## Abstract

Copepods swim either continuously by vibrating their feeding appendages or erratically by repeatedly beating their swimming legs, resulting in a series of small jumps. The two swimming modes generate different hydrodynamic disturbances and therefore expose the swimmers differently to rheotactic predators. We developed an impulsive stresslet model to quantify the jump-imposed flow disturbance. The predicted flow consists of two counter-rotating viscous vortex rings of similar intensity, one in the wake and one around the body of the copepod. We showed that the entire jumping flow is spatially limited and temporally ephemeral owing to jump-impulsiveness and viscous decay. In contrast, continuous steady swimming generates two well-extended long-lasting momentum jets both in front of and behind the swimmer, as suggested by the well-known steady stresslet model. Based on the observed jump-swimming kinematics of a small copepod *Oithona davisae*, we further showed that jump-swimming produces a hydrodynamic disturbance with much smaller spatial extension and shorter temporal duration than that produced by a same-size copepod cruising steadily at the same average translating velocity. Hence, small copepods in jump-swimming are in general much less detectable by rheotactic predators. The present impulsive stresslet model improves a previously published impulsive Stokeslet model that applies only to the wake vortex.

## 1. Introduction

Planktonic organisms create water flows at the scale of their body as they move and feed. They generate feeding/swimming currents by rapidly beating their flagella, cilia or cephalic appendages more or less continuously. Or they may make instantaneous, short-lasting jumps to relocate, escape predators or attack prey (e.g. [1–5]), which in some cases may generate toroidal flow structures in their wake, as has been described for copepods [6–8]. Finally, even when zooplankters do not move their appendages, cilia or flagella, they may sink passively, leading to flow passing around their body. It is of great relevance to investigate these small-scale biogenic fluid dynamical phenomena as they are related to and have implications for the various survival tasks of the plankters, such as feeding, nutrient uptake, predator avoidance, mating and signalling (for reviews, see [9–16]). Specifically, zooplankters with different feeding and motility behaviours generate different hydrodynamic disturbances and therefore expose themselves differently to rheotactic predators. Quantitative characterization of these different hydrodynamic disturbances allows a mechanistic understanding of the (probably) different levels of predation risk faced by these zooplankters.

Planktonic copepods, arguably the most abundant metazoans in the ocean [17], swim either by continuously vibrating their feeding appendages, which results in a rather constant propulsion, or by using their swimming legs, which leads to sequences of small jumps. Continuous swimming by vibrating the feeding appendages is common among calanoid copepods that generate a feeding current. In contrast, ambush-feeding copepods, mainly among the small cyclopoid copepods, do not generate a feeding current, but swim by sequentially striking the swimming legs posteriorly and by repeating this sequence at short intervals (e.g. [18,19]). This leads to a very unsteady, erratic motion, which probably generates a hydrodynamic disturbance much different from that of a continuously swimming copepod. The purpose of this study is to compare the hydrodynamic signal generated by these two fundamentally different propulsion mechanisms in zooplankton, using copepods as an example.

Several analytical solutions from classical fluid dynamics have previously been applied to investigate low-Reynolds-number flows created by small continuously moving plankters. For example, the solution of the Stokes flow associated with a steady point force was used to model the feeding current created by a negatively buoyant, stationary/hovering copepod [20]. The same authors also modelled the flow around a steadily sinking copepod using the well-known Stokes solution for the flow associated with a steadily translating sphere. Later, an analytical model of a negatively buoyant swimming copepod (hovering and free-sinking as special cases) was developed based on a linear combination of the Stokes flow around a stationary solid sphere and the Stokes flow owing to a point force external to the same sphere [21,22]. This model separates the copepod main body resistance from the thrusting effect of the beating appendages and therefore has the ability to reproduce the major feeding/swimming current patterns that have been observed so far.

However, these solutions to the steady Stokes flow equations cannot be applied to the intrinsic unsteady flows associated with plankters that swim by jumping because these are of an impulsive nature. Jumping plankters impart only brief and localized momentum to the surrounding water during short-lasting power strokes. The localized impulsive forcing exerted by the thrusting appendages (e.g. copepods) or membranelles (e.g. the jumping ciliate *Mesodinium rubrum* [2]) will stop as soon as the power strokes are completed and the flow so created will decay because of viscosity.

