702
Modes and scaling in aquatic locomotion
Steven Vogel1
Biology Department, Duke University, Durham, NC 27708, USA
Synopsis Organisms spanning a 107-fold range in length of the body engage in aquatic propulsion—swimming; they
do so with several kinds of propulsors and take advantage of several different fluid mechanical mechanisms.
A hierarchical classification of swimming modes can impose some order on this complexity. More difficult are the issues
surrounding the different kinds of propulsive devices used by different organisms. These issues can be in part exposed
by an examination of how speeds and accelerations scale with changes in body length, both for different lineages
of swimmers and for all swimmers collectively. Clearly, fluid mechanical factors impose general rules and constraints;
just as clearly, these only roughly anticipate actual scaling. Indeed, collections of data on scaling can serve as
useful correctives for assumptions about functional mechanisms. They can also reveal size-dependent constraints on
biological designs.
Introduction
All powered flight in nature operates in essentially
the same manner. One or two pairs (four pairs in
copulating dragonflies) of broad lateral appendages
move transversely to the direction of the animal’s
motion. Their reciprocating motion generates fluidmechanical lift, from which the animal derives both
lift and thrust. With the possible (or partial)
exception of the smallest insects, all active fliers
operate at Reynolds numbers high enough for
effective lift-based propulsion (Walker and
Westneat 2000), roughly above 10. With the possible
exception of some extinct forms, all operate at
Reynolds numbers below about 1,000,000, a limit
most likely imposed by the scaling of wing loading
and minimum flying speed (Chatterjee et al. 2007).
Aquatic locomotion contrasts sharply with this
simple situation. Many lineages, far more than the
four for flight, have invented powered swimming.
The range of Reynolds numbers ranges from about
106 (bacteria) to 108 (large whales)—14, rather than
five, orders of magnitude. Most notably, the modes
of propulsion are far more diverse, arguably more
diverse than those yet explored by human technology. Explaining the difference presents no apparent
problem. It turns on the combination of a greater
underlying diversity of aquatic animals, as opposed
to terrestrial ones, and the difference in density of
water and air, one nearly that of the organisms
themselves, the other far lower and demanding
continuous production of lift. What matters here
are the implications of that diversity of modes of
aquatic locomotion, in particular the physical constraints that determine which mode is used and by
whom, and how modes, speeds, and accelerations
change with size.
As a cautionary note, one must bear in mind the
existence of biological as well as physical constraints;
animals of necessity carry what might be termed
‘‘ancestral baggage.’’ Some designs require less modification of preexisting structure or instructions.
For instance, muscle-wrapped pipes of digestive or
circulatory function might provide preadaptive routes
to jet propulsion. Some groups of animals appear
ignorant of potentially useful devices; thus, arthropods
do not use cilia for locomotion. More difficult to
delineate are developmental constraints, but these
must nonetheless play some limiting role.
Simple criteria of maximum speed or locomotory
efficiency provide no sure criteria of functional superiority. The latter must depend on the overall
ecological context in which an animal lives. Thus,
niche-dependent biological factors may favor maximization of acceleration or maneuverability; or
minimization of structural investment, of acoustic or
hydrodynamic disturbance, or of visual profile. In
short, the incomplete success of fluid-dynamic criteria
to rationalize the distribution of locomotory modes
cannot be blamed solely on limited understanding of
the fluid dynamics of aquatic locomotion!
From the symposium ‘‘Going with the Flow: Ecomorphological Variation across Aquatic Flow Regimes’’ presented at the annual meeting of the
Society for Integrative and Comparative Biology, January 2–6, 2008, at San Antonio, Texas.
1
E-mail: [email protected]
Integrative and Comparative Biology, volume 48, number 6, pp. 702–712
doi:10.1093/icb/icn014
Advanced Access publication April 11, 2008
ß The Author 2008. Published by Oxford University Press on behalf of the Society for Integrative and Comparative Biology. All rights reserved.
For permissions please email: [email protected].
703
Scaling of swimming
Modes of aquatic locomotion
One might begin with some classification of how
animals move about within, or on, bodies of water.
The following scheme makes no pretense of being
definitive nor does it carry canonical aspirations
(other schemes are described by Daniel and Webb
1987). A number of rather odd modes, most notably
among the protists, have been ignored.
(I) Drag-based swimming
(A) Low-Re modes in which drag varies directly
with viscosity, velocity, and some characteristic length.
(1) Flagellar, with either bacterial or eukaryotic flagella that are long relative to the
body and few in number.
(2) Ciliary, with similar appendages but
short relative to the body and more
numerous.
(a) With the body entirely or nearly entirely
covered with cilia.
(b) With cilia limited to tracts and covering
less than half the body.
(3) Setal paddles: movable appendages with
passive hair-like protrusions.
(B) Moderate and high Re modes in which drag
varies with density, some characteristic area,
and approximately the square of velocity.
(1) Paddling with a single bilateral pair of
appendages.
(2) Paddling with serially arranged bilaterally
paired appendages.
(a) Moving with simultaneous sweeps.
(b) Moving with rear-first metachronal
waves of sweeps.
(II) Lift-based swimming
(A) With paired lateral propulsors, e.g., wings,
fins.
(B) With a single caudal propulsor such as tail or
flukes.
(C) By passing waves posteriorly along an elongate trunk.
(III) Direct-reaction swimming—with pulsating jet or
paired jets.
(IV) Interfacial swimming
(A) Using surface tension for both support and
propulsion.
(1) By pushing downward and rearward on
the surface.
(2) By reducing surface tension posteriorly
with surfactant.
(B) Using a hull that displaces its own weight of
water.
(1) And
propelling
paddling.
with
drag-based
(2) And propelling with lift-based hydrofoils.
(C) Using dynamic support contingent on
movement.
(1) Support by aquaplaning.
