Rowing jellyfish contract to maintain neutral buoyancyPatricia J. Yanga, Matthew Lemonsa, David L. Hua,b,* a Schools of Mechanical Engineering , Georgia Institute of Technology, Atlanta, GA 30332, USAb Schools of Biology, Georgia Institute of Technology, Atlanta, GA 30332, USA

A  R  T  I  C  L  E      I  N  F  O A  B  S  T  R  A  C  T

 

Article history:Received 23 October 2017Received in revised form 10 March 2018Accepted 14 March 2018Available online 20 March 2018  

Keywords:JellyfishThrustBackground flow

 

Jellyfish are easily carried by ocean currents, yet most studies on jellyfish focus on the kinematicsin a quiescent fluid. In this experimental and theoretical study, we focus on rowing jellyfish, thosethat partially contract their bells and paddle in a relaxed manner. We film six species of rowingjellyfish in a range of background flow speeds at the Georgia Aquarium. Each species has a uniquecontraction frequency, which is independent of both the body orientation and the backgroundflow speed. Our mathematical model reveals that jellyfish contract to offset their sinking. Thisbehavior is invariant: Despite the background flow conditions, jellyfish contract as if they areoriented upright in a quiescent fluid. Our study suggests that jellyfish operate in open-loop withoutfeedback from their environment.

 

©2018 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical andApplied Mechanics. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). 

 

Ocean currents vary in speed from regular currents of 0.08m/s to a storm of 2.5 m/s [1, 2], which affect a number of aquaticorganisms. Jellyfish are particularly susceptible because theyswim relatively slowly by contracting and relaxing their bell-shaped bodies. The magnitude of jellyfish contraction is associ-ated with two propulsive modes: jetting and rowing. A jettingjellyfish fully contracts its bell, yet a rowing jellyfish only par-tially contracts its bell and paddle in a more relaxed manner [3].In 1983, Daniel proposed that the bell contraction is the sourceof thrust for jetting jellyfish [4, 5]. Later studies accept thisconcept and further state that jetting jellyfish contract at the nat-ural frequency of the bell, which reduces the energy cost ofswimming [6-8] and maximize propulsive efficiency [9]. Besidesjetting jellyfish, rowing jellyfish have been investigated by visual-izing the full cycle of contraction [10-13]. However, these studieshave been focused on jellyfish in a quiescent fluid. The goal ofthis study is to determine how background flow influences thekinematics of jellyfish.

Early theoretical studies have classified the forces on jellyfishas thrust, drag, added mass force, and inertia force [4]. The fourforces provide a theoretical framework in later jellyfish studies[9, 14], which neglect buoyancy force for simplicity. In 1981,

Mills observed that most jellyfish sink in water. Jellyfish main-tain themselves at a preferred depth by assuming a characterist-ic swimming pattern [15].

In this study, we incorporate the jellyfish sinking into a math-ematical model and discuss its contribution to the kinematics ofjellyfish. We first report the morphology of jellyfish and its kin-ematics in background flow. We then present the mathematicalmodel for jellyfish swimming and compare its prediction to ourobservations. We lastly discuss the implications of our work andsuggest directions for future research.

½j

We model the shape of jellyfish as a section of a hemi-ellips-oid with density , diameter D, height H, and mass mj as illus-trated in Fig. 2(a). Previous researchers have observed that mostjellyfish are negatively buoyant [15, 16]. However, none havemeasured the density directly because measuring the volume ofthe soft and fragile jellyfish is challenging. Gemmell assumesthat density of jellyfish Aurelia aurita is

½j = 1:025 g=cm3; (1)

and examine the sinking of static jellyfish in a numerical model[12]. Since the density of seawater is 1.020 g/cm3, jellyfish A.aurita are 0.5% denser than water [12]. For simplicity, weassume that all jellyfish species have the same density.

In 2007, Dabiri compiled the morphology of 660 jellyfish spe-

 

  

* Corresponding author. E-mail address: hu@me.gatech.edu (D.L. Hu).

Theoretical & Applied Mechanics Letters 8 (2018) 147-152

 

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http://dx.doi.org/10.1016/j.taml.2018.03.0012095-0349/© 2018 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under theCC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

cies [11]. Using Dabiri's and our data, the relationship betweenjellyfish bell height H and diameter D may be written

 

H = 0:81D 0:72¡

N = 761;R 2 = 0:83¢; (2)

where H and D are both in cm. The height of the bell increasesdisproportionally with body size as shown in Fig. 2(d).Consequently, small jellyfish are prolate (bullet-shaped), whilelarge jellyfish are oblate (plate-shaped), as reported in previousstudies [3, 17].