There are few previous attempts to develop theoretical hydrodynamic models for the low-Reynolds-number unsteady flow associated with jumping plankton, even though jumping is a very common motility mode among both protozoan [2,4] and metazoan plankton [19,20,23–25]. Here, we develop such a model by taking into account the two characteristics of the jumping flow: namely, impulsiveness and viscous decay. We develop our model with jumping copepods in mind, but the model has a wider application. It is well known that jumping copepods leave toroidal flow structures in their wake [7,8], and we previously developed an impulsive Stokeslet model for such viscous wake vortex rings [6]. However, an additional vortex ring develops around the decelerating body, and, in a typical repositioning jump of a millimetre-size copepod, the two rings are of similar intensity and opposite direction (figure 1*a*; [6]). We here present an impulsive stresslet model that describes the entire flow field and includes both vortex rings generated by the jumping copepod. We first compare the two theoretical models, and then apply the present impulsive stresslet model to observations of the kinematics of swimming in a small cyclopoid copepod, *Oithona davisae*. We compare the modelled flow for a jump-swimming copepod to the flow generated by a steadily cruising one and show that the hydrodynamic disturbance generated by the former has a much smaller spatial extension and temporal duration than that generated by the latter, suggesting that jump-swimming is a hydrodynamically quiet propulsion mode.

## 2. Methods

### 2.1. Video observation

We observed the swimming behaviour of males of the cyclopoid copepod *O. davisae* (prosome length 0.3 mm) using high-speed video (400 or 1900 frames s^{−1}). Males, but not females, of this species frequently swim when searching for females [26], and they swim using their swimming legs, not their feeding appendages. Animals were taken from our continuous culture and placed in approximately 100 ml aquaria. Light from a 50 W halogen lamp was guided through a collimator lens and the aquarium towards the camera, thus providing shadow images of the copepods. The camera (Phantom v.4.2 Monochrome) was equipped with lenses to produce fields of view of approximately 3 × 3 mm^{2} when filming at the high frame rate, or approximately 15 × 15 mm^{2} when filming at the lower frame rate. At the low frame rate and low magnification, we placed a mirror in the diagonal of the aquarium; by following both the image and the mirror image of the animals, we could reconstruct their three-dimensional movement paths. When filming at the higher magnification, we selected sequences where the animal was moving in the view plane perpendicular to the camera.

Selected video sequences (*n* = 18) showing one copepod moving were analysed frame by frame, either semi-automatically using ImageJ or automatically using the particle-tracking software LabTrack (DiMedia, Kvistgård, Denmark). The position of the animal was followed over time, allowing us to describe the temporal variation in move speed. At the higher magnification, we also noted the duration of the power stroke of the swimming legs in four to six beat cycles per animal.

Observations of the flow field generated by the repositioning jumps of a different copepod (*Acartia tonsa*) and visualized using particle image velocimetry (PIV) were taken from Kiørboe *et al*. [6] for comparison with model predictions.

### 2.2. The impulsive stresslet model

Using PIV, we have recently measured the flow fields imposed by copepods that performed repositioning jumps [6]. Being approximately axisymmetric, a typical such measured flow field consists of two counter-rotating viscous vortex rings of similar intensity, one in the wake and one around the body of the copepod (figure 1*a*). The wake vortex is generated owing to the momentum applied almost impulsively to the water by the rapid backward kick of the swimming legs; almost simultaneously but slightly delayed, the counter-rotating body vortex is generated owing to the oppositely directed momentum of equal magnitude paid back to the water by the forward moving copepod body. Here we consider the two rings as a whole system and describe their behaviour using an impulsive stresslet model (figure 1*b*).

