Propulsion efficiency and imposed flow fields of a copepod jump

Planktonic copepods are a group of abundant, evolutionarily successful and ecologically important multicellular organisms found in the ocean and in fresh water (e.g. Mauchline, 1998; Miller, 2004). Most, if not all, of them are capable of jumping by sequentially striking the swimming legs posteriorly. Such jumps fall into four general categories as distinguished by their functioning: (1) powerful and rapid escape jumps for getting away from approaching danger (e.g. Singarajah, 1969; Strickler, 1975; Fields and Yen, 1997; Suchman, 2000; Buskey et al., 2002); (2) short and less powerful repositioning jumps for frequent relocation in the water column (e.g. Tiselius and Jonsson, 1990; Svensen and Kiørboe, 2000; Paffenhöfer and Mazzocchi, 2002); (3) accurately directed and carefully maneuvered attack jumps for capturing prey (e.g. Jiang and Paffenhöfer, 2008; Kiørboe et al., 2009); and (4) sequences of many consecutive small swimming jumps for traveling through the water (e.g. Strickler, 1975; Morris et al., 1990). All the jumps generate hydrodynamic disturbances, and a rheotactic predator may detect a jumping copepod by sensing the hydrodynamic disturbance the copepod generates (e.g. Strickler and Bal, 1973; Visser, 2001). Quantitative characterization of the jump-associated hydrodynamic disturbances allows a mechanistic understanding of the jump-related predation risk faced by the copepods. In spite of being a very common motility mode in copepods, jumping is generally believed to be energetically costly. Thus, it is also meaningful to assess directly the mechanical energy expenditure and propulsion efficiency of jumping by copepods.

We have recently examined the flow fields imposed by copepods that perform relatively weak repositioning jumps (Kiørboe et al., 2010a; Jiang and Kiørboe, in press). Flow visualization (particle image velocimetry; PIV) and simple analytical models have demonstrated that the 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. These studies have suggested that the predation risk of ambush-feeding copepods that perform frequent repositioning jumps, and of copepods that swim by repetitive small jumps is less than that of copepods that produce a feeding current or cruise steadily through the water. Much less is known about the hydrodynamics of the powerful escape jumps. Although such jumps may allow a copepod to escape a predator, they also pose a risk: the jump-generated flow disturbance may inform the predator about the position of the copepod prey, and may elicit an attack (Williamson and Vanderploeg, 1988; Yen and Strickler, 1996). Therefore, the speed and distance of a jump becomes crucial. The specific force production during an escape jump is unusually high compared with that reported for any other organism (Lenz et al., 2004; Kiørboe et al., 2010b), but the propulsion efficiency is unknown. The simple analytical models are not applicable to escape jumps and PIV observations are not possible because, at the magnification required, the jump distance and the flow field will exceed the field of view. Schlieren (i.e. deflection of light by flow density gradient) observations have revealed a series of toroidal flow structures in the wake of an escaping copepod, probably frozen at several initial instants since the start of jumping (e.g. Yen and Strickler, 1996). However, nothing is known about the spatial configuration and temporal evolution of the flow field associated with these escape jumps, especially for the period following the completion of jumping, or of the energetics and efficiency. This is the focus of the present study.

Computational fluid dynamics (CFD) simulation has emerged as a suitable and useful approach for investigating small-scale hydrodynamic phenomena occurring at individual plankter scale and their interplay with plankton behavior and ecology (e.g. Jiang et al., 1999; Jiang et al., 2002; Gilmanov and Sotiropoulos, 2005; Jiang and Strickler, 2007; Latz et al., 2008; Musielak et al., 2009). Here we extend the CFD simulation approach to the investigation of the flow field associated with a jumping copepod. We validate the simulation approach using the PIV flow data measured for copepod repositioning jumps (Kiørboe et al., 2010a), and then apply the validated approach to simulate the flow field associated with copepod escape jumps. We use previously measured copepod jump kinematics (Kiørboe et al., 2010a; Kiørboe et al., 2010b) to drive our simulations. Based on the simulation data, we calculate Froude propulsion efficiency for both the repositioning jumps and the escape jumps.

