (counterclockwise, out of the page) would tend to cause a gyrotactic plankter to swim in the negative x direction (to the left), and positive vorticity (clockwise, into the page) would tend to cause a gyrotactic plankter to swim in the positive x direction (to the right). Instead, we observe anti-gyrotactic behavior, and find that the larvae are on-average swimming opposite the expected tilt direction by vorticity. There is some variation in the slope of the relationships across turbulence levels, but they all tend to overlap within the uncertainty bounds.

To demonstrate this effect, we include snapshots of individuals tilting against the vorticity at the high turbulence level in Fig. 3. The flow relative to the larva in both cases has strong negative $\omega_y$ (counterclockwise), which would tend to tilt the bottom-heavy larvae to the left. However, in both cases, the larvae are observed swimming to the right with positive $u_r$ values, demonstrating anti-gyrotactic behavior.

This result is further illustrated with a conceptual model in Fig. 1D-F. The larva in its stable, upright posture is shown for reference in Fig. 1D, and a larva tilting against the flow rotation is depicted in Fig. 1F. Because the larva is bottom-heavy, when tilted to the right, a gravitational torque exists that would tend to rotate the larvae counterclockwise (as drawn). In addition, the flow's vorticity also induces a counterclockwise torque on the larva. If the system is in equilibrium, then the larva would have to actively impose a torque to rotate clockwise, opposing the flow and gravitational torques. This suggests that we are observing an active orientational behavior, as the larval orientation cannot be fully explained without it.

We further assess this observed correlation by calculating the Pearson correlation coefficient $\rho$ between $\omega_y$ and $u_r$, which we report in Table 3 for each scenario. We find that all scenarios have statistically significant negative correlations, which agrees with the binned data plot in Fig. 2. However, we find that the correlation coefficients are low for the early-stage larvae, where $\rho$ ranges from -0.1 to -0.26, indicating high variability in the data. The late-stage larvae show stronger correlations, where $\rho$ ranges from -0.38 to -0.41. This suggests that the late-stage larvae show a more consistent anti-gyrotactic response when compared to the early-stage larvae. And yet, the $\rho$ values in all cases are rather modest, which is to be expected both due to the inherent variability in studying living organisms and the fact that the larvae are responding to a three-dimensional flow but we are only able to measure the flow and their behavior in two dimensions. Thus, this anti-gyrotactic orientational correlation is only observed in the average sense. Overall, we find that the larvae are responding actively to the flow in a way which could promote
surfing, and that we would expect the late-stage larvae to be better surfers than the early-stage larvae given their stronger response to the vorticity.

## Simulated surfers actively orient to oppose fluid vorticity

We next present results from the simulations of plankton in turbulence using the two different models of swimming behavior: passive gyrotaxis and active surfing, as described in the methods. To compare between the simulations and experiments, we nondimensionalize the data. We normalize the fluid vorticity using the Kolmogorov time scale $\tau_{\eta} = \sqrt{V/\epsilon}$. In the experimental data, we normalize $u_r$ with the maximum vertical swimming velocity of the larvae observed in still water $w_r$ as a proxy for orientation. The observed max $w_r$ values were 0.18 and 0.33 cm s⁻¹ for the early- and late-stage larvae respectively (as reported in (DiBenedetto et al., 2022)). In the simulations, we have direct access to the plankton orientation, so we plot their orientation projected onto one horizontal axis

$$p_x = u_r / V_{swim}.$$

The results from the simulation are plotted in Fig. 4(left), where we see that the simulated surfer orientation indeed also shows a strong anti-correlation with vorticity, resembling the trend observed in experiments in Fig. 4(right). When comparing to the gyrotactic plankton orientation, which is also plotted in Fig. 4(left), we see the opposite orientational behavior, as expected.

These simulations again support the result that the larvae in the experiment were not exhibiting traditional passive gyrotaxis, and instead were actively rotating to oppose the vorticity. The simulations plotted were from surfers with the optimal time horizon $\tau$, which is the free parameter in the surfing model and describes how surfing plankton should weight their response to the velocity gradient. The similarity in the plots also indicates that the larval orientational behavior response to vorticity has good agreement with that of optimal surfers, i.e. surfers with the optimal time horizon.