#### 2.2.1. Equations of the impulsive stresslet

An impulsive stresslet consists of two point momentum sources of equal magnitude (*ρI*, where *ρ* is the mass density of the fluid and *I* is the hydrodynamic impulse), acting synchronously in opposite direction and separated by distance *ɛ* (figure 1*b*). Each momentum source acts impulsively for a very short period of time formally represented by the Dirac delta function *δ*(*t*). The definition of the strength of the impulsive stresslet is *M* = lim_{ɛ→0,} _{I}_{→∞} *I**ɛ* = const. with [*M*] = *L*^{5}*T*^{−1} for three-dimensional flows. Vorticity (*ω*_{ϕ}) and streamfunction (*ψ*_{ϕ}) for the above-described flow have been obtained in Stokes approximation [27],
2.1aand
2.1b where *ν* is the kinematic viscosity, and the error function
The solution is axisymmetric and independent of the azimuthal coordinate *ϕ* in the cylindrical polar coordinate system (*x*, *r*, *ϕ*) (figure 1*b*). The components of velocity in the meridian plane are given by
2.2aand
2.2bwhere *u* and *v* are the velocity components in the axial (*x*-) and the radial (*r*-) direction, respectively,

#### 2.2.2. Viscous decay

Integrating *ω*_{ϕ} (equation (2.1*a*)) over one half of the meridian plane (e.g. *x* ≥ 0) where the vorticity is one-signed leads to a simple formula for viscous decay of circulation of the one-signed vorticity,
2.3

The hydrodynamic impulse of the one-signed vorticity satisfies 2.4

It is obvious that *Γ*_{x}_{≥0}(*t*) + *Γ*_{x}_{<0}(*t*) = 0 and *I*_{x}_{≥0}(*t*) + *I*_{x}_{<0}(*t*) = 0.

#### 2.2.3. Asymptotic analysis for finding spatial extension and temporal duration of the flow

At small time, the flow far field (i.e. *ξ* ≫ 1) is approximately irrotational and behaves as
2.5aand
2.5b

The associated velocity magnitude is 2.6

From equation (2.6), two lengths (denoted and ) are formed to define the size of the domain over which the flow velocity magnitude is greater than a threshold velocity *U**: setting *r* = 0 leads to
2.7aand setting *x* = 0 leads to
2.7b

Asymptotic analysis at large time (i.e. *ξ* → 0) is too complicated to provide a formula for calculating the time *t** after which the whole flow field is below *U**. However, dimensional analysis together with numerical calculation using equation (2.2*a*,*b*) suggests a scaling relationship,
2.8a
2.8b
2.8cwhere *S* is the area of influence, defined as the area in the meridian plane within which the flow velocity magnitude is greater than *U**. A rheotactic predator is likely to perceive the prey-induced flow velocity magnitude [28] and, hence, this area is a measure of the encounter cross-section of the copepod prey. The scaling relationship has been numerically determined and plotted (figure 2*a*); the hydrodynamic signal quantified as the area within which the induced flow velocity magnitude exceeds *U** initially attains its maximum (i.e. *S**) and then decays until it dies out completely after *t**. Both *S** and *t** depend on *M*/*U** only.

#### 2.2.4. Translation of the vortices

The impulsive stresslet initially sets up an irrotational flow everywhere except at its application point, where there exist two opposite-signed vorticity singularities (figure 3*a*). Both singularities diminish and the associated vorticities diffuse away from the application point as time goes on (figure 3*b–d*; electronic supplementary material, appendix movie S1). The coordinates of the two vorticity maxima evolve as
2.9aand
2.9band the two flow stagnation points (vortex centres) occur at
2.10aand
2.10b

Therefore, the vorticity maxima separate from the flow stagnation points as time goes on.

### 2.3. Analysis of the particle image velocimetry flow data using the impulsive stresslet model

To test whether the flow predicted by the impulsive stresslet model is a good approximation of the flow field associated with a copepod repositioning jump, we fitted by nonlinear regression the decay phase of the observed wake vortex circulation [6] to equation (2.3) subsequent to a virtual time origin *t*_{0},
2.11

We then compared the fitted impulsive stresslet strength, *M*_{fitted}, with that directly calculated from jump kinematics,
2.12where *U*_{max} is the maximum speed attained by the copepod, *D*_{jump} is the distance travelled by the copepod during a jump, the copepod body volume, *V*_{copepod}, is 4/3*πη*^{2}*a*^{3}, and *a* is half the prosome length, *η* = 0.38 is the copepod aspect ratio, assuming the shape of a prolate spheroid with the long axis equal to the prosome length, *L*, and the short axes are equal to *η* × *L*.