(2) Support by repeated ‘‘slapping’’ against
the surface.
The low Reynolds-number world
For practical purposes, thrust cannot be produced by
movement of some propulsor normal to the direction of progression. Thus, while a diversity
of schemes find use, all depend on production of
drag in the direction opposite that of progression, in
effect antidrag or thrust. That effective motion
commonly pairs with some form of recovery
motion, either alternately, as in conventional ciliary
beating, or simultaneously, as with the 2D or 3D
progression of waves along the lengths of eukaryotic
flagella. Not only is some recovery motion usually
necessary, but an uncongenial fluid-mechanical
regime limits drag-minimization during that recovery; the limit occurs roughly at a recovery drag half
as great as the thrusting antidrag.
Among the eukaryotes, the old division between
flagellates (Mastigophora) and ciliates (Ciliophora)
survives our contemporary realization that their
propulsive organelles use the same operative
mechanism. Flagellates are considerably smaller and
ordinarily have two to four relatively long organelles.
Ciliates are typically larger, with surfaces wholly or
largely covered with a dense pelage of relatively short
organelles.
What precludes both lift-based propulsion and an
agreeably high ratio of useful drag to recovery drag
is, of course, viscosity. The latter acts as a purely
dissipative agent, just as friction does between solids,
except that it yields less willingly to technological
remedies such as lubrication. In this viscous domain,
drag (good and bad) varies directly both with a
characteristic length of a body and with the speed of
motion. That suggests some simple scaling rules.
(1) For a flagellate, assume a propulsive force
proportional to flagellar length. If both body drag
and net flagellar antidrag vary directly with length,
704
Fig. 1 Body length versus swimming speed for flagellates and
ciliates (from Brennen and Winet 1977) and for Vibrio harveyi
(from Mitchell et al. 1995).
then speed should be constant, as will be the relative
investment in such swimming machinery. (2) For a
ciliate, assume a propulsive force proportional to
surface area. Thrust should then vary with length
squared, while resistance should vary only with
length to the first power. Thus, the bigger organisms
should be faster, with speed increasing directly with
body length.
Neither expectation quite matches reality. As one
can see from Fig. 1, while ciliates go faster than
flagellates, at least in this sampling, neither group
shows a clear relationship between size and speed. For
the combined collection, speed does seem to increase
with size, as just suggested for ciliates, but with an
exponent of 0.73 rather than 1.00 (51.00 with 95%
confidence interval). The figure strongly suggests
that the common assertion that, where sizes overlap,
ciliates go faster (10 times so is often cited), deserves
reappraisal—or quiet oblivion. The figure agrees with
one’s impression from direct observation of living
material that flagellates, even allowing for artifacts of
observational magnification, can be quite speedy.
The two groups do seem to differ in speed relative
to body length, as shown in Fig. 2, which uses the
same data set. Larger is relatively slower for both, but
they lie on different regression lines, with ciliates
marginally better in the area of size overlap. Especially large examples of each are slow, perhaps a
general phenomenon.
A marine bacterium, Vibrio harveyi (Mitchell et al.
1995), while reaching impressive speeds of over
100 mm/s, does not appear out of line with the data
for protists and the implication that larger is slower,
relative to body length. Another bacterium,
Bdellovibrio bacteriovorus, has been reported to go
as fast, ‘‘colliding’’ with its prey (another bacterial
S. Vogel
Fig. 2 The data from Fig. 1, now plotted as body length versus
length-specific swimming speed.
cell)—an odd choice of gerund by Wikipedia for
something with negligible momentum!
One might ask two interrelated questions. First,
why does speed relative to length fail to keep pace
with increase in size? Second, if cilia yield no speed
advantage, why are the larger forms covered with
cilia rather than utilizing a few flagella? Many short
appendages bring with them potential inefficiencies
of interaction and a requirement for a steeper
velocity gradient (to which viscosity does not take
kindly) at the surfaces of the organisms. One
common answer for both questions notes the constant diameter of cilia and flagella (unless bundled;
rare when used for locomotion) of about 0.25 mm.
Longer ones protruding normal to a flow will be
more prone to passive bending and will, as a result,
face limits on practical motion speeds. Similarly, the
limited flexural stiffness of the organelles will limit
effective transmission of force back to the parent
body, which reduces their effectiveness. The same
argument can rationalize the limited use of prokaryotic flagella, only 20 nm in diameter, either as selfproduced or symbiotic (as the spirochaetes on
Mixotricha, for instance) propulsors for all but the
very small.
Another explanation points to a peculiar advantage of that steep velocity gradient. By limiting the
extent of fluid disturbance around the organism,
both predators and prey receive less information
about the organism’s presence; see, for instance, a
tracing (Fig. 3) of a pair of photographs by Wu
(1977). A third explanation is that the steeper gradient gives a swimming organism better diffusive
access to dissolved material in its vicinity.
These microorganisms (as well as spermatozoa)
can, and often do, increase swimming speed by
forming tight swarms, just as dense groups of
705
Scaling of swimming
Fig. 3 Comparison of flow patterns (as streaks representing
equal elapsed times) for a ciliate, Paramecium, swimming (above)
and in gravitational free fall, both through clouds of particles.
The figure is a rough trace of photographs presented, without
further explanation, by Wu (1977).
nonmotile particles sink or rise more rapidly than do
isolated ones. The effect can be substantial; Mitchell
et al. (1995) measured speeds up to 440 mm/s for
groups of Vibrio harveyi but only up to 140 mm/s for
individuals (Re ¼ 4 104). The effect, notable at
low Reynolds numbers, amounts to a converse of the
wide hydrodynamic disturbance, whereby conspecifics draw each other along rather than drawing
predators. It apparently does not reach significance
at somewhat higher Reynolds number; Jiang et al.
