j. r. soc. interface-2011-jiang-rsif.2010

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doi: 10.1098/rsif.2010.0481published online 5 January 2011J. R. Soc. Interface

Houshuo Jiang and Thomas KirboeThe fluid dynamics of swimming by jumping in copepods

Supplementary data

l

http://rsif.royalsocietypublishing.org/content/suppl/2011/01/04/rsif.2010.0481.DC2.htm"Data Supplement"

lhttp://rsif.royalsocietypublishing.org/content/suppl/2011/01/04/rsif.2010.0481.DC1.htm

"Data Supplement"

References

ref-list-1http://rsif.royalsocietypublishing.org/content/early/2011/01/10/rsif.2010.0481.full.html#

This article cites 39 articles, 11 of which can be accessed free

P


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The fluid dynamics of swimming byjumping in copepods

Houshuo Jiang1,* and Thomas Kirboe2

1Department of Applied Ocean Physics and Engineering, Woods Hole OceanographicInstitution, Woods Hole, MA 02543, USA

2National Institute for Aquatic Resources, Technical University of Denmark, Kavalergarden 6,2920 Charlottenlund, Denmark

Copepods swim either continuously by vibrating their feeding appendages or erratically byrepeatedly beating their swimming legs, resulting in a series of small jumps. The two swimmingmodes generate different hydrodynamic disturbances and therefore expose the swimmers differ-ently 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 vortexrings of similar intensity, one in the wake and one around the body of the copepod. Weshowed that the entire jumping flow is spatially limited and temporally ephemeral owing to jump-impulsiveness and viscous decay. In contrast, continuous steady swimming generatestwo well-extended long-lasting momentum jets both in front of and behind the swimmer, assuggested by the well-known steady stresslet model. Based on the observed jump-swimmingkinematics of a small copepod Oithona davisae, we further showed that jump-swimmingproduces a hydrodynamic disturbance with much smaller spatial extension and shorter tem-poral duration than that produced by a same-size copepod cruising steadily at the sameaverage translating velocity. Hence, small copepods in jump-swimming are in general muchless detectable by rheotactic predators. The present impulsive stresslet model improves apreviously published impulsive Stokeslet model that applies only to the wake vortex.

Keywords: copepod jump; viscous vortex ring; impulsive stresslet; impulsiveStokeslet; hydrodynamic camouflage; non-dimensional jump number

1. INTRODUCTION

Planktonic organisms create water flows at the scale oftheir body as they move and feed. They generate feed-ing/swimming currents by rapidly beating theirflagella, cilia or cephalic appendages more or lesscontinuously. Or they may make instantaneous, short-lasting jumps to relocate, escape predators or attackprey (e.g. [1 5]), which in some cases may generate tor-oidal flow structures in their wake, as has beendescribed for copepods [6 8]. Finally, even when zoo-

plankters do not move their appendages, cilia orflagella, they may sink passively, leading to flow passingaround their body. It is of great relevance to investigatethese small-scale biogenic fluid dynamical phenomenaas they are related to and have implications for thevarious survival tasks of the plankters, such as feed-ing, nutrient uptake, predator avoidance, matingand signalling (for reviews, see [9 16]). Specifically,zooplankters with different feeding and motilitybehaviours generate different hydrodynamic disturb-ances and therefore expose themselves differently torheotactic predators. Quantitative characterization ofthese different hydrodynamic disturbances allows a

mechanistic understanding of the (probably) differentlevels of predation risk faced by these zooplankters.

Planktonic copepods, arguably the most abundantmetazoans in the ocean [17], swim either by continu-ously vibrating their feeding appendages, whichresults in a rather constant propulsion, or by usingtheir swimming legs, which leads to sequences of smalljumps. Continuous swimming by vibrating the feedingappendages is common among calanoid copepodsthat generate a feeding current. In contrast, ambush-

feeding copepods, mainly among the small cyclopoidcopepods, do not generate a feeding current, but swimby sequentially striking the swimming legs posteriorlyand by repeating this sequence at short intervals(e.g. [18,19]). This leads to a very unsteady, erraticmotion, which probably generates a hydrodynamic dis-turbance much different from that of a continuouslyswimming copepod. The purpose of this study is tocompare the hydrodynamic signal generated by thesetwo fundamentally different propulsion mechanisms inzooplankton, using copepods as an example.

Several analytical solutions from classical fluiddynamics have previously been applied to investigate

low-Reynolds-number flows created by small continu-ously moving plankters. For example, the solution ofthe Stokes flow associated with a steady point forcewas used to model the feeding current created by a

*Author for correspondence ([email protected]).

Electronic supplementary material is available at http://dx.doi.org/10.1098/rsif.2010.0481 or via http://rsif.royalsocietypublishing.org.

