The performance of swimming may change dramaticallyover the growth of an aquatic animal. Despite ourunderstanding of the broad-scale hydrodynamic differences inthe swimming of animals spanning many orders of magnitudein body length (Daniel et al., 1992; Lighthill, 1975; Wu, 1977),we cannot predict how ontogenetic changes in the size, shapeand motion of the body influence the speed and energetic costof swimming within individual species. The purpose of thepresent study was to examine the scaling of hydrodynamicforces in the jellyfish Aurelia auritain order to understand howsuch ontogenetic changes affect swimming performance.

The ontogenetic scaling of swimming performance

Although it is generally appreciated that a fully grownaquatic animal will swim faster than when it was smaller, theprecise relationship between speed and body size over a lifehistory is dictated by scale-dependent hydrodynamics. Muchof our understanding for this scale dependency comes fromcomparisons between species that differ in body mass by manyorders of magnitude (e.g. Daniel et al., 1992; Lighthill, 1975;

Wu, 1977). These comparisons illustrate that thrust isgenerated primarily by viscous force at the size ofspermatozoa, inertial force at the size of adult fish and acombination of these forces at intermediate sizes. Such broadcomparisons are useful for understanding the major fluid forcesthat play a role in the hydrodynamics of a growing animal butcannot provide predictive explanations for how swimmingperformance (e.g. speed and cost of transport) should changeover the ontogeny of individual species.

Ontogenetic changes in swimming kinematics have beenmost thoroughly explored in larval fish (e.g. Batty, 1984;Fuiman, 1993; Fuiman and Webb, 1988; Hale, 1999; Hunter,1972; Hunter and Kimbrell, 1980; Osse and van den Boogaart,2000), which propel themselves by lateral tail undulation.During routine swimming, larval fish generally beat their tailswith greater length-specific amplitude but propel themselves atlower speed than adults of the same species. Although it isappreciated that force generated by the inertia of waterincreases in importance relative to viscous force as fish growlarger (Fuiman and Batty, 1997; McHenry et al., 2003; Muller

4125The Journal of Experimental Biology 206, 4125-4137© 2003 The Company of Biologists Ltddoi:10.1242/jeb.00649

It is not well understood how ontogenetic changes in themotion and morphology of aquatic animals influence theperformance of swimming. The goals of the present studywere to understand how changes in size, shape andbehavior affect the hydrodynamics of jet propulsion in thejellyfish Aurelia aurita and to explore how such changesaffect the ontogenetic scaling of swimming speed and costof transport. We measured the kinematics of jellyfishswimming from video recordings and simulated thehydrodynamics of swimming with two computationalmodels that calculated thrust generation by paddle and jetmechanisms. Our results suggest that thrust is generatedprimarily by jetting and that there is negligible thrustgeneration by paddling. We examined how fluid forcesscaled with body mass using the jet model. Despite anontogenetic increase in the range of motion by the belldiameter and a decrease in the height-to-diameter ratio,we found that thrust and acceleration reaction scaled with

body mass as predicted by kinematic similarity. However,jellyfish decreased their pulse frequency with growth, andspeed consequently scaled at a lower exponential rate thanpredicted by kinematic similarity. Model simulationssuggest that the allometric growth in Aurelia resultsin swimming that is slower, but more energeticallyeconomical, than isometric growth with a prolate bellshape. The decrease in pulse frequency over ontogenyallows large Aurelia medusae to avoid a high cost oftransport but generates slower swimming than if theymaintained a high pulse frequency. Our findings suggestthat ontogenetic change in the height-to-diameter ratioand pulse frequency of Aurelia results in swimming thatis relatively moderate in speed but is energeticallyeconomical.

Key words: swimming, morphology, scaling, jetting,scyphomedusae, locomotion, jellyfish, Aurelia aurita.

Summary

Introduction

The ontogenetic scaling of hydrodynamics and swimming performance injellyfish (Aurelia aurita)

Matthew J. McHenry1,* and Jason Jed2

1The Museum of Comparative Zoology, Harvard University, 26 Oxford St, Cambridge, MA 02138, USAand2The Department of Integrative Biology, University of California, Berkeley, CA 94720, USA

*Author for correspondence (e-mail: mchenry@fas.harvard.edu)

Accepted 6 August 2003

4126

et al., 2000; Webb and Weihs, 1986), few investigators havetested whether a hydrodynamic model is capable of predictingthe scaling of swimming performance in fish (althoughVlymen, 1974 is an exception). Such an approach would allowan investigator to explore the relative contribution of inertialand viscous forces to thrust and drag and to evaluate whetheralternative larval morphology or tail kinematics could improveon swimming performance.

Although the scaling of swimming performance is not aswell characterized for aquatic invertebrates as for larval fish,some investigators have used a combination of modeling andexperimentation in order to understand the hydrodynamicmechanisms that explain the scaling of performance ininvertebrate species. Using such an approach, Williams (1994)demonstrated that the serial addition of developing limbs alongthe abdomen of the brine shrimp (Artemia sp.) initially doesnot contribute to propulsion when larvae are small, but theadditional limbs generate thrust in later life history stages whenunsteady forces play a greater role in the hydrodynamics ofswimming. Using a combination of kinematics and forcemeasurements, Nauen and Shadwick (2001) found that the tailof the spiny lobster (Panulirus interruptus) generates most ofits force with a paddle mechanism and that maximum forceproduction scales according to a paddle model. Dadswell andWeihs (1990) determined that giant scallops (Placopectenmagellanicus) swim with the greatest speed at a medium bodysize range, when they attain the highest thrust-to-weight ratioof their life history.

Jellyfish are a potentially useful group for exploring theontogenetic scaling of hydrodynamics because their swimmingis easily modeled. Daniel (1983) proposed a mathematicalmodel that suggested that the hydrodynamics of prolate (bullet-shaped) hydromedusae are dominated by the thrust generatedby jetting, the acceleration reaction (i.e. the force generated byaccelerating the water around the body) and the drag resistingthe forward motion of the body. This model replicatedobserved oscillations in swimming speed (Daniel, 1983), andColin and Costello (2002) found the model to accuratelypredict body acceleration in prolate, but not oblate (plate-shaped), jellyfish. They proposed that oblate jellyfish generatethrust primarily by paddling the flexible margins of their bellinstead of using a jet mechanism. We tested this hypothesisby comparing measurements of speed in Aurelia, an oblatejellyfish, with predictions from mathematical models ofswimming that assume thrust generation by either paddling orjetting.

Geometric and kinematic similarity

Changes in the shape or relative motion of an animal’s bodyduring growth should be reflected in the allometric scaling ofmorphometric and kinematic parameters (Huxley, 1932;McMahon, 1984). An allometric relationship is defined asa deviation from isometry (Schmidt-Nielsen, 1984) andtherefore requires the formulation of an a priori null hypothesisas predicted by isometry. In complex biomechanical systems,these predictions are not always obvious and therefore merit

careful consideration (e.g. Hernandez, 2000; Nauen andShadwick, 2001; Quillin, 2000; Rome, 1992).