(2002) suggested no interactive benefit for copepods
at Re ¼ 3. (While perhaps analogous in effect, its
fluid-mechanical basis is distinct from that of
formation flight in birds.) Clearly, group advantage
should be taken into account when considering their
ecologies.
In short, flagellar and ciliary swimming systems do
not scale well. Eukaryotic flagella have a minimum
length of about 5 mm, set by the minimum possible
radius of curvature of a flagellum of about 2 mm
(Sleigh and Blake 1977). Ciliated swimmers of
whatever lineage do not ordinarily exceed a few
millimeter in length, most likely due to the nearly
fixed flexural stiffness of organelles that cannot
increase in diameter.
Nonetheless, the basic physical scheme proves
useful for larger animals. Thus, microcrustaceans and
aquatic insects use muscle to power locomotory
systems analogous to that of a stroke-and-recover
ciliate such as Paramecium or a breast-stroking
flagellate such as Chlamydomonas. A cilium may
alter its flexural stiffness between power and recovery
strokes; setae accomplish the same thing passively,
typically through the arrangements of their articulations. Thus, a seta-bearing appendage can be swung
back and forth with setae extended during one
half-stroke and flexed during the other, with at least
as high a drag ratio between the two half-strokes.
The system may be less easily reversible than a ciliary
one, but it runs into no fundamental limitation by
size. For very low Reynolds numbers, a plane of
splayed setae works adequately. At somewhat larger
scale, leakiness (sieving, to put it positively; Koehl
1995) can be kept in check with closer-fitting
or flattened appendages, as noted by Nachtigall
(1980) for large aquatic beetles such as Dytiscus
(Re ¼ 15,000). With the addition of webbing,
no Reynolds number should be too high for effective,
analogous, drag-based paddling. Indeed, it gets
better as recovery strokes get less wasteful
(Williams 1994), i.e., as the ratio of power-stroke
drag to recovery-stroke drag increases. Only the
development of better, lift-based, propulsive schemes
limits its use at high Reynolds numbers.
The extensive use of such setae by microcrustaceans under conditions not much different from
those of large ciliates might, of course, represent
making the best of an odd disability. As noted
earlier, cilia and flagella are remarkably uncommon
among arthropods despite their near-ubiquity elsewhere. We know of some classic 9 þ 2 flagellar
structures, but nonmotile ones, in their sense organs,
and a few properly motile flagella in insect sperm
(Alexander 1979).
Size and speed among the
‘‘not-so-small’’
At moderate and high Reynolds numbers, larger
animals generally swim faster than do smaller ones,
706
much as we saw for very low-Re swimmers. Does the
behavior follow some obvious scaling rule? If thrust
varies with area (muscle cross-section) and drag
varies with area as well, then top speed, V, should
vary with L0, i.e., length squared over length squared,
which is to say that it should be size invariant. Also,
speed relative to length should vary inversely with
length. The big fish should not be able to catch the
small one, particularly if the small one has any
advantage in maneuverability. This clearly clashes
with reality. It assumes a constant drag coefficient,
tacitly alluding to bluff bodies. With effective
streamlining, drag might vary with L1.5. In effect,
the drag coefficient, rather than remaining constant
over the range of Reynolds number, then varies
inversely with its square root and thus with (LV)0.5.
If so, then speed should be proportional to the
cube root of length; V ! L0.33 (or V/L ! L0.67).
Now, the large fish catches the small one unless the
small one remains cryptic, hides, outmaneuvers the
large one, or in some way gains advantage from
schooling behavior.
The exponent of 0.33 has at least rough (very
rough) empirical support. For instance, one can
derive a scaling exponent of 0.59 (r2 ¼ 0.61) for
maximum speed from the data collected by
Domenici and Blake (1997) for fishes ranging from
0.049 to 0.63 m in body (Fig. 4). Videler and Nolet
(1990) gave an exponent of 0.27 for optimal-fortransport-cost speeds (rather than maximum speeds)
of vertebrate swimmers, from goldfish (0.027 m)
to gray whales (11.5 m), but Gallon et al. (2007)
noted that the relationship does not hold for marine
mammals taken as a specific group. Including marine
Fig. 4 Body length versus maximum reported swimming speed
for aquatic vertebrates—fishes, mammals and penguins. Data
from Birky and Field 1966; Lang and Pryor 1966; Hartman 1979;
Davis et al. 1985; Fish et al. 1988; Hui 1988; Bose and Lien 1989;
Nowak 1991; Videler and Wardle 1991; Fish 1992, 1993, 1998,
2002; Domenici and Blake 1997; Rohr et al. 2002; Gallon et al.
2007.
S. Vogel
mammals (and penguins), I get the same exponent,
0.43 (r2 ¼ 0.70) as for fish alone (Fig. 4). Looking
only at mammals and penguins, I get 0.35, but with
wide scatter (r2 ¼ 0.27).
Alternatively, one might compare a particularly
speedy copepod (Cyclops, L ¼ 0.002 m, V ¼ 0.4 m/s;
Strickler 1977) with a fish of similar impulsive
(speed-bursting) behavior (pike, L ¼ 0.4 m, V ¼ 4 m/s;
Harper and Blake 1990). That selection gives an
exponent of 0.56, well above 0.33, but consistent
with the regression for fishes.
In short, speed increases with size with lower
scaling exponents (around 0.43–0.59) than noted for
microorganisms (0.73). So, speed relative to body
length (properly ‘‘length-specific speed’’) seems to
drop rather more steeply with size for these larger
forms. In this macroscopic domain, the aquatic
vertebrates that use lift-based propulsion take honors
for top speeds expressed as body lengths per time
(‘‘length-specific speed,’’ properly), whether they use
lateral or caudal propulsors. One should note,
however, the lack of data for large pelagic cephalopods using jet propulsion, although it appears
unlikely that they could top the speeds of
10–15 m/s reported for the best cetaceans.
