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First published online May 21, 2007
Journal of Experimental Biology 210, 1868-1873 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.003772
Morphological diversity of medusan lineages constrained by animal–fluid interactions
1 Graduate Aeronautical Laboratories and Bioengineering, California
Institute of Technology, Pasadena, CA 91125, USA
2 Environmental Sciences, Roger Williams University, Bristol, RI 02809,
USA
3 Biology, Providence College, Providence, RI 02918, USA
* Author for correspondence (e-mail: jodabiri{at}caltech.edu)
Accepted 11 March 2007
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Summary |
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Key words: locomotion, biomechanics, fluid dynamics, medusae
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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; Vogel,
2003). Central to understanding the importance of evolved
morphological features of these animals is the determination of how the animal
interacts with the surrounding fluid (i.e. water or air). Jellyfish are the
earliest known metazoans to use muscle power for swimming
(Valentine, 2004). As
cnidarians, they are limited in the morphological structures available for
propulsion (Gladfelter,
1972a,b;
Chapman, 1974). Nonetheless,
they are often identified as highly proficient jet propellers
(Donaldson et al., 1980;
Mackie, 1980). Jet propulsion
is an energetically expensive, but highly effective, mode of propulsion
(Daniel, 1985).
Prevailing models of jet propulsion describe the forces that medusae use
for swimming to be generated solely during the swimming power stroke. The jet
propulsive force is generated by the contraction of the circular muscle fibers
lining the surface of the subumbrellar cavity, exerting pressure
(force/subumbrellar area) on the fluid in the cavity and forcing it out of the
bell. Following bell contraction, a single vortex ring is formed in the wake
(termed the `starting' vortex) and the momentum imparted to the fluid during
this power stroke provides the force available for forward motion
(Fig. 1A; see also Movie 1 in
supplementary material) (Dabiri et al.,
2006). Therefore, the force available for thrust is directly
related to, and even less than (e.g. due to necessary elastic storage and
mechanical losses) (DeMont and Gosline,
1988a; DeMont and Gosline,
1988b; DeMont and Gosline,
1988c), the force of the contracting muscle fibers.
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). In
addition, they limit the force that medusae can generate for jet propulsion
because the muscular contractile forces required to achieve jet propulsion do
not scale favorably with increasing medusa size
(Bonner, 1965;
Valentine, 2004).
Specifically, the mass of water expelled from a medusa bell with diameter
D scales as D3, whereas the muscle force required
to accelerate that mass only scales as D1 [i.e. muscle
force
muscle cross-sectional area=muscle thickness (a
constant)xmuscle sheet width (D) (Gladfelter, 1974;
Chapman, 1974)]. Therefore the
force required for jet propulsion increases with animal size more rapidly than
the available physiological force.
In this paper, we use models that compare the forces produced by medusae to the forces required for propulsion, in order to investigate how fluid interactions may constrain medusan bell morphology across the extant species. The developed analytical models are validated by comparison with in situ field and laboratory observations of freely swimming Aurelia aurita and juvenile Aequorea victoria medusae, respectively. These species have been selected for wake studies because they represent distinct regions of medusan morphospace. In particular, Aurelia is typically larger and more oblate (flatter) while juvenile Aequorea is smaller and prolate (torpedo-shaped). To complement the studies of wake kinematics, we also examine morphological data collected from measurements of 660 extant species reported in the literature. The following section describes the experimental and analytical methods.
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Materials and methods |
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)
using a 720x480-pixel CCD camera connected to a PC. The wake flow was
made visible by injecting fluorescein dye around the medusa prior to bell
contraction. Measurements of Aurelia aurita (Linnaeus 1746), an oblate scyphomedusa, were collected from a marine lake (145 hectares, maximum depth 46 m) on the island of Mljet, Croatia, located in the Adriatic Sea (latitude: 42.75°N, longitude: 17.55°E) during July 2003. All of the video was recorded in shallow water (<20 m) by SCUBA using natural light. Video was recorded at 30 frames s–1 on miniDV videotape using a Sony DCR VX2000 camera with a zoom lens contained within an Amphibico underwater housing (Amphibico, Inc., Montreal, QC, Canada). A second diver injected 20 µl pulses of concentrated fluorescein dye into the water at specific locations around the medusae.
