Coral tentacle elasticity promotes an out-of-phase motion that improves mass transfer

Abstract

Corals rely almost exclusively on the ambient flow of water to support their respiration, photosynthesis, prey capture, heat exchange and reproduction. Coral tentacles extend to the flow, interact with it and oscillate under the influence of waves. Such oscillating motions of flexible appendages are considered adaptive for reducing the drag force on flexible animals in wave-swept environments, but their significance under slower flows is unclear. Using in situ and laboratory measurements of the motion of coral tentacles under wave-induced flow, we investigated the dynamics of the tentacle motion and its impact on mass transfer. We found that tentacle velocity preceded the water velocity by approximately one-quarter of a period. This out-of-phase behaviour enhanced mass transfer at the tentacle tip by up to 25% as compared with an in-phase motion. The enhancement was most pronounced under flows slower than 3.2 cm s⁻¹, which are prevalent in many coral-reef environments. We found that the out-of-phase motion results from the tentacles' elasticity, which can presumably be modified by the animal. Our results suggest that the mechanical properties of coral tentacles may represent an adaptive advantage that improves mass transfer under the limiting conditions of slow ambient flows. Because the mechanism we describe operates by enhancing convective processes, it is expected to enhance other fitness-determining transport phenomena such as heat exchange and particle capture.

1. Introduction

Sessile organisms depend on the flow of the surrounding fluid to provide them with nutrients, oxygen and prey, excrete waste products, and exchange gametes [1–5]. In corals and other passive suspension feeders, the interaction with the water can determine the rates and efficiencies of respiration [6,7], photosynthesis [3,4], nutrient uptake [8–10], particle capture [11], heat exchange [12] and the ability of corals to recover from bleaching events [13]. In the competition-dominated coral-reef ecosystem, enhancement of these physiological processes is particularly beneficial. Corals have developed specific mechanisms, such as pulsating polyps, symbiosis with sleep-swimming fish and ciliary motions that improve mass transfer by facilitating water flow [5,14–16].

The flow regime in many coral reefs is dominated by oscillatory flows generated by surface waves. Studies that examined the effects of surface waves on the flow field inside the complex reef geometry have shown that oscillatory flows improve the exchange of dissolved material (mass transfer) as compared with unidirectional flows [17,18]. However, in those studies corals were treated as solid structures, and the interaction of the flow with the flexible living tissue was generally ignored. In other coastal ecosystems such as kelp forests and macrophyte beds, it is evident that the stipes and stalks of these flexible organisms sway and bend with the wave-induced flows [19,20]. This is also true for coral tentacles, which are hydrodynamic structures that inflate and extend to the flow via pressurization of the coelenteron [21].

The research of flexible organisms in wave-swept environments focused mostly on modelling the dynamics of their oscillatory motion [22–26], and on the breaking and dislodging of organisms under high hydrodynamic loads [22,27]. In this context, reduction of drag force has generally been considered to be adaptive, and is often achieved by reducing the cross-sectional area exposed to the flow [28] by means of shape reconfiguration [23,29–32] or by dwelling in dense groups [33]. An increase in the relative velocity between the tissue and the water is expected to increase both the drag forces and mass transfer rates. It is not clear, however, whether the movement of flexible appendages contributes to that increase.

Here, we used underwater particle image velocimetry (PIV) to simultaneously measure the flow field and the kinematics of flexible tentacles. The in situ measurements were obtained at the coral reef in Eilat, Israel, and focused on tentacles of the coral Dipsastraea favus when the flow was dominated by waves (figure 1). We quantified the phase difference between the water velocity and the tentacle velocity and asked whether this could increase the rate of mass transfer into the coral tissue. We complemented the field measurements with laboratory experiments and numerical simulations in order to (i) characterize the relative motion of the tentacle and the water around it, (ii) evaluate the mechanical properties of the tentacle that are responsible for the observed motion, and (iii) develop a proxy of the contribution of this motion to the rate of mass transfer into the coral.

Figure 1.