A look at the whole size range
Swimming eukaryotic organisms range in size from
a few micrometers to a few tens of meters in body
length, seven orders of magnitude. Figure 5 combines
the data just discussed in a single plot, filling the gap
Fig. 5 Body length versus swimming speed for all eukaryotes;
previously cited data plus ones for jellyfish, chaetognaths, scallops,
cephalopods, microcrustaceans, and aquatic insects. Data from
sources cited in Figs. 1 and 4, plus Vlymen 1970; Nachtigall and
Bilo 1975; Strickler 1975; Trueman 1975; Gruffydd 1976;
Nachtigall 1977; Weihs 1977; Morton 1980; O’Dor 1982; Daniel
1985; Daniel and Meyhöfer 1989; Craig 1990; Cromarty et al.
1991; Spanier et al. 1991; Jordan 1992; Cheng and DeMont 1996;
Nauen and Shadwick 1999; Bartol et al. 2001; Colin and Costello
2002; McHenry et al. 2003.
Scaling of swimming
between ciliates and fishes with figures for invertebrates—jellyfish, chaetognaths, cephalopods, crustaceans, and aquatic insects. A regression of the entire
set of 129 entries yields a scaling exponent for
speed versus body length of 0.93, with the remarkably high r2 of 0.965. That stands in sharp (and
statistically significant) contrast with the figures
obtained from those earlier regressions, again emphasizing the pitfalls of facile interpretation of such
exponents. Apparently the scaling factors, the
y-intercepts on logarithmic plots, interact somewhat
peculiarly with the scaling exponents, the slopes on
logarithmic plots. Still, caution is advisable; the high
exponent may in part reflect shift from a mode (dragbased) that works best at low relative speeds to modes
(jetting and, especially, lift-based) that work best at
relatively high speeds.
That graph may carry a further message. The slope
for the largest of the animals looks less steep than
that for smaller ones, with a transition at a body
length of about 0.1 m. Those below that length
scale with an exponent of 1.09 (r2 ¼ 0.93); those in
the more limited range above it scale with the
significantly different exponent of 0.45 (r2 ¼ 0.64).
Something appears to shift at that length that
transcends choice of locomotory mode or ancestry.
An admittedly hazardous guess points to a basic
hydrodynamic transition as responsible. A length of
0.1 m and a speed of 10 body lengths per second
corresponds to a Reynolds number of 100,000. That
lies at the lower end of the range in which the
boundary layers on ordinary flat plates and wellstreamlined bodies shift from laminar to turbulent.
Drag coefficients shift from dependence on Re0.5 to
Re0.2; thus, drag increases more strongly with increasing size or speed; accordingly, propulsive cost also
increases more strongly with increasing size or speed.
Much has been made of whether locomoting
animal bodies experience that shift, with expectations, data, and analyses going back to Gray’s
paradox, broached in the 1930s. Few animal bodies
near the transition range come particularly close to
the minimum drag of flat plates parallel to flow,
ideal streamlining. That may not matter greatly, since
data for body drag reflect overall body behavior with
no regard for how velocity gradients vary at specific
locations. Interference between the drag of passive
structures and the thrust of propulsive structures will
be enormous when the two elements are either
closely linked or identical. It may be significant that
this empirical test of scaling yields a halved exponent
for speed versus length, as one would expect in a
shift from Cd ! Re0.5 to Cd ! Re0.2.
707
General patterns for size, speed, and mode
Lift-based locomotion works only at Reynolds
numbers high enough for circulation to develop
around airfoils and hydrofoils. Jet-based locomotion
becomes impractical at very low Reynolds numbers
as the effects of viscosity become disablingly pernicious. Submerged swimmers at moderate and high
Reynolds numbers move in a realm in which nature
faces unusually few constraints, and they enjoy two
modes in addition to the ever-available drag-based
system. In the data sets just cited, we have lumped
all three. What generalizations might safely be
asserted about the way the modes associate with
such things as habitat, trophic circumstance, and
size? The most facile notion, largely supported by
both observation and theory, is that drag-based
locomotion does best for acceleration from a start.
Typically, large appendages sweep rearward, with
high values of drag almost from the start—drag will
increase with the square of speed and develop with
little or no lag. Furthermore, to that high drag is
added the rearward momentum of both the actual
mass and the apparent additional mass of
those moving appendages. Lift, in contrast, does
not develop immediately, especially without an
analog of the interaction between insect wings
separating at the beginning of their downstroke.
Even assuming immediate achievement of full speed
by the appendage, several chord-lengths of travel
elapse before the full value of lift is achieved. The
virtual mass of the appendages can contribute little,
inasmuch as they move cross-wise to flow.
The downsides of drag, of course, are its much
lower propulsion efficiency (not, for the most part,
the necessity of a recovery stroke) and lower effective
speed maxima. I made a crude quantitative case for
the difference (Vogel 1994); it was properly developed by Walker and Westneat (2000). Fish (1996)
goes into specific mammalian cases with experimentally derived numbers. Paddling (as performed by,
for instance, muskrats), yields propulsion efficiencies
of around 33%, while those of lift-based swimming
in pinnipeds and cetaceans routinely exceeds 80%.
Jetting may do better than lift-production for
acceleration, but it ordinarily does less well than
drag-production. While squid appear to accelerate
with alacrity, the data do not quite bear out one’s
impression, with abdomen-flipping decapod crustaceans and C-starting fishes of the same size doing
much better. Thus, squid and cuttlefish ranging in size
from 39 to 230 mm in mantle length accelerate at an
average of 12.3 m/s2 (6.4–20 m/s2, n ¼ 4) (Trueman
and Packard 1968; Packard 1969; Johnson et al. 1972;
708
O’Dor and Hoar 2000). In contrast, decapods and
fish ranging from 31 to 270 mm in length accelerate
at an average of 61.4 m/s2 (15–110 m/s2, n ¼ 8)
(Webb 1975, 1979, 1983; Daniel and Meyhöfer 1989;
Nauen and Shadwick 1999; Spierts and Van Leeuwen
1999).