### 2.4. Brief description of a previously published impulsive Stokeslet model

The impulsive stresslet model was worked out after we had published Kiørboe *et al.*'s [6] paper, in which we constructed an impulsive Stokeslet model to describe the wake vortex ring left behind a repositioning-by-jumping copepod. Unlike the impulsive stresslet model, the impulsive Stokeslet model did not consider the interaction between the wake vortex ring and the body vortex ring. Here, we recapitulate some equations from the impulsive Stokeslet model for comparing the two models (figure 2).

An impulsive Stokeslet consists of a point momentum source of magnitude *ρI* acting impulsively for a very short period of time, formally represented by *δ*(*t*) (figure 1*c*). [*I*] = *L*^{4}*T*^{−1} for three-dimensional flows. Similar to the impulsive stresslet model, the scaling relationship for the area of influence, *S*, is
2.13a
2.13b
2.13c

The scaling relationship has been numerically determined and plotted (figure 2*a*). The model parameter *I* (the hydrodynamic impulse) can be estimated as
2.14

## 3. Results

### 3.1. Jump-swimming kinematics

Jump-swimming consists of a sequence of small jumps: the swimming legs are pushed posteriorly in power strokes, one after the other, and returned all at the same time, and the sequence is repeated at an average frequency of 21.5 Hz (table 2 and electronic supplementary material, appendix movie S2). This leads to a highly fluctuating velocity: short bursts with speeds of up to 30–50 mm s^{−1} during power strokes of 7.2 ms duration on average and interrupted by long pauses, resulting in an average propulsion speed of 8 mm s^{−1} (table 2 and figure 4). The individual jumps resemble repositioning, attack or escape jumps in this species [29], but they differ in the phase lag between individual swimming legs: near 90° in escape and attack jumps, but only between 30 and 60° for the swimming jumps (i.e. the legs are closer to one another) (figure 5). As a consequence, the average forward movement is only a little more than 1 body length for a swimming jump (0.375 mm or 1.25 body lengths; table 2), whereas it is near 2 body lengths for repositioning, escape and attack jumps [29].

### 3.2. The impulsive stresslet flow field

The imposed flow predicted by the impulsive stresslet model consists of two counter-rotating viscous vortex rings that expand and decay as exact mirror images about the *r*-axis (figure 3 and electronic supplementary material, appendix movie S1). This is consistent with our PIV observation of the flow field around a copepod during a repositioning jump in that the spatial extension and temporal evolution of the two vortex rings, one in the wake and one around the forward moving body, are similar [6].

The impulsive stresslet model provides a satisfactory least-squares fit to the decaying phase of the observed circulation of the wake vortex (figure 6), and the stresslet strengths estimated from such fits of 12 observed flow fields, *M*_{fitted} (equation (2.11)), are in good 1 : 1 correspondence with stresslet strengths estimated from the kinematics of copepod jumps, *M*_{measured} (equation (2.12)) (figure 7 and table 1). Thus, equation (2.12) allows for using the measured jump kinematics to estimate the impulsive stresslet strength, *M*, and therefore determine the entire jumping flow. The model also provides a satisfactory least-squares fit to the decaying phase of the observed circulation of the vortex around the copepod body, similar to the fit to the circulation of the wake vortex but with a time lag slightly shorter than the beat duration of the power stroke (figure 6).

For both vortex rings associated with a copepod repositioning jump, the positions of vorticity maximum separate increasingly from the flow stagnation points as time goes on [6]. Such separation, which is mainly due to viscous diffusion, is predicted by equations (2.9*a*,*b*) and (2.10*a*,*b*) and is a characteristic feature of viscous vortex rings.

## 4. Discussion

### 4.1. The flow field

Schlieren observations have long revealed the toroidal flow structures left by jumping copepods (e.g. [30]). We previously developed a theoretical hydrodynamic model (impulsive Stokeslet) that described the wake flow [6]; here, we have presented a model that includes the flow that develops around the body of the jumping copepod (impulsive stresslet). It is relevant to here first compare the two models.