## Plankton preferentially sample the flow

While we have shown how the orientational behaviors of the larvae resemble surfing behaviors, to truly confirm whether the larvae are surfing, we need to assess whether they preferentially sample the flow. In other words, we measure the average fluid vertical-velocity experienced by the larvae relative to a fluid tracer. Measuring the effect of preferential sampling is not trivial in a closed, jet-stirred turbulence tank; this is because even though the tank was designed to have zero mean flow, small mean flows can set up in each experimental trial that are on the order of the velocities of interest. To account for this, we can consider each experimental trial and compare the average fluid velocity sampled by the larvae to the average fluid velocity in the tank in each trial. To be consistent and avoid any bias due to the fluid interpolation onto the larvae, we measure the fluid velocity interpolated to random points in the flow using the same interpolation method. The difference between these two values we call $\Delta\langle w_f \rangle$ which is a
measure of preferential sampling. When $\Delta\langle w_f \rangle = 0$, the larvae are not preferentially sampling the flow, when $\Delta\langle w_f \rangle > 0$, it indicates the larvae oversample upwelling vertical velocities relative to the mean fluid velocity, and when $\Delta\langle w_f \rangle < 0$ it indicates the opposite.

We plot $\Delta\langle w_f \rangle$ for each experimental trial in Figure 1. First we consider the late-stage larvae, and we see that $\Delta\langle w_f \rangle$ is close to zero for the low turbulence trials, and that $\Delta\langle w_f \rangle$ tends to increase for most of the medium and high turbulence trials, indicating preferential sampling of upwelling flow. The maximum of these values exceeds 2 mm s⁻¹ which is more than twice the average vertical swimming speed (measured as relative velocity $w_r$) for the late-stage larvae in the high turbulence level (where the mean $w_r$ was measured to be 0.37 mm s⁻¹ as reported in Table 1). In contrast, the early-stage larvae do not show strong evidence of preferential sampling, as the values of $\Delta\langle w_f \rangle$ in Figure 1 are all close to zero regardless of the background turbulence level. The values do skew negative in the high turbulence trials. Therefore, these observations indicate that only the late-stage larvae may be successfully surfing the turbulence in our experiment.

## Larval response to fluid velocity gradient promotes surfing

In the original surfing model, the surfers use information from the full velocity gradient. However, so far we have only considered how the larvae respond to the vorticity. In this section, we further examine the relative importance of the different components of the velocity gradient for both the simulated surfers and the larvae observations.

As previously discussed, the simulated surfers tend to tilt into the vorticity, however we additionally observe that they also tend to align with the strain axis of maximal extension (see (Monthiller, 2022) for more details). We test whether the larvae in our experiments are similarly aligning with the strain by plotting their horizontal relative velocity as a function of the strain angle and calculating the Pearson correlation coefficients $\rho$, included in the Supplemental Information. Although we can only make 2-dimensional measurements in the lab, we do not observe any strong correlation between the two variables. For example, in the medium and high turbulence levels, $|\rho| \le 0.1$ for both sets of larvae. This suggests that the larvae do not have a strong orientational response to the fluid strain in these experiments. This lack of correlation between the larval relative velocity and the strain angle is unsurprising given how the larvae sense the flow: the larvae have statocysts which can measure rotational motion and translational accelerations, but the larvae sense strain via deformation to their ciliated velum (Budelmann, 1989) which may not be as sensitive to the flow gradients.

To test the efficiency of surfing by responding only to the vorticity, we conducted additional simulations to test how restricting the sensed information affects plankton's surfing ability. Instead of using the full velocity gradient in Eqn 5, we tested using only the rate of strain tensor (the symmetric portion of $\nabla u_f$) or only the rate of rotation
tensor (the skew-symmetric portion of $\nabla u_f$). The surfing strategy depends on a free parameter $\tau$ which relates to the time horizon for which the flow velocity gradient $\nabla u_f$ measured by the plankton can be considered almost constant. The optimal value of this parameter depends both on the swimming speed and the flow, as discussed in (Monthiller et al., 2022). We plot the net vertical velocity due to both swimming and preferential sampling normalized by the simulated swim speed as a function of the surfing time horizon in Figure for the three different cases. We assess surfing performance as the degree to which this normalized vertical transport

$$ (\langle w_r + w_f \rangle / V_{swim}) \text{ exceeds unity, where } \langle w_r + w_f \rangle / V_{swim} = 1 \text{ denotes when the plankton experience no surfing effect and are transported at their swimming speed } V_{swim} \text{ on average. We see that while surfers perform best when given the full velocity gradient, those responding solely to the rate of rotation outperform those responding solely to the rate of strain within the optimal surfing range } (\tau \lesssim 4\tau_\eta \text{ in this case}). This suggests that plankton can still benefit from surfing even if they only respond to fluid vorticity, and that sensing rotation is much more important to surfing than sensing strain. At high } \tau / \tau_\eta \text{ we see that surfing performance shows a marked decrease.} $$

This is due to the fact that larger time horizons $\tau$ underestimate the importance of the unsteadiness in the turbulence velocity gradient; the optimal $\tau$ is thought to be related to the correlation timescale of the turbulence signal experienced by the plankter (Monthiller et al., 2022).