Jetting excels in top speeds, briefly achieved at the
expense of propulsion efficiency, in animals such as
cephalopods that have narrow jetting orifices.
Anderson and DeMont (2000) give a maximum
efficiency of 56% for squid; other calculations are
lower. Eliciting top speeds from squid confined in
tanks is notoriously difficult (O’Dor, personal communication). Anecdotal reports of squid of modest
size landing on the decks of ships with railings of
known heights above the water’s surface demand
launch speeds well over 5 m/s and perhaps 20 body
lengths per second. Jellyfish, with broader orifices
and thus ejection speeds closer to propulsion speed,
achieve better efficiencies but relatively lower speeds
(Daniel 1985; Colin and Costello 2002). Weihs
(1977) suggested jetting to be more fluid mechanically efficient than one might think. Perhaps jetting
has an advantage too easily missed by a purely
adaptationist outlook (noted earlier), its simple
origin, inasmuch as the underlying device, muscle
around a tube or chamber, could scarcely be more
widespread.
Thus, drag-based swimming accelerates best from
rest, a lift-based mode is best for efficient steady
swimming, and jetting works best for briefly achieving high speed. Each does what it does best at some
obvious sacrifice of the strong points of the others.
This basic trichotomization, however, must be
handled with care and discretion; one cannot
simply divide animals according to the mode each
uses. A wide range of fishes, for instance, start up in
a drag-based mode and then proceed with lift-based
propulsion. Squid, famous for jet propulsion, in fact
do most of their routine moving with undulating
or beating, and thus lift-based, fins (Bartol et al.
2001). Even jellyfish, described as exclusively jet
propelled, may—in particular the more oblate
ones—paddle with the margins of their umbrellas,
a drag-based mode (Colin and Costello 2002; Dabiri
et al. 2005). We may build purely drag-based,
lift-based, and jetting craft (but note that a jet
engine uses lift internally); nature’s traditions are less
constrained.
The scaling of acceleration from a start
Often, starting with alacrity must count for as much
as speed, either in chasing down prey or when
S. Vogel
escaping a predator. Part of the process of course
consists of minimizing neuro-muscular delay, resulting in the evolution of such things as the famous
giant nerve fibers of squid. Here, we will take such
factors for granted and look solely at acceleration.
That variable gets far less attention than does speed,
although a substantial literature, alluded to earlier,
considers the initial tail-flipping evasive maneuver
of decapod crustaceans and the so-called C-start
in fishes. We need more data, particularly for species
in which acceleration might be of especial behavioral
and ecological significance—the cats rather than the
dogs of the aquatic world.
How should acceleration scale on essentially
dimensional grounds? For initial acceleration, drag
will not matter; only mass resists acceleration. Again,
to the mass of the animal must be added an apparent
additional mass that reflects the inescapable
necessity of accelerating water backward as the
animal accelerates forward, the so called acceleration
reaction, reflected in what is usually termed the
‘‘virtual mass.’’ Given the size range and the scatter
of available data, no distinction between virtual and
intrinsic masses is needed. Force equals mass times
acceleration; force (as muscle cross section or
equivalent) scales as length squared, and mass
scales as length cubed, so acceleration should be
inversely proportional to length or inversely proportional to the cube root of body mass, as noted by,
among others, Nauen and Shadwick (1999).
I asserted an analogous scaling rule for acceleration
of ballistic biological projectiles, from spores to
jumping mammals, but it rested on quite different
biological factors (Vogel 2005).
How does acceleration actually scale? Figure 6 gives
a plot for a wide diversity of cases. Domenici and
Blake (1997) gave a more complete tabulation
for medium-sized fish; Fisher and Hogan (2007) provided additional information about small reef fish;
inclusion of their other data has little effect here.
No particular regularity appears evident, either with
size or kind of animal; I have deliberately avoided
giving a scaling exponent for fear that it might be cited
as at least tacitly meaningful. Certainly, the inverse
relationship of acceleration and body length gains
no support. At best one has evidence that accelerational ability might be traded off against other factors,
the latter varying in identity and importance.
Nonetheless, size should matter, both to the
practicality and to the utility of high values of
acceleration. For small organisms, our naı̈ve scaling
argument implies that high acceleration ought to be
practical. The argument, though, ignores two subtle,
but seriously limiting, factors. First, severe viscous
709
Scaling of swimming
Fig. 6 Body length versus maximum reported acceleration for
metazoans. Data from Hughes 1958; Trueman and Packard 1968;
Packard 1969; Vlymen 1970; Johnson et al. 1972; Strickler 1975;
Webb 1975, 1976, 1978, 1979, 1983; Lehman 1977; Kayan et al.
1978; Kils 1979; Donaldson et al. 1980; Videler and Weihs 1982;
Bone and Trueman 1983; Daniel 1983, Daniel and Meyhöfer
1989; Craig 1990; Harper and Blake 1990; Cromarty et al. 1991;
Spanier et al. 1991; Jordan 1992; Cheng and DeMont 1996;
Brainerd et al. 1997; Nauen and Shadwick 1999; Spierts and Van
Leeuwen 1999; Goldbogen et al. 2006.
action at low Reynolds numbers causes an extra
volume and thus mass of water to move with the
organism. Thus, the acceleration reaction mentioned
earlier may be doubly exaggerated by viscosity; an
accelerating body will have extra volume and will
displace additional volume. Most authors (see, for
instance, Daniel 1984) cite factors for virtual mass
that presume inviscid flows, and these may underestimate the phenomenon considerably. Second,
when a body changes speed in a viscous fluid,
a curious temporal term may be relevant. Flow
patterns take time to become established, especially
where the disturbance field of a body extends far
from its surface. For an accelerating body, the
pattern will have steeper velocity gradients and
greater shear forces than would be anticipated in a
quasi-steady analysis; this last retardation of acceleration has been termed the ‘‘Bassett term’’
(Michaelides 1997; Koehl et al. 2003).
