Surfing on Turbulence: A Strategy for Planktonic Navigation

In marine plankton, many swimming species can perceive their environment with flow sensors. Can they use this flow information to travel faster in turbulence? To address this question, we consider plankters swimming at constant speed, whose goal is to move upward. We propose a robust analytical behavior that allows plankters to choose a swimming direction according to the local flow gradients. We show numerically that such plankters can “surf” on turbulence and reach net vertical speeds up to twice their swimming speed. This new physics-based model suggests that planktonic organisms can exploit turbulence features for navigation.

Figure 1

Plankters can exploit velocity gradients to “surf” on the flow: (a) 2D Taylor-Green vortex flow; (b) 3D turbulent flow. We compare the trajectories of surfers (red) to those of bottom-heavy swimmers (blue), which always swim upward. We also show trajectories of passive particles (black). In (a), arrows show the swimming direction. In (b), the gray line shows the depth of the initial positions and circles show the average final vertical position for the same turbulent flow.

Figure 2

Effective upward velocity [$V_{\mathrm{eff}}$, defined by Eq. (2)] as a function of the swimming velocity ($V_{\mathrm{swim}}$) for a surfer ($\widehat{\mathit{n}}={\widehat{\mathit{n}}}_{\mathrm{surf}}$ with optimal time horizon $\tau ={\tau }^{*}$) and for a bottom-heavy swimmer ($\widehat{\mathit{n}}=\widehat{\mathit{z}}$), here assuming instantaneous reorientation time [Eq. (1)]. Velocities are normalized either by the Kolmogorov velocity [$u_{\eta }$, Eq. (6)] (bottom $x$ axis) or by the root-mean-square velocity $u_{\mathrm{rms}}$ (top $x$ axis). The inset presents the same data where the effective upward velocity is normalized by the swimming velocity. The solid line represents $V_{\mathrm{eff}}=V_{\mathrm{swim}}$. Shaded areas correspond to 95% confidence intervals.

Figure 3

Influence of the time horizon on the surfing strategy. (a) Effect of the time horizon [$\tau$, Eq. (5)] on the effective velocity [$V_{\mathrm{eff}}$, Eq. (2)], for different swimming velocities $V_{\mathrm{swim}}$. Shaded area represents the 95% confidence interval. Solid lines represent a fit with Chebyshev polynomials of degree 3. (b) Correlation time ${\tau }_{\mathrm{corr}}$, defined in Eq. (7), and optimal time horizon ${\tau }^{*}$ as a function of swimming velocity (${\tau }^{*}$ is evaluated using the fitted polynomial).

Figure 4

(a) Performance ($V_{\mathrm{eff}}/V_{\mathrm{swim}}$) of surfers using a time dependent $\tau$ as a function of the dimensionless constant $\alpha$ [Eq. (8)] for different swimming velocities $V_{\mathrm{swim}}$. Solid lines represent a fit with Chebyshev polynomials of degree 3. (b) $V_{\mathrm{eff}}/V_{\mathrm{swim}}$ of surfers and bottom-heavy swimmers as a function of the reorientation time ${\tau }_{\text{align}}$. $V_{\mathrm{swim}}=u_{\eta }$. Shaded area represents the 95% confidence interval.

Figure 5

Expected vertical migration speed (effective vertical velocity, $V_{\mathrm{eff}}$, Eq. (2), relative to swimming velocity $V_{\mathrm{swim}}$) as a function of the turbulence dissipation rate $\epsilon$ [34]. We consider three typical plankters: a copepod, an invertebrate larva, and a dinoflagellate, whose characteristics are given in Table 1. Two strategies are compared: the proposed surfing strategy (red) and bottom-heavy swimmers (blue) orienting upwards due to gravity. In the upper panel, we indicate the range of turbulence intensity for different marine habitats (data from [35]) and the corresponding range of Kolmogorov time ${\tau }_{\eta }$ and Kolmogorov velocity $u_{\eta }$, Eq. (6).