If a jellyfish grows isometrically, then its bell height,h, scaleslinearly with bell diameter, d, and medusae at all sizes will begeometrically similar (i.e.h∝ d). Geometric similarity impliesthat the volume of the body scales as d3, which means that belldiameter scales with body mass (m) as m1/3 (assuming constanttissue density). During swimming, the shape of the bell changeswith time. Bell height rapidly increases and bell diameterrapidly decreases over a pulse phase (Gladfelter, 1973). Toachieve kinematic similarity (Quillin, 1999), a jellyfish mustmaintain the speed of height change in proportion to the speedof diameter change at all body sizes. In geometrically similarjellyfish, kinematic similarity is maintained if the pulsefrequency, duty factor (the proportion of the propulsive cyclespent pulsing) and range of motion remain constant. Kinematicsimilarity also requires that a jellyfish moves through the waterat a speed that is directly proportional tod and therefore scalesas m1/3. This form of kinematic similarity has been observed infish made to swim steadily at the same frequency (Bainbridge,1958).

Geometric and kinematic similarity suggest predictions forthe scaling of hydrodynamic forces in jetting jellyfish. Dragscales with the area of the bell (∝ d2) and the square ofswimming speed (∝ d2; Daniel, 1983), which suggests that thisforce scales as d4 and m4/3. Thrust scales with the inverse ofthe area of the subumbrellar opening through which water jets(∝ d–2) and the square of the rate of change in bell volume(∝ d6), suggesting that thrust also scales as d4 and m4/3. Theacceleration reaction varies with the volume of the body (∝ d3)and its rate of change in velocity (∝ d, assuming sinusoidalchanges in velocity), which implies that this force also scalesas d4 and m4/3. Since all three hydrodynamic forces arepredicted to scale with mass in the same way, we predict thatthe hydrodynamics of swimming should not change if jellyfishmaintain geometric and kinematic similarity. Conversely,scaling that deviates from this null hypothesis implies that thehydrodynamics of jetting changes with size. These scalingpredictions assume that the jellyfish operate at relatively high,inertia-dominated, Reynolds numbers.

The present study

We pursued three objectives in order to address howontogenetic changes affect the swimming performance ofAurelia. (1) The hydrodynamics of swimming were modeledwith paddle and jet mechanisms in separate simulations inorder to test which mechanism more accurately predictsswimming speed. (2) The scaling of relationships ofparameters that play a role in the dynamics of swimming weremeasured. (3) The performance of swimming in Aurelia wascompared with that predicted for model jellyfish exhibitingdifferent patterns of growth.

Aurelia is a marine scyphozoan with an oblate medusa stagethat spans over an order of magnitude in bell diameter(Meinkoth, 1995). This large change in body size makesAurelia an ideal system for examining the scaling of

M. J. McHenry and J. Jed

4127Scaling of hydrodynamics in jellyfish

swimming performance. The swimming of medusae such asAurelia is thought to affect their position in the water columnand thereby influence dispersal and movement into areas ofhigh prey density and favorable environmental conditionswithin the plankton (Buecher and Gibbons, 1999; Johnson etal., 2001; Mutlu, 2001; Nicholas and Frid, 1999). The bellpulsing used by Aurelia to swim also facilitates mass transportand prey capture by increasing with flow over the bell andtentacles (Costello and Colin, 1994; Daniel, 1983; Mills,1981).

Materials and methodsKinematics and morphometrics

We measured the shape and swimming movement ofmedusae of Aurelia aurita (L.) from video recordings madeat the Monterey Bay Aquarium, Pacific Grove, CA, USA.Medusae ranging in bell diameter from 1.57·cm to 9.51·cmwere held in aquaria containing natural seawater at 16°C.These tanks were sufficiently large (15·cm deep × 17·cm ×15·cm wide for bell diameters less than 4·cm, and 61·cm deep× 65·cm × 28·cm wide for bell diameters greater than 4·cm)to avoid wall effects when individual animals were videorecorded (30·frames·s–1; Panasonic PV-S62D SVHS-CCamcorder) in the tank’s center (Vogel, 1981). The meantissue density,ρtissue, of each jellyfish was calculated as theratio of measured body mass and the body volume, which wasfound by water displacement in a graduated cylinder.

The motion of jellyfish was measured by following themovement of landmarks on the bell using NIH Image on anApple Macintosh G3 computer (Fig.·1A). We tracked themovement of the exumbrellar and subumbrellar surfaces alongthe central axis and defined the bell margin as the ring offlexible tissue running from the distal margin of the bell (wherethe tentacles begin) to a proximal line of high bending. Fromthese data, we calculated the bell height,h, as the distancebetween the exumbrellar surface and the opening of thesubumbrellar cavity along the central axis; the cavity height,hcav, as the distance between the subumbrellar surface and theopening of the subumbrellar cavity; the diameter of the bell, d,as the distance between lateral surfaces of the bell; and themargin angle, θ, as the angle between the central axis and themargin of the bell (Fig.·1A). We described the passive shapeof the bell by measuring the diameter,drest, and height, hrest,of each animal while at rest from video images. The height-to-diameter ratio for each individual was calculated as thequotient of these quantities. The pulse frequency, f, and theduty factor, q, which is the ratio of the period of the pulse phaseand the period of the whole propulsive cycle, were measuredover a duration of three to five propulsive cycles. The range ofvalues in margin angle, θrange, cavity height, hrange, and belldiameter, drange, were also recorded.

We described the scaling of individual kinematic andmorphological parameters,y, with body mass,m, using thescaling constant, a, and scaling exponent, b, of an exponentialfunction (y=amb). These values were found from the interceptand slope of a reduced major axis regression fit to log-transformed data. We rejected the null hypothesis, bo, in caseswhere predictions fell outside of the lower, L1, and upper, L2,95% confidence intervals about the slope of the regression(Rayner, 1985; Sokal and Rohlf, 1995). This form of Type IIregression was appropriate because the scale and dimensionsof our independent and dependent variables were not equal andthe error for each variable was unknown. The coefficient ofdetermination, r2, was used to assess the degree of variationexplained by reduced major axis regressions.