In addition, one has difficulty envisioning how a
system propelled by flagella or cilia can generate,
especially high accelerations. Thus, even a protozoan
covered with cilia that swing suddenly and synchronously would face the difficulty of propelling fluid at
any distance with minimal delay.
About the smallest creatures that appear to be
acceleration specialists are the microcrustaceans, in
particular the copepods. With antennules spread
wide and with rapidly acting muscle rather than
slower ciliary action, they do remarkably well—for
their sizes—as one can see in Fig. 6. One advantage,
pointed out by Strickler (1977), is that by getting up
to rather high speed quickly, they push their
Reynolds numbers up to the extent that they carry
less water with them and are less likely to push prey
on ahead of ingestion range. In this way, they may
also be taking a converse advantage of the higher
shear implicit in the Bassett term.
For very large swimmers, though, that basic
scaling argument may be immediately relevant.
Both lunge feeding by fin whales and piscivory by
dolphins should benefit from accelerative ability,
but neither achieves values that approach those of,
say, the escape reactions of crayfish and lobsters
or the C-starts of fish.
In short, high values of acceleration may be the
prerogative of middle-sized, muscle-powered animals
of body lengths from about a millimeter to something short of a meter. One does wish for additional
data on large fish, more cetaceans, and perhaps some
aquatic reptiles. To some extent, acceleration specialists can be identified by their shapes, at least among
fish propelled mainly by caudal fins and probably
among aquatic mammals. Minimizing drag relative
to body volume produces a relatively bulbous body.
Minimizing acceleration reaction, by demanding that
less water be accelerated, produces a more elongate
body, that of such forms as pike and barracuda.
Incidentally, I know of no specific indication that
any accelerating swimmer makes much use of power
amplification, preloading tendon or some other
elastic as does every jumping flea or grasshopper
(Vogel 2005).
Caveat emptor
Several cautionary matters need the emphasis of lastword status. First, in no case did our admittedly
näive argument yield a particularly good prediction
for the value of an empirical scaling exponent.
That should point up the limitations of such
arguments, the underlying complexity of the situations, and the limitations of our understanding.
Second is the disquieting dependence of the
scaling exponents on how the data were categorized.
Thus, exponents derived from linear regressions of
double-logarithmic plots of data for different animals
derived from different studies deserve more skepticism than we tend to give published numbers.
Third, plots typically lump data from creatures
for which an extreme value of the variable in
question has major adaptive value with ones for
whom, adaptively speaking, other variables matter
more. Sometimes, a line on such a plot that describes
how the extreme values change with size may have
710
S. Vogel
greater biological relevance; such a line, though,
usually loads a heavy burden of trust on a few values.
For instance, it might be more informative to
compare swimming speeds at which animals move
most of the time; unfortunately such data appear in
few published accounts. And maximum speeds, as
used here, often have problems of credibility. Thus,
I have omitted a reported speed of 20 m/s for a tuna
because of a persuasive theoretical objection by
Iosilevskii and Weihs (2008).
Finally, the casual adoption of data from diverse
sources commonly conceals wide error ranges,
mathematical sleights of hand, and experimental
inconsistencies. Like watching the manufacture
of sausages, examining the origin of scaling exponents may (perhaps should) limit our taste for them.
Colin SP, Costello JH. 2002. Morphology, swimming
performance and propulsive mode of six co-occurring
hydromedusae. J Exp Biol 205:427–437.
Craig DA. 1990. Behavioural hydrodynamics of Cloeon
dipterum larvae (Ephemeroptera:Baetidae). J N Am
Benthol Soc 9:346–357.
Cromarty SI, Cobb JS, Kass-Simon G. 1991. Behavioral
analysis of the escape response in the juvenile lobster
Homarus americanus over the molt cycle. J Exp Biol
158:565–581.
Dabiri JO, Colin SP, Costello JH, Gharib M. 2005. Flow
patterns generated by oblate medusan jellyfish: field
measurements and laboratory analyses. J Exp Biol
208:1257–1265.
Daniel TL. 1983. Mechanics and energetics of medusan jet
propulsion. Can J Zool 61:1406–1420.
Acknowledgments
I thank Richard Blob and Gabriel Rivera for
organizing the symposium. The symposium was
sponsored by four divisions of the Society for Integrative and Comparative, Ecology and Evolution,
Vertebrate Morphology, Comparative Biomechanics,
and Invertebrate Zoology, as well as the support
of Vision Research (www.visionresearch.com),
EmicroScribe (www.emicroscribe.com), and the
National Science Foundation (IOS-0733441).
Daniel TL. 1984. Unsteady aspects of aquatic locomotion.
Am Zool 24:121–134.
Daniel TL. 1985. Cost of locomotion: unsteady medusan
swimming. J Exp Biol 119:149–164.
Daniel TL, Meyhöfer E. 1989. Size limits in escape locomotion
of Carridean shrimp. J Exp Biol 143:245–265.
Daniel TL, Webb PW. 1987. Physical determinants of
locomotion. In: Dejours P, Bolis L, Taylor CR,
Weibel ER, editors. Comparative physiology: life in water
and on land. Padova (Italy): Liviana Press. p. 343–369.
Davis RW, Williams TM, Kooyman GL. 1985. Swimming
metabolism of yearling and adult harbor seals Phoca
vitulina. Physiol Zool 58:590–596.
References
Alexander RM. 1979. The
Cambridge University Press.
Cheng J-Y, DeMont ME. 1996. Hydrodynamics of scallop
locomotion: unsteady fluid forces on clapping shells. J Fluid
Mech 317:73–90.
invertebrates.