J. R. Soc. Interface

doi:10.1098/rsif.2010.0481

Published online

Received2 September 2010Accepted 7 December 2010 1 This journal is q 2011 The Royal Society

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negatively buoyant, stationary/hovering copepod [20].The same authors also modelled the flow around a stea-dily sinking copepod using the well-known Stokessolution for the flow associated with a steadily translat-ing sphere. Later, an analytical model of a negativelybuoyant swimming copepod (hovering and free-sinkingas special cases) was developed based on a linear combi-

nation of the Stokes flow around a stationary solidsphere and the Stokes flow owing to a point force exter-nal to the same sphere [21,22]. This model separates thecopepod main body resistance from the thrusting effectof the beating appendages and therefore has the abilityto reproduce the major feeding/swimming currentpatterns that have been observed so far.

However, these solutions to the steady Stokes flowequations cannot be applied to the intrinsic unsteadyflows associated with plankters that swim by jump-ing because these are of an impulsive nature. Jumpingplankters impart only brief and localized momentumto the surrounding water during short-lasting power

strokes. The localized impulsive forcing exerted by thethrusting appendages (e.g. copepods) or membranelles(e.g. the jumping ciliate Mesodinium rubrum [2]) willstop as soon as the power strokes are completed andthe flow so created will decay because of viscosity.

There are few previous attempts to develop theoreti-cal hydrodynamic models for the low-Reynolds-numberunsteady flow associated with jumping plankton, eventhough jumping is a very common motility modeamong both protozoan [2,4] and metazoan plankton[19,20,2325]. Here, we develop such a model bytaking into account the two characteristics of the jump-ing 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 knownthat jumping copepods leave toroidal flow structures intheir wake [7,8], and we previously developed an impul-sive Stokeslet model for such viscous wake vortex rings[6]. However, an additional vortex ring develops aroundthe decelerating body, and, in a typical repositioningjump of a millimetre-size copepod, the two rings are ofsimilar intensity and opposite direction ( figure 1a; [6]).We here present an impulsive stresslet model thatdescribes the entire flow field and includes both vortexrings generated by the jumping copepod. We first com-pare the two theoretical models, and then apply the

present impulsive stresslet model to observations of thekinematics of swimming in a small cyclopoid copepod,Oithona davisae. We compare the modelled flow for ajump-swimming copepod to the flow generated by a stea-dily cruising one and show that the hydrodynamicdisturbance generated by the former has a much smallerspatial extension and temporal duration than that gener-ated by the latter, suggesting that jump-swimming is ahydrodynamically quiet propulsion mode.

2. METHODS

2.1. Video observation

We observed the swimming behaviour of males ofthe cyclopoid copepod O. davisae (prosome length0.3 mm) using high-speed video (400 or 1900 frames s21).

Males, but not females, of this species frequentlyswim when searching for females [26], and they swimusing their swimming legs, not their feeding appen-

dages. Animals were taken from our continuousculture and placed in approximately 100 ml aquaria.Light from a 50 W halogen lamp was guided througha collimator lens and the aquarium towards thecamera, thus providing shadow images of the cope-pods. The camera (Phantom v.4.2 Monochrome) wasequipped with lenses to produce fields of view ofapproximately 3 3 mm2 when filming at the highframe rate, or approximately 15 15 mm2 when filmingat the lower frame rate. At the low frame rate and lowmagnification, we placed a mirror in the diagonal of theaquarium; by following both the image and the mirrorimage of the animals, we could reconstruct their

three-dimensional movement paths. When filmingat the higher magnification, we selected sequenceswhere the animal was moving in the view planeperpendicular to the camera.

x

r

O

copepodcopepod

(a)

(b)

(c)

rId(X) d(t)

x

r

O

rId(X-e/2) d(t)

x

r

O

rId(X+e/2) d(t)

Figure 1. (a) Schematic illustrating the general vorticity struc-ture of the flow field imposed by a repositioning-by-jumpingcopepod, as shown in a meridian plane of a cylindrical polar

coordinate system (x, r, f), where f is the azimuthal coordi-nate, r the radial coordinate and the positive axial (x-)direction coincides with the jump direction. The vorticitysign is determined by the conventional right-hand rule. Theblue patch indicates the wake vortex of negative vorticityand the red patch indicates the vortex of positive vorticitythat is around the copepod body. (b) The impulsive stressletmodel. (c) The impulsive Stokeslet model.

2 Swimming by jumping in copepods H. Jiang and T. Kirboe


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Selected video sequences (n 18) showing one cope-pod moving were analysed frame by frame, either semi-automatically using IMAGEJ or automatically usingthe particle-tracking software LABTRACK (DiMedia,Kvistgard, Denmark). The position of the animal wasfollowed over time, allowing us to describe the temporalvariation in move speed. At the higher magnification,

we also noted the duration of the power stroke of theswimming legs in four to six beat cycles per animal.Observations of the flow field generated by the repo-

sitioning jumps of a different copepod (Acartia tonsa)and visualized using particle image velocimetry (PIV)were taken from Kirboe et al. [6] for comparison withmodel predictions.