Hydrodynamic forces

As in Daniel (1983), we modeled the hydrodynamics ofjellyfish swimming as the sum of thrust, T, drag, D, theacceleration reaction force, A, and the force acting to changethe inertia of the body, F. This model is expressed in anequation of motion as:

T + D + A + F = 0·. (1)

Drag was calculated with the following equation (Batchelor,1967):

D = 0.5cbellρwatersbelluU·, (2)

where sbell is the instantaneous projected area of the bell(sbell=0.25πd2, where d varies with time), u and U are theinstantaneous speed and velocity of the body, respectively,ρwateris the density of seawater, and cbell is the drag coefficientof the bell. We assumed that the drag on the bell was equal to

Fig.·1. Morphometrics and hydrodynamic modeling. (A) The filleddots show the landmarks used to reconstruct the shape of the bodyand provide measurements for the parameters noted. (B,C) The twohydrodynamic models used in the present study are illustrated withthe drag, D, and thrust, T, vectors predicted to be generated as thebell contracts and the bell margin adducts. (B) The jet modelassumes thrust to be generated by a jet, T jet. (C) The paddle modelassumes thrust is generated by paddling, Tpaddle.

Bell height

Bell diameter

Cavity height

Bell margin

Margin angle

T jet

Exumbrellar surfaceSubumbrellar surface

D

A

B C

Tpaddle

D

Paddle model (T=Tpaddle)Jet model (T=T jet)

4128

that of a hemisphere, which has a greater drag coefficient whenthe body is moving backward (i.e. U<0) than when movingforward (i.e. U>0) (Hoerner, 1965):

The acceleration reaction was calculated with the followingequation (Daniel, 1983):

A = αρwaterv(∆U/∆t)·, (4)

where α is the added mass coefficient, v is the volume of thesubumbrellar cavity, and t is time. Assuming a hemi-ellipsoidshape to the subumbrellar cavity (as in Daniel, 1985), theinstantaneous cavity volume was calculated as follows:

v=πd2hcav/6·, (5)

where the cavity height and diameter of the bell vary with time.The added mass coefficient varies with the shape of the bell(Daniel, 1985):

α=(2h/d)1.4·. (6)

In separate simulations, we modeled the thrust generation byjet and paddle mechanisms (Fig.·1B,C). The thrust generatedby jetting, T jet, was calculated with the following equation(Daniel, 1983):

T jet = (ρwater/sbell)(∆v/∆t)2·. (7)

Negative thrust (i.e. thrust acting to impede forward motion)was generated by this force whenever the cavity volumeincreased. The thrust generated by paddling, Tpaddle, wascalculated using the following equation:

Tpaddle= sin(θ)0.5cmarginρwatersmarginwW·, (8)

wherecmargin is the drag coefficient for the bell margin, w andW are the speed and velocity, respectively, of the bell marginat the midpoint between its proximal base and distal tip, andsmargin is the area of the inside surface of the margin. Since themargin of the bell is a flattened strip of tissue, we modeled itas a flat plate oriented perpendicular to flow (cmargin=1.98;Hoerner, 1965). Note that the thrust generated by paddling mayalso contribute to drag at instances where the velocity of themargin is directed in the aboral direction. We calculated thearea of the margin with the following equation for the area ofa ring:

smargin= πl(drest– l)·, (9)

wherel is the length of the bell margin. These hydrodynamicforces acted against the force to change the inertia of the body,which was calculated with the following equation:

F = m(∆U/∆t)·. (10)

Computational modeling

Numerical solutions for swimming speed were calculatedusing the equation of motion (equation·1) with the measuredkinematics of the bell and its margin as input variables. Speedwas calculated with a variable order Adams–Bashforth–

Moulton solver for integration (Shampine and Gordon, 1975)programmed in MATLAB (version 6.0; Mathworks). Ourkinematic equations used a sawtooth function, k(t), whichincreased linearly over time, t, from values of 0 to 1 overthe duration of the pulse phase, then decreased linearlyfrom 1 to 0 over the recovery phase. This function providedan input to the following equations, which we used todescribe the motion of the margin angle, bell height and belldiameter:

θ(t) = θrangecos[2πk(t)f]·, (11)

h(t) = hrangesin[2πk(t)f] + hrest·, (12)

d(t) = drange{cos[2πk(t)f] – 1} + drest·. (13)

The mean (time-averaged) speed was calculated fromsimulations using both the jet model (i.e. T=T jet) and thepaddle model (i.e. T=Tpaddle) for each jellyfish that we hadmeasure kinematic and morphometric data. Measurements ofmean speed were taken over the duration of five propulsivecycles that followed the three cycles that were necessary forthe models to reach steady state.

The accuracy of models was tested by comparing the meanswimming speed measured for each jellyfish with the meanspeed predicted by simulations run with the same bell andmargin kinematics. Experimental sequences were rejected ifthe mean swimming speed of individual pulses differed bymore than 10%. We determined the relationship betweenmeasured speed and the speed predicted by simulations witha major axis regression (Rayner, 1985). A model wasconsidered perfectly accurate if the 95% confidence intervalsof the slope of this regression, e, included a value of 1.Furthermore, we tested whether models accurately predictedthe scaling of speed with body mass by finding the reducedmajor axis regression for predictions of speed for both models.Models were considered accurate if the 95% confidenceintervals of predictions included the measured values for thescaling constant and scaling exponent (Sokal and Rohlf,1995).

In addition to measuring speed, the hydrodynamic cost oftransport, THCOT, was calculated to assess the performance ofeach simulation. The hydrodynamic cost of transport is ameasure of the total energy invested in jetting to propel a unitmass of the body over a unit distance. It was calculated withthe following equation:

where i is the index for each of the n measurements ofspeed and the magnitude of thrust, and x is the net distancetraversed over the duration of a swimming sequence. Thismeasure of energetic economy neglects internal costs andtherefore provides a minimum estimate of metaboliceconomy.

(14)THCOT = ,

ui|T i|∆t

mx

^n

i=1

(3)cbell =0.42 for U>0

1.17 for U<0.

M. J. McHenry and J. Jed

4129Scaling of hydrodynamics in jellyfish

In order to examine the effects of bell shape and body masson swimming performance, we ran numerous simulations ofour jet model over a range of height-to-diameter ratio and bodymass values. Using our data for the scaling of kinematics, weanimated the bell with the kinematics appropriate to the bodymass used in each simulation. To investigate the effects ofontogenetic changes in behavior on performance, we ransimulations at varying pulse frequency and body mass values.For these simulations, we used the measured values for bellheight, bell diameter and pulse frequency specific to bodymass. From the results of these simulations, we generatedparameter maps describing the effects of body mass, pulsefrequency and the height-to-diameter ratio on swimming speedand the hydrodynamic cost of transport. From these parametermaps, we calculated the mean speed and hydrodynamic cost oftransport predicted for jellyfish that are geometrically andkinematically similar over the range of measured body massvalues.