Owing to fluid viscosity, the flow fields of both models decay immediately after the application of the impulsive forcing. The scaled area of influence, *S*/*S**, decreases monotonically and in a very similar way to a function of the scaled time, *t*/*t**, in the two models (figure 2*a*), indicating that viscous decay is the primary effect in both models.

To compare the two models more explicitly, we applied both to an average one-beat-cycle repositioning jump of the copepod *A. tonsa* from our previous PIV observations (table 1). For small signal threshold velocities (i.e. *U** < 0.5 mm s^{−1}), the signal disappearance time, *t**, is smaller for the impulsive stresslet model than for the impulsive Stokeslet model (figure 2*b*). This difference is due to partial mutual vorticity cancellation by the two counter-rotating viscous vortices, which is ignored by the impulsive Stokeslet model. For the same reason, the wake flow velocity distribution, *U*(*x*), directly behind the application point of the forcing (i.e. *x* = 0) at the time immediately after the onset of the forcing (i.e. *t* = 0^{+}) is predicted to decline faster by the stresslet than by the Stokeslet model (figure 2*c*). Thus, on top of viscous decay, which is the primary effect, partial mutual cancellation of oppositely signed vorticities in the impulsive stresslet model causes the imposed flow to attenuate faster, both spatially and temporally. Finally, the hydrodynamic impulse, *I*, of the wake vortex, estimated from the attenuation of vortex circulation using PIV observations and the impulsive Stokeslet model, somewhat underestimates the maximum density-specific momentum of the jumping copepod (fig. 2*d* in [6]), while the impulsive stresslet strength, *M*, shows a better 1 : 1 correspondence between that estimated from circulation decay and measured directly from jump kinematics (figure 6 and table 1). Again, this difference can be ascribed to the ignorance of partial vorticity cancellation by the impulsive Stokeslet model.

Thus, the impulsive stresslet model represents an improvement over our previous impulsive Stokeslet model, both by considering the interaction between the two vortex rings and, mainly, by describing the entire imposed flow field, not only the wake flow. The impulsive stresslet model indeed does provide a good approximation of the entire flow associated with a copepod repositioning jump as evidenced by the good correspondence both to PIV observations (figures 6 and 7) as well as to computational fluid dynamics (CFD) simulations [31]. The strength of the impulsive stresslet is the only parameter involved, and can be estimated as the product of the body volume, maximum jump speed and total jump distance of the copepod. Therefore, accurate measurement of jump kinematics is sufficient to estimate the entire flow field. Both spatial extension and temporal duration of the imposed hydrodynamic signal scale with (*M*/*U**)^{1/2} (equation (2.8)).

Because the scaling equation (2.13) of the impulsive Stokeslet model cannot be directly applied to the entire repositioning-by-jumping flow, in our previous work we had to fit the PIV-measured flow velocity data to the general scaling that relates the initial area of influence, *S*, to (*I*/*U**)^{2/3} in order to determine the coefficient in front of the scaling [6]. Based on this, we predicted that the imposed hydrodynamic signal is much less for an ambush-feeding than for a cruising or hovering copepod for small individuals in a time-averaged sense. In the following, we use the more accurate impulsive stresslet model to show that this prediction holds true at any instant in time.

### 4.2. Swimming by jumping in *Oithona* is hydrodynamically quiet swimming

The flow fields generated by motility or feeding currents are likely to shape, to a certain degree, the feeding and motile behaviour of planktonic organisms. We have previously discussed how the induced flow field may serve as a signal to predators, and how different feeding behaviours consequently may expose zooplankters differently to predation risk depending on their size and on the predation landscape [6]. Continuous feeding currents and steady cruising generate momentum jets, while unsteady jumps generate vortex rings. The latter attenuate spatially much faster than the two former flow fields and are, in addition, by default intermittent. Thus, the impulsive stresslet model predicts that the flow velocity induced by a jumping plankter decreases with distance, *d*, as *d*^{−4} (equation (2.6)). (Note that the impulsive Stokeslet predicts *d*^{−3}.) For a cruising neutrally buoyant plankter (modelled as a steady stresslet; appendix A) or a negatively buoyant hovering one (modelled as a steady Stokeslet), induced flow speeds decline with, respectively, *d*^{−2} and *d*^{−1} (e.g. [22]). Despite the higher induced peak flow velocity magnitudes, jumps may thus produce significantly weaker hydrodynamic signals to predators over a large range of body sizes. This is illustrated by the hydrodynamic signals generated by the two different mechanisms of propulsion in copepods, erratic jump-swimming and continuous cruising, as predicted by the impulsive stresslet model and a steady stresslet model, respectively: the flow velocity magnitude as well as the deformation rate of the wake flow attenuate much faster spatially for jump-swimming *O. davisae* males than for a like-sized copepod cruising steadily at the same average velocity, and both flow velocity and deformation rate are smaller for the jump-swimmer just 1–2 body lengths (0.5 mm) behind the copepod (figure 8*a*,*b*).