High-speed video observations of the kinematics of copepod jumps (i.e. the time-dependent copepod jump velocity) and PIV observations of associated flow fields were taken from Kiørboe et al. (Kiørboe et al., 2010a; Kiørboe et al., 2010b). Two species of copepod were studied: Acartia tonsa Dana 1849 (prosome length 0.7–1.1 mm) and Calanus finmarchicus Gunnerus 1765 (3 mm). Copepods jump by sequentially striking the four (A. tonsa) or five (C. finmarchicus) pairs of swimming legs posteriorly (Fig. 1). This pushes the copepod forward. Subsequently the swimming legs are recovered simultaneously. This beat cycle may be repeated after a few milliseconds. Repositioning jumps consist of one to several beat cycles, and escape jumps may consist of many beat cycles. The duration of the power stroke varies between 2 and 10 ms for A. tonsa and 5 and 20 ms for C. finmarchicus (both are absolute ranges), and the jump speed varies inversely with the stroke duration and may be up to ∼500 mm s^–1 for A. tonsa and ∼800 mm s^–1 for C. finmarchicus.

At the onset of a jump and immediately before the striking of the swimming legs, a real copepod wraps its two antennules (A1) around its body, presumably to reduce drag. Modeling the copepod body as a prolate spheroid is therefore regarded as a good approximation. We simulated the flow field created by a prolate spheroidal model copepod of long axis equal to copepod body (prosome) length (2a) and short axes equal to ϵ×2a, where ϵ is the aspect ratio of the copepod. The model copepod was assumed to jump forward along its long axis (i.e. the x-axis; Fig. 2). Thus only a meridian plane was included as the computational domain (i.e. the finite geometric domain on which the flow field is to be solved numerically). A cylindrical polar coordinate system was used with r being the radial distance from the x-axis (Fig. 2). Symmetry boundary condition was specified on the upper boundary. Pressure-outlet boundary conditions were specified on both the left and right boundary. Because of axisymmetry, the flow patterns are identical in all planes containing the x-axis, independent of the azimuthal coordinate (φ). This is consistent with observational evidence that mushroom-like toroidal flow structures are left behind jumping copepods (Yen and Strickler, 1996; van Duren and Videler, 2003; Kiørboe et al., 2010a).

The computational domain is 90a in the x-direction and 30a in the r-direction (Fig. 2 upper panel). The domain was discretized into 6000 quadrilateral control volumes (CVs) immediately adjacent to the copepod body and into ∼35,000 triangular CVs that were stretched radially outward from the outer boundary of the quadrilateral CVs. The model copepod body and the quadrilateral CVs travel together (Fig. 2) according to an observed time-dependent copepod jump velocity [U(t)]. U(t) was obtained by applying a Savitzky–Golay smoothing filter (Press et al., 2007) to the data from Kiørboe et al. for repositioning jumps and from Kiørboe et al. for escape jumps (Kiørboe et al., 2010a; Kiørboe et al., 2010b) (Fig. 3). This filtering approach advantageously tends to preserve the relative maxima, minima and peak widths of the observed velocity time series. The outer zone consisting of the triangular CVs (i.e. the deforming mesh zone; Fig. 2) was remeshed every time step in order to accommodate the combined motion of the model copepod body (as an internal solid-wall boundary) and the adjacent quadrilateral CV zone.

As to the numerical schemes, the highly accurate third-order MUSCL (monotone upstream-centered schemes for conservation laws) scheme was used for spatial interpolation. The body force weighted scheme was selected as the discretization method for pressure. The PISO (pressure-implicit with splitting of operators) scheme was used for pressure–velocity coupling. Temporal discretization was a first-order implicit scheme. For the jumping phase when the model copepod was moving, the time step was τ₁/50 where τ₁ is the sample time interval of the jump kinematics data obtained using the high speed camera. Three kinds of copepod jumps were simulated, namely, the repositioning jumps of A. tonsa with τ₁=1/1000 s, the escape jumps of A. tonsa with τ₁=1/3596 s, and the escape jumps of C. finmarchicus with τ₁=1/2200 s. Using such a small time step was also a requirement by the dynamic (moving or deforming) mesh model in ANSYS FLUENT. At the end of the jumping phase, when the model copepod was no longer moving (in an approximate sense), the dynamic mesh model was switched off to allow for a much larger time step, e.g. τ₂/200 where τ₂=(ϵa)²/ν (a viscous time scale based on half the body width of the model copepod). After the jumping phase the simulation was still continued for 20–40 τ₂ for the repositioning jumps and 100–130 τ₂ for the escape jumps. As a result, the time-dependent flow field at the decaying phase was also obtained. Good comparison between the simulated results and the PIV flow data measured for copepod repositioning jumps (see Results and discussion) validated the CFD simulation approach used.

For a power stroke, the mean thrust () was calculated as the time average (over the power stroke duration) of T(t), which had been obtained at every time step by solving Eqn 2 and the flow field equations jointly.

