## SUMMARY

It is not well understood how ontogenetic changes in the motion and
morphology of aquatic animals influence the performance of swimming. The goals
of the present study were to understand how changes in size, shape and
behavior affect the hydrodynamics of jet propulsion in the jellyfish
*Aurelia aurita* and to explore how such changes affect the ontogenetic
scaling of swimming speed and cost of transport. We measured the kinematics of
jellyfish swimming from video recordings and simulated the hydrodynamics of
swimming with two computational models that calculated thrust generation by
paddle and jet mechanisms. Our results suggest that thrust is generated
primarily by jetting and that there is negligible thrust generation by
paddling. We examined how fluid forces scaled with body mass using the jet
model. Despite an ontogenetic increase in the range of motion by the bell
diameter 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,
and speed consequently scaled at a lower exponential rate than predicted by
kinematic similarity. Model simulations suggest that the allometric growth in
*Aurelia* results in swimming that is slower, but more energetically
economical, than isometric growth with a prolate bell shape. The decrease in
pulse frequency over ontogeny allows large *Aurelia* medusae to avoid a
high cost of transport but generates slower swimming than if they maintained a
high pulse frequency. Our findings suggest that ontogenetic change in the
height-to-diameter ratio and pulse frequency of *Aurelia* results in
swimming that is relatively moderate in speed but is energetically
economical.

## Introduction

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.

### The ontogenetic scaling of 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.

### Geometric and kinematic similarity

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
*d*^{3}, 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 *d*
and 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 (∝*d*^{2}) and the square of swimming speed
(∝*d*^{2}; Daniel,
1983), which suggests that this force scales as
*d*^{4} 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 (∝*d*^{6}), suggesting that thrust also scales as
*d*^{4} and *m*^{4/3}. The acceleration reaction
varies with the volume of the body (∝*d*^{3}) and its rate
of change in velocity (∝*d*, assuming sinusoidal changes in
velocity), which implies that this force also scales as *d*^{4}
and *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.

### The present study

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 *Aurelia*
was 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).

## Materials and methods

### Kinematics and morphometrics

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, *L*_{1},
and upper, *L*_{2}, 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,
*r*^{2}, was used to assess the degree of variation explained
by reduced major axis regressions.

### Hydrodynamic forces

As in Daniel (1983), we
modeled the hydrodynamics of jellyfish swimming as the sum of thrust,
**T**, drag, **D**, the acceleration reaction force, **A**, and the
force acting to change the inertia of the body, **F**. This model is
expressed in an equation of motion as:
1
Drag was calculated with the following equation
(Batchelor, 1967):
2
where *s*_{bell} is the instantaneous projected area of the
bell (*s*_{bell}=0.25π*d*^{2}, where
*d* varies with time), *u* and **U** are the instantaneous
speed and velocity of the body, respectively, ρ_{water} is the
density of seawater, and *c*_{bell} is the drag coefficient of
the bell. We assumed that the drag on the bell was equal to that of a
hemisphere, which has a greater drag coefficient when the body is moving
backward (i.e. **U**<0) than when moving forward (i.e. **U**>0)
(Hoerner, 1965):
3
The acceleration reaction was calculated with the following equation
(Daniel, 1983):
4
where α is the added mass coefficient, *v* is the volume of the
subumbrellar cavity, and *t* is time. Assuming a hemi-ellipsoid shape
to the subumbrellar cavity (as in Daniel,
1985), the instantaneous cavity volume was calculated as follows:
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):
6
In separate simulations, we modeled the thrust generation by jet and paddle
mechanisms (Fig. 1B,C). The
thrust generated by jetting, **T**_{jet}, was calculated with the
following equation (Daniel,
1983):
7
Negative thrust (i.e. thrust acting to impede forward motion) was generated by
this force whenever the cavity volume increased. The thrust generated by
paddling, **T**_{paddle}, was calculated using the following
equation:
8
where *c*_{margin} is the drag coefficient for the bell margin,
*w* and **W** are the speed and velocity, respectively, of the bell
margin at the midpoint between its proximal base and distal tip, and
*s*_{margin} is the area of the inside surface of the margin.
Since the margin of the bell is a flattened strip of tissue, we modeled it as
a flat plate oriented perpendicular to flow (*c*_{margin}=1.98;
Hoerner, 1965). Note that the
thrust generated by paddling may also contribute to drag at instances where
the velocity of the margin is directed in the aboral direction. We calculated
the area of the margin with the following equation for the area of a ring:
9
where *l* is the length of the bell margin. These hydrodynamic forces
acted against the force to change the inertia of the body, which was
calculated with the following equation:
10

### Computational modeling

Numerical solutions for swimming speed were calculated using the equation
of motion (equation 1) with the measured kinematics of the bell and its margin
as input variables. Speed was calculated with a variable order
Adams-Bashforth-Moulton solver for integration
(Shampine and Gordon, 1975)
programmed in MATLAB (version 6.0; Mathworks). Our kinematic equations used a
sawtooth function, *k*(*t*), which increased linearly over time,
*t*, from values of 0 to 1 over the duration of the pulse phase, then
decreased linearly from 1 to 0 over the recovery phase. This function provided
an input to the following equations, which we used to describe the motion of
the margin angle, bell height and bell diameter:
11
12
13
The mean (time-averaged) speed was calculated from simulations using both the
jet model (i.e. **T**=**T**_{jet}) and the paddle model (i.e.
**T**=**T**_{paddle}) for each jellyfish that we had measure
kinematic and morphometric data. Measurements of mean speed were taken over
the duration of five propulsive cycles that followed the three cycles that
were necessary for the models to reach steady state.

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 addition to measuring speed, the hydrodynamic cost of transport,
*T*_{HCOT}, was calculated to assess the performance of each
simulation. The hydrodynamic cost of transport is a measure of the total
energy invested in jetting to propel a unit mass of the body over a unit
distance. It was calculated with the following equation:
14
where *i* is the index for each of the *n* measurements of speed
and the magnitude of thrust, and *x* is the net distance traversed over
the duration of a swimming sequence. This measure of energetic economy
neglects internal costs and therefore provides a minimum estimate of metabolic
economy.

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.

## Results

### Jet versus paddle propulsion

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, *L*_{1}=-0.015,
*L*_{2}=0.019, *N*=19;
Fig. 4A). Although the jet
model tended to overestimate measured speed (*e*=1.86,
*L*_{1}=1.28, *L*_{2}=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).

### The scaling of morphology, kinematics, hydrodynamics and performance

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
(*r*^{2}=0.05).

### The effects of morphology and behavior on performance

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^{-1}
m^{-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^{-1}
m^{-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^{-1}
m^{-1}).

## Discussion

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).

### The hydrodynamics of jellyfish swimming

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 *Aurelia*
suggests 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 scaling of swimming performance

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, *Aurelia*
avoids 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 *Aurelia*
without 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.

### Performance and growth

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
*L*_{1}- lower limit of 95% confidence interval
*L*_{2}- 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

## ACKNOWLEDGEMENTS

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.

- © The Company of Biologists Limited 2003