The likely size of the smallest energy-containing eddies of small-scale oceanic turbulence is *L*_{e} = 2*π* (*ν*^{3}/*E*)^{1/4}, and the associated turbulent shear is (e.g. [32]). If the kinematic viscosity *ν* ∼ 10^{−6} m^{2} s^{−l} and the turbulent kinetic energy dissipation rate *E* ∼ 10^{−7} W kg^{−1}, then *L*_{e} ∼ 10 mm and *Δ*_{t} ∼ 0.18 s^{−1}. Thus, the hydrodynamic signal owing to jump-swimming by the male copepod *O. davisae* is probably overwhelmed by the smallest turbulent eddies.

Rheotactic predators are more likely to respond to the flow velocity than to the deformation rate of the flow generated by their prey [28], and the area of influence (*S*, within which the imposed flow velocities exceed a threshold velocity, *U**) of the moving copepod is therefore a measure of its hydrodynamic ‘visibility’ to such predators (encounter cross section). *S* is substantially smaller for the jump-swimming copepod than for a same-size copepod cruising steadily through the water at the same average translating velocity (figure 8*c*). This applies at any one point in time but the discrepancy becomes even more pronounced when expressed as time-averaged signals; the area of influence owing to steady cruising is approximately five times larger than the time-averaged area of influence owing to jump-swimming (figure 8*c*).

Thus, despite the higher induced peak flow velocity magnitudes but because of the rapid spatial and temporal attenuation as well as the intermittent nature, jumps may produce significantly weaker hydrodynamic signals to predators. This suggests that the adoption of erratic jump-swimming is an adaptation to moving quietly and reducing detection by rheotactic predators. This is consistent with the observation of Yen [33] that a rheotactic predatory copepod (*Euchaeta elongata*) had higher predation rates on the cruising copepod *Pseudocalanus* sp. than on the similarly sized jumping copepod *Acartia clausii*. For those copepods that swim by means of the feeding appendages, ‘swimming’ may thus be interpreted as a ‘by-product’ of feeding.

### 4.3. Hydrodynamic ‘camouflage’

An interesting question is why *O. davisae* males in jump-swimming reduce the phase lag between individual swimming legs and consequently the jump distance relative to that of repositioning, escape and attack jumps. We suggest that a jump-swimming copepod adjusts its jump length to equal the distance that the vortex around its body travels within one jump interval because this generates the best hydrodynamic camouflage. The vortex travelling distance is calculated according to equation (2.9*a*) and the observed average jump interval *T* (=1/21.5 s) as ∼ 0.35 mm, which is indeed close to the average jump distance observed, 0.38 mm (table 2). Figure 9 shows the flow field consequence if the jump distance is roughly equal to the vortex travelling distance. The centre of the vortex ring pairs of the next jump occurs at the vorticity maxima of the previous vortex around the copepod body. This way, vortex ring pairs are regularly spaced and do not deviate much from their canonical form; it looks like a train of equally distanced stepping stones; one disappears at the back and the other emerges in front. In doing so, predators may misread the vortex rings as rings created by physical processes (turbulence), because the background flow field is likely to be made up of many such viscous vortices. This is because any unbounded flow that has net linear momentum (or momentum pair) eventually decays to the unique vortex ring solutions of the Stokes equations [34,35].