Mathematical derivation of morphological diversity models
The proposed morphological diversity model is based on an expression for
the net time-averaged locomotive force required for swimming:
 | (1) |
The force FJ during bell contraction is calculated
using the model of Daniel (Daniel,
1983):
 | (2) |
D2/4 is the oral cavity exit area and
V is the volume of the oral cavity. Approximating the oral cavity
volume as a hemiellipsoid, its volume is related to the bell height H
and diameter D as V=(/6)HD2. Hence
its time-derivative is:
 | (3) |
 | (4) |
;
Dabiri et al., 2006):
 | (5) |
is the vortex ring circulation, a measure of its strength. The vortex
ring enclosed area AV can be approximated by the oral
cavity area A=D2/4, since the vortex is
formed at the bell margin and remains attached to the body until the end of
the relaxation phase. The vortex ring circulation can be calculated by
applying the slug model (Didden,
1979) to the flow at the bell margin, where the characteristic
flow velocity is given by the bell motion dD/dt, i.e.
(dD/dt)2TR.
Combining these results in Eqn
5:
 | (6) |
 | (7) |
 | (8) |
FM),
 | (9) |
M, subumbrellar muscle sheet
thickness 
M, and muscle sheet width, which has been observed
to be roughly one half of the bell diameter D (Gladfelter, 1972):
 | (10) |
,b;
Ford and Costello, 2000). For
simplicity and generality, we currently utilize the function
(D/4)(3+cos
t), where is the swimming
frequency in rad s–1 and the swimming cycle occurs over a
period of duration T, i.e.
0tT=2/. Functions that more faithfully
represent the bell motion of each species could also be used here. However, we
will show that this simple approximation is sufficient to explain the observed
trends in the morphological data without appealing to the detailed kinematics
of each species. Substituting this function into
Eqn 9 gives:
 | (11) |
) is given by the
time average of [D(dD/dt)]2 over the
duration of one swimming cycle. For the trigonometric function used presently,
g() is given by:
 | (12) |
 | (13) |
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Results |
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, FM must be greater than or equal
to FJ. The parameter FM is the product
of muscle cross-section area and the isometric stress of the muscle tissue;
therefore, its magnitude is a function of bell size and shape. Likewise, for
jet propulsion, FJ is related to the volume of the
subumbrellar cavity, the oral cavity exit area, and their respective changes
with bell size and shape (Daniel,
1983). Fig. 2A
plots the size-limiting curve FM=FJ
for medusa shape (quantified by the fineness ratio f=bell height
H/bell diameter D) versus bell diameter D
for various swimming frequencies. This curve is generated by plotting
Eqn 11 while neglecting the
first term in parenthesis, which arose due to the presence of the stopping
vortex. These limiting upper-bound curves illustrate that for most bell shapes
(i.e. fineness ratios), medusae larger than 10 cm cannot produce a sufficient
muscle force to swim via jet propulsion.
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), apparently violating the predicted size limits for jetting
medusae. The present field observations of Aurelia show that oblate
medusae (i.e. low fineness ratios) create more complex wake structures than
those of more prolate jetting medusae (i.e. high fineness ratios;
Fig. 1A), and swim with a
combined jet-paddling mode of propulsion (also termed rowing;
Fig. 1B; see also Movie 2 in
supplementary material) (Dabiri et al.,
2005). In jet-paddling medusae, the contraction phase generates a
starting vortex similar to that of traditional jetting medusae. However,
during the relaxation phase the paddling motion of the bell causes the
formation of a second, stopping vortex ring with opposite rotational
orientation relative to the starting vortex.
Including the effect of the stopping vortex in
Eqn 1 and using average
physiological and kinematic values
(Gladfelter, 1972a;
Gladfelter, 1972b;
Anderson and Schwab, 1981;
Bone and Trueman, 1982) for
M (160 kPa), M (3.5 µm) and
(, 2 and 4 rad s–1),
Fig. 2B plots the predicted
morphological distribution of fineness ratio versus bell diameter for
all medusae swimming at various bell contraction frequencies (i.e.
Eqn 11). The new model predicts
that the bell diameter for oblate jet-paddling medusae is not constrained by
physiological limits, as prolate jetting medusae are. In other words,
according to the model, medusae with lower fineness ratios create sufficient
stopping vortices during bell relaxation to effectively reduce the forces
required for locomotion. Consequently, oblate medusae of any size are able to
generate sufficient muscle forces to swim. Interestingly, the model predicts
that across all medusan lineages there exists a universal critical fineness
ratio fCRIT below which medusae of any bell diameter can
exist,
fCRIT=
0.265.