The wet mass of jellyfish mj is complied from previous literat-ure [18-20]. Its relationship to diameter D may be written

mj = 0:08D 2:77¡

N = 22;R 2 = 0:99¢; (3)

where D is in cm and mj is in grams. This trend indicates that themass increases disproportionally with body size as shown in Fig.2(e). Larger jellyfish are lighter than expected from isometry(mj~D3). To understand the impact of morphology on theswimming kinematics, we turn to the filming of jellyfish.

We film six species of jellyfish at the Georgia Aquarium, in-cluding Aurelia aurita, Chrysaora plocamia, Chrysaora colorata,Chrysaora pacifica of the Order Semaeostomeae and Phyllorhizapunctata and Catostylus mosaicus of the Order Rhizostomeae(Fig. 1(a) to (f)). The jellyfish are of a broad range of size from 2

 

0

1

2

3

4

0 5 10 15 20

a b

e

c

d f

g

D (cm)

ΔD (

cm)

0

0.5

1.0

1.5

2.0

0 5 10 15 20

f (H

z)

D (cm)

Chrysaora colorata

Chrysaora pacifica

Aurelia aurita

Chrysaora plocamia

Phyllorhiza punctata

Catostylus mosaicus

h

 

¢D

Fig. 1.   a Aurelia aurita. b Chrysaora plocamia. c Chrysaora colorata. d Phyllorhiza punctata. Courtesy of Marco Fumasoni in Wikimedia Com-mons. e Chrysaora pacifica. f Catostylus mosaicus. Courtesy of Steven Johnson in Wikimedia Commons. g-h The relationship between the dia-meter of jellyfish D and g the contraction amplitude , h the contraction frequency f. Symbols represent experimental measurements, thedashed line represents best fit to the data, and the solid line represents the theoretical prediction.

148 P.J. Yang et al. / Theoretical & Applied Mechanics Letters 8 (2018) 147-152

to 20 cm, which are identified as rowing jellyfish in previous lit-erature [3]. The jellyfish are held in aquaria with constantly cir-culating background flow as illustrated in Fig. 2(c). The aquari-um staff do not permit us to turn off the flow off and state that allspecies sink to the bottom of the tanks except C. mosaicus. Thisanecdotal sinking for five of the six species is qualitatively con-sistent with our assumption that jellyfish have a greater densitythan water. We measure the flow speed by tracing particles nextto the jellyfish but not near the jet from the bell. For each spe-cies, we adjust the background flow speed in the range of 1 to 8cm/s. Overall, we conduct 24 experiments of jellyfish swimmingin different flow conditions.

Jellyfish are carried by background flow in most experiments.The velocity of background flow is labeled as uflow and the velo-city of jellyfish swimming is labeled as ujelly. Here we investigateonly cases in which jellyfish swim parallel to background flow.There are three cases because jellyfish can be pushed by back-ground flow in one of two orientations. In addition, the jellyfishcan successfully swim upstream.

When jellyfish swim with the current, they are pushed by theflow as shown in Fig. 3(a). Half of the species, including P.punctata, C. plocamia, and C. mosaicus, have swimming speedsequal to the current speed. When jellyfish are pushed by the cur-rent in the inverted orientation, they reach a range of speeds (0to 6 cm/s), as shown in Fig. 3(b). Jellyfish in both orientationsare pushed by background flow. Conversely, if the jellyfish swimagainst the current, the speed of jellyfish is limited to less than 2cm/s as in Fig. 3(c). The strength of the jellyfish is insufficient toswim strongly against the current.

¢Df¢D

We quantify the bell contraction of jellyfish by amplitude and frequency f. The product of amplitude and frequency, ,indicates the speed of the bell contraction. Among the six spe-cies, only P. punctata has an abnormally high product of amp-litude and frequency, which ranges from 1 to 2.5 cm/s. The re-maining species have slow bell speeds of less than 1 cm/s asshown in Fig. 3(d). Since bell speeds are less than speeds ofbackground flow (1 to 8 cm/s), it explains that our jellyfish are

futile swimmers.

¢D N = 24

The bell contraction is a function of body size, which is inde-pendent of jellyfish swimming and background flow. Figure 4shows amplitude and frequency of the bell contraction acrossjellyfish swimming speeds and background flow speeds. Thecontraction amplitude is constant for only one species A. aurita,

= 0.55 ± 0.10 cm ( ), while that of the remaining fivespecies has no clear relation with the speeds of either jellyfishswimming and background flow, as in Fig. 4(a)-(b). Instead, theamplitude of contraction is a function of body size. The scalingof contraction amplitude is proportional to the diameter of bell,

¢D = 0:25D 0:9 cm¡

N = 104;R 2 = 0:79¢: (4)

Jellyfish contract their bell with the amplitude of 25% of theirdiameter as in Fig. 1(g).

N = 18N = 18

N = 24 N = 15N = 11

The frequency of contraction is much more consistent thanthe amplitude. Specifically, the frequency is a separate constantfor each species, as shown in Fig. 4(c)-(d). The frequencies arealso remarkably consistent, and they have a standard deviationaround 10 percent of the average: 0.49 ± 0.06 Hz ( ) for C.colorata, 0.40 ± 0.04 Hz ( ) for C. pacifica, 0.75 ± 0.09 Hz( ) for A. aurita, 0.30±0.03 Hz ( ) for P. punctata,and 0.45±0.06 Hz ( ) for C. mosaicus. It is noteworthy thatthe frequency is maintained independent of the direction ofbackground flow.

The frequency of contraction is also a function of the bodysize. The scaling of frequency is inversely proportional to thediameter of bell as shown in Fig. 1(h),

f = 1:1D¡0:51 Hz¡

N = 27;R 2 = 0:68¢: (5)

The frequency is similar to McHenry's previous scaling on onespecies A. aurita ranging in the diameter from 1.6 to 9.5 cm: f =25D-0.88 Hz (N = 25, R2 = 0.83) [10]. Our exponent indicates asimilar trend but the pre-factor is 25 times less. Theinconsistency might be due to the selection of species, since we

 

D (cm) D (cm)

D

H v

mj

w

uflowujelly

ab c

0.01

0.1

1

10

100

0.01 0.1 1 10 100 1000

H (

cm)

Current work Previous work

0.001

0.1

10

1000

100000

0.1 1 10 100

mj (

g)

d e

ΔD

 

Fig. 2.   a-b Schematic of a The dimensions of jellyfish and the trapped water. b The contraction of jellyfish. c The circulating water in the jelly-fish tank of speed uflow, and the jellyfish successfully swimming upstream with speed ujelly. d The relationship between the diameter of jellyfish Dand the height of jellyfish H. e The relationship between the diameter of jellyfish D and the wet mass of jellyfish mj.

P.J. Yang et al. / Theoretical & Applied Mechanics Letters 8 (2018) 147-152 149

 

0

2

4

6

8

0 2 4 6 8

|uje

lly| (

cm/s

) |u

jell

y| (

cm/s

)

|uje

lly| (

cm/s

) |u

jell

y| (

cm/s

)

ujelly

ujelly

ujelly

|uflow| (cm/s) |uflow| (cm/s)

|uflow| (cm/s) fΔD (cm/s)

0

2

4

6

8

0 2 4 6 8

0

2

4

6

8

0 2 4 6 8

Chrysaora colorata

Chrysaora pacifica

Aurelia aurita

Chrysaora plocamia

Phyllorhiza punctata

Catostylus mosaicus

a b

c d

0

2

4

6

8

0 1 2 3

uflow

uflow

uflow

 

f¢D

Fig. 3.   a-c The relationship between the speed of background flow |uflow| and the speed of jellyfish |ujelly| in three categories. a Jellyfish pushedby the current. b Jellyfish unsuccessfully swim against the current. c Jellyfish successfully swim against the current. d The relationship betweenthe product of contraction frequency and amplitude and the speed of jellyfish |ujelly|.