Beyond just vorticity, we can further break down the plankton responses to the individual velocity gradient components in the experiments. If the larvae were indeed actively orienting to increase their upward transport, we would expect them to respond more to the vertical velocity shear $dw_f / dx$, rather than the horizontal velocity shear $du_f / dz$. This distinction comes from the fact that even though both shear components will rotate the larva, only the former is important to surfing. To elaborate, the surfing strategy optimizes the swimming direction in order to preferentially sample upward fluid velocity. Therefore, it is more important for the surfer to respond to changes in the vertical velocity (e.g. $dw_f / dx$ in this case) than to changes in the horizontal velocity (e.g. $du_f / dz$). Mathematically, this can be further justified by noticing that the surfing strategy in Eqn 5 can be approximated to first order as $\mathbf{p}(t) \approx \mathbf{z} + \tau \nabla w_f$ when $\mathbf{n}_s = \mathbf{z}$ (Monthiller, 2022). If the larvae were actively rotating in order to stay upright, rather than to surf, we might expect equal responses to both velocity gradients. We assess the response by again calculating the Pearson correlation coefficients $\rho$ for each shear velocity gradient component, reported in Table 4. In all but one case we find higher magnitudes of $\rho$ when comparing $u_r$ with $dw_f / dx$ than with $du_f / dz$. This suggests that the larvae tend to have a stronger orientational response to $dw_f / dx$ which is consistent with the surfing model.
DISCUSSION

By carefully observing plankton in turbulence and comparing their swimming behavior to simulations, we have begun to disentangle the coupled effects of their passive and active orientational responses to the fluid flow. The veliger larvae of *Crepidula fornicata* are an ideal model organism due to their smooth but varied swimming behaviors, their intermediate Reynolds number, their bottom-heaviness and their approximately spherical shape. By analyzing their horizontal relative velocity as a function of the flow around them, we have found evidence that suggests these plankton are capable of actively exploiting turbulent flows to enhance their vertical transport, agreeing with the theoretical “surfing” strategy rather than passive gyrotaxis alone.

We have presented several pieces of evidence that support the active surfing hypothesis. The first is the observed orientational response to the local vorticity. This rotational response must be active as passive reorientation alone cannot result in rotation opposing fluid vorticity, thereby strongly suggesting active behavioral control. For example, at low Reynolds numbers the torque on a rigid object imposed by the fluid vorticity will always be in the direction of the vorticity, no matter the shape, because the resistance tensor that relates vorticity and viscous torque is positive definite. Thus, in contrast to passive gyrotactic stabilization, the anti-gyrotactic behavior of the larvae we observe must be active because it is in the opposite direction of the vorticity. Previous observations of *C. fornicata* larvae in still water have shown that they do actively vary their swimming direction and speed (DiBenedetto et al., 2021) and can respond to turbulent accelerations by varying their vertical swimming speed (DiBenedetto et al., 2022). These active variations in swimming direction in response to the local flow are consistent with our findings. Finally, while we have described the larvae as approximately spherical in shape, their shells are somewhat flattened and become more asymmetric as they develop. This ontogenetic change in morphology could potentially influence their hydrodynamic response. However, at low Reynolds number, rotation by vorticity is independent of shape, and strain-induced rotation only occurs for elongated shapes (Jeffery, 1922). Therefore, shape changes alone cannot explain the observed responses to vorticity.

The next piece of evidence is that the larvae show minimal orientational response to the local fluid strain rate. The surfing strategy described by (Monthiller et al., 2022) demonstrates optimal surfing occurs when plankton respond to the full velocity gradient. However, simulations show that the rotational component (vorticity) contributes significantly more to successful surfing than the strain component (as shown in Figure). This finding aligns with our observations of larvae primarily responding to vorticity. We note that plankton shape likely influences sensitivity to strain. Elongated plankton naturally tend towards alignment with the extensional rate of strain tensor, enhancing surfing transport passively as has been shown for gyrotactic plankton (Borgnino et al., 2018; Lovecchio et al., 2019). Thus, more elongated plankton may benefit from combining an active response to vorticity with a passive strain alignment, further enhancing their transport. Although our study organism is approximately spherical, future studies involving elongated plankton could provide deeper insights into these combined effects.