2.2. The impulsive stresslet model

Using PIV, we have recently measured the flow fieldsimposed by copepods that performed repositioningjumps [6]. Being approximately axisymmetric, a typical

such measured flow field consists of two counter-rotatingviscous vortex rings of similar intensity, one in the wakeand one around the body of the copepod (figure 1a). Thewake vortex is generated owing to the momentumapplied almost impulsively to the water by the rapidbackward kick of the swimming legs; almost simul-taneously but slightly delayed, the counter-rotatingbody vortex is generated owing to the oppositely directedmomentum of equal magnitude paid back to the waterby the forward moving copepod body. Here we considerthe two rings as a whole system and describe their behav-iour using an impulsive stresslet model (figure 1b).

2.2.1. Equations of the impulsive stresslet. An impulsivestresslet consists of two point momentum sources ofequal magnitude (rI, where r is the mass density ofthe fluid and I is the hydrodynamic impulse), actingsynchronously in opposite direction and separatedby distance 1 (figure 1b). Each momentum sourceacts impulsively for a very short period of time formallyrepresented by the Dirac delta function d(t). Thedefinition of the strength of the impulsive stresslet isM lim

10, I1 I1 const. with [M] L5T21 for

three-dimensional flows. Vorticity (vf) and stream-function (cf) for the above-described flow have beenobtained in Stokes approximation [27],

vf Mxr32p3=2nt7=2

ej2 2:1a

and

cf 3Mxr2

2p3=2x2 r25=2

ffiffiffiffip

p2

erfj jej2 1 23j2

;

2:1bwhere j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 r2=4ntp ; n is the kinematic viscosity,and the error function

erfj 2= ffiffiffiffipp j

0 ey2

dy:

The solution is axisymmetric and independent of theazimuthal coordinate f in the cylindrical polar

coordinate system (x, r, f) ( figure 1b). The componentsof velocity in the meridian plane are given by

u;1

r

@cf@r

M2p3=2

x

x2 r27=22x2 3r2C r2D 2:2a

and

v; 1r

@cf@x

M2p3=2

r

x2 r27=2r2 4x2C x2D; 2:2b

where u and v are the velocity componentsin the axial (x-) and the radial (r-) direction,respectively,

C 3 ffiffiffiffip

p2

erfj jej2 1 23j2

and

D 4j5 ej2 :

2.2.2. Viscous decay. Integrating vf (equation (2.1a))over one half of the meridian plane (e.g. x! 0) wherethe vorticity is one-signed leads to a simple formulafor viscous decay of circulation of the one-signedvorticity,

Gx!0t ;1

0

10

vfdxdr M8p3=2

nt

3=2

: 2:3

The hydrodynamic impulse of the one-signedvorticity satisfies

Ix!0t ; p1

0

10

vfr2 dxdr Mffiffiffiffiffiffiffiffiffiffi

4pntp : 2:4

It is obvious that Gx!0(t) Gx,0(t) 0 andIx!0(t) Ix,0(t) 0.

2.2.3. Asymptotic analysis for finding spatial extensionand temporal duration of the flow. At small time, the

flow far field (i.e. j) 1) is approximately irrotationaland behaves as

u 3M4p

x2x2 3r2x2 r27=2

2:5a

and

v 3M4p

r4x2 r2x2 r27=2

: 2:5b

The associated velocity magnitude is

U; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 v2p

3M

4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4x

4

r4

p

x2 r23:

2:6

From equation (2.6), two lengths (denoted Rx

and Rr) are formed to define the size of the domain

Swimming by jumping in copepods H. Jiang and T. Kirboe 3




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over which the flow velocity magnitude is greater than athreshold velocity U*: setting r 0 leads to

Rx 3

2p

M

U

1=4; 2:7a

and setting x 0 leads to

Rr 34pMU

1=4

: 2:7b

Asymptotic analysis at large time (i.e. j 0) is toocomplicated to provide a formula for calculating thetime t* after which the whole flow field is below U*.However, dimensional analysis together with numericalcalculation using equation (2.2a,b) suggests a scalingrelationship,

S

S f t

t

; 2:8a

S % 0:476pRxRr % 0:868M

U

1=22:8b

and t % 0:068n

M

U

1=2; 2:8c

where S is the area of influence, defined as the area inthe meridian plane within which the flow velocity mag-nitude is greater than U*. A rheotactic predator is likelyto perceive the prey-induced flow velocity magnitude

[28] and, hence, this area is a measure of the encountercross-section of the copepod prey. The scaling relation-ship has been numerically determined and plotted(figure 2a); the hydrodynamic signal quantified as thearea within which the induced flow velocity magnitudeexceeds U* initially attains its maximum (i.e. S*) andthen decays until it dies out completely after t*. BothS* and t* depend on M/U* only.