 

|ujelly| (cm/s) |uflow| (cm/s)

|ujelly| (cm/s) |uflow| (cm/s)

fΔD

(cm

/s)

f ΔD

(cm

/s)

Chrysaora colorata

Chrysaora pacifica

Aurelia aurita

Chrysaora plocamia

Phyllorhiza punctata

Catostylus mosaicus

a b

c d

0

0.5

1.0

1.5

2.0

1.0

2.0

0 2 4 6 8

f (H

z)

0

0.5

1.5

0 2 4 6 8

f (H

z)

0

1

2

3

4

0 2 4 6 80

1

2

3

4

0 2 4 6 8

 

f¢DFig. 4.   a-b The relationship between the contraction amplitude and a the speed of jellyfish |ujelly| b the speed of background flow |uflow|. c-d The relationship between the contraction frequency f and c the speed of jellyfish |ujelly| d the speed of background flow |uflow|.

150 P.J. Yang et al. / Theoretical & Applied Mechanics Letters 8 (2018) 147-152

have six species and each of them is of a relatively small range ofsize. We now turn to a mathematical model of jellyfishswimming and show that our prediction of the frequency is moreconsistent with Eq. (5).

We present a hydrodynamic model for jellyfish swimmingsteadily at velocity ujelly under background flow at flow velocityuflow. Since jellyfish undergo periodic contractions, their swim-ming is unsteady on the time scale of a single period. Neverthe-less, here we focus on an average swimming speed across mul-tiple cycles, a state we consider as steady. Moreover, we only dis-cuss the cases of jellyfish swimming with or against the current.Therefore, our model is steady-state one-dimensional.

When jellyfish swim, the forces on jellyfish are composed oftheir inertia, the added mass force, thrust by jet creation fromcontraction, drag, and buoyancy,

F inertia = F add¡mass+ F drag+ F thrust + F buoyancy: (6)

½j ½j

½V ½

½Cd

½ ¢A¢A

¢A

¡½j Vj g½j Vj g

½ ½j

The inertia force is the product of jellyfish mass and the ac-celeration of swimming, Finertia = Vj dujelly/dt, where is thedensity of jellyfish and Vj is the volume of jellyfish. The addedmass force is the time rate change of water contained in the bell,Fadd-mass = dujelly/dt, where is the density of water and V isthe volume of the bell. We include the drag and thrust forcesfrom Daniel [4, 5]. The drag force is Fdrag = (ujelly-uflow)2A/2,where Cd is the drag coefficient and A is the cross sectional area.The thrust force is Fthrust = - |w|2 , where w is the velocity ofthe jet relatively to the bell and is the change of cross-sec-tional area of the bell. The speed of the jet is the ratio of volumet-ric flowrate and the change of cross-sectional area, |w| = fV/ ,where f is the frequency of contraction. Previous models neglectthe buoyancy force. The weight of jellyfish is . Bouyancyis the weight of the displacement water . The net buoyancyforce is the Fbuoyancy = ( - )Vjg. In the case when jellyfish swimupwards, we rewrite Eq. (6) to obtain:

(½j Vj + ½V)dujelly

dt=¡ 1

2½Cd (ujelly ¡ u°ow)

2 A +½

¢AV 2f 2

+(½¡ ½j)Vj g: (7)

Equation (7) is one of the four cases since the orientations ofboth jellyfish and background flow may be upwards ordownwards. It is impossible for jellyfish to contract at a singlefrequency and satisfy all four cases.

We propose a new and remarkably simple rule for jellyfishmotion: Jellyfish contract to prevent sinking as if they are ori-ented upright in a quiescent fluid. Since the motion is steadystate, we neglect accelerations. Moreover, the drag force isnegibile because there is no fluid motion. Together, Eq. (7) sim-plifies to:

½

¢AV 2f 2 + (½¡ ½j)Vj g = 0: (8)

We rearrange rewrite Eq. (8) to express the frequency,

f =

sµ½j

½¡ 1

¶µVj

V

¶µ¢A

V

¶g: (9)

The allometry of jellyfish provides estimation of each term onthe right hand side of Eq. (9). The volume of bell scales as

¢A ¢D½j

½j

¢A¢A

V~HD2~D2.72. The change of cross-sectional area scales as~D ~D1.9. For simplicity, we assume that the density of

jellyfish, , is constant. Thus, the volume of jellyfish, which isthe ratio of mass and density, scales as Vj = mj/ ~mj~D2.77.Substituting these expressions in Eq. (9) yields that the ratio ofvolumes is Vj/V~D0.05 and area-volume ratio is /V~D-0.82.Thus, the area-volume ratio, /V, is the dominant term in Eq.(9). All together, the frequency of contraction is f~D-0.39. Theexponent is of the same order as that in our experiments (f~D-

0.51). We conclude larger jellyfish contract more slowly becauseof their lower area-volume ratios.