(a,b) An example of a D. favus colony with extended tentacles. The area marked by a rectangle is enlarged in (b). Tentacle extension greatly increases the surface area of the polyp, even in corals where polyps and tentacles are small. (c) The underwater PIV set-up that was developed to measure the flow field above the coral at a depth of approximately 5 m in the natural reef of Eilat, Israel. (d) An instantaneous flow field above D. favus tentacles measured in the laboratory. White arrows represent the measured water velocity, and the orange arrow represents the tentacle horizontal velocity. For clarity, only 4% of the vectors that were measured are shown. (Online version in colour.)

2. Results

(a). Field and laboratory experiments

We measured the in situ time-dependent location of the tentacle tips to obtain the tip velocity in two D. favus colonies, and the velocity field of the water above them under natural wave-induced flow conditions (figure 2a). The experiment was conducted during the night, when the D. favus tentacles were extended and when the flow was dominated by waves with low background mean current. In all our field measurements (n = 39; electronic supplementary material, table S2), the velocity of the tentacle tip preceded the water velocity signal, generating an out-of-phase motion. The phase difference between the signals of the velocity of the tentacles and the velocity of the water was Δφ = −57° ± 19° (mean ± s.d.; figure 2b). We repeated these measurements in a wave tank for five additional D. favus colonies (electronic supplementary material, movie S1 and figure 2c) under sinusoidal flow conditions, and found that the phase difference was nearly approximately one-quarter of a period and with a narrower distribution compared with the field measurements, Δφ = −95° ± 11° (n = 76; electronic supplementary material, table S3; figure 2d).

Figure 2.

The phase difference between the horizontal velocity of the tentacle and the horizontal velocity of the water in the coral D. favus. (a) Horizontal velocities of the tentacle tip (orange crosses) and the water (blue circles) measured as a function of time during one of the field experiments. The oscillation frequency in this case was f = 0.28 Hz and the phase difference was Δφ = −65°. The continuous line represents a moving average filter (applying a span of five). (b) The distribution of phase difference during 39 field experiments. (c) The horizontal velocities of the tentacle tip (orange crosses) and the water (blue circles) as a function of time, measured in a laboratory experiment. The frequency of oscillation in this case was f = 0.9 Hz and the phase difference was Δφ = −81°. The number of data points shown for the water and tentacle velocities (originally 180 per second) was reduced for plotting purposes. (d) The distribution of phase difference during 76 laboratory experiments. See ‘observations and flow measurements’ in the electronic supplementary material for additional details about the field and laboratory experiments. (Online version in colour.)

(b). Tentacle dynamic model

We hypothesized that the mechanical properties of the tentacle are responsible for the out-of-phase motion. In order to quantify these properties and investigate the dynamics underlying the phase difference phenomenon, we modelled the tentacle as a mass–spring–damper torsional system (box 1; see ‘Tentacle dynamic model’ in the electronic supplementary material). We used the measured flow field above the D. favus tentacles in the laboratory and their kinematics in order to calculate the hydrodynamic torques exerted on a single tentacle (τ in equation (2.1), box 1). Using equation (2.1), we curve fitted the spring and damping coefficients in each of the 76 experiments. The estimated 95% bootstrap confidence interval for the spring coefficient was 8.7 ≤ κ ≤ 14 nN m rad⁻¹ and the damping coefficient was 0.074 ≤ C ≤ 0.27 nN m s rad⁻¹ (electronic supplementary material, figure S6). These results indicate that the kinematics of the tentacle are adequately predicted by a mass–spring–damper torsional model with a non-zero spring coefficient κ.

Box 1. The torsional mass–spring–damper model.

The tentacle is modelled as a cylinder of length L_T, with uniform density and diameter D_T, attached to the polyp's body with a torsional spring and a damper. The tentacle oscillates due to the hydrodynamic torques applied by the water. The dynamics of the tentacle is described by

$$I\ddot{\theta }+C\dot{\theta }+\kappa (\theta -{\theta }_0)=\tau ,$$ 2.1

where θ(t) = arctan (x_tip/z_tip) is the time-dependent bending angle of the tentacle, θ₀ (rad) is the angle of the tentacle when the water is still and no hydrodynamic torques are applied, $\dot{\theta }(t)$ is the angular velocity, $\ddot{\theta }(t)$ is the acceleration, I (N m s² rad⁻¹) is the mass moment of inertia, C (N m s rad⁻¹) is the damping coefficient, κ (N m rad⁻¹) is the spring coefficient and τ (N m) is the sum of hydrodynamic torques, calculated using the Morison equation (see ‘Tentacle dynamic model’ in the electronic supplementary material).

