Using kinematic and mechanical experiments, we have shown how fisher spiders, Dolomedes triton (Araneae, Pisauridae), can generate horizontal propulsive forces using their legs. This horizontal thrust is provided primarily by the drag of the leg and its associated dimple as both move across the water surface. Less important sources of resistance are surface tension and bow waves. The relative contributions of drag, surface tension and bow waves were examined in several different ways. In one experiment, we measured the forces acting on a leg segment as water flowed past it in non-turbulent flow; the bow wave was not present at leg relative velocities below 0.2 m s−1 and thus cannot play a role in thrust production at low leg speeds. In a second experiment, we varied the surface tension by altering the concentration of ethanol from 0 % to 9 % in the experimental water tank. At a constant dimple depth, force varied little with changes in surface tension, a result consistent with the hypothesis that drag is the primary source of resistance. In addition, however, as surface tension decreased from 0.072 to 0.064 N m−1, the power exponent of the relationship between force and velocity (as measured by the exponent of the power function relating the two variables) increased; at lower surface tensions, down to 0.054 N m−1, the power exponent of the relationship between force and velocity decreased. These results suggest an influence of surface tension (albeit still secondary to drag) in generating horizontal resistance to leg movement. We also measured flow disturbance in the water downstream from a leg segment and confirmed that, even at velocities well below 0.2 m s−1, the leg-cum-dimple transferred momentum to the water, which is a clear indication that drag is a contributor to the resistance encountered by a spider’s leg. Finally, modeling the leg-cum-dimple as a circular cylinder generates values of drag that account for 75–98 % of the measured leg force when the dimple is 0 or 1 mm deep. These results not only elucidate the primary mechanism of propulsion for D. triton and other similar-sized arthropods, such as adult water striders (Gerridae), but also suggest that the formerly enigmatic locomotion of very small water-walking organisms (e.g. first-instar water striders) can be understood in the same way.
While experimental studies of legged locomotion on land have led to important generalizations regarding power requirements, gait transitions, maximum velocities and the effects of body size (e.g. Alexander, 1977, 1989; Blickhan and Full, 1993; Farley and McMahon, 1992; Farley and Taylor, 1991; Fedak et al. 1982; Full, 1990; Full and Tu, 1991; Full et al. 1995; Kram et al. 1997; Schmidt-Nielsen, 1984), the mechanical principles governing legged locomotion on the surface of the water remain relatively unexplored (Denny, 1993; Vogel, 1994). A notable exception to this dearth of information is work on basilisk lizards, Basiliscus basiliscus, of 2–200 g body mass, which support themselves on the water surface by using their feet to generate upward forces from surface impact (‘slap’) and the formation of an air cavity (‘stroke’) (Glasheen and McMahon, 1996a,b). This vertical slap-and-stroke mechanism, however, is unlikely to be used by the small (body mass <1 g) arthropods that propel themselves on the water surface using primarily horizontal leg movements. Some other physical phenomena must be at work, and experimental elucidation of the primary propulsive mechanism is the goal of this study.
For legged arthropods moving horizontally on land, the feet produce a friction force that transfers momentum from the animal to the substratum, accelerating both horizontally (in opposite directions). This propulsive force varies with changes in behavior. During running, 3.1 g death-head cockroaches, Blaberus discoidalis, produce single-leg, horizontal ground reaction forces of 0.005 N; during righting, those forces increase to 0.034 N (Full et al. 1995). Larger still were the peak static forces, 0.160 N, produced by the metathoracic leg as the animal attempted to squeeze through a constricted opening (Full and Ahn, 1995). When adjusted for body mass (kg), these measurements provide an empirical basis for the possible range of horizontal forces (1.6–51.6 N kg−1) that might be measured in arthropods of different sizes moving on the water surface.
A few groups of uniramian arthropods, notably water striders (Hemiptera, Gerridae) and fisher spiders (Araneae, Pisauridae), inhabit the surface of fresh water and locomote using feet supported but not wetted by the water surface. While the locomotion of water striders and similar insects has been studied descriptively (Anderson, 1976; Bowdan, 1978) and ethologically (Barnes and Barth, 1991), the mechanics of moving on the water surface has not been explored, except for some theoretical ruminations (Denny, 1993; Vogel, 1994) and cinematographic analyses (Anderson, 1976). The water-surface locomotion of the fisher spider Dolomedes sp. has been described in general (McAlister, 1959) and with regard to the variety of modes used (Deshefy, 1981), the details of the stepping patterns during rowing (Shultz, 1987) and the special conditions surrounding prey capture (Gorb and Barth, 1994). In spite of these efforts, we have no experimental data, and no single-leg force measurements, with which to test theoretical models of propulsion on the water surface.
The locomotion of a water strider or a fisher spider on the water surface is interesting for two additional reasons. First, the same hydrophobicity of the animal’s exoskeleton and cuticular hairs that makes possible support of the animal by the water’s surface tension also makes terrestrial-style locomotion difficult on the water surface. The very small amount of interaction between water molecules and the waxy molecules of the animal’s surface reduces friction to a small fraction of the friction that would anchor the animal’s foot to a solid substratum. Second, the weight of the animal, when supported by the water’s surface tension, pushes the points of contact downwards, creating dimples in the water surface. When the legs are moved backwards to thrust the body forwards, it is the legs with their accompanying dimples that move backwards, not just the legs themselves. In that respect, one might want to consider the leg-cum-dimple to be hull-like in its interaction with the fluid medium (Fig. 1).