ResultsJet versus paddle propulsion

During the pulse phase of the propulsive cycle, Aureliamedusae rapidly decreased their bell diameter, increased bellheight and adducted their bell margin as the body increased inspeed (Fig.·2). The body slowed as the bell diameter graduallyincreased, height decreased and the margin abducted during therecovery phase. Using the kinematics of the bell and margin,thrust predicted by the jet model was generally over an orderof magnitude greater than thrust from paddling (Fig.·3). Thelow thrust generated by paddling resulted in predictedswimming speeds that fell short of measurements (Fig.·4). Thiswas reflected in a major axis regression between measured andpredicted speed having a slope with 95% confidence intervalswell outside a value of 1 (e=0.002, L1=–0.015, L2=0.019,N=19; Fig.·4A). Although the jet model tended to overestimatemeasured speed (e=1.86, L1=1.28, L2=3.02, N=19; Fig.·4B), itprovided a more accurate estimate of speed than the paddle

RecoveryPulse

–π/4

0

π/4

0 0.5 1.0 1.5 2.0 2.5 3.0

1.27 s 1.40 s 1.53 s 1.67 s 1.80 s 1.93 s 2.07 s

A

B

C

D

E

Time (s)

Sw

imm

ing

spee

d (c

m s

–1)

Mar

gin

angl

e (

rad)

Bel

l di

amet

er

and

heig

ht (

cm)

0

1.0

2.0

3.0

0

1.0

2.0

1.27 s1.27 s1.27 s 1.40 s 1.53 s 1.67 s 1.80 s 1.93 s 2.07 s

Fig.·2. Representativekinematics of swimming inAurelia aurita. (A) Videoframes for one propulsivecycle. (B) The positions oflandmarks (filled dots) wereused to reconstruct the outersurface of the bell (thickline) and the inner surface ofthe bell cavity (thin line).(C) Measurements for thebell height (blue line) anddiameter (red line) areshown during the pulse(gray band) and recovery(white band) phases of thepropulsive cycle. (D) Themargin angle and (E) theswimming speed are shownover time.

4130

model. The jet model predicted oscillations in speed thatclosely approximated the variation of speed measured in bothlarge (Fig.·4C) and small (Fig.·4D) jellyfish. Furthermore, thejet model was found to better predict the scaling of swimmingspeed (see below).

The scaling of morphology, kinematics, hydrodynamics andperformance

Our morphometric data suggest that Aurelia medusae donot maintain geometric similarity over ontogeny. Jellyfish ofhigh body mass had a bell height that was disproportionatelysmall (b=0.27; Fig.·5A) and a bell diameter that wasdisproportionately large (b=0.40) compared with jellyfish oflow body mass (Fig.·5B; Table·1). This resulted in a significantdecrease in the height-to-diameter ratio with increasing bodymass (b=–0.16; Fig.·5C). Furthermore, the length of the bellmargin scaled by a factor significantly greater than isometry(b=0.43; Table·1).

Aurelia medusae of different body sizes moved withdifferent swimming kinematics. Large jellyfish pulsed at alower frequency than small jellyfish (b=–0.35), but theymaintained similar values for duty factor and the range of

margin angle at all sizes (Fig.·6; Table·1). The range of belldiameter scaled by a greater factor (b=0.42) than predicted bykinematic similarity, but the range of bell height did not(bo=0.33; Table·1). Despite the changes in pulse frequency andthe range of bell diameter with body mass, accelerationreaction and thrust scaled with body mass as predictedby kinematic similarity (bo=1.33; Table·1; Fig.·7A,B). Bycontrast, drag scaled by a factor (b=0.87) significantly lowerthan predicted by kinematic similarity (Table·1; Fig.·7C).

Despite pulsing at a lower frequency, larger jellyfish movedwith a faster swimming speed than did smaller jellyfish(Fig.·8A). However, the increase in speed with body mass wassignificantly lower (b=0.17) than predicted by kinematicsimilarity (Table·1). The jet model consistently overestimatedthe speed of swimming, which was reflected in its scalingconstant (a=–1.4) being significantly greater than themeasured value. However, the scaling factor predicted forspeed by the jet model was statistically indistinguishablefrom the measured value (b=0.17; Fig.·8A; Table·2). Thepaddle model greatly underestimated speed (a=–3.0), butoverestimated the scaling factor of speed (bpaddle=0.33;Table·2). The hydrodynamic cost of transport decreased with

M. J. McHenry and J. Jed

–π/4

0

π/4

0 2.0 4.0 6.0 8.0 10.0

A

B

C

D

E

Time (s)

Pre

dict

ed

spee

d (c

m s

–1)

Mar

gin

angl

e (ra

d)

Bel

l di

amet

er

and

heig

ht (

cm)

0

10.0

Hyd

rody

nam

ic

forc

e (m

N)

–5.0

0

5.0

0

3.0

5.0

2.0

1.0

D

T jetTpaddle

T=Tpaddle

T=T jet

Fig.·3. A computationalsimulation of jellyfishswimming. (A) A schematicof the body illustrates itsshape at the beginning andimmediately following thepulse phase (gray bar) foreach propulsive cycle.(B) The jet model calculatedthrust production fromchanges in bell height (blueline) and diameter (red line)and (C) the paddle modelcalculated thrust fromchanges in margin angle.(D) The force predicted by jetthrust (T jet; green line), anddrag (D; violet line) during asimulation of the jet model.The paddle thrust (Tpaddle;orange line) was generatedin a separate simulation ofthe paddle model, and thedrag generated during thissimulation is not shown.(E) The changes inswimming speed predicted bypaddle and jet models.

4131Scaling of hydrodynamics in jellyfish

body mass (b=–0.62), but we formulated no nullhypothesis for its scaling factor (Table·1; Fig.·8B)and its variation was not well described by areduced major axis regression (r2=0.05).

The effects of morphology and behavior onperformance

We examined the effects of allometric andisometric growth on swimming performance withparameter maps for body mass and the height-to-diameter ratio (Fig.·9A,B). These parameter mapssuggest that relatively large and prolate jellyfishswim faster than small and oblate jellyfish. IfAurelia maintained a prolate bell shape (as seen inthe early medusa stage) over their entire ontogeny,then their mean swimming speed would be faster(2.7·cm·s–1) than the speed that we observed(1.9·cm·s–1) but they would swim with a higherhydrodynamic cost of transport (0.05·J·kg–1·m–1,compared with 0.04·J·kg–1·m–1). If theymaintained an oblate bell shape (as seen in the latemedusa stage), then jellyfish would swim slightlyslower (1.6·cm·s–1) than what we observed butwould have a lower hydrodynamic cost oftransport (0.03·J·kg–1·m–1, Fig.·9A,B).