An order of magnitude analysis of equation (2.1) showed that the horizontal component of the drag torque was similar in magnitude to the tentacle spring torque and that both were larger than the other torques. The damping torque was an order of magnitude smaller and the tentacle inertia was negligible. We therefore simplified equation (2.1) by dropping the inertia term $I\ddot{\theta }$. In all the laboratory experiments, the hydrodynamic torque τ showed a good fit to a sine function (average R² = 0.95 and never below 0.85), and was therefore represented as

$$\tau ={\tau }_0\sin \left(\frac{2\pi }{T}t\right)+c_{\tau },$$ 2.2

where τ₀ (N m) is the amplitude of the sine wave, T (s) is its period, t (s) is time and c_τ (N m) is the torque caused by the unidirectional flow component. Solving the simplified form of equation (2.1)

$$C\dot{\theta }+\kappa (\theta -{\theta }_0)=\tau$$ 2.3

yields the following solution for the tentacle angular velocity $\dot{\theta }(t)$:

$$\dot{\theta }=\left(\sqrt{A^2+B^2}\right)\text{sin}\left(\frac{2\pi }{T}t+\frac{\pi }{2}-\text{atan}2(A,\hspace{0.17em}B)\right),$$ 2.4

where A = (4π²Cτ₀)/(4π²C² + κ²T²) and B = (2πτ₀κT)/(4π²C² + κ²T²). Since the hydrodynamic torque τ is in-phase with the water velocity, a comparison between the hydrodynamic torque (equation (2.2)) and the tentacle angular velocity (equation (2.4)) reveals that the phase difference between the tentacle and the water velocities is Δφ = −π/2 + atan2(A, B). Because atan2(A, B) is bounded between − π/2 and π/2, the theoretical phase difference can change between −π and 0. The tentacle motion in the case of zero damping (a solution of equation (2.3) where C = 0) results in a phase difference of Δφ = −π/2_, similar to the mean phase difference we measured in the laboratory (electronic supplementary material, figure S4C,D). An analysis of the effect of κ, C and T on the phase difference (electronic supplementary material, figure S6) indicates that the phase difference approaches −π/2 when the spring coefficient κ is dominant. The sign of C determines the direction of its deviation from −π/2 such that for C > 0 the phase difference ranges −π/2 < Δφ < 0, and for C < 0 it is −π/2 < Δφ < −π. The observations and the result of equation (2.4) suggest that the out-of-phase motion is ubiquitous in elastic organisms and appendages under wave-induced flow, including other coral tentacles, appendages of sessile marine organisms, macrophytes and sea grass. Moreover, kinematics in which flexible organisms are ‘out of sync’ with temporal flow changes can be expected in other terrestrial and aquatic environments, such as forests, grass meadows, agricultural crops, riverbeds and lagoons. Such a response can occur due to wind gusts and wave surges even when the motion is not purely harmonic and a single phase difference is impossible to define. Examples of such out-of-phase motions are the flapping motion of macroalgae [34], salt-marsh sedges [24] and kelp [35]. When the flow oscillates in a harmonic motion, the model of the tentacle as a torsional mass–spring–damper system provides an opportunity to determine in situ and in vivo the mechanical properties of appendages.

(c). Numerical solutions

We hypothesized that the out-of-phase motion is likely to increase the rate of transport of dissolved material between the coral tentacle and the water. To test this hypothesis, we used a two-dimensional numerical simulation of the flow and concentration fields around a cross-section of an oscillating vertical cylinder, representing the observed tentacle motion, in a wave-induced flow field (see ‘Numerical modelling’ in the electronic supplementary material for details about the mesh discretization and distortion, convergence study and the time-stepping of the numerical study). The simulation was repeated for a variety of tentacle and water velocity amplitudes (U_T and U_f) encompassing the entire range measured in the laboratory with the D. favus coral (electronic supplementary material, figure S12). For each set of conditions, the following phase differences were artificially imposed: Δφ = −180°, −135°, −90°, −45° and 0°. Dissolved oxygen was used as a tracer and its concentration away from the tentacle was set to 0.2 mol m⁻³, as typically found during the night in the Gulf of Eilat [14,36]. By applying a zero-concentration boundary condition on the tentacle tissue, the tentacle served as a perfect sink for oxygen. The results were analysed when the concentration field reached a periodic steady state (after 50 periods of 1 s each).