Cambridge:
Anderson EJ, DeMont ME. 2000. The mechanics of locomotion in the squid Loligo pealei: locomotory function and
unsteady hydrodynamics of the jet and intramantle
pressure. J Exp Biol 203:2851–2863.
Bartol IK, Patterson MR, Mann R. 2001. Swimming
mechanics and behavior of the shallow-water brief squid
Lolliguncula brevis. J Exp Biol 204:3655–3682.
Birky CW, Field B. 1966. Swimming speed of a Pacific
bottlenose porpoise. Science 151:588–589.
Bone Q, Trueman ER. 1983. Jet propulsion in salps (Tunicata:
Thalicea). J Zool 201:481–506.
Bose N, Lien J. 1989. Propulsion of a fin whale (Balaenoptera
physalis): why the fin whale is a fast swimmer. Proc R Soc
Lond B 237:175–200.
Brainerd EL, Page BN, Fish FE. 1997. Opercular jetting during
fast-starts by flatfishes. J Exp Biol 200:1179–1188.
Brennen C, Winet H. 1977. Fluid mechanics of propulsion by
cilia and flagella. Annu Rev Fluid Mech 9:339–398.
Chatterjee S, Templin RJ, Campbell KE Jr. 2007. The
aerodynamics of Argentavis, the world’s largest flying bird
from the Miocene of Argentina. Proc Natl Acad Sci USA
104:12398–12403.
Domenici P, Blake RW. 1997. The kinematics and
performance of fish fast-start swimming. J Exp Biol
200:1165–1178.
Donaldson S, Mackie GO, Roberts AO. 1980. Preliminary
observations on escape swimming and giant neurons in
Aglantha digitale (Hydromedusae: Trachylina). Can J Zool
58:549–552.
Fish FE. 1992. Aquatic locomotion. In: Tomasi T, Horton T,
editors. Mammalian energetics: interdisciplinary views of
metabolism and reproduction. Ithaca (NY): Comstock
Publ. Assoc. p. 34–63.
Fish FE. 1993. Influence of hydrodynamic design and
propulsive mode on mammalian swimming energetics.
Aust J Zool 42:69–101.
Fish FE. 1996. Transitions from drag-based to liftbased propulsion in mammalian swimming. Am Zool
36:628–641.
Fish FE. 1998. Comparative kinematics and hydrodynamics
of odontocete cetaceans: morphological and ecological
correlates with swimming performance. J Exp Biol
201:2867–2877.
Fish FE. 2002. Speed. In: Perrin WF, Würsig B,
Thewissen JGM, editors. Encyclopedia of marine mammals.
San Diego (CA): Academic Press. p. 1161–1163.
Scaling of swimming
Fisher R, Hogan JD. 2007. Morphological predictors of
swimming speed: a case study of pre-settlement juvenile
coral reef fishes. J Exp Biol 210:2436–2443.
Fish FE, Innes S, Ronald K. 1988. Kinematics and estimated
thrust production of swimming harp and ringed seals.
J Exp Biol 137:157–173.
Gallon SL, Sparling CE, Georges J-Y, Fedak MA, Biuw M.
2007. How fast does a seal swim? Variations in behaviour
under differing foraging conditions. J Exp Biol
210:3285–3294.
Goldbogen JA, Calambokidis J, Shadwick RE, Oleson EM,
McDonald MA, Hildebrand JA. 2006. Kinematics of foraging dives and lunge-feeding in fin whales. J Exp Biol
209:1231–1244.
Gruffydd LD. 1976. Swimming in Chlamys islandica in
relation to current speed and an investigation of hydrodynamic lift in this and other scallops. Norw J Zool
24:365–378.
Harper DG, Blake RW. 1990. Fast-start performance of
rainbow trout Salmo gairdneri and northern pike Esox
lucius. J Exp Biol 150:321–342.
Hartman DS. 1979. Ecology and behavior of the manatee
(Trichechus manatus) in Florida. Special Publication, Am
Soc Mamm 5:1–153.
Hughes GM. 1958. The co-ordination of insect movements.
J Exp Biol 35:567–583.
711
McHenry MJ, Azizi E, Strother JA. 2003. The hydrodynamics of locomotion at intermediate Reynolds
numbers: undulatory swimming in ascidian larvae
(Botrylloides sp.). J Exp Biol 206:327–343.
Michaelides EE. 1997. Review—the transient equation of
motion for particles, bubbles, and droplets. J Fluids Engin
119:233–247.
Mitchell JG, Pearson L, Bonazinga A, Dillon S, Khouri H,
Paxinos R. 1995. Long lag times and high velocities in the
motility of natural assemblages of marine bacteria.
Appl Environ Microbiol 61:877–882.
Morton B. 1980. Swimming in Amusium pleuronectes
(Bivalvia: Pectinidae). J Zool 190:375–404.
Nachtigall W. 1977. Swimming mechanics and energetics of
locomotion of variously sized water beetles—Dytiscidae,
body length 2 to 35 mm. In: Pedley TJ, editor. Scale
effects in animal locomotion. London: Academic Press.
p. 269–283.
Nachtigall W. 1980. Mechanics of swimming in water-beetles.
In: Elder HY, Trueman ER, editors. Aspects of animal
movement. Cambridge: Cambridge University Press.
p. 107–124.
Nachtigall W, Bilo D. 1975. Hydrodynamics of the body of
Dytiscus marginalis. In: Wu TY-T, Brokaw CJ, Brennan C,
editors. Swimming and flying in nature. New York: Plenum
Press. p. 585–595.
Iosilevskii G, Weihs D. 2008. Speed limits on swimming of
fishes and cetaceans. J Roy Soc Interface 5:329–338.
Nauen JC, Shadwick RE. 1999. The scaling of acceleratory
aquatic locomotion: body size and tail-flip performance of
the California spiny lobster Panulirus interruptus. J Exp Biol
202:3181–3193.