According to our parameter maps of pulsefrequency and body mass, larger jellyfish with arelatively high pulse frequency move at fasterspeeds but with a dramatically greaterhydrodynamic cost of transport compared withsmaller jellyfish swimming with lower pulsefrequencies (Fig.·9C,D). If Aurelia maintained ahigh pulse frequency (as seen in the early medusastage) over ontogeny, they would swim faster(19.0·cm·s–1) but with a much greaterhydrodynamic cost of transport (1.39·J·kg–1·m–1)than the swimming that results from the changesin pulse frequency that we observed (with a speedof 2.0·cm·s–1 and a hydrodynamic cost of transportof 0.07·J·kg–1·m–1; Fig.·9C,D). Alternatively, ifthese jellyfish maintained a low pulse frequency(as seen in the late medusa stage) over ontogeny,then they would swim slower (0.01·cm·s–1) butwith a much lower hydrodynamic cost of transport(0.01·J·kg–1·m–1).

DiscussionThe results of our experiments and mathematical

simulations suggest that Aurelia swims by jet propulsion andthat the relative magnitude of thrust, the acceleration reactionand drag changes over ontogeny (Figs·4,·7,·8). This change inhydrodynamics is the apparent result of an ontogeneticdecrease in pulse frequency, which causes swimming speedand drag to scale below the factors predicted by kinematicsimilarity. By changing the shape of the bell to a more oblate

morphology and by decreasing pulse frequency during growth,Aurelia avoids a relatively high cost of transport whileswimming at a moderately high speed over their ontogeny(Fig.·9).

The hydrodynamics of jellyfish swimming

Prior to the present study, there was evidence suggesting thatAurelia does not generate thrust by jetting. Colin and Costello(2002) reported that Daniel’s (1983) model of jettingunderestimated maximum acceleration in hydromedusae withan oblate bell shape similar to that of Aurelia. Furthermore,Aurelia lacks the velum used by hydromedusae to force the

0

1.0

2.0

3.0

4.0

0 1.0 2.0 3.0 4.00 1.0 2.0 3.0 4.0

Measured speed (cm s–1)

Paddle model

Paddle model

Jet model

Jet model

Pred

icte

d sp

eed

(cm

s–1) Bell diameter=10 cm

J1

J2

J1J2

J1

J2

Measured

Spee

d (c

m s–1

)

0

1.0

2.0

3.0

4.0

5.0

Spee

d (c

m s–1

)

0

0.5

1.0

1.5

2.0

2.5

0 1.0 2.0 3.0 4.0

Time (s)

Time (s)

0 0.2 0.4 0.6 0.8 1.0

A B

C

D

Fig.·4. An experimental test of paddle and jetting models for the hydrodynamics ofswimming in Aurelia aurita. The relationship between the measured and predictedmean swimming is shown for the paddle model (A) and the jet model (B). Theblack line has a slope of 1, which represents a perfectly accurate prediction ofmeasured speed (N=18). The diameter of points is proportional to the bell diameterof jellyfish (see legend for scale). Representative large (J1; bell diameter, 8.5·cm)and small (J2; bell diameter, 1.8·cm) jellyfish are highlighted in black. The speedpredictions for each model are compared with measurements of speed for J1 (C)and J2 (D).

4132

water ejected from the bell cavity through a small hole.Lacking this membrane, it appeared unlikely that Aureliatransports water from the subumbrellar cavity as a cylindricaljet, which violates an assumption of the jet model (Daniel,1983). Furthermore, flow visualization around the bodies ofoblate jellyfish like Aureliasuggests that vortices are generatedin close proximity to the bell during the pulse phase, which isan observation consistent with the hypothesis that the margingenerates thrust (Colin and Costello, 2002; Costello and Colin,1994). However, our results explicitly refute this hypothesisand lend support to the idea that thrust is generated by jetting(Figs·4,·8; Table·2).

Although not perfectly accurate, a jet model provides the

best approximation of the hydrodynamics of swimming inoblate jellyfish like Aurelia. Unlike the paddle model, the jetmodel consistently predicted oscillations in speed with timethat followed measured changes (Fig.·4C,D) and mean speedvalues that were within measured values by a factor of three(Fig.·4A,B). Furthermore, the jet model accurately predictedthe scaling factor for swimming speed (Fig.·8A; Table·2).These successful predictions suggest that Aurelia generatesthrust by jetting, as found in other jellyfish species (Colin andCostello, 2002; Daniel, 1983) and that the same hydrodynamicprinciples operate for all jellyfish species, regardless of bellmorphology. However, the jet model did generally predictspeeds that were greater than measured values (Figs·4B,·8A;Table·2), which is a result consistent with previously reportedhigh estimates of maximum acceleration (Colin and Costello,2002).

Discrepancies between performance measurements andjet model predictions suggest a need for refinement of ourunderstanding of the hydrodynamics of oblate jellyfish. Daniel(1983) calculated the drag on the bell of prolate jellyfish usingthe drag coefficient for a sphere, but the shape of oblate speciesmay be better approximated by a hemisphere, as in the present

M. J. McHenry and J. Jed

10–2

10–1

100

10–2

10–1

100

Hei

ght/d

iam

eter

Bel

l di

amet

er (m

)

10–3

10–2

10–1

Bel

l he

ight

(m

)

C

B

A

–410

–310

–210

–1

Body mass (kg)

10

b=0.27, bo=0.33

b=0.40, bo=0.33

b=–0.16, bo=0

Fig.·5. The scaling of bell morphology with body mass. Linearregressions found by reduced major axis (black line; b) weresignificantly different (see Table·1 for statistics) from the nullhypothesis (green line; bo), which predicts geometric similarity inbell height (A), bell diameter (B) and the ratio of height to diameter(C). The gray silhouettes in C approximate the bell shape of smalland large medusae.

10–1

100

101

10–2

10–1

100

Pul

se fr

eque

ncy

(Hz)

Dut

y fa

ctor

–4 10–3 10–2 10–1

Body mass (kg)

10

A

Bb=–0.35, bo=0

bo=0

Fig.·6. The scaling of bell kinematics with body mass. The nullhypothesis (green line; bo), of kinematic similarity predicted constantvalues in duty factor and pulse frequency. (A) Duty factor wasindependent of body mass, as predicted by kinematic similarity.(B) The reduced major axis regression (black line; b) shows adecrease in pulse frequency with increasing body mass (see Table·1for statistics).