The simulations reveal that as the tentacle oscillates, a region of low concentration flips from side to side (figure 3; electronic supplementary material, movies S2 and S3). In the case of moving in-phase, the tentacle spent most of the time progressing towards this oxygen-depleted region (electronic supplementary material, movie S2 and figure 3a–d), and in cases where the tentacle was moving out of phase, it spent a larger fraction of the period moving away from this region (electronic supplementary material, movie S3; figure 3e–h). These two scenarios resulted in different flow patterns and concentration fields around the moving tentacles and, therefore, in different mass transfer rates into the tentacle (figure 4a). For example, mass transfer per period (calculated during a single wave period using electronic supplementary material, eq. S32) for a tentacle moving in-phase with a velocity amplitude of 2.2 cm s⁻¹ under a water velocity amplitude of 4 cm s⁻¹ was 8.3 nmol m⁻¹ (leftmost point in figure 4b). For these velocity amplitudes, the mass transfer per period and unit length of tentacle increased by up to approximately 25% when the tentacle was moving out of phase, reaching a flux of more than 10 nmol m⁻¹ at Δφ = −90°, −135° and −180° (figure 4b). Unlike our simulated tentacles, the relative velocity of the tentacle is expected to diminish towards the tentacle base, and consequently also the mass transfer rate. However, as the tentacle tip is farthest away from the colony and exposed to the ambient flow, it can be assumed that a substantial part of the mass transfer to the tentacle occurs near the tentacle tip. Therefore, our simulations are designed as a comparative test of the effects of out-of-phase motion on mass transfer rates, and not to accurately estimate the mass transfer to the entire polyp.

Figure 3.

Two-dimensional concentration fields around an oscillation tentacle while moving in-phase (a–d) and out-of-phase (e–h). Oxygen concentration around the tentacle ranges between zero at the tentacle interface (blue) and 0.2 mol m⁻³ (red) away from the tentacle. Insets in a and e show the velocities of the tentacles (red) and ambient flow (blue), with vertical dashed lines marking the relative time (t/T) when each snapshot was taken: t/T = 0 (a,e), t/T = 0.25 (b,f), t/T = 0.5 (c,g) and t/T = 0.75 (d,h). White arrows represent the water velocity and black arrows represent the tentacle velocity. The arrow length is proportional to the velocity magnitude. The ambient flow velocity in a, c, e and g was low, hence only a few arrows can be seen. In all the in-phase and out-of-phase simulations, the tentacle diameter was 0.5 mm, the water velocity amplitude was U_f = 3.3 cm s⁻¹, the tentacle velocity was U_T = 1.2 cm s⁻¹ and the period of oscillation was T = 1 s. (Online version in colour.)

Figure 4.

Results of the mass transfer analysis. (a) Rate of dissolved oxygen transferred into the tentacle (left y axis) as a function of time, plotted for three phase differences (0° dashed line, −90° dotted line and −180° dash-dot line). The solid blue line depicts the water horizontal velocity (right y axis). These mass transfer rates are taken from the simulations presented in figure 3, electronic supplementary material, figure S13 and movies S2 and S3. (b) Mass of oxygen absorbed by the tentacle, integrated over one period of oscillation, M (nmol m⁻¹), as a function of phase difference. The amplitude of the water velocity was U_f = 4 cm s⁻¹, the tentacle velocity was U_T = 2.2 cm s⁻¹ and the period of oscillation was T = 1 s. (c) Mass absorbed by the tentacle M (nmol m⁻¹) as a function of the relative velocity amplitude, defined as a function of the phase difference and the water and tentacle velocity amplitudes $U_r=\sqrt{U_f^2-2U_f{U}_T\cos (\mathrm{\Delta }\phi )+U_T^2}$. The colours indicate n = 10 simulation sets in which each set is defined by a different combination of U_f and U_T values (electronic supplementary material, table S6 and figure S12) for a range of phase difference: Δφ = 0°, −45°, −90°, −135° and −180°. The red symbols correspond to the same set shown in (b). Dashed black line represents a fit to the sigmoid function, $M={\text{e}}^{126.5U_r}/(1+{\text{e}}^{126.5U_r-1.724})+4.861$ for all the data points (where U_r is in m s⁻¹; R² = 0.99). The sigmoid curve asymptoticly approaches to M = 10.5 nmol m⁻¹. Each dotted coloured line is a sigmoidal function of the same form, $M={\text{e}}^{a{U}_r}/(1+{\text{e}}^{a{U}_r+b})+c$, fitted separately for each simulation set marked with the same colour. The black crossed square represents the mass transfer into a stationary tentacle given a steady unidirectional flow field with a velocity of 3.2 cm s⁻¹ over one period of 1 s. The result indicates a lower mass transfer rate as compared to the result of the oscillatory flow with the same relative velocity amplitude. (Online version in colour.)

