Integrative and Comparative Biology Advance Access originally published online on April 11, 2008
Integrative and Comparative Biology 2008 48(6):702-712; doi:10.1093/icb/icn014
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Modes and scaling in aquatic locomotion
Biology Department, Duke University, Durham, NC 27708, USA
Correspondence: 1E-mail: svogel{at}duke.edu
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Synopsis |
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Top
Synopsis
IntroductionOrganisms 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.
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Introduction |
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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 fluid-mechanical 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 10–6 (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!
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Modes of aquatic locomotion |
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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.- Drag-based swimming
- Low-Re modes in which drag varies directly with viscosity, velocity, and some characteristic length.
- Flagellar, with either bacterial or eukaryotic flagella that are long relative to the body and few in number.
- Ciliary, with similar appendages but short relative to the body and more numerous.
- With the body entirely or nearly entirely covered with cilia.
- With cilia limited to tracts and covering less than half the body.
- With the body entirely or nearly entirely covered with cilia.
- Setal paddles: movable appendages with passive hair-like protrusions.
- Flagellar, with either bacterial or eukaryotic flagella that are long relative to the body and few in number.
- Moderate and high Re modes in which drag varies with density, some characteristic area, and approximately the square of velocity.
- Paddling with a single bilateral pair of appendages.
- Paddling with serially arranged bilaterally paired appendages.
- Moving with simultaneous sweeps.
- Moving with rear-first metachronal waves of sweeps.
- Moving with simultaneous sweeps.
- Paddling with a single bilateral pair of appendages.
- Low-Re modes in which drag varies directly with viscosity, velocity, and some characteristic length.
- Lift-based swimming
- With paired lateral propulsors, e.g., wings, fins.
- With a single caudal propulsor such as tail or flukes.
- By passing waves posteriorly along an elongate trunk.
- With paired lateral propulsors, e.g., wings, fins.
- Direct-reaction swimming—with pulsating jet or paired jets.
- Interfacial swimming
- Using surface tension for both support and propulsion.
- By pushing downward and rearward on the surface.
- By reducing surface tension posteriorly with surfactant.
- By pushing downward and rearward on the surface.
- Using a hull that displaces its own weight of water.
- And propelling with drag-based paddling.
- And propelling with lift-based hydrofoils.
- And propelling with drag-based paddling.
- Using dynamic support contingent on movement.
- Support by aquaplaning.
- Support by repeated "slapping" against the surface.
- Support by aquaplaning.
- Using surface tension for both support and propulsion.
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The low Reynolds-number world |
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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, 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 (<1.00 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.
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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.
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A marine bacterium, Vibrio harveyi (Mitchell et al. 1995
), while reaching impressive speeds of over 100 µm/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 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 µm. 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 self-produced 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.
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These microorganisms (as well as spermatozoa) can, and often do, increase swimming speed by forming tight swarms, just as dense groups of 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 µm/s for groups of Vibrio harveyi but only up to 140 µm/s for individuals (Re = 4 x 10–4). 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 = 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 µm, set by the minimum possible radius of curvature of a flagellum of about 2 µm (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 = 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 + 2 flagellar structures, but nonmotile ones, in their sense organs, and a few properly motile flagella in insect sperm (Alexander 1979).
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Size and speed among the "not-so-small" |
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At moderate and high Reynolds numbers, larger animals generally swim faster than do smaller ones, 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 L–0.67). 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 = 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-for-transport-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 mammals (and penguins), I get the same exponent, 0.43 (r2 = 0.70) as for fish alone (Fig. 4). Looking only at mammals and penguins, I get 0.35, but with wide scatter (r2 = 0.27).
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Alternatively, one might compare a particularly speedy copepod (Cyclops, L = 0.002 m, V = 0.4 m/s; Strickler 1977
) with a fish of similar impulsive (speed-bursting) behavior (pike, L = 0.4 m, V = 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.
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A look at the whole size range |
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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 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 (drag-based) that works best at low relative speeds to modes (jetting and, especially, lift-based) that work best at relatively high speeds.
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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 = 0.93); those in the more limited range above it scale with the significantly different exponent of 0.45 (r2 = 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 well-streamlined bodies shift from laminar to turbulent. Drag coefficients shift from dependence on Re–0.5 to Re–0.2; 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 Re–0.5 to Cd Re–0.2.
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General patterns for size, speed, and mode |
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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 = 4) (Trueman and Packard 1968; Packard 1969; Johnson et al. 1972; 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 = 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.
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The scaling of acceleration from a start |
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Often, starting with alacrity must count for as much as speed, either in chasing down prey or when 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.
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Nonetheless, size should matter, both to the practicality and to the utility of high values of acceleration. For small organisms, our naïve scaling argument implies that high acceleration ought to be practical. The argument, though, ignores two subtle, but seriously limiting, factors. First, severe viscous 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).
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Caveat emptor |
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Several cautionary matters need the emphasis of last-word 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 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.
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Acknowledgments |
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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).
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Footnotes |
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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.
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