The final piece of evidence is the observed preferential sampling of the flow. The late-stage larvae exhibited preferential sampling of upward fluid velocities at medium and high turbulence levels, effectively doubling their net upward velocity relative to their average swimming speed, in close agreement with predictions from surfing simulations. This is consistent with the observation of the late-stage larvae having a stronger anti-vorticity rotational
response than the early-stage larvae. The absence of preferential sampling for the early-stage larvae does not necessarily indicate an inability to surf, given the other evidence. The early-stage plankton did typically swim up faster than the late-stage larvae (as was also previously reported in (DiBenedetto et al., 2022)). Due to the relatively small size of the tank, the larvae can quickly reach and accumulate near the top of the tank. Consequently, our data from observations in the center of the tank may be biased toward slower swimmers or larvae preferentially sampling downwelling regions of the flow. These experimental constraints mean that we cannot definitely assess whether the early-stage larvae were able to effectively surf the flow.

We can compare the late-stage larvae preferential sampling values to the swimming speeds to assess how large the surfing effect would be on their net transport. We reported that the late-stage larvae had an average upward swim speed of less than 0.1 cm s⁻¹ in all turbulence levels (although with instantaneous maximum speeds up to 0.33 cm s⁻¹) (DiBenedetto et al., 2022). This suggests that surfing can more than double their effective upward velocity by enabling the sampling of higher speed fluid velocities up to and over 0.2 cm s⁻¹ on average. Note that the total larval transport in this case is due to the sum of the average swim speed and the average preferential sampling speed. This is similar to the magnitude of increase reported in the optimal surfing simulations (Monthiller et al., 2022) which found that optimal surfing could effectively double the net speed of swimming plankton.

Finally, although we hypothesize that active surfing behavior is causing the larvae to preferentially sample the flow, finite-sized particles can also preferentially sample turbulence due to inertial effects (e.g. see (Balachandar and Eaton, 2010)). However, we argue that the observed preferential sampling is not due to inertial effects, because even in the highest turbulence level, the highest Stokes number $St$ of the plankton is order $10^{-3}$. We estimate Stokes number here by taking the ratio of $St = \tau_p / \tau_\eta$ where $\tau_p = (\rho_p - \rho_f) d^2 / 18\mu$ is the Stokes relaxation timescale of a sphere where $\rho_p$ is the density of the larvae and $\rho_f$ is the density of seawater. We use the diameter of the larvae and a density of 1.18 g cm⁻³ based off of oyster larvae reported in (Fuchs et al., 2015b). We find that the early and late stage larvae have $\tau_p = 0.002, 0.005$ seconds respectively. Inertial preferential sampling is typically strongest when Stokes numbers approach unity and is minimal for $St \ll 1$ (Balachandar and Eaton, 2010), so we do not expect it to play an important role here given the small Stokes number values of $St \le 10^{-3}$.

In summary, our results demonstrate that relying only on passive reorientation models to describe plankton responses to turbulence is often insufficient, especially for the veliger larvae considered in this study. Although bottom-heavy plankton can passively stabilize their orientation, this does not preclude them from also actively controlling it. In addition to enabling surfing, this active orientational behavior also allows plankton to maintain upright posture under higher turbulence levels than passive stability alone would permit. Thus, understanding the interplay between active and passive behavioral responses to turbulence is important to fully characterize plankton transport.
Acknowledgements

We thank Erik Anderson (Grove City College) for the use of his turbulence tank, Anthony Pires (Dickinson College) for his assistance in culturing the plankton, and Karl Helfrich (Woods Hole Oceanographic Institution) for his help in designing and performing the experiments. We thank all three for discussions that helped inform this work. M.H.D. was supported by a postdoctoral scholarship from Woods Hole Oceanographic Institution. The project was supported by NSF OCE-1948623. R.M. and C.E. have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 834238).

Competing interests

The authors declare no competing or financial interests.