½j

¿

¢A ¢D

We go beyond scaling analysis to further simplify Eq. (9). Weapproximate the shape of jellyfish as a hemi-ellipsoid withheight H and diameter D as shown in Fig. 2(a). The hemi- ellips-oid is composed of two parts: the volume of jellyfish and thevolume of the bell. The volume of jellyfish is the ratio of massand density, Vj = mj/ . The jellyfish is mainly composed by thintissues, thus the volume of jellyfish body is much less than theencapsulation of the hemi-ellipsoid, Vj V. The volume of thebell is approximated as that of the hemi-ellipsoid, V ≈ D2H/6.The change of cross-sectional area is ≈ D . Thus, the fre-quency of contraction is

f =6

DH

sµ½j ¡ ½

½½j

¶mj g¢D

D: (10)

With the morphology of jellyifsh, Eq. (10) can be written f =1.3D-0.39, which corresponds to the solid line in Fig. 1(g). Thisline agrees well with our experiments, shown by the dashed line(f = 1.1D-0.51). Therefore, we conclude that our analysis capturesthe kinematics of jellyfish swimming.

The most surprising finding of our study is that jellyfishmaintain constant frequency independent of their orientationunder background flow. Since the drag force depends on orient-ation. Our result suggests that the drag force is too small to influ-ence their frequency. A previous study has also shown that dragforce is disproportionally less than other forces [10]. We derivethe full governing equation for steady-state jellyfish motion, butdo not attempt to solve it because a single contraction frequencycannot solve the equation in all flow directions. The influence ofall involving forces has to be investigated case-by-case.

The scaling law presented in Eq. (5) is limited to rowing jelly-fish. P. punctata has an anomalously high frequency for its bodysize. The species may satisfy a different scaling law, which mightbe further investigated by field experiments such as those byKatija [21], since these jellyfish are difficult to raise in captivity.Although we only have six species of rowing jellyfish from 2 to 20cm, our mathematical model might be valid for more species.

We conduct an experimental and theoretical study of rowingjellyfish swimming under background flow at various speeds. Wediscover that each species has a unique contraction frequency,which is independent of both the speed of background flow andthe orientation of jellyfish. We present a mathematical model ofjellyfish oriented upright in a quiescent fluid. Their contractionfrequency in these conditions matches well with observed fre-quencies, suggesting jellyfish do not change kinematics withtheir environment.

Below we present our experimental methods for filming andtracking jellyfish. The jellyfish are housed in the Jellyfish lab atthe Georgia Aquarium in Atlanta. We use Sony HDR-XR200 to

P.J. Yang et al. / Theoretical & Applied Mechanics Letters 8 (2018) 147-152 151

film jellyfish in the tanks present in the lab. Each tank has an ad-justable flow which enters the tank through a bar with multipleexits over the width of the tank. We place the camera in line withthe bar so that the camera is perpendicular to background flow.We film jellyfish with a known reference length on the surface ofthe tank. At each background flow setting, we obtain close up re-cording on jellyfish and zoom-out recording on background flowover the whole tank. The staff bring jellyfish to the surface of thetank and measure the diameter of them.

We use open-source software Tracker to analyze jellyfishover a full cycle of movement, which is defined as one full ex-pansion of the bell to another. We track three points on the jelly-fish at the full expansions and the full contractions: one on thetop and two on the edge of the bell. These three points set thespeed of jellyfish swimming, the contraction amplitude, and thecontraction frequency. We split the direction of jellyfish swim-ming into four categories: with the current, with the current andinverted, against the current, and perpendicular to the current.

We visualize background flow by small particles (1 to 3 mmin diameter) in the tanks. The particles are zooplankton anddead skin cells from jellyfish. These particles are uniformly dis-tributed in every tank. We use open-source software Tracker tomeasure the speed of a particle as that of background flow.

Acknowledgements

We acknowledge our funding sources Army Research Office63207-EG, our photographer (S. Zan), and hosts at the GeorgiaAquarium (L. Camp, M. Smith, K. Stone, and A. Arnold). Wethank S. Kim and C. Apicella for their early contributions.

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