4133Scaling of hydrodynamics in jellyfish

study. Furthermore, neither the present study nor Daniel (1983)considers the drag generated by the tentacles. Although weintuitively understand that the acceleration reaction shouldvary with the Reynolds number of a swimming animal (Daniel

et al., 1992; Jordan, 1992; McHenry et al., 2003), we lackan equation that describes the relationship between theacceleration reaction coefficient (equation·6) and Reynoldsnumber. Furthermore, flow visualization studies (Colin and

Table 1. Scaling with body mass,m

y=amb

Dependent variable, y a b L1 L2 bo r2

Morphometrics (N=25)drest(m) –0.4 0.40* 0.36 0.44 0.33 0.94hrest(m) –1.5 0.27* 0.23 0.31 0.33 0.87hrest/drest –1.1 –0.16* –0.19 –0.12 0 0.73srest –0.9 0.80* 0.72 0.89 0.66 0.94l (m) –1.0 0.43* 0.36 0.50 0.33 0.84

Kinematics (N=25)f (Hz) –1.0 –0.35* –0.41 –0.29 0 0.83q –0.63 –0.11* –0.15 –0.06 0 0.02drange(m) –0.80 0.42* 0.36 0.49 0.33 0.87hrange(m) –1.2 0.38 0.30 0.46 0.33 0.66θrange (rad) 0.98 0.29* 0.17 0.42 0 0.03

Hydrodynamics (N=25) (jet model)Dmax (N) –2.2 0.76* 0.68 1.1 1.33 0.74Tmax (N) –0.22 1.3 1.1 1.5 1.33 0.84Amax (N) –0.50 1.3 1.1 1.5 1.33 0.87

Performance (N=18)U (m s–1) –1.4 0.17* 0.10 0.24 0.33 0.53THCOT (J·kg–1·m–1) –3.2 –0.60* –0.92 –0.31 0.05

a, scaling constant; b, scaling exponent (asterisks denote significant difference from bo); L1 lower limit of 95% confidence interval; L2 upperlimit of 95% confidence interval; bo, null hypothesis; drest, resting bell diameter; hrest, resting bell height; srest, projected area of the bell at rest;l, length of the bell margin; f, pulse frequency; q, duty factor; drange, range of bell diameter; hrange, range of bell height; θrange, range of marginangle; Dmax, maximum drag; Tmax maximum thrust; Amax maximum acceleration reaction; U, swimming speed; THCOT, hydrodynamic cost oftransport.

10–5

10–4

10–3

10–2

10–1

Max

imum

forc

e (N

)

10–6

10–3 10–2 10–1

Acceleration reaction DragThrust

–410 10–3 10–2 10–1–410 10–3 10–2 10–1–410

Body mass (kg)

A CBbo=1.33 bo=1.33 b=0.87, bo=1.33

Fig.·7. The scaling ofhydrodynamic forcespredicted by the jetmodel. Maximumacceleration reaction(A) and thrust (B)scaled as predicted bykinematic similarity(green line; bo), butdrag (C) increasedwith a lower exponent(black line; b) than thenull hypothesis (seeTable·1 for statistics).

4134

Costello, 2002; Costello and Colin, 1994) suggest that the jetof oblate jellyfish has a more complex wake pattern than thecylindrical column of water assumed in the present study.

Future models should become more predictive as theyincorporate more of this hydrodynamic complexity.

The scaling of swimming performance

The hydrodynamic changes over the growth of Aurelia arereflected in the different scaling relationships of the three fluidforces considered (Fig.·7). Although maximum values of thrustand the acceleration reaction scale at rates consistent withkinematic similarity, swimming speed scales at a rate belowthat predicted by kinematic similarity (Fig.·8A). Therefore,larger jellyfish swim disproportionately slower than smallerjellyfish, and drag consequently scales at a lower factor thanpredicted by kinematic similarity (Fig.·7C). This low scalingfactor for drag occurs despite the fact that the projected areaof the bell scales with a higher factor than predicted bygeometric similarity (Table·1). According to our model, thescaling of speed is influenced by the ontogenetic decrease inpulse frequency (Fig.·6B). When pulse frequency was heldconstant in our simulations, speed increased rapidly withincreasing body mass (Fig.·9C). This suggests that it is the rate,not the magnitude, of force production that caused speed anddrag to scale by a factor below that predicted by kinematicsimilarity.

Although fish swim by lateral undulation instead of jetting,their swimming speed is also strongly dependent on thefrequency of the propulsive cycle. Speed increases linearlywith tail beat frequency in a diversity of species (Jayne andLauder, 1995; Long et al., 1994, 1996; Webb, 1986). However,when fish of different size within a species are made to swimat the same frequency, they move at the same length-specificspeed (Bainbridge, 1958). Such kinematic similarity couldexist in jellyfish if they maintained the same pulse frequencyover a range of body size.

Our results suggest that ontogenetic changes in the bodyshape and pulse frequency of Aurelia influence a tradeoffbetween swimming speed and the cost of transport. Bychanging from a prolate to oblate bell shape, Aurelia growsfrom a shape of relative high speed and high cost to one of lowspeed and low cost (Fig.·9A,B). However, the difference inmean performance between allometric and isometric growth issubtle, when compared to the effects of pulse frequency. Byreducing pulse frequency over ontogeny, Aurelia avoids thedramatically high cost of transport predicted for large jellyfishthat maintain a high pulse frequency (Fig.·9D), but thisdecrease in frequency comes at a cost to speed (Fig.·9C).

M. J. McHenry and J. Jed

10–2

10–1

100

Hyd

rody

nam

ic c

ost o

f tra

nspo

rt (

J kg–1 m

–1)

10–3

10–4

10–2

10–1

Spee

d (m

s–1 )

10–3

B

A

10–3

10–2

10–1

Body mass (kg)

–410

b=0.17, bo=0.33

b=–0.62

bjet=0.26, bpaddle=0.33

Fig.·8. The scaling of swimming performance. (A) The nullhypothesis (green line; bo) was significantly different from thereduced major axis regression for measurements of swimming speed(black line; b). The measured scaling factor was not significantlydifferent from predictions for the jet model (blue line; bjet), but wassignificantly different for the paddle model (orange line; bpaddle; seeTable·2 for statistics). (B) There was no null hypothesis predicted forthe hydrodynamic cost of transport (see Table·1 for statistics).

Table 2. Scaling of swimming speed predicted by mathematical models

Jet model Paddle model

u=amb Measured Predicted L1 L2 Predicted L1 L2

b 0.17 0.26 0.13 0.39 0.33* 0.19 0.48a –1.4 –1.1* –1.2 –1.0 –3.0* –3.1 –2.9

u, swimming speed;m, body mass; a, scaling constant; b, scaling exponent (asterisks denote significant difference from measurement); L1

lower limit of 95% confidence interval; L2 upper limit of 95% confidence interval; N=18.

4135Scaling of hydrodynamics in jellyfish

Our results support the hypothesis that medusae change theirbehavior and morphology to maintain a moderate speed whileavoiding a high cost of transport. However, it is difficult toweigh the relative importance of speed and the cost of transportto the ecology and evolution of Aurelia without anunderstanding of how these aspects of performance influence

fitness (Koehl, 1996). Furthermore, ontogenetic changes inbehavior and morphology may influence other aspects oforganismal performance that have a greater effect on fitnessthan does locomotion. For this reason, it would be interestingto explore how these ontogenetic changes affect prey captureand mass transport.