Author contributions

Conceptualization: M.H.D., R.M., C.E., L.S.M.; Data curation: M.H.D., R.M.; Formal analysis: M.H.D., R.M., C.E.;
Funding acquisition: M.H.D., C.E., L.S.M.; Investigation: M.H.D., R.M., C.E., L.S.M.; Methodology: M.H.D.,
R.M., C.E., L.S.M.; Project administration: M.H.D., C.E., L.S.M.; Resources: C.E., L.S.M.; Software: R.M.;
Supervision: C.E., L.S.M.; Validation: M.H.D., R.M., C.E.; Visualization: M.H.D., R.M.; Writing – original draft:
M.H.D., R.M.; Writing – review & editing: M.H.D., R.M., C.E., L.S.M.

Funding

Open Access funding provided by Princeton University Library. Deposited in PMC for immediate release.

Data and resource availability

Experimental data are available from figshare: https://figshare.com/s/d2adfbdf1cbc07bf09f7. All relevant data and
details of resources can be found within the article and its supplementary information.

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Figures

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Plankton diagram and kinematic schematics. (A) Diagram of veliger larvae of *Crepidula fornicata* (not drawn to scale) with representative relative locations of center of buoyancy (C.O.B.) and center of mass (C.O.M.). (B) In still water, plankton relative velocity **u**<sub>r</sub> is the sum of the plankton swimming velocity **u**<sub>s</sub> and gravitational sinking velocity **u**<sub>g</sub>. The plankton is assumed to swim in the direction it is oriented, as denoted by the unit vector **p** in (d). (C) The total plankton velocity in turbulence can be divided into the sum of the plankton relative velocity **u**<sub>r</sub> and fluid velocity interpolated onto the plankton **u**<sub>f</sub>. (D) A larva drawn in its stable upright posture. (E) Gyrotactic behavior with passive stabilization resulting in tilting in the direction of flow rotation. (F) Conceptual model for active stabilization behavior resulting in tilting opposing flow rotation consistent with surfing.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>**Fig. 2. Larval response to the instantaneous vorticity.** Horizontal relative velocity $u_r$ for larvae swimming up is binned by vorticity $\omega_y$, and plotted by background turbulence level. (Left) Early-stage larvae and (Right) late-stage larvae data plotted separately. Error bars on bins denote 95% confidence intervals found by bootstrapping the data. Bins are equally spaced in vorticity for each turbulence level (with a spacing of 0.125 s⁻¹ for the low turbulence level and 0.2 s⁻¹ for the medium and high levels); bins are only shown that have a minimum of 75 data points. Corresponding correlation coefficients reported in Table 3.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Observations of larvae in turbulence in high vorticity regions, which corresponds to the conceptual model in Fig. 1(F). (Left) Early-stage larva and (Right) late-stage larva snapshots shown under high turbulence forcing. The vector field denotes fluid flow relative to the larval velocity vector: **u**<sub>f</sub> − **u**<sub>p</sub>. Larval velocity **u**<sub>p</sub> and larval relative velocity **u**<sub>r</sub> are plotted from the centroid of each larva. Scale bar shows 5 mm s<sup>−1</sup>. Images are taken from raw experimental footage.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Comparison of the response to instantaneous vorticity between simulations and experiments. (Left) Simulation are evaluated for a swimmer of swimming velocity $V_{swim} = u_{\eta}$ where $u_{\eta}$ is the Kolmogorov velocity scale given by $u_{\eta} = (\nu\epsilon)^{1/4}$. The surfer has been provided its optimal time horizon $	au = 2\tau_{\eta}$ and the gyrotactic swimmer is evaluated with a reorientation timescale $B = \tau_{\eta}$. (Right) The data of Fig. 2 is reported here but renormalized to be compared with simulations. The still water maximum $w_r$ values are used for normalizing the data. Early-stage are dashed and late-stage are solid lines.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5. Average fluid velocity sampled by larvae minus the average fluid velocity sampled by random points in the tank, as a function of the average background turbulence dissipation. Each data point represents one experimental trial. Positive values indicate enhanced upward transport and negative values indicate reduced transport.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Vertical migration performance of simulated surfers in turbulence as function of the surfing time horizon $\tau$ for surfers responding to different parts of the full velocity gradient $\nabla u_f$. Simulations are evaluated for a swimmer of swimming velocity $V_{swim} = u_\eta$ where $u_\eta$ is the Kolmogorov velocity scale. Note that the optimal $\tau = 2\tau_\eta$ corresponds to the peak in the surfing enhancement.</figcaption></figure>