It is well established that many physiological processes in corals are limited by mass transfer rates. For example, measurements of oxygen concentration indicate near-anoxic conditions close to the coral tissue at night, and over-saturation of oxygen during the day, which have been shown to limit dark respiration and photosynthesis, respectively [14,37]. Other commodities such as nitrogen [9,10], phosphate [8,38] and prey [11,39,40] may be in short supply in the oligotrophic coral reef, and their intake by corals was shown to be positively correlated with the flow velocity. Our findings suggest that an increase in mass transfer rate of 25% would increase respiration and photosynthesis, and other fitness-determining processes, providing considerable adaptive benefits for the corals in the competitive reef environment.

How does the out-of-phase motion increase the mass transfer rates? We found that an increase in the relative velocity between the simulated tentacle and the water [U_r (cm s⁻¹)], which is a direct result of the phase difference, is responsible for augmenting the mass transfer rate. We plotted the mass transferred per period [M (nmol m⁻¹)] as a function of the relative velocity amplitude in ten sets of simulations. In each set, we tested a different combination of water and tentacle velocity amplitudes for the five phase differences (figure 4c). Mass transfer per period increased sharply with relative velocity, U_r (fitted to a sigmoidal curve R² = 0.99; $M={\text{e}}^{126.5U_r}/(1+{\text{e}}^{126.5U_r-1.724})+4.861$; where U_r is in m s⁻¹), but only up to a relative velocity amplitude of 3.2 cm s⁻¹, where the mass transfer per period reached 9.97 nmol m⁻¹ (95% of its maximal value of 10.46 nmol m⁻¹ predicted by the sigmoidal curve). Our inference from this curve is that the contribution of the out-of-phase motion to mass transfer persists as long as the ambient flow does not exceed a critical value that generates an already high concentration gradient at the tentacle interface. Water velocities of 3.2 cm s⁻¹ and slower are characteristic in fringing reefs such as the reef at the current study site in the Red Sea. The average current speed measured at our study site between February and August 1997 was 4.6 ± 1.9 cm s⁻¹ ([41], see also [42,43]). Furthermore, currents slower than 3.2 cm s⁻¹ persisted for greater than 47% of that time at the height of the coral tentacles (unpublished data from [41]; electronic supplementary material, figure S5). Measurements of flow speed at the height of the coral colonies have shown that this flow velocity distribution is characteristic of other reef locations such as Discovery Bay in Jamaica and St Croix in US Virgin Islands [6,39,44,45]. Moreover, even in reef sites which are characterized with higher flow velocities, there are niches in which slow flow velocities are dominant. We note that the maximum mass transfer rate calculated for cases of fast unidirectional velocities (10 and 20 cm s⁻¹) around an upright, stationary tentacle was 10.5 nmol m⁻¹ during a time duration of one period. This rate is equivalent to the maximum mass transfer predicted by the sigmoidal curve shown in figure 4c. For the same case with a flow velocity of 3.2 cm s⁻¹, the mass transfer rate was lower than predicted by the curve (see crossed square in figure 4c). Finally, we note that as the phase difference increases, the higher relative velocity between the water and the tentacle leads to an increase of drag force per unit length (see ‘Drag force’ section in the electronic supplementary material).