Thus, the locomotion of a water strider or fisher spider across a surface with which it has little micro-scale contact must depend upon the ability of the rowing legs to encounter or create resistance to their motion. Three possible mechanisms of resistance are presented below. In each case, the mathematical relationships are provided not for calculation of the predicted forces, but for the purpose of making testable predictions about how the water’s resistive force should vary with experimental manipulation of the leg-cum-dimple, its motion and the properties of the fluid.
Bow wave resistance
As a hull moves at constant velocity through the water surface, it generates a standing bow wave that offers resistance to its forward motion. If the leg-cum-dimple operates as a hull, the resistance to the motion of a leg may result from the formation of a bow wave and the water surface’s tendency to flatten under the influence of gravity and capillarity (Fig. 2A). For water striders, Denny (1993) stated the hypothesis as follows: “The answer appears to be that when the striders’ middle pair of legs is swept backward, the legs create surface waves. The drag of moving against these waves provides the ‘purchase’ needed by the animal to skate across the water.” The efficacy of this method of generating resistance depends upon the ability of the moving leg to generate a wave, and that ability is a function, in part, of the leg’s velocity. Because of the interaction between surface tension, γ (in N m−1 or J m−2), and the force caused by gravitational acceleration, g (m s−2), which, respectively, form small capillary and larger gravity waves, no waves at all will be produced below a minimum hull speed, cmin (Denny, 1993):
where ρ is the density of the water (kg m−3). If γ is 0.0728 J m−2 (pure water at 20 °C) and ρ is 998 kg m−3 (fresh water at 20 °C), then cmin is 0.23 m s−1.
Surface tension resistance
Another possible consequence of surface tension that we initially considered was assymetry in γ as the leg moved and distorted the surface. If, by pushing on the surface, the leg were able to compress the leading and stretch the trailing surface, we might expect Fs,a and Fs,p to be unbalanced, thus providing a net positive Fs. If the surface behaved like a thin rubber membrane, this might be a reasonable hypothesis. However, the comparison with an elastic membrane is unjustified on theoretical grounds and, because no new water surface is actually created under these circumstances, this hypothesis is untenable (Denny, 1994).
In the experiments described below, we sought to determine, by measuring Ft on an isolated leg, which of these sources of resistance (equation 8; Fig. 2; Table 1) was most important in the aquatic locomotion of the fisher spider, D. triton. To accomplish this, we manipulated three independent variables: U, the velocity of the foot moving across the water surface, h, the depth of the leg-cum-dimple, and γ, surface tension. In order to determine the physiologically relevant range of leg velocities and locomotor forces for our isolated leg experiments, we also measured the locomotor kinematics of live fishing spiders. Finally, we measured the presence of waves and flow pattern disturbance as supplementary evidence regarding the importance of a resistive bow wave (equations 1 and 2) and sub-surface momentum transfer via drag (equation 3), respectively.
Materials and methods
Dolomedes triton (Walckenaer) (Fig. 1; Araneae, Pisauridae) are large spiders (adult females: body mass ⩽1.2 g, leg length ⩽5 cm) that inhabit marshes and the edges of ponds and streams throughout much of North America (Gertsch, 1979). The subjects for these experiments were collected from small ponds in Mississippi, USA, and held in our laboratory (at 22–25 °C) in 4 l plastic aquaria containing fresh water (approximately 2 cm deep) and an inverted clay flower pot to provide a solid substratum. We fed the spiders assorted insects and changed their water approximately once a week. We used 19 intact spiders of a range of instars in the videographic portion of this study. The exuviae of newly molted adult spiders and leg segments of killed adults were stored in dry air; from this sample, we selected distal portions (tarsus plus tibia) of legs III, varying in length from 10 to 18 mm and in maximum (proximal) width from 1 to 1.5 mm for the isolated leg segment experiments. These segment lengths roughly approximated the lengths of the propulsive (in contact with the water) portions of the legs of the intact spiders that we video-taped moving across the water surface (see Results). All observations and experiments were conducted at laboratory temperatures between 22 and 25 °C.
Kinematics of live spiders
Because most of the motion involved in the spiders’ movement across the surface of water occurs in the horizontal plane, we video-taped their locomotion from directly above. The arena enclosing a test spider consisted of a white porcelain-surfaced tray, a smooth, circular plastic barrier to prevent the spider’s escape, and a layer of water at least twice as deep as the deepest dimple we had observed for a spider of the size of the test spider. The arena was lit with an incandescent projector bulb, mounted at 45 ° above and to one side of the video-taped part of the arena, which provided an intense point source. We adjusted the video camera’s lens aperture to obtain sufficient depth of field to allow both the spider and its shadow (on the porcelain surface of the arena) to be in sharp focus.
During a trial, we placed the test spider in the arena, recorded its movements at 1000 images s−1 using a high-speed video system (Kodak, model EktaPro EM-1000) and stored the images in S-VHS format (Sony, model 9500 MDR). For analysis, we used only motion sequences that met the following criteria: (1) the spider’s initial velocity was between 0 and 0.02 m s−1, (2) the motion of the spider during the filmed sequence was in a straight line, (3) the spider was in contact with the water at all times (i.e. it did not leap or gallop), and (4) the sequence included both the power stroke and enough of the subsequent glide to allow an accurate determination of the rate of deceleration.
We analysed the spider’s motion, image by image, in the horizontal plane by overlaying each digitally paused video frame (Sony, model 9500 MDR) with a computer-generated x–y cursor grid (National Institutes of Health software Image, version 1.55 f) by means of a video scan convertor and synchronizer (Mass Microsystem model Colorspace II installed in an Apple Corporation computer, model Centris 650). Coordinates of the anