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

The performance of swimming may change dramatically over the growth of an aquatic animal. Despite our understanding of the broad-scale hydrodynamic differences in the swimming of animals spanning many orders of magnitude in body length (Daniel et al.,1992; Lighthill,1975; Wu, 1977),we cannot predict how ontogenetic changes in the size, shape and motion of the body influence the speed and energetic cost of swimming within individual species. The purpose of the present study was to examine the scaling of hydrodynamic forces in the jellyfish Aurelia aurita in order to understand how such ontogenetic changes affect swimming performance.

Although it is generally appreciated that a fully grown aquatic animal will swim faster than when it was smaller, the precise relationship between speed and body size over a life history is dictated by scale-dependent hydrodynamics. Much of our understanding for this scale dependency comes from comparisons between species that differ in body mass by many orders of magnitude (e.g. Daniel et al.,1992; Lighthill,1975; Wu, 1977). These comparisons illustrate that thrust is generated primarily by viscous force at the size of spermatozoa, inertial force at the size of adult fish and a combination of these forces at intermediate sizes. Such broad comparisons are useful for understanding the major fluid forces that play a role in the hydrodynamics of a growing animal but cannot provide predictive explanations for how swimming performance (e.g. speed and cost of transport) should change over the ontogeny of individual species.

Ontogenetic changes in swimming kinematics have been most 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 tails with greater length-specific amplitude but propel themselves at lower speed than adults of the same species. Although it is appreciated that force generated by the inertia of water increases in importance relative to viscous force as fish grow larger (Fuiman and Batty,1997; McHenry et al.,2003; Muller et al.,2000; Webb and Weihs,1986), few investigators have tested whether a hydrodynamic model is capable of predicting the scaling of swimming performance in fish (although Vlymen, 1974 is an exception). Such an approach would allow an investigator to explore the relative contribution of inertial and viscous forces to thrust and drag and to evaluate whether alternative larval morphology or tail kinematics could improve on swimming performance.

Although the scaling of swimming performance is not as well characterized for aquatic invertebrates as for larval fish, some investigators have used a combination of modeling and experimentation in order to understand the hydrodynamic mechanisms that explain the scaling of performance in invertebrate species. Using such an approach, Williams(1994) demonstrated that the serial addition of developing limbs along the abdomen of the brine shrimp(Artemia sp.) initially does not contribute to propulsion when larvae are small, but the additional limbs generate thrust in later life history stages when unsteady forces play a greater role in the hydrodynamics of swimming. Using a combination of kinematics and force measurements, Nauen and Shadwick (2001) found that the tail of the spiny lobster (Panulirus interruptus) generates most of its force with a paddle mechanism and that maximum force production scales according to a paddle model. Dadswell and Weihs(1990) determined that giant scallops (Placopecten magellanicus) swim with the greatest speed at a medium body size range, when they attain the highest thrust-to-weight ratio of their life history.

Jellyfish are a potentially useful group for exploring the ontogenetic scaling of hydrodynamics because their swimming is easily modeled. Daniel(1983) proposed a mathematical model that suggested that the hydrodynamics of prolate (bullet-shaped)hydromedusae are dominated by the thrust generated by jetting, the acceleration reaction (i.e. the force generated by accelerating the water around the body) and the drag resisting the forward motion of the body. This model replicated observed oscillations in swimming speed(Daniel, 1983), and Colin and Costello (2002) found the model to accurately predict body acceleration in prolate, but not oblate(plate-shaped), jellyfish. They proposed that oblate jellyfish generate thrust primarily by paddling the flexible margins of their bell instead of using a jet mechanism. We tested this hypothesis by comparing measurements of speed in Aurelia, an oblate jellyfish, with predictions from mathematical models of swimming that assume thrust generation by either paddling or jetting.

Changes in the shape or relative motion of an animal's body during growth should be reflected in the allometric scaling of morphometric and kinematic parameters (Huxley, 1932; McMahon, 1984). An allometric relationship is defined as a deviation from isometry(Schmidt-Nielsen, 1984) and therefore requires the formulation of an a priori null hypothesis as predicted by isometry. In complex biomechanical systems, these predictions are not always obvious and therefore merit careful consideration (e.g. Hernandez, 2000; Nauen and Shadwick, 2001; Quillin, 2000; Rome, 1992).

If a jellyfish grows isometrically, then its bell height, h,scales linearly with bell diameter, d, and medusae at all sizes will be geometrically similar (i.e. h∝d). Geometric similarity implies that the volume of the body scales as d³, which means that bell diameter scales with body mass(m) as m^1/3 (assuming constant tissue density). During swimming, the shape of the bell changes with time. Bell height rapidly increases and bell diameter rapidly decreases over a pulse phase(Gladfelter, 1973). To achieve kinematic similarity (Quillin,1999), a jellyfish must maintain the speed of height change in proportion to the speed of diameter change at all body sizes. In geometrically similar jellyfish, kinematic similarity is maintained if the pulse frequency,duty factor (the proportion of the propulsive cycle spent pulsing) and range of motion remain constant. Kinematic similarity also requires that a jellyfish moves through the water at a speed that is directly proportional to dand therefore scales as m^1/3. This form of kinematic similarity has been observed in fish made to swim steadily at the same frequency (Bainbridge,1958).

Geometric and kinematic similarity suggest predictions for the scaling of hydrodynamic forces in jetting jellyfish. Drag scales with the area of the bell (∝d²) and the square of swimming speed(∝d²; Daniel,1983), which suggests that this force scales as d⁴ and m^4/3. Thrust scales with the inverse of the 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 m^4/3. The acceleration reaction varies with the volume of the body (∝d³) and its rate of change in velocity (∝d, assuming sinusoidal changes in velocity), which implies that this force also scales as d⁴and m^4/3. Since all three hydrodynamic forces are predicted to scale with mass in the same way, we predict that the hydrodynamics of swimming should not change if jellyfish maintain geometric and kinematic similarity. Conversely, scaling that deviates from this null hypothesis implies that the hydrodynamics of jetting changes with size. These scaling predictions assume that the jellyfish operate at relatively high,inertia-dominated, Reynolds numbers.

We pursued three objectives in order to address how ontogenetic changes affect the swimming performance of Aurelia. (1) The hydrodynamics of swimming were modeled with paddle and jet mechanisms in separate simulations in order to test which mechanism more accurately predicts swimming speed. (2)The scaling of relationships of parameters that play a role in the dynamics of swimming were measured. (3) The performance of swimming in Aureliawas compared with that predicted for model jellyfish exhibiting different patterns of growth.

Aurelia is a marine scyphozoan with an oblate medusa stage that spans over an order of magnitude in bell diameter(Meinkoth, 1995). This large change in body size makes Aurelia an ideal system for examining the scaling of swimming performance. The swimming of medusae such as Aurelia is thought to affect their position in the water column and thereby influence dispersal and movement into areas of high prey density and favorable environmental conditions within the plankton(Buecher and Gibbons, 1999; Johnson et al., 2001; Mutlu, 2001; Nicholas and Frid, 1999). The bell pulsing used by Aurelia to swim also facilitates mass transport and prey capture by increasing with flow over the bell and tentacles(Costello and Colin, 1994; Daniel, 1983; Mills, 1981).

We measured the shape and swimming movement of medusae of Aurelia aurita (L.) from video recordings made at the Monterey Bay Aquarium,Pacific Grove, CA, USA. Medusae ranging in bell diameter from 1.57 cm to 9.51 cm were 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 video recorded (30 frames s^-1; Panasonic PV-S62D SVHS-C Camcorder) in the tank's center(Vogel, 1981). The mean tissue density, ρ_tissue, of each jellyfish was calculated as the ratio of measured body mass and the body volume, which was found by water displacement in a graduated cylinder.

The motion of jellyfish was measured by following the movement of landmarks on the bell using NIH Image on an Apple Macintosh G3 computer(Fig. 1A). We tracked the movement of the exumbrellar and subumbrellar surfaces along the central axis and defined the bell margin as the ring of flexible tissue running from the distal margin of the bell (where the tentacles begin) to a proximal line of high bending. From these data, we calculated the bell height, h, as the distance between the exumbrellar surface and the opening of the subumbrellar cavity along the central axis; the cavity height, h_cav, as the distance between the subumbrellar surface and the opening of the subumbrellar cavity; the diameter of the bell, d,as the distance between lateral surfaces of the bell; and the margin angle,θ, as the angle between the central axis and the margin of the bell(Fig. 1A). We described the passive shape of the bell by measuring the diameter, d_rest, and height, h_rest, of each animal while at rest from video images. The height-to-diameter ratio for each individual was calculated as the quotient of these quantities. The pulse frequency, f, and the duty factor, q, which is the ratio of the period of the pulse phase and the period of the whole propulsive cycle,were measured over a duration of three to five propulsive cycles. The range of values in margin angle, θ_range, cavity height, h_range, and bell diameter, d_range,were also recorded.

We described the scaling of individual kinematic and morphological parameters, y, with body mass, m, using the scaling constant, a, and scaling exponent, b, of an exponential function (y=am^b). These values were found from the intercept and slope of a reduced major axis regression fit to log-transformed data. We rejected the null hypothesis, b_o,in cases where 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 II regression was appropriate because the scale and dimensions of our independent and dependent variables were not equal and the error for each variable was unknown. The coefficient of determination, r², was used to assess the degree of variation explained by reduced major axis regressions.

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

In order to examine the effects of bell shape and body mass on swimming performance, we ran numerous simulations of our jet model over a range of height-to-diameter ratio and body mass values. Using our data for the scaling of kinematics, we animated the bell with the kinematics appropriate to the body mass used in each simulation. To investigate the effects of ontogenetic changes in behavior on performance, we ran simulations at varying pulse frequency and body mass values. For these simulations, we used the measured values for bell height, bell diameter and pulse frequency specific to body mass. From the results of these simulations, we generated parameter maps describing the effects of body mass, pulse frequency and the height-to-diameter ratio on swimming speed and the hydrodynamic cost of transport. From these parameter maps, we calculated the mean speed and hydrodynamic cost of transport predicted for jellyfish that are geometrically and kinematically similar over the range of measured body mass values.

During the pulse phase of the propulsive cycle, Aurelia medusae rapidly decreased their bell diameter, increased bell height and adducted their bell margin as the body increased in speed(Fig. 2). The body slowed as the bell diameter gradually increased, height decreased and the margin abducted during the recovery phase. Using the kinematics of the bell and margin, thrust predicted by the jet model was generally over an order of magnitude greater than thrust from paddling(Fig. 3). The low thrust generated by paddling resulted in predicted swimming speeds that fell short of measurements (Fig. 4). This was reflected in a major axis regression between measured and predicted speed having a slope with 95% confidence intervals well outside a value of 1(e=0.002, L₁=-0.015, L₂=0.019, N=19; Fig. 4A). Although the jet model tended to overestimate measured speed (e=1.86, L₁=1.28, L₂=3.02, N=19; Fig. 4B), it provided a more accurate estimate of speed than the paddle model. The jet model predicted oscillations in speed that closely approximated the variation of speed measured in both large (Fig. 4C) and small (Fig. 4D) jellyfish. Furthermore, the jet model was found to better predict the scaling of swimming speed (see below).

Our morphometric data suggest that Aurelia medusae do not maintain geometric similarity over ontogeny. Jellyfish of high body mass had a bell height that was disproportionately small (b=0.27; Fig. 5A) and a bell diameter that was disproportionately large (b=0.40) compared with jellyfish of low body mass (Fig. 5B; Table 1). This resulted in a significant decrease in the height-to-diameter ratio with increasing body mass(b=-0.16; Fig. 5C). Furthermore, the length of the bell margin scaled by a factor significantly greater than isometry (b=0.43; Table 1).

Aurelia medusae of different body sizes moved with different swimming kinematics. Large jellyfish pulsed at a lower frequency than small jellyfish (b=-0.35), but they maintained similar values for duty factor and the range of margin angle at all sizes(Fig. 6; Table 1). The range of bell diameter scaled by a greater factor (b=0.42) than predicted by kinematic similarity, but the range of bell height did not(b_o=0.33; Table 1). Despite the changes in pulse frequency and the range of bell diameter with body mass, acceleration reaction and thrust scaled with body mass as predicted by kinematic similarity (b_o=1.33; Table 1; Fig. 7A,B). By contrast, drag scaled by a factor (b=0.87) significantly lower than predicted by kinematic similarity (Table 1; Fig. 7C).

Despite pulsing at a lower frequency, larger jellyfish moved with a faster swimming speed than did smaller jellyfish(Fig. 8A). However, the increase in speed with body mass was significantly lower (b=0.17)than predicted by kinematic similarity(Table 1). The jet model consistently overestimated the speed of swimming, which was reflected in its scaling constant (a=-1.4) being significantly greater than the measured value. However, the scaling factor predicted for speed by the jet model was statistically indistinguishable from the measured value(b=0.17; Fig. 8A; Table 2). The paddle model greatly underestimated speed (a=-3.0), but overestimated the scaling factor of speed (b_paddle=0.33; Table 2). The hydrodynamic cost of transport decreased with body mass (b=-0.62), but we formulated no null hypothesis for its scaling factor(Table 1; Fig. 8B) and its variation was not well described by a reduced major axis regression(r²=0.05).

We examined the effects of allometric and isometric growth on swimming performance with parameter maps for body mass and the height-to-diameter ratio(Fig. 9A,B). These parameter maps suggest that relatively large and prolate jellyfish swim faster than small and oblate jellyfish. If Aurelia maintained a prolate bell shape (as seen in the 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 higher hydrodynamic cost of transport (0.05 J kg^-1 m^-1,compared with 0.04 J kg^-1 m^-1). If they maintained an oblate bell shape (as seen in the late medusa stage), then jellyfish would swim slightly slower (1.6 cm s^-1) than what we observed but would have a lower hydrodynamic cost of transport (0.03 J kg^-1m^-1, Fig. 9A,B).

According to our parameter maps of pulse frequency and body mass, larger jellyfish with a relatively high pulse frequency move at faster speeds but with a dramatically greater hydrodynamic cost of transport compared with smaller jellyfish swimming with lower pulse frequencies(Fig. 9C,D). If Aurelia maintained a high pulse frequency (as seen in the early medusa stage) over ontogeny, they would swim faster (19.0 cm s^-1)but with a much greater hydrodynamic cost of transport (1.39 J kg^-1m^-1) than the swimming that results from the changes in pulse frequency that we observed (with a speed of 2.0 cm s^-1 and a hydrodynamic cost of transport of 0.07 J kg^-1 m^-1; Fig. 9C,D). Alternatively, if these 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) but with a much lower hydrodynamic cost of transport (0.01 J kg^-1m^-1).

The results of our experiments and mathematical simulations suggest that Aurelia swims by jet propulsion and that the relative magnitude of thrust, the acceleration reaction and drag changes over ontogeny (Figs 4, 7, 8). This change in hydrodynamics is the apparent result of an ontogenetic decrease in pulse frequency, which causes swimming speed and drag to scale below the factors predicted by kinematic similarity. 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 while swimming at a moderately high speed over their ontogeny(Fig. 9).

Prior to the present study, there was evidence suggesting that Aurelia does not generate thrust by jetting. Colin and Costello(2002) reported that Daniel's(1983) model of jetting underestimated maximum acceleration in hydromedusae with an oblate bell shape similar to that of Aurelia. Furthermore, Aurelia lacks the velum used by hydromedusae to force the water ejected from the bell cavity through a small hole. Lacking this membrane, it appeared unlikely that Aurelia transports water from the subumbrellar cavity as a cylindrical jet, which violates an assumption of the jet model(Daniel, 1983). Furthermore,flow visualization around the bodies of oblate jellyfish like Aureliasuggests that vortices are generated in close proximity to the bell during the pulse phase, which is an observation consistent with the hypothesis that the margin generates thrust (Colin and Costello, 2002; Costello and Colin, 1994). However, our results explicitly refute this hypothesis and 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 in oblate jellyfish like Aurelia. Unlike the paddle model, the jet model consistently predicted oscillations in speed with time that followed measured changes(Fig. 4C,D) and mean speed values that were within measured values by a factor of three(Fig. 4A,B). Furthermore, the jet model accurately predicted the scaling factor for swimming speed(Fig. 8A; Table 2).These successful predictions suggest that Aurelia generates thrust by jetting, as found in other jellyfish species (Colin and Costello, 2002; Daniel,1983) and that the same hydrodynamic principles operate for all jellyfish species, regardless of bell morphology. However, the jet model did generally predict speeds that were greater than measured values (Figs 4B, 8A; Table 2), which is a result consistent with previously reported high estimates of maximum acceleration(Colin and Costello, 2002).

Discrepancies between performance measurements and jet model predictions suggest a need for refinement of our understanding of the hydrodynamics of oblate jellyfish. Daniel (1983)calculated the drag on the bell of prolate jellyfish using the drag coefficient for a sphere, but the shape of oblate species may be better approximated by a hemisphere, as in the present study. Furthermore, neither the present study nor Daniel(1983) considers the drag generated by the tentacles. Although we intuitively understand that the acceleration reaction should vary with the Reynolds number of a swimming animal (Daniel et al., 1992; Jordan, 1992; McHenry et al., 2003), we lack an equation that describes the relationship between the acceleration reaction coefficient (equation 6) and Reynolds number. Furthermore, flow visualization studies (Colin and Costello,2002; Costello and Colin,1994) suggest that the jet of oblate jellyfish has a more complex wake pattern than the cylindrical column of water assumed in the present study. Future models should become more predictive as they incorporate more of this hydrodynamic complexity.

The hydrodynamic changes over the growth of Aurelia are reflected in the different scaling relationships of the three fluid forces considered(Fig. 7). Although maximum values of thrust and the acceleration reaction scale at rates consistent with kinematic similarity, swimming speed scales at a rate below that predicted by kinematic similarity (Fig. 8A). Therefore, larger jellyfish swim disproportionately slower than smaller jellyfish, and drag consequently scales at a lower factor than predicted by kinematic similarity (Fig. 7C). This low scaling factor for drag occurs despite the fact that the projected area of the bell scales with a higher factor than predicted by geometric similarity (Table 1). According to our model, the scaling of speed is influenced by the ontogenetic decrease in pulse frequency (Fig. 6B). When pulse frequency was held constant in our simulations, speed increased rapidly with increasing body mass (Fig. 9C). This suggests that it is the rate, not the magnitude, of force production that caused speed and drag to scale by a factor below that predicted by kinematic similarity.

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

Our results suggest that ontogenetic changes in the body shape and pulse frequency of Aurelia influence a tradeoff between swimming speed and the cost of transport. By changing from a prolate to oblate bell shape, Aurelia grows from a shape of relative high speed and high cost to one of low speed and low cost (Fig. 9A,B). However, the difference in mean performance between allometric and isometric growth is subtle, when compared to the effects of pulse frequency. By reducing pulse frequency over ontogeny, Aureliaavoids the dramatically high cost of transport predicted for large jellyfish that maintain a high pulse frequency (Fig. 9D), but this decrease in frequency comes at a cost to speed(Fig. 9C).

Our results support the hypothesis that medusae change their behavior and morphology to maintain a moderate speed while avoiding a high cost of transport. However, it is difficult to weigh the relative importance of speed and the cost of transport to the ecology and evolution of Aureliawithout an understanding of how these aspects of performance influence fitness(Koehl, 1996). Furthermore,ontogenetic changes in behavior and morphology may influence other aspects of organismal performance that have a greater effect on fitness than does locomotion. For this reason, it would be interesting to explore how these ontogenetic changes affect prey capture and mass transport.

The effect that the shape of an organism has on its mechanical performance is likely to change during growth because biomechanical forces are typically scale dependent. Therefore, changes in shape may reflect a shift in the mechanical demands of larger structures. Alternatively, morphology may change in order to meet other physiological demands or because of developmental constraints. In the interest of understanding why organisms change or preserve shape during growth, it is useful to compare the ontogenetic changes in morphology that organisms exhibit to alternate patterns of growth. For example, Gaylord and Denny(1997) used a mathematical model to examine how different patterns of growth in seaweeds (Eisenia arborea and Pterygophora californica) affected their susceptibility to breakage in intertidal habitats. They found that the allometric growth exhibited by these organisms results in lower material stress than if the seaweeds maintained their juvenile shape over their life history. Similarly, we found that jellyfish have a lower energetic cost of swimming by changing their bell shape over ontogeny than if they maintained the prolate shape of juveniles (Fig. 9B). In aquatic animals, ontogenetic change in performance may be strongly influenced by changes in behavior. For example, we found that changes in pulse frequency have an even stronger effect on performance than the changes in bell shape (Fig. 9).

It would be difficult to explore alternate patterns of growth or to tease apart the individual effects of morphological and kinematic 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 by measurements. Therefore, the integration of theoretical and experimental approaches should greatly facilitate our understanding of how ontogenetic changes in morphology or behavior affect organismal performance.

List of symbols
 
• a

  scaling constant

 
• A

  acceleration reaction force

 
• b

  scaling exponent

 
• b_o

  null hypothesis

 
• c_bell

  drag coefficient for the bell

 
• c_margin

  drag coefficient for the bell margin

 
• d

  bell diameter

 
• d_range

  range of bell diameter

 
• d_rest

  resting bell diameter

 
• D

  drag

 
• e

  slope of major axis regression

 
• f

  pulse frequency

 
• F

  force to change body inertia

 
• h

  bell height

 
• h_cav

  height of subumbrellar cavity

 
• h_range

  range of bell height

 
• h_rest

  resting bell height

 
• i

  index of measurements

 
• k

  sawtooth function

 
• l

  length of the bell margin

 
• L₁

  lower limit of 95% confidence interval

 
• L₂

  upper limit of 95% confidence interval

 
• m

  body mass

 
• n

  number of measurements

 
• q

  duty factor

 
• s_bell

  projected area of the bell

 
• s_margin

  area of the bell margin

 
• t

  time

 
• T

  thrust

 
• T_HCOT

  hydrodynamic cost of transport

 
• T_jet

  thrust generated by a jet

 
• T_paddle

  thrust generated by a paddle

 
• u

  swimming speed

 
• U

  body velocity

 
• v

  volume of the subumbrellar cavity

 
• w

  speed of the bell margin

 
• W

  velocity of the bell margin

 
• x

  net distance

 
• y

  dependent variable

 
• α

  added mass coefficient

 
• θ

  margin angle

 
• θ_range

  range of margin angle

 
• ρ_tissue

  density of tissue

 
• ρ_water

  density of seawater

We thank M. Koehl and T. Daniel for their guidance and advice, and A. Summers, W. Korff, J. Nauen and J. Strother for their suggestions. This work was supported with an NSF predoctoral fellowship and grants-in-aid of research from the American Society of Biomechanics, Sigma Xi, the Department of Integrative Biology (U.C. Berkeley) and the Society for Integrative and Comparative Biology to M.J.M. Additional support came from grants from the National Science foundation (# OCE-9907120) and the Office of Naval Research(AASERT # N00014-97-1-0726) to M. Koehl.