2.2.4. Translation of the vortices. The impulsive stress-let initially sets up an irrotational flow everywhere

except at its application point, where there exist twoopposite-signed vorticity singularities (figure 3a).Both singularities diminish and the associated vortici-ties diffuse away from the application point as timegoes on (figure 3b d; electronic supplementarymaterial, appendix movie S1). The coordinates of thetwo vorticity maxima evolve as

xvt +ffiffiffiffiffiffiffi

2ntp

2:9aand

rvt ffiffiffiffiffiffiffi

2ntp

; 2:9b

and the two flow stagnation points (vortex centres)occur at

xut +1:86326ffiffiffiffiffint

p 2:10a

and

ru

t ffiffiffi2

pjxu

tj

2:63505 ffiffiffiffiffintp

:2:10b

Therefore, the vorticity maxima separate from the

flow stagnation points as time goes on.

0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6

t/t*

S/S*

t*(s)

0.1

100.001

0.01

1

0.1

1.0

0.1

10(c)

(b)

(a)

8 6 4 2 0

1.0U*(mm s1)

U(mms

1)

x(mm)

0.8 1.0

Figure 2. Comparisons of the impulsive stresslet model (redlines) versus the impulsive Stokeslet model (blue lines).(a) Scaled area of influence, S/S*, as a function of scaledtime, t/t*, plotted for the impulsive stresslet model (equation(2.8)) and for the impulsive Stokeslet model (equation (2.13))with symbols defined therein. (b) Signal disappearance time,t*, plotted against signal threshold velocity, U*. (c) Wakeflow velocity distribution, U(x), directly behind the appli-cation point of the forcing (i.e. x 0) at the timeimmediately after the onset of the forcing (i.e. t 0 whenthe signal is the strongest). Note that, in (b), both the x-and the y-axis are in the logarithmic scale and that, in (c),the y-axis is in the logarithmic scale. (b,c) The average one-beat-cycle repositioning jump of the copepod A. tonsa

(table 1); the model parameter, I (the hydrodynamicimpulse), is calculated from equation (2.14) for the impulsiveStokeslet model, and M(the impulsive stresslet strength) fromequation (2.12) for the impulsive stresslet model. It is con-cluded that viscous decay is the primary effect in bothmodels, and that the flow field declines faster both temporallyand spatially for the impulsive stresslet model than for theimpulsive Stokeslet model, owing to partial mutual vorticitycancellation by the two counter-rotating viscous vortices inthe impulsive stresslet model.





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(a) (b)

(c) (d)

10.00

10

0

2

4

6

8

10

5x (mm)

0 5 10 10 5x (mm)

0 5 10

7.0005.0002.0001.0000.7000.5000.2000.1000.0700.0500.0200.0100.0070.0050.0020.001

10.007.0005.0002.0001.0000.7000.5000.2000.1000.0700.0500.0200.0100.0070.0050.0020.001

vorticity

(s1)

r(mm)

0

2

4

6

8

10

r(mm)

0

2

4

6

8

10

r(mm)

0

2

4

6

8

10

r(mm)

Azimuthal vorticity (filled contours) + streamfunction (line contours) Azimuthal vorticity (filled contours) + streamfunction (line contours)

Azimuthal vorticity (filled contours) + streamfunction (line contours)

M= 20.0 mm5s

1, velocity field, time = 0s M= 20.0 mm

5s

1, velocity field, time = 0.02s

M= 20.0 mm5s

1, velocity field, time = 0.30s M= 20.0 mm

5s

1, velocity field, time = 1.00s

Azimuthal vorticity (filled contours) + streamfunction (line contours)

Figure 3. Time evolution of flow velocity and vorticity fields calculated from equations (2.1a,b) and (2.2a,b) for M 20 mm5 s21

(typical for repositioning jumps by the copepod A. tonsa). (a) time 0 s, (b) time 0.02 s, (c) time 0.30 s, and (d) time1.00 s. Shown in the upper panel of each of the four blocks are uniform length velocity vectors (visualizing the flow directiononly) overlapped with flow velocity magnitude contours labelled by values in mm s21. The lower panel shows streamfunctionline contours overlapping filled colour contours of azimuthal vorticity.

Table 1. Analysis summary for 12 A. tonsa jumps. All raw data are from Kirboe et al. [6]. Djump is the distance travelled bythe copepod during a jump; Umax is the maximum speed attained by the copepod; Mmeasured is the impulsive stresslet strengthcalculated from measured jump kinematics using equation (2.12); Mfitted is the impulsive stresslet strength estimated from a fitof equation (2.11) to the decaying phase of the wake vortex; the Reynolds number, Re, is calculated as Gmax/n, where Gmax is

the maximum circulation of the wake vortex and n is the kinematic viscosity of sea water. All jumps consisted of one beat cycleof the swimming legs, except jump no. 69 (two beat cycles) and jump no. 73-2 (three beat cycles). For those one-beat-cyclejumps, the average Djump is 2.12 L (L, prosome length) and the average Umax is 135 L s

21.

jump no. copepod size prosome length (mm) Djump (mm) Umax (mm s21) Mmeasured (mm

5 s21) Mfitted (mm5 s21) Re

12 0.97 2.34 173 27.9 24.2 1417 1.08 2.36 177 39.8 63.6 1620-1 1.04 3.01 192 49.1 30.7 1120-2 1.04 2.44 192 39.9 26.8 1126-1 0.93 1.69 90 9.2 16.2 429-1 1.13 1.79 84 16.5 14.8 434 0.99 1.14 81 6.7 7.6 549 0.7 1.32 78 2.7 5.0 2

58 1.11 2.50 125 32.4 30.7 1083 1.03 2.73 161 36.4 25.4 2769 0.72 2.33 163 10.7 12.6 473-2 1.12 3.97 157 66.4 68.5 23





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supplementary material, appendix movie S2). This

leads to a highly fluctuating velocity: short burstswith speeds of up to 3050 mm s21 during powerstrokes of 7.2 ms duration on average and interruptedby long pauses, resulting in an average propulsionspeed of 8 mm s21 (table 2 and figure 4). The individualjumps resemble repositioning, attack or escape jumps inthis species [29], but they differ in the phase lag betweenindividual swimming legs: near 908 in escape and attackjumps, but only between 30 and 608 for the swimming jumps (i.e. the legs are closer to one another)(figure 5). As a consequence, the average forward move-ment is only a little more than 1 body length for aswimming jump (0.375 mm or 1.25 body lengths;

table 2), whereas it is near 2 body lengths for reposition-ing, escape and attack jumps [29].

3.2. The impulsive stresslet flow field

The imposed flow predicted by the impulsive stressletmodel consists of two counter-rotating viscous vortexrings that expand and decay as exact mirror imagesabout the r-axis (figure 3 and electronic supplementarymaterial, appendix movie S1). This is consistent withour PIV observation of the flow field around a copepodduring a repositioning jump in that the spatial exten-sion and temporal evolution of the two vortex rings,

one in the wake and one around the forward movingbody, are similar [6].

The impulsive stresslet model provides a satisfac-tory least-squares fit to the decaying phase of theobserved circulation of the wake vortex (figure 6),and the stresslet strengths estimated from such fitsof 12 observed flow fields, Mfitted (equation (2.11)),are in good 1 : 1 correspondence with stressletstrengths estimated from the kinematics of copepodjumps, Mmeasured (equation (2.12)) (figure 7 andtable 1). Thus, equation (2.12) allows for usingthe measured jump kinematics to estimate theimpulsive stresslet strength, M, and therefore deter-

mine the entire jumping flow. The model alsoprovides a satisfactory least-squares fit to the decay-ing phase of the observed circulation of the vortexaround the copepod body, similar to the fit to the

circulation of the wake vortex but with a time lagslightly shorter than the beat duration of thepower stroke (figure 6).

For both vortex rings associated with a copepodrepositioning jump, the positions of vorticity maximumseparate increasingly from the flow stagnation points astime goes on [6]. Such separation, which is mainly due

to viscous diffusion, is predicted by equations (2.9a,b)and (2.10a,b) and is a characteristic feature of viscousvortex rings.

4. DISCUSSION

4.1. The flow field

Schlieren observations have long revealed the toroidalflow structures left by jumping copepods (e.g. [30]).We previously developed a theoretical hydrodynamicmodel (impulsive Stokeslet) that described the wakeflow [6]; here, we have presented a model that includes

the flow that develops around the body of the jumpingcopepod (impulsive stresslet). It is relevant to here firstcompare the two models.

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

To compare the two models more explicitly, weapplied both to an average one-beat-cycle repositioning jump of the copepod A. tonsa from our previous

PIV observations (table 1). For small signal thresholdvelocities (i.e. U*, 0.5 mm s21), the signal disappear-ance time, t*, is smaller for the impulsive stressletmodel than for the impulsive Stokeslet model(figure 2b). This difference is due to partial mutual vor-ticity cancellation by the two counter-rotating viscousvortices, which is ignored by the impulsive Stokesletmodel. For the same reason, the wake flow velocity dis-tribution, U(x), directly behind the application point ofthe forcing (i.e. x 0) at the time immediately after theonset of the forcing (i.e. t 0) is predicted to declinefaster by the stresslet than by the Stokeslet model(figure 2c). Thus, on top of viscous decay, which is

the primary effect, partial mutual cancellation of oppo-sitely signed vorticities in the impulsive stresslet modelcauses the imposed flow to attenuate faster, bothspatially and temporally. Finally, the hydrodynamicimpulse, I, of the wake vortex, estimated from theattenuation of vortex circulation using PIV obser-vations and the impulsive Stokeslet model, somewhatunderestimates the maximum density-specific momen-tum of the jumping copepod (fig. 2d in [6]), while theimpulsive stresslet strength, M, shows a better 1 : 1 cor-respondence between that estimated from circulationdecay and measured directly from jump kinematics(figure 6 and table 1). Again, this difference can

be ascribed to the ignorance of partial vorticitycancellation by the impulsive Stokeslet model.