During a power stroke, the instantaneous mechanical power [P(t)] applied by the swimming legs was calculated as the volume integral of [F(x, r, t)•u(x, r, t)] over the volume where the swimming leg forcing F(x, r, t) was applied (i.e. the blue patch in the lower panel of Fig. 2); here u(x, r, t) was the CFD-simulated instantaneous flow velocity vector field. For this power stroke, the CFD-simulated total mechanical work [W_CFD] was then calculated as the time integral of P(t) over τ.

The flow field associated with a weak repositioning jump consists of two counter-rotating viscous vortex rings of similar intensity, one in the wake of the copepod and one around the body. This was first revealed by a PIV observational study (Kiørboe et al., 2010) and then confirmed by a theoretical hydrodynamic model (Jiang and Kiørboe, in press). This can also be reproduced by the present CFD simulation approach. The simulated flow velocity field and vorticity field compare favorably with those obtained from the PIV observation (Fig. 4), despite some discrepancies that might be due to the three-dimensionality of a real copepod jump versus the model assumption of axisymmetry, especially at the initial stage of the jump when the leg forcing is somewhat three-dimensional (compare Fig. 4A with B).

The CFD-simulated results show that, at the initial stage of a repositioning jump, the spatial extension of the imposed flow field does not reach its maximum, even at the time when the thrust is at the maximum (compare Fig. 5A with B,C). It takes (a very short) time for the flow to reach its maximum spatial extension (Fig. 6) because the water surrounding the copepod receives momentum, although in opposite directions, during both the accelerating and decelerating phase of the jump. At the time when the jump stops the vortex around the copepod body is stronger than the wake vortex because of the existence of a thin boundary layer surrounding the copepod body that hosts vorticity with sign opposite to that of the vortex around the body (Fig. 5B). However, in the decaying phase, the body vortex and the wake vortex quickly become similar in intensity because of viscous diffusion and mutual cancellation of the opposite-signed vorticities (Fig. 5C). The CFD-simulated circulation of the wake vortex also compares favorably with the PIV-measured circulation of the wake vortex, and the decaying phase of both conforms to the theoretical viscous vortex decay predicted from the impulsive stresslet model of Jiang and Kiørboe (Jiang and Kiørboe, in press) (Fig. 7). The CFD-simulated normalized area of influence follows the trend of the theoretical prediction from the impulsive stresslet model (Fig. 6). However, the CFD estimates of the areas of influence of different threshold velocities do not exactly collapse on to each other; rather, areas of larger threshold velocities decay faster than those of smaller threshold velocities, and they all decay slightly faster than the theoretical prediction. These discrepancies may be due to the fact that the model copepod body was included in the CFD simulations whereas the impulsive stresslet model neglects the body effect and use the Stokes approximation (i.e. zero Reynolds number). Nevertheless, these comparisons validate the present CFD simulation approach, and therefore we can confidently apply the validated approach to simulate the flow field associated with copepod escape jumps (for which no PIV flow data are available) and use the CFD output data to calculate several useful quantities that are often difficult to measure directly for both repositioning jumps and escape jumps.

A repositioning jump may involve a few kicks of the swimming legs (e.g. A. tonsa jump 73-2; Table 1, Fig. 7B). The CFD results show that all kicks create vorticity of the same sign as the leading wake vortex, which is associated with the first kick. However, only the leading wake vortex retains its integrity; the other wake vorticities are close to the opposite-signed body vortex (because a repositioning jump is of short distance) and therefore disintegrate quickly by vorticity cancellation. As a result, at the time when the jump stops, the vorticity configuration is still similar to that of a single-kick repositioning jump (not shown).

An escape jump is more powerful (fast power stroke) and may involve many beat cycles. However, at the initial stage of an escape jump, the spatial extension of the imposed flow field is still very limited even at the time when the thrust is largest (Fig. 8A, Fig. 9A); this is similar to a repositioning jump at the same stage (Fig. 5A). At the time when the copepod completes an escape jump sequence and stops (Fig. 8B, Fig. 9B), the associated flow field consists of two dominant vortex structures: one leading wake vortex ring located near the position where the copepod has executed its first power stroke of the swimming legs, and one vortex surrounding the copepod body. These two vortex structures are of opposite sign but have similar spatial extension. Between these two vortex structures lies an elongated patch of vorticity of the same sign as that of the vortex around the copepod body. Along the edge of this elongated vorticity patch is a train of isolated blobs of vorticity of opposite sign (i.e. of the same sign as that of the leading wake vortex ring); each of these vorticity blobs corresponds to a power stroke of the swimming legs (subsequent to the first power stroke). However, this train of isolated vorticity blobs is short-lived; the vorticity blobs quickly disintegrate through vorticity cancellation with the nearby elongated vorticity patch of the opposite sign. As a result, in the decay phase the flow field evolves into one leading wake vortex ring plus a deformed vortex structure of opposite sign that surrounds the copepod body and trails behind (Fig. 8C, Fig. 9C). In other words, the major difference in terms of imposed flow disturbance between a slow repositioning jump and a fast escape jump is the elongated vorticity shed from the body in the escape jump. This is also evidenced by the fact that the total circulation of all the wake vortices eventually collapses with the time evolution of the circulation of the leading wake vortex (Fig. 10). The time evolution of the circulation of the leading wake vortex fits well with the theoretical prediction from the impulsive Stokeslet model of Kiørboe et al. (Kiørboe et al., 2010a) (Fig. 10). This is because the leading wake vortex ring is sufficiently isolated from the rest of the vorticity structures. It is therefore suggested that the impulsive Stokeslet model can be used to describe the behavior of the leading wake vortex ring, even for powerful escape jumps. This level of detailed description of the spatiotemporal evolution of the flow field associated with a copepod escape jump sequence has not been available previously.