<table><caption>**Table 1.** Experimental overview</caption><thead><tr><th></th><th colspan="4">Turbulence level</th></tr><tr><th></th><th>None</th><th>Low</th><th>Medium</th><th>High</th></tr></thead><tbody><tr><td>Re<sub>λ</sub></td><td>-</td><td>27</td><td>52</td><td>65</td></tr><tr><td>η (cm)</td><td>-</td><td>0.14</td><td>0.11</td><td>0.082</td></tr><tr><td>ε (cm<sup>2</sup> s<sup>-3</sup>)</td><td>-</td><td>0.0025</td><td>0.0081</td><td>0.022</td></tr><tr><td>Early stage &lt;w<sub>r</sub>&gt; (mm s<sup>-1</sup>)</td><td>0.69±0.02</td><td>0.037±0.018</td><td>0.48±0.023</td><td>1.4±0.22</td></tr><tr><td>Late stage &lt;w<sub>r</sub>&gt; (mm s<sup>-1</sup>)</td><td>0.24±0.03</td><td>−0.046±0.031</td><td>0.58±0.031</td><td>0.37±0.030</td></tr></tbody></table>

The table lists the average Reynolds number $Re_\lambda$, Kolmogorov length scale $\eta$, and dissipation rate $\epsilon$ for the three turbulence levels and still water. Also included is the average vertical relative velocity of the larvae $\langle w_r \rangle$ in each condition with 95% confidence intervals.
<table><caption>**Table 2.** Simulations overview</caption><thead><tr><th>Re<sub>λ</sub></th><th>n</th><th>L/η</th><th>T<sub>L</sub>/τ<sub>η</sub></th><th>u<sub>rms</sub>/u<sub>η</sub></th></tr></thead><tbody><tr><td>11</td><td>128</td><td>35</td><td>19</td><td>1.8</td></tr></tbody></table>

The table lists flow parameters of the homogeneous isotropic turbulence simulations performed using *Snoopy*. *η*, *τ*<sub>η</sub>, *u*<sub>η</sub> characterise the Kolmogorov scales and *T*<sub>L</sub> = *L* / *u*<sub>rms</sub> the integral time scale.

<table><caption>**Table 3.** Pearson correlation coefficient *ρ* for horizontal relative velocity *u*<sub>r</sub> correlated against vorticity *ω*<sub>y</sub> as plotted in Fig. 2</caption><thead><tr><th>Turbulence level</th><th>Low</th><th>Medium</th><th>High</th></tr></thead><tbody><tr><td>Early stage</td><td>N = 5036 ρ = -.10</td><td>N = 3995 ρ = -.18</td><td>N = 6791 ρ = -.26</td></tr><tr><td>Late stage</td><td>N = 1243 ρ = -.41</td><td>N = 2102 ρ = -.38</td><td>N = 3675 ρ = -.38</td></tr></tbody></table>

All *P*-values <0.001.

<table><caption>**Table 4.** Pearson correlation coefficient *ρ* for horizontontal relative velocity *u*<sub>r</sub> correlated against the shear velocity gradients *du*<sub>*f*</sub> / *dz* and -*dw*<sub>*f*</sub> / *dx*, where *ω*<sub>*y*</sub> = *du*<sub>*f*</sub> / *dz* - *dw*<sub>*f*</sub> / *dx*</caption><thead><tr><th>Turbulence level</th><th>Low</th><th>Medium</th><th>High</th></tr></thead><tbody><tr><td>Early stage <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">du<sub>f</sub></div><div style="padding-top: 1px;">dz</div></div></td><td>ρ = -0.03*</td><td>ρ = -0.14</td><td>ρ = -0.15</td></tr><tr><td>- <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">dw<sub>f</sub></div><div style="padding-top: 1px;">dx</div></div></td><td>ρ = -0.20</td><td>ρ = -0.07</td><td>ρ = -0.23</td></tr><tr><td>Late stage <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">du<sub>f</sub></div><div style="padding-top: 1px;">dz</div></div></td><td>ρ = -0.26</td><td>ρ = -0.14</td><td>ρ = -0.25</td></tr><tr><td>- <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">dw<sub>f</sub></div><div style="padding-top: 1px;">dx</div></div></td><td>ρ = -0.36</td><td>ρ = -0.34</td><td>ρ = -0.32</td></tr></tbody></table>

All *P*-values <0.001 expect for * where *P* = 0.015.