### 4.4. Jet or vortex: the jump number

If intermittent jump-swimming is a means of moving quietly and because the formation of viscous vortex rings is essential to this effect, it becomes relevant to decide what degree of intermittency is required for the induced flow field to be characterized by viscous vortex rings (figure 3) rather than by momentum jets (appendix A). Sozou [36] showed that the Stokes flow developing subsequent to the instantaneous application of a constant point force initially has a dipole-like structure (i.e. develops a vortex ring), and this applies as long as the dimensionless parameter . This inspires the definition of a dimensionless ‘jump number’ to characterize the impulsiveness of the jump behaviour,
4.1where *τ* is the duration of the power stroke and *L* is the prosome length. *N*_{jump} is the ratio of two time scales (beat duration and viscous time scale), and for values of order 1 or less, the induced flow will be in the form of a viscous vortex ring, while for very large values the induced flow will be in the form of a jet. Using CFD numerical simulations, we have shown that viscous vortex rings do form from copepod jumps for the jump numbers of the order of 1 or less [31]. The duration of the power stroke does not have a very clear scaling with size, as suggested from the data compiled for various zooplankters, but the jump number increases dramatically with declining size (figure 10). This suggests that there is a critical minimum size below which plankters cannot create vortices and hence move quietly by jump-swimming. From equation (4.1) and assuming a power stroke of 1–10 ms duration as typical for plankters, this threshold is of order 0.05–0.2 mm. Thus, most copepods can create vortices, copepod nauplii are questionable and protozoa cannot. This is consistent with observations of toroidal flow structures generated by copepods [6–8], and with the lack of such structures for the jumping ciliate *M. rubrum* [37].

### 4.5. Viscous versus inviscid vortex rings in the context of animal propulsion

Copepod jumps generate viscous vortex rings whose properties are strikingly different from the inviscid vortex rings typically considered in the context of animal propulsion (e.g. [38–40]). In a viscous vortex ring, owing to viscous diffusion, the vorticity maximum point separates increasingly from the flow stagnation point as time goes on [34]. In an inviscid vortex ring (e.g. Hill's spherical vortex; [41]), both points move through the fluid with the same constant velocity and the form of the vortex ring does not change with time. A (nearly inviscid) vortex ring generated from the traditional piston–cylinder arrangement is due to rolling-up of a vortex sheet that separates at the nozzle edge (e.g. [42]); when long-lasting momentum as well as fluid volume flux are issued out from the piston–cylinder arrangement, there exists a vortex ring formation process that physically limits the size of the generated vortex ring [43,44]. Here, in contrast, viscous vortex rings are generated because of short-lasting localized momentum forcing only (e.g. [45]), and therefore there is no such vortex ring formation process.

Both the impulsive stresslet model and the impulsive Stokeslet model provide tractable theoretical frameworks for describing not only the copepod jumping flows but also the impulsively created flows by many other ecologically important marine organisms, including most zooplankton, small fish larvae and even krill operating in the low-Reynolds-number regime (*Re* typically ranging from 0.1 to several hundreds). Both models may also find their applications in small insect hovering flights, which are of a similar low-Reynolds-number range. The biological and ecological implications of viscous vortices have hitherto been very little studied.

## Acknowledgements

This work was supported by National Science Foundation grants NSF OCE-0352284 and IOS-0718506 and an award from WHOI's Ocean Life Institute to H.J. and by grants from the Danish Research Council for independent research and the Niels Bohr Foundation to T.K. We are grateful to two anonymous reviewers whose constructive comments have greatly improved this manuscript.

## Appendix A. Stresslet model for the swimming current of a neutrally buoyant copepod

Stresslet (e.g. [46]) flow equations are written in a cylindrical polar coordinate system (*x*, *r*, *ϕ*) where the positive *x*-direction coincides with swimming direction,
A 1aand
A 1bwhere *Q* is the stresslet strength (in dimensions of force times distance). The associated velocity magnitude is
A 2and two lengths are formed,
A 3aand
A 3b

The scaling for area of influence, *S*, is
A 4

Also, vorticity and streamfunction are A 5aand A 5b

A calculation example for the swimming current created by a neutrally buoyant copepod is shown in figure 11.

- Received September 2, 2010.
- Accepted December 7, 2010.

- This Journal is © 2011 The Royal Society