In terms of the area of influence, the escape jump by a copepod of 0.67 mm prosome length (A. tonsa Jump 90) created comparable, yet stronger, hydrodynamic signals to those of the repositioning jump by a copepod of 1.11 mm prosome length (A. tonsa Jump 58). (Fig. 8 and Fig. 5 were plotted using the same length scale, so that the two areas of influence can be easily compared.) The former copepod achieved a much higher jumping velocity than the latter (compare Fig. 10A and Fig. 7A) even though it was smaller. The escape jump by a C. finmarchicus of 3.0 mm prosome length (C. finmarchicus Jump 212) created much stronger hydrodynamic signals than the two Acartia jumps, differing by orders of magnitudes in terms of both spatial extension and temporal duration. (Note that Fig. 9 was plotted using a different length scale from that used for Fig. 8 and Fig. 5.) An escape jump in general creates stronger and longer lasting hydrodynamic signals than a repositioning jump. For example, the escape of the 3-mm C. finmarchicus left a unidirectional flow trail with velocity vectors pointing to the location of the copepod; the trail was still persistent ∼4.5 s after the completion of the jump and the length of the trail was ∼5 cm with flow velocity magnitudes >350 μm s^–1 (Fig. 9C). Potentially, this allows a rheotactic predator to pursue its escaping prey at rather long distances, because the flow velocity threshold for rheotactic predator responses are in the order of 100 μm s^–1 (reviewed by Kiørboe, in press).

In terms of the spatial configuration and temporal evolution of the jump-associated flow field, the difference between a repositioning jump and an escape jump sequence can have consequences depending on whether or not an effective hydrodynamic ‘camouflage’ can be achieved. The flow field associated with a repositioning jump has a near mirror image of the wake vortex and the vortex around the copepod body (Fig. 5C), and therefore a rheotactic predator a short distance away may have to choose which side it should attack; this may reduce the probability at which the copepod prey gets caught. Such a near mirror image does not exist in the flow field associated with a copepod escape jump sequence. On the contrary, the flow velocity vectors behind the copepod are all pointing towards the location of the jumping copepod over almost the entire jumping path and over a long time period (Fig. 8C, Fig. 9C), and the flow structure may therefore efficiently inform a rheotactic predator about the position of the copepod. Thus, the escape jumps should only be executed as the last resort to avoid immediate predation. Also, as pointed out by Jiang and Kiørboe, viscous vortex rings created by copepod repositioning jumps may aid the copepod in hiding among those viscous vortex rings that are created by physical processes (turbulence) (Jiang and Kiørboe, in press). This is because the background flow field is likely to be made up of many such viscous vortices, as any unbounded flow that has net linear momentum (or momentum pair) eventually decays to the unique vortex ring solutions of the Stokes equations (Phillips, 1956; Stanaway et al., 1988; Shariff and Leonard, 1992; Afanasyev, 2004). This is, however, unlikely to be the case for the flow field associated with a copepod escape jump sequence, because the elongated, long-lasting, unidirectional flow trail behind the copepod may be more likely to be ‘read’ by a rheotactic predator as biologically generated.

The mean thrust reported in Table 1 for each power stroke ranges from 1.66×10^–6 to 1.58×10^–4 N, varying greatly for different jump types, copepod body sizes and maximum jump velocities. The values for the escape jumps are similar to those measured experimentally (Alcaraz and Strickler, 1988; Lenz et al., 2004) or estimated from a previous simpler model (Kiørboe et al., 2010b). As pointed out by Kiørboe et al. the force production generated by the swimming legs, when normalized by the mass of the muscles, is much higher than reported for the muscle motors of any other organisms (Kiørboe et al., 2010b).