If these medusae–fluid interactions are influential in constraining
the evolution of bell morphology we would expect to observe a similar
relationship between bell diameter and fineness ratio for most extant medusan
species. A medusan morphospace, compiled from average bell diameter and height
values of all of the medusae published or illustrated in monographs describing
hydromedusae (Kramp, 1959;
Kramp, 1961;
Kramp, 1968), scyphomedusae
(Mayer, 1910), or
siphonophores (Pugh, 1999),
reveals that the relationship between bell fineness and diameter is not random
and that bell shape is highly dependent on bell size
(Fig. 3; see Table S1 in
supplementary material). At small bell diameters (<50 mm), bell shape is
highly variable between oblate and prolate forms. However, at larger bell
diameters (>200 mm), only oblate bell forms exist in nature. As a result,
there is a large region of potential bell size and shape combinations
unoccupied by any medusa.
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Finally, plotted on a log–log scale with H on the ordinate
axis and D2+C/D on the abscissa, the model (i.e.
Eqn 13) predicts that the
morphological data for bell height versus diameter should lie on a
line with slope=1/2, i.e.
logHlog(D2+C/D).
Fig. 5 compares this model with
the morphological data. The agreement is reasonable, although a least-squares
fit to the data suggests a smaller slope of 0.37. Most of the discrepancy
occurs at large bell diameters, where the model assumption that
dH/dt
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Discussion |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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), is directly
proportional to the amount of rotational motion in the medusa wake. The
stopping vortex interacts with the contraction-phase starting vortex of the
next swimming cycle through mutual cancellation of their opposite-signed
vorticity and through the velocity fields that they induce upon one another
(i.e. Biot–Savart induction)
(Saffman, 1992). These
interactions reduce the total rotational motion in the water behind the animal
(Fig. 1B; see also Movie 2 in
supplementary material and Figs 2 and 3)
(Dabiri et al., 2005). Hence,
motion-canceling between the starting and stopping vortices in the
jet-paddling swimming mode acts to reduce the kinetic energy lost in the wake,
thereby increasing the swimming efficiency. This is in contrast with
conventional jet-propulsion, which suffers from reduced hydrodynamic
efficiency due to the large quantity of kinetic energy that is deposited into
the wake during thrust production (Vogel,
2003) (cf. Fig.
1A). Interestingly, a similar energy-recovery mechanism has been
identified in the swimming of bony fishes
(Ahlborn et al., 1991;
Ahlborn et al., 1997). However,
the existence of such energy-saving behaviors has not previously been
appreciated in lineages as primitive as medusae.
The developed model is especially useful because of the small number of
input parameters required to make predictions. Nonetheless, it relies on a
quasi-steady approximation of transient swimming dynamics and muscle
mechanics. There may potentially exist variations in muscle performance across
medusan lineages due to differences in muscle myosin isoforms, twitch
durations, shortening velocities, sarcomere geometries, etc. Models explicitly
incorporating these effects have been shown to require a large number of input
parameters (Daniel, 1995),
which detracts from the goals of the present model. However,
Fig. 4B shows that even if the
combined effect of these variations were to change the nominal physiologically
available force computed above by an order of magnitude (i.e.
0.1FM or 10FM), the predicted
morphological distribution would be relatively unaffected. Mathematically,
this robustness of the model follows from the relatively weak (i.e.
square-root) dependence of the limiting curves on the physiological force (cf.
Eqn 11). To be sure, an order of
magnitude variation in the physiological force FM is
highly unlikely, given that cnidarians are limited to a single cell layer of
muscle. We hypothesize that the outliers in the data exhibit departures from
the assumed bell kinematics rather than the bell mechanics. However, the
present data are insufficient to resolve this question conclusively.
In conclusion, the present model and the supporting morphological data indicate that animal–fluid interactions provide organising principles for the most primitive form of muscle-powered locomotion. The physiological limits of medusan force generation have determined the permissible range of medusan design solutions to swimming in a fluid environment, and in general, two solutions have emerged: medusae may be either small and jet-propelled or oblate and propelled by jet-paddling.
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Acknowledgments |
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Footnotes |
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References |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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