10–3

10–2

10–1

10–1

Body mass (kg)

Hei

ght/d

iam

eter

–410

Mean speed over

ontogeny(cm s–1)

10.00

2.7

1.91.6

20.0Speed (cm s–1)A

10–3

10–2

10–1

10–1

Body mass (kg)

–410

Mean THCOT over ontogeny

(J kg–1 m–1)

0.200 0.40

Hydrodynamic cost of transport (J kg–1 ml–1)B

0.05

0.040.03

–110

010

10–3

10–2

10–1

Body mass (kg)

Pul

se f

requ

ency

(H

z)

Pul

se f

requ

ency

(H

z)

–410

Mean speed over

ontogeny(cm s–1)

10.00 20.0

Speed (cm s–1)C

10–3

10–2

10–1

Body mass (kg)

–410

Mean THCOT over ontogeny

(J kg–1 m–1)

0.200 >0.40

Hydrodynamic cost of transport (J kg–1 ml–1)D

1.39

0.05

0.01

010

19.0

2.0

1.4

Hei

ght/d

iam

eter

Fig.·9. Parameter maps show how the scaling of the height-to-diameter ratio and pulse frequency with body mass is predicted to affectswimming speed and the hydrodynamic cost of transport. Gray lines follow patterns of ontogenetic change, and the performance values to theright of these lines are the mean performance predicted over ontogeny. (A,B) Gray silhouettes show the shape of the bell at high (prolate) andlow (oblate) height to diameter values. The thin, broken, horizontal line shows isometric scaling where the jellyfish maintain a prolatemorphology at all sizes, the solid line follows the allometric scaling that we measured (Fig.·5), and the thick, broken, horizontal line shows theisometric scaling where jellyfish maintain an oblate morphology at all sizes. (C,D) The thin, broken, horizontal line follows a constant highvalue of pulse frequency at all body sizes, the solid line tracks the change in frequency that we measured (Fig.·6), and the thick, broken,horizontal line maintains a constant low value for pulse frequency at all body sizes. (D) Notice that we have coded values for the hydrodynamiccost of transport (THCOT) exceeding 0.40·J·kg–1·m–2 as light green in order to maintain a scale that results in visible typography for the rest ofthis parameter map.

4136

Performance and growth

The effect that the shape of an organism has on itsmechanical performance is likely to change during growthbecause biomechanical forces are typically scale dependent.Therefore, changes in shape may reflect a shift in themechanical demands of larger structures. Alternatively,morphology may change in order to meet other physiologicaldemands or because of developmental constraints. In theinterest of understanding why organisms change or preserveshape during growth, it is useful to compare the ontogeneticchanges in morphology that organisms exhibit to alternatepatterns of growth. For example, Gaylord and Denny (1997)used a mathematical model to examine how different patternsof growth in seaweeds (Eisenia arboreaand Pterygophoracalifornica) affected their susceptibility to breakage inintertidal habitats. They found that the allometric growthexhibited by these organisms results in lower material stressthan if the seaweeds maintained their juvenile shape over theirlife history. Similarly, we found that jellyfish have a lowerenergetic cost of swimming by changing their bell shape overontogeny than if they maintained the prolate shape of juveniles(Fig.·9B). In aquatic animals, ontogenetic change inperformance may be strongly influenced by changes inbehavior. For example, we found that changes in pulsefrequency have an even stronger effect on performance thanthe changes in bell shape (Fig.·9).

It would be difficult to explore alternate patterns of growthor to tease apart the individual effects of morphological andkinematic parameters without the use of a mathematical model.In order for such models to accurately predict performance,they require accurate parameter values that are provided bymeasurements. Therefore, the integration of theoretical andexperimental approaches should greatly facilitate ourunderstanding of how ontogenetic changes in morphology orbehavior affect organismal performance.

List of symbolsa scaling constantA acceleration reaction forceb scaling exponentbo null hypothesiscbell drag coefficient for the bellcmargin drag coefficient for the bell margind bell diameterdrange range of bell diameterdrest resting bell diameterD drage slope of major axis regressionf pulse frequencyF force to change body inertiah bell heighthcav height of subumbrellar cavityhrange range of bell heighthrest resting bell heighti index of measurements

k sawtooth functionl length of the bell marginL1 lower limit of 95% confidence intervalL2 upper limit of 95% confidence intervalm body massn number of measurementsq duty factorsbell projected area of the bellsmargin area of the bell margint timeT thrustTHCOT hydrodynamic cost of transportT jet thrust generated by a jetTpaddle thrust generated by a paddleu swimming speedU body velocityv volume of the subumbrellar cavityw speed of the bell marginW velocity of the bell marginx net distancey dependent variableα added mass coefficientθ margin angleθrange range of margin angleρtissue density of tissueρwater density of seawater

We thank M. Koehl and T. Daniel for their guidance andadvice, and A. Summers, W. Korff, J. Nauen and J. Strotherfor their suggestions. This work was supported with an NSFpredoctoral fellowship and grants-in-aid of research from theAmerican Society of Biomechanics, Sigma Xi, theDepartment of Integrative Biology (U.C. Berkeley) and theSociety for Integrative and Comparative Biology to M.J.M.Additional support came from grants from the NationalScience foundation (# OCE-9907120) and the Office of NavalResearch (AASERT # N00014-97-1-0726) to M. Koehl.

ReferencesBainbridge, R. (1958). The speed of swimming fish as related to size

and to the frequency and amplitude of the tail beat. J. Exp. Biol.35, 109-133.

Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. New York:Cambridge University Press.

Batty, R. (1984). Development of swimming movements and musculature oflarval herring (Clupea harengus). J. Exp. Biol.110, 217-229.

Buecher, E. and Gibbons, M. J.(1999). Temporal persistence in the verticalstructure of the assemblage of planktonic medusae in the NW MediterraneanSea. Mar. Ecol. Prog. Ser.189, 105-115.

Colin, S. P. and Costello, J. H.(2002). Morphology, swimming performanceand propulsive mode of six co-occurring hydromedusae. J. Exp. Biol.205,427-437.

Costello, J. H. and Colin, S. P.(1994). Morphology, fluid motionand predation by the scyphomedusa Aurelia aurita. Mar. Biol. 121, 327-334.

Dadswell, M. J. and Weihs, D. (1990). Size-related hydrodynamiccharacteristics of the giant scallop, Placopecten magellanicus(Bivalvia:Pectinidae). Can. J. Zool.68, 778-785.

Daniel, T., Jordan, C. and Grunbaum, D. (1992). Hydromechanicsof swimming. In Advances in Comparative and Environmental

M. J. McHenry and J. Jed

4137Scaling of hydrodynamics in jellyfish

Physiology, vol. 11 (ed. R. McN. Alexander), pp. 17-49. London:Springer-Verlag.

Daniel, T. L. (1983). Mechanics and energetics of medusan jet propulsion.Can. J. Zool.61, 1406-1420.