The W_CFD reported in Table 1 for each beat cycle ranges from 2.67×10^–9 to 9.84×10^–7 J, also varies significantly for different jump types, copepod body sizes and maximum jump velocities. For the three simulated repositioning jumps the cost of the jump corresponds to between 4 and 14% of the metabolic rate over the power stroke duration of the jump [metabolic rate computed from empirical size–metabolism relationships in Ikeda et al. (Ikeda et al., 2001)]. At a jump frequency of 1 Hz, which is high (Kiørboe et al., 2010a), the time-averaged cost of repositioning jumps is less than 1% of the total metabolic expenditure. The much more powerful escape jumps are of course energetically more expensive, corresponding to more than 400% of the instantaneous metabolic rate; however, such jumps are rare and similarly accounts for only a minute fraction of the metabolism of the copepod. These considerations suggest that jumps come at a very small energetic cost, consistent with most of the earlier estimates (e.g. Vlymen, 1970; van Duren and Videler, 2003).

The total mechanical work is the mechanical work expended in moving the swimming legs through the water. Of the W_CFD, ∼50–93% was used to overcome the hydrodynamic resistance (including the added mass effect) acting on the accelerating body of the copepod during the power stroke durations, and ∼5–47% was used to accelerate the copepod body itself (Table 1). These two parts should be regarded as the useful mechanical work done by the copepod during the power stroke. Together they account for a significant share of the total mechanical work; the Froude propulsion efficiency was very high, ranging between 0.94 and 0.98 (Table 1). This result appears counter to our expectation because the calculated values of the Froude propulsion efficiencies are significantly greater than the known values (0.2–0.78) estimated for small swimmers as reviewed by Vogel (Vogel, 2003). We suggest that this difference is due to the unsteady nature of jumps as opposed to the steady cruising motion of the small swimmers examined by Vogel (Vogel, 2003) (see below). However, Morris et al. also obtained a low Froude propulsion efficiency (0.34) for the escape jump of the copepod Pleuromamma xiphias with the Reynolds number (>1000) similar to one of our simulated jumps (C. finmarchicus Jump 212; Table 1) (Morris et al., 1985). This discrepancy is probably due to the different methods in calculating the hydrodynamic resistance acting on the accelerating copepod body. We directly calculated the hydrodynamic resistance from the area integral of pressure and shear stress over the copepod body surface, whereas empirical drag laws were used by Morris et al. (Morris et al., 1985). We note that the empirical drag laws were derived for well-developed steady-state boundary layer conditions, which might not be suitable for the copepod jump where only a very thin and highly unsteady boundary layer develops in the course of the few milliseconds power stroke duration.

We thank Dr Sean P. Colin for discussions and comments on the manuscript.

 
• a

  half the prosome length

 
• A(t)

  forcing magnitude at the tip of the swimming leg

 
• CFD

  computational fluid dynamics

 
• CV

  control volume

 
• D(t)

  instantaneous hydrodynamic resistance

 
• F

  field of body force

 
• I_fitted,CFD

  fitted hydrodynamic impulse

 
• L_leg

  typical length of the swimming legs

 
• M

  strength of the impulsive stresslet

 
• m_copepod

  body mass of the model copepod

 
• M_fitted,CFD

  fitted impulsive stresslet strength

 
• MUSCL

  monotone upstream-centered schemes for conservation laws

 
• PISO

  pressure-implicit with splitting of operators

 
• PIV

  particle image velocimetry

 
• r

  radial distance from the x-axis

 
• R_r*

  length scale for determining the scaling of area of influence

 
• R_x*

  length scale for determining the scaling of area of influence

 
• S

  area of influence

 
• S(t)

  time series of the area of influence

 
• t

  time

 
• t*

  time scale for determining the scaling of area of influence

 
• 

  mean thrust

 
• T(t)

  instantaneous thrust

 
• u

  CFD-simulated velocity vector field

 
• U(t)

  jump velocity

 
• U*

  velocity threshold

 
• UDF

  user defined function

 
• W_CFD

  CFD-simulated total mechanical work

 
• Γ(t)

  time series of the wake vortex circulation

 
• ϵ

  aspect ratio of the copepod

 
• η

  Froude propulsion efficiency

 
• ν

  fluid kinematic viscosity

 
• ρ

  fluid density

 
• τ

  power stroke duration

 
• τ₁

  sample time interval of the jump kinematics data

 
• φ

  azimuthal coordinate

 
• ω_φ(x, r, t)

  instantaneous azimuthal vorticity field