(d). Broader implications

In this study, we combined field and laboratory experiments to demonstrate that out-of-phase motion is characteristic of coral tentacles under oscillating flow conditions (figure 2). Using a torsional mass–spring–damper model, we have shown that the spring properties of the tentacle are dominant in controlling the out-of-phase motion. The analytical solution of equation (2.3) suggests that a phase difference is a general property of flexible appendages motion under oscillatory flows.

To further test the generality of the phase difference, we conducted additional measurements with the sea anemone A. diaphana and the coral Galaxea fascicularis. Field measurements of phase differences in five A. diaphana colonies under wave-dominated flows revealed an average (±s.d.) phase difference of Δφ = −77° ± 24° (electronic supplementary material, figure S4B and table S2). Laboratory measurements in a wave tank under a sinusoidal flow field with three G. fascicularis colonies showed an average phase difference of Δφ = −91° ± 11° (electronic supplementary material, figure S4D and table S4). These phase differences are similar to those observed for D. favus in the laboratory (figure 2d).

The increase in mass transfer to the flexible coral tentacles is driven by the enhancement of convective processes. Thus, an increase in the relative water velocity between the tentacle and the water is expected to improve the rates of other transport-limited physiological processes such as photosynthesis, respiration, heat exchange and particle capture. Using dimensional analysis, we show that the mass transfer is a function of five non-dimensional quantities (electronic supplementary material, eq. S43): (i) the Reynolds number, which represents the ratio between the inertial and viscous terms; (ii) the Keulegan–Carpenter number that is often used in oscillatory flows to represent the ratio between the drag and inertial terms; (iii) the Peclét number, which is the ratio between convective and diffusive fluxes; (iv) the aspect ratio of the cylinder; and (v) the mass fraction of dissolved oxygen in the ambient water (for full definitions see electronic supplementary material). The flow conditions we tested in the numerical simulations were drag-dominated (35 < KC < 169) with an intermediate Reynolds number (9 < Re < 44), and convection dominated mass transfer (Pe > 400). For given flow and transport regimes that are within these ranges, an out-of-phase motion of other flexible appendages (e.g. [24,34,35]) is expected to be found and enhance mass transfer.

Many species of corals and sea anemones are capable of extending, retracting and altering the length of their tentacles in response to currents, prey availability and light intensity [46,47]. It is therefore probable that the mechanical properties of the tentacle, including the spring coefficient κ, can be modified by controlling the internal pressure of the coelenteron. We suggest that altering these properties could provide a mechanism by which corals are able to affect the degree of phase difference to their benefit (electronic supplementary material, figure S6). For example, when ambient velocities are low and nutrient flux is limited, an increase in the phase difference may be beneficial. However, when the mass transfer is high due to the ambient velocity, a decrease in phase difference as a strategy of drag reduction would be preferred. In general, a reduction of the drag forces is considered beneficial, and often critical to their survival in the wave-swept environment. The bending ability of submerged aqueous vegetation offers an example of such a drag reduction mechanism [22]. Similarly, sea anemones bend and retract their tentacles under unidirectional tidal flows [28]. Here we report on a mechanism that increases, rather than decreases, the drag in order to enhance mass transfer for the benefit of the organism. We suggest that along the spectrum of flow speeds experienced by benthic organisms, drag reduction mechanisms may be dominant in strong flows in which organisms are limited by mechanical failure, whereas mechanisms that enhance drag and therefore enhance mass transfer will be prevalent under slower flows, in which the organisms are limited by mass transfer.

Ethics

Specimens were collected and maintained in the laboratory under a permit granted by the Israel Nature and Parks Authority.

Data accessibility

Additional data are available at Dryad Digital Repository: https://doi.org/10.5061/dryad.sqv9s4n0s [48].

Authors' contribution

D.M. carried out the laboratory experiments, modelling of tentacle dynamics, numerical mass transfer simulations, and analyses of the laboratory and field experiments. R.H. and U.S. initiated and coordinated the research. All the authors participated in the design of the experiments, analyses of the results and writing the manuscript. All authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests

We declare no competing interests.

Funding

This work was supported by the Israel Science Foundation (grant nos 620/07 and 1487/10).