Jiang H, Osborn TR, Meneveau C. 2002. Hydrodynamic
interaction between two copepods: a numerical study.
J Plankton Res 24:235–253.
Nowak RM. 1991. Walker’s mammals of the world.
5th edition. Baltimore (MD): Johns Hopkins University
Press.
Johnson WP, Soden PD, Trueman ER. 1972. A study in jet
propulsion: an analysis of the motion of the squid Loligo
vulgaris. J Exp Biol 56:155–165.
O’Dor RK. 1982. Respiratory metabolism and swimming
performance of the squid, Loligo opalescens. Can J Fish
Aquat Sci 39:580–587.
Jordan CE. 1992. A model of rapid-start swimming at
intermediate Reynolds number: undulatory locomotion in
the chaetognath Sagitta elegans. J Exp Biol 163:119–137.
O’Dor RK, Hoar JA. 2000. Does geometry limit squid growth?
ICES J Mar Sci 57:8–14.
Hui CA. 1988. Penguin swimming. II. Energetics and
behavior. Physiol Zool 61:344–350.
Kayan VP, Kozlov LF, Pyatetskil VE. 1978. Kinematic
characteristics of the swimming of certain aquatic animals.
Fluid Dynamics 13:641–646.
Kils U. 1979. Swimming speed and escape capacity
of Antarctic krill, Euphausia superba. Meeresforsch
27:264–266.
Koehl MAR. 1995. Fluid flow through hair-bearing
appendages: feeding, smelling, and swimming at low and
intermediate Reynolds number. Soc Exp Biol Symp
49:157–182.
Koehl MAR, Jumars PA, Karp-Boss L. 2003. Algal biophysics.
In: Norton AD, editor. Out of the past. Belfast (UK): Brit
Phycol Assoc. p. 115–130.
Packard A. 1969. Jet propulsion and the giant fiber response
of Loligo. Nature 221:875–877.
Rohr JJ, Fish FE, Gilpatrick JW Jr. 2002. Maximum swim
speeds of captive and free-ranging delphinids: critical
analysis of extraordinary performance. Mar Mam Sci
18:1–19.
Sleigh MA, Blake JR. 1977. Methods of ciliary propulsion
and their size limitations. In: Pedley TJ, editor. Scale
effects in animal locomotion. London: Academic Press.
p. 243–256.
Spanier E, Weihs D, Almog-Shtayer G. 1991. Swimming of
the Mediterranean slipper lobster. J Exp Mar Biol Ecol
145:15–31.
Lang TG, Pryor K. 1966. Hydrodynamic performance of
porpoises (Stenella attenuata). Science 152:531–533.
Spierts ILY, Van Leeuwen JL. 1999. Kinematics and muscle
dynamics of C- and S-starts of carp (Cyprinus carpio L.).
J Exp Biol 202:393–406.
Lehman JT. 1977. On calculating drag characteristics for
decelerating zooplankton. Limnol Oceanogr 22:170–172.
Strickler JR. 1975. Swimming of planktonic Cyclops species
(Copepoda, Crustacea): pattern, movements & their control.
712
S. Vogel
In: Wu TY-T, Brokaw CJ, Brennan C, editors. Swimming and
flying in nature. New York: Plenum Press. p. 599–613.
Walker JA, Westneat MW. 2000. Mechanical performance of
aquatic rowing and flying. Proc R Soc Lond B 267:1875–1881.
Strickler JR. 1977. Observation of swimming performances of
planktonic copepods. Limnol Oceanogr 22:165–170.
Webb PW. 1975. Acceleration performance of rainbow trout
Salmo gairdneri and green sunfish Lepomis cyanellus. J Exp
Biol 63:451–465.
Trueman ER. 1975. The locomotion of soft-bodied animals.
London: Edward Arnold.
Trueman ER, Packard A. 1968. Motor performance of some
cephalopods. J Exp Biol 49:495–507.
Videler JJ, Nolet BA. 1990. Costs of swimming measured at
optimum speed: scale effects, differences between swimming styles, taxonomic groups and submerged and surface
swimming. Comp Biochem Physiol 97A:91–99.
Videler JJ, Wardle CS. 1991. Fish swimming stride by stride:
speed limits and endurance. Rev Fish Biol Fisheries
1:23–40.
Videler JJ, Weihs D. 1982. Energetic advantage of burstand-coast swimming of fish at high speeds. J Exp Biol
97:169–178.
Vlymen WJ. 1970. Energy expenditure of swimming copepods. Limnol Oceanogr 15:348–356.
Vogel S. 1994. Life in moving fluids. 2nd edition. Princeton
(NJ): Princeton University Press.
Vogel S. 2005. Living in a physical world. III. Getting up to
speed. J Biosciences 30:303–312.
Webb PW. 1976. The effect of size on the fast-start
performance of rainbow trout Salmo gairdneri and a
consideration of piscivorous predator-prey interactions.
J Exp Biol 65:157–177.
Webb PW. 1978. Fast-start performance and body form in
seven species of teleost fish. J Exp Biol 74:211–226.
Webb PW. 1979. Mechanics of escape responses in crayfish
(Orconectes virilis). J Exp Biol 79:245–263.
Webb PW. 1983. Speed, acceleration and manoeuverability
of two teleost fishes. J Exp Biol 102:115–122.
Weihs D. 1977. Periodic jet propulsion of aquatic creatures.
Fortschr Zool 24:171–175.
Williams TA. 1994. A model of rowing propulsion and the
ontogeny of locomotion in Artemia larvae. Biol Bull
187:164–173.
Wu TY. 1977. Hydrodynamics of swimming at low Reynolds
numbers. In: Nachtigall W, editor. Physiology of movement—biomechanics. Stuttgart (Germany): Gustav Fischer.
p. 149–169.