Daniel, T. L. (1985). Cost of locomotion: Unsteady medusan swimming. J.Exp. Biol.119, 149-164.

Fuiman, L. A. (1993). Development of predator evasion in Atlantic herring,Clupea harengusL. Anim. Behav.45, 1101-1116.

Fuiman, L. A. and Batty, R. S. (1997). What a drag it is getting cold:partitioning the physical and physiological effects of temperature on fishswimming. J. Exp. Biol.200, 1745-1755.

Fuiman, L. A. and Webb, P. W. (1988). Ontogeny of routine swimmingactivity and performance in zebra danios (Teleostei: Cyprinidae). Anim.Behav.36, 250-261.

Gaylord, B. and Denny, M. W.(1997). Flow and flexibility. I. Effects of size,shape and stiffness in determining wave forces on the stipitate kelps Eiseniaarboreaand Pterygophora californica. J. Exp. Biol.200, 3141-3164.

Gladfelter, W. G. (1973). A comparative analysis of the locomotory systemof medusoid Cnidaria. Helgol. Wiss. Meeresunters25, 228-272.

Hale, M. (1999). Locomotor mechanics during early life history: effects ofsize and ontogeny on fast-start performance of salmonid fishes. J. Exp. Biol.202, 1465-1479.

Hernandez, L. P. (2000). Intraspecific scaling of feeding mechanics in anontogenetic series of zebrafish, Danio rerio. J. Exp. Biol.203, 3033-3043.

Hoerner, S. F.(1965). Fluid-Dynamic Drag. Brick Town, NJ: Hoerner FluidDynamics.

Hunter, J. R. (1972). Swimming and feeding behavior of larval anchovyEngraulis mordax. Fish. Bull.70, 821-838.

Hunter, J. R. and Kimbrell, C. A. (1980). Early history of pacific mackerel,Scomber japonicus. Fish. Bull.78, 89-101.

Huxley, J. S.(1932).Problems of Relative Growth. New York: Dial Press.Jayne, B. C. and Lauder, G. V.(1995). Speed effects on midline kinematics

during steady undulatory swimming of largemouth bass, Micropterussalmoides. J. Exp. Biol. 198, 585-602.

Johnson, D. R., Perry, H. M. and Burke, W. D.(2001). Developing jellyfishstrategy hypotheses using circulation models. Hydrobiology451, 213-221.

Jordan, C. E. (1992). A model of rapid-start swimming at intermediateReynolds number: undulatory locomotion in the chaetognath Sagittaelegans. J. Exp. Biol.163, 119-137.

Koehl, M. A. R. (1996). When does morphology matter? Annu. Rev. Ecol.Syst.27, 501-542.

Lighthill, J. (1975). Mathematical Biofluidynamics. Philadelphia: Society forIndustrial and Applied Mathematics.

Long, J. H., Hale, M. E., McHenry, M. J. and Westneat, M. W.(1996).Functions of fish skin: flexural stiffness and steady swimming of longnosegar Lepisosteus osseus. J. Exp. Biol.199, 2139-2151.

Long, J. H., McHenry, M. J. and Boetticher, N. C.(1994). Undulatoryswimming: how traveling waves are produced and modulated in sunfish(Lepomis gibbosus). J. Exp. Biol.192, 129-145.

McHenry, M. J., Azizi, E. and Strother, J. A. (2003). The hydrodynamics

of undulatory swimming at intermediate Reynolds numbers in ascidianlarvae (Botrylloidessp.). J. Exp. Biol.206, 327-343.

McMahon, T. A. (1984). Muscles, Reflexes, and Locomotion. Princeton, NJ:Princeton University Press.

Meinkoth, N. A. (1995). National Audubon Society Field Guide to NorthAmerican Seashore Creatures. New York: Alfred A. Knopf, Inc.

Mills, C. E. (1981). Diversity of swimming behaviors in hydromedusae asrelated to feeding and utilization of space. Mar. Biol. 64, 185-189.

Muller, U. K., Stamhuis, E. J. and Videler, J. J.(2000). Hydrodynamics ofunsteady fish swimming and the effects of body size: comparing the flowfields of fish larvae and adults. J. Exp. Biol.203, 193-206.

Mutlu, E. (2001). Distribution and abundance of moon jellyfish (Aureliaaurita) and its zooplankton food in the Black Sea. Mar. Biol. 138, 329-339.

Nauen, J. and Shadwick, R.(2001). The dynamics and scaling of forceproduction during the tail-flip escape response of the California spiny lobsterPanulirus interruptus. J. Exp. Biol.204, 1817-1830.

Nicholas, K. R. and Frid, C. L. J. (1999). Occurrence of hydromedusae inthe plankton off Northumberland (western central North Sea) and the roleof planktonic predators. J. Mar. Biol. Assoc. UK79, 979-992.

Osse, J. W. M. and van den Boogaart, J. G. M.(2000). Body size andswimming types in carp larvae: effects of being small. Neth. J. Zool.50,233-244.

Quillin, K. J. (1999). Kinematic scaling of locomotion by hydrostatic animals:ontogeny of peristaltic crawling by the earthworm Lumbricus terrestris. J.Exp. Biol.202, 661-674.

Quillin, K. J. (2000). Ontogenetic scaling of burrowing forces in theearthworm Lumbricus terrestris. J. Exp. Biol.203, 2757-2770.

Rayner, J. M. V. (1985). Linear relations in biomechanics – the statistics ofscaling functions. J. Zool.206, 415-439.

Rome, L. (1992). Scaling of muscle fibers and locomotion. J. Exp. Biol.168,243-252.

Schmidt-Nielsen, K. (1984). Scaling: Why is Animal Size so Important?Cambridge: Cambridge University Press.

Shampine, L. F. and Gordon, M. K.(1975). Computer Solution of OrdinaryDifferential Equations: The Initial Value Problem. San Francisco: W. H.Freeman.

Sokal, R. R. and Rohlf, F. J.(1995). Biometry. New York: W. H. Freeman& Co.

Vlymen, W. J. (1974). Swimming energetics of the larval anchovy Engraulismardax. Fish. Bull.72, 885-899.

Vogel, S.(1981). Life in Moving Fluids. Princeton: Princeton University Press.Webb, P. W. (1986). Kinematics of lake sturgeon, Acipenser fulvescens, at

cruising speeds. Can. J. Zool.64, 2137-2141.Webb, P. W. and Weihs, D.(1986). Functional locomotor morphology of

early life-history stages of fishes. Trans. Am. Fish. Soc.115, 115-127.Williams, T. A. (1994). A model of rowing propulsion and the ontogeny of

locomotion in Artemia larvae. Biol. Bull. 187, 164-173.Wu, T. Y. (1977). Introduction to the scaling of aquatic animal locomotion.

In Scale Effects in Animal Locomotion(ed. T. J. Peldley), pp. 203-232.London: Academic Press.