Thus, the impulsive stresslet model represents animprovement over our previous impulsive Stokeslet

0

1020

30

40

50

60

70

50 100time after initiation of jump (ms)

ve

locity(mms

1)

150

Figure 4. Velocity fluctuations of a swimming O. davisaemale,here observed during about 200 ms (time within field of view).The observation period was only a small portion of the jumpsequence that may have lasted for seconds. Red diamonds, rawdata; blue line, smoothing filtered.





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model, both by considering the interaction betweenthe two vortex rings and, mainly, by describing theentire imposed flow field, not only the wake flow.The impulsive stresslet model indeed does provide agood approximation of the entire flow associated witha copepod repositioning jump as evidenced by thegood correspondence both to PIV observations(figures 6 and 7) as well as to computational fluiddynamics (CFD) simulations [31]. The strength of theimpulsive stresslet is the only parameter involved, andcan be estimated as the product of the body volume,

maximum jump speed and total jump distance of thecopepod. Therefore, accurate measurement of jumpkinematics is sufficient to estimate the entire flowfield. 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 impulsiveStokeslet model cannot be directly applied to the entirerepositioning-by-jumping flow, in our previous work wehad to fit the PIV-measured flow velocity data to thegeneral scaling that relates the initial area of influence,S, to (I/U*)2/3 in order to determine the coefficient infront of the scaling [6]. Based on this, we predictedthat the imposed hydrodynamic signal is much less foran ambush-feeding than for a cruising or hovering cope-

pod for small individuals in a time-averaged sense. Inthe following, we use the more accurate impulsive stress-let model to show that this prediction holds true at anyinstant in time.

0.2 mm

(a) (b)

Figure 5. Frozen high-speed video images of O. davisae (a) swimming by jumping or (b) performing a repositioning jump. Notethat the phase lag between the swimming legs is approximately 308 for the swimming copepod, but approximately 908 for the

jumping individual. The individual in (b) is a female carrying eggs.

t(ms)0

2

4

6

8

10

12

14

30 60 90 120 150 180

0

40

80

120

160

200

240

U(mms

1)

G(mm

2s

1)

Figure 6. Time evolutions of PIV-measured circulations of the wake vortex and the vortex around the copepod body for A. tonsa jump no. 58 from Kirboe et al. [6]. Also plotted are the fits of equation (2.11) to the decaying phase of the circulation data ofboth vortices. Besides those presented in the main text, this figure suggests that, for repositioning jumps (including component

jumps in jump-swimming), the short time interval that the wake vortex leads the body vortex is simply controlled by the durationof the power stroke. The shorter the duration of the power stroke, the better the synchronization between the wake vortex and thebody vortex, and therefore the more accurate the flow described by the impulsive stresslet model. Also, the shorter the duration ofthe power stroke, the faster the spatial and temporal attenuation of the flow field. The average power stroke duration of jump-swimming by the male O. davisae is 7.2 ms, in contrast to 10 ms for the repositioning jumps by this species. Blue squares, wakevortex circulation, measured; blue line, wake vortex circulation, fitted; red squares, body vortex circulation, measured; red line,body vortex circulation, fitted; black line, copepod velocity.





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4.2. Swimming by jumping in Oithona ishydrodynamically quiet swimming

The flow fields generated by motility or feeding currentsare likely to shape, to a certain degree, the feeding andmotile behaviour of planktonic organisms. We have pre-viously discussed how the induced flow field may serveas a signal to predators, and how different feedingbehaviours consequently may expose zooplankters dif-

ferently to predation risk depending on their size andon the predation landscape [6]. Continuous feeding cur-rents and steady cruising generate momentum jets,while unsteady jumps generate vortex rings. Thelatter attenuate spatially much faster than the twoformer flow fields and are, in addition, by default inter-mittent. Thus, the impulsive stresslet model predictsthat the flow velocity induced by a jumping plankterdecreases with distance, d, as d24 (equation (2.6)).(Note that the impulsive Stokeslet predicts d23.) Fora cruising neutrally buoyant plankter (modelled as asteady stresslet; appendix A) or a negatively buoyanthovering one (modelled as a steady Stokeslet), induced

flow speeds decline with, respectively, d22

and d21

(e.g.[22]). Despite the higher induced peak flow velocitymagnitudes, jumps may thus produce significantlyweaker hydrodynamic signals to predators over a largerange of body sizes. This is illustrated by the hydro-dynamic signals generated by the two differentmechanisms of propulsion in copepods, erratic jump-swimming and continuous cruising, as predicted bythe impulsive stresslet model and a steady stressletmodel, respectively: the flow velocity magnitude aswell as the deformation rate of the wake flow attenuatemuch faster spatially for jump-swimming O. davisaemales than for a like-sized copepod cruising steadily at

the same average velocity, and both flow velocity anddeformation rate are smaller for the jump-swimmer just 12 body lengths (0.5 mm) behind the copepod(figure 8a,b).

0 1

20

40

60

80

100

120

2 3

t/T

50.001

0.01

0.1

1

10

4 3 2

x (mm)

D(

s1)

0.001

0.01

0.1

1

10(a)

(b)

(c)

U(mms

1)

1 0

S/Scopepod

4 5 6 7

Figure 8. Comparisons of swimming by jumping in the malecopepod O. davisae versus a same-size neutrally buoyant cope-pod cruising steadily at the same averaged translating velocity.(a) Distribution of wake flow velocity, U(x), and (b) distributionof wake flow deformation rate, D(x), directly behind the appli-cation point of the forcing (i.e. x 0). Note that, in both (a)and (b), the y-axis is in the logarithmic scale. For swimmingby jumping, as modelled by an impulsive stresslet, the flowquantities are those at the time immediately after the onset ofthe impulsive forcing (i.e. t 0 when the flow is the strongest).The impulsive stresslet strength, M, is calculated from equation(2.12) using the average jump-swimming kinematics (i.e. Umax40 mm s21 and Djump 0.375 mm; table 2) and copepod bodyvolume Vcopepod 4/3 p (L/2)

3 h2 (where the prosome length

L 0.3 mm and the aspect ratio h 0.5). The flow fieldimposed by the copepod cruising steadily at the same averagetranslating velocity (i.e. Uaverage 8 mm s

21) is calculatedusing the steady stresslet model (appendix A) of stressletstrength Q 6pmae Uaverage (2ae), where ae L/2 h2/3.(c) Normalized area of influence, S/Scopepod, within which atranslating copepod imposes flow velocities exceeding a chosenthreshold velocity, U* (0.1 mm s21), as a function of normal-ized time, t/T, where Scopepod 0.5 p(L/2)

2 hand the average jump interval T 1/21.5 s is based on the average jump fre-quency of 21.5 Hz. The dashed blue line is for the time-averagedarea of influence owing to swimming by jumping. Because ofthe linearity of the impulsive stresslet model, the jump-swimmingflow field is calculated as the linear combination of a series of

impulsive stresslets separated spatially by Djump and temporallyby T. (a,b) Blue jump-swimming, impulsive stresslet; red steadyswimming, stresslet. (c) Red line, steady swimming at the sameaverage velocity; blue line, a typical jump sequence.

0

10

20

30

40

50

60

70

10 20 30

Mmeasured (mm5 s1)

Mfitted

(mm

5s

1)

40 50 60 70

Figure 7. Scatter plot ofMfitted versus Mmeasured. Fitted valuesof the impulsive stresslet strength, M, were obtained by fittingequation (2.11) to the observed temporal decay of wake vortexcirculation of the 12 repositioning jumps in the copepodA. tonsa (table 1), including two multiple-beat-cycle jumps

(marked by triangles). Measured values of M were calculatedfrom observed jump kinematics according to equation (2.12).All raw data are from Kirboe et al. [6].





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The likely size of the smallest energy-containingeddies of small-scale oceanic turbulence is Le 2p(n3/E)1/4, and the associated turbulent shear isDt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipE=10n

p(e.g. [32]). If the kinematic viscosity

n 1026 m2 s2l and the turbulent kinetic energydissipation rate E 1027 W kg21, then Le 10 mm

and Dt

0.18 s21

. Thus, the hydrodynamic signalowing to jump-swimming by the male copepod O. davi-sae is probably overwhelmed by the smallest turbulenteddies.

Rheotactic predators are more likely to respond tothe flow velocity than to the deformation rate of theflow generated by their prey [28], and the area of influ-ence (S, within which the imposed flow velocities exceeda threshold velocity, U*) of the moving copepod istherefore a measure of its hydrodynamic visibility to

such predators (encounter cross section). S is substan-tially smaller for the jump-swimming copepod thanfor a same-size copepod cruising steadily through thewater at the same average translating velocity

(c)

(b)

(a)

10.007.0005.0002.0001.0000.7000.5000.2000.1000.0700.0500.0200.0100.0070.0050.0020.001

10.007.0005.0002.0001.0000.7000.5000.2000.1000.0700.0500.0200.0100.0070.005

0.0020.001

vorticity

(s1)

time=4T0.0023s

time=4T

time=4T+0.0023s

20

0.4

0.8

1.2

r(mm)

0

0.4

0.8

1.2

r(mm)

0

0.4

0.8

1.2

r(mm)

0

0.4

0.8

1.2

r(mm)

0

0.4

0.8

1.2

r(mm)

0

0.4

0.8

1.2

r(mm)

azimuthal vorticity (filled contours) + streamfunction (line contours)



velocity field

velocity field

velocity field

1 0x (mm)

1 2

2 1 0 1 2

2 1 0 1 2

Figure 9. Modelled instantaneous flow velocity and vorticity field imposed by a swimming-by-jumping male O. davisae at: (a) ashort time (approx. 2.3 ms) just before the initiation of the next jump, (b) the initiation of the jump (the time instant indicatedby the red arrow in figure 8c), and (c) a short time (approx. 2.3 ms) just after the initiation of the jump (see figure 8 legend for

computation details). Shown in the upper panel of each of the three blocks are uniform length velocity vectors (visualizing theflow direction only) overlapped with flow velocity magnitude contours labelled by values in mm s21. The lower panel showsstreamfunction line contours overlapping filled colour contours of azimuthal vorticity. In this way, the geometry of the individualvortex rings is least deformed; the copepod does not achieve minimized signal strength but optimized flow geometry (that of thecanonical viscous vortex ring) for individual jumping vortices.





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below which plankters cannot create vortices and hencemove quietly by jump-swimming. From equation (4.1)and assuming a power stroke of 110 ms duration astypical for plankters, this threshold is of order0.050.2 mm. Thus, most copepods can create vortices,copepod nauplii are questionable and protozoa cannot.This is consistent with observations of toroidal flowstructures generated by copepods [6 8], and with thelack of such structures for the jumping ciliateM. rubrum [37].

4.5. Viscous versus inviscid vortex rings in thecontext of animal propulsion

Copepod jumps generate viscous vortex rings whoseproperties are strikingly different from the inviscidvortex rings typically considered in the context ofanimal propulsion (e.g. [38 40]). In a viscous vortexring, owing to viscous diffusion, the vorticity maximumpoint separates increasingly from the flow stagnationpoint as time goes on [34]. In an inviscid vortex ring(e.g. Hills spherical vortex; [41]), both points movethrough the fluid with the same constant velocity andthe form of the vortex ring does not change withtime. A (nearly inviscid) vortex ring generated fromthe traditional pistoncylinder arrangement is due torolling-up of a vortex sheet that separates at the

nozzle edge (e.g. [42]); when long-lasting momentumas well as fluid volume flux are issued out from thepiston cylinder arrangement, there exists a vortexring 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]), andtherefore there is no such vortex ring formation process.

Both the impulsive stresslet model and the impulsiveStokeslet model provide tractable theoretical frame-works for describing not only the copepod jumpingflows but also the impulsively created flows by manyother ecologically important marine organisms, includ-ing most zooplankton, small fish larvae and even krilloperating in the low-Reynolds-number regime (Retypi-cally ranging from 0.1 to several hundreds). Bothmodels may also find their applications in small insecthovering flights, which are of a similar low-Reynolds-number range. The biological and ecological impli-cations of viscous vortices have hitherto been verylittle studied.

This work was supported by National Science Foundationgrants NSF OCE-0352284 and IOS-0718506 and an awardfrom WHOIs Ocean Life Institute to H.J. and by grantsfrom the Danish Research Council for independent researchand the Niels Bohr Foundation to T.K. We are grateful totwo anonymous reviewers whose constructive commentshave greatly improved this manuscript.

APPENDIX A. STRESSLET MODEL FORTHE SWIMMING CURRENT OF ANEUTRALLY BUOYANT COPEPOD

Stresslet (e.g. [46]) flow equations are written in acylindrical polar coordinate system (x, r, f) where

10.00

100

2

4

6

8

10

r(mm)

0

2

4

6

8

10equal length velocity vectors + velocity magnitude contours (in mm s1)


r(mm)

5 0

x (mm)

5 10

7.0005.0002.000

1.0000.7000.5000.2000.1000.0700.0500.0200.0100.0070.0050.0020.001

10.007.0005.0002.0001.0000.7000.5000.2000.100

0.0700.0500.0200.0100.0070.0050.0020.001

vorticity

(s1)

Figure 11. Stresslet flow velocity and vorticity fields calculated from equations (A 1a,b) and (A 5a,b) for Q Fh2/3 L 1.167 10211 N m. Here Q is calculated by assuming the neutrally buoyant copepod (aspect ratio h 0.38 and prosome lengthL 1 mm) to swim at the same terminal sinking velocity as a same-size negatively buoyant copepod of excess densityDr 30 kg m23 (i.e. F 2.225 1028 N, a typical value for excess weight of a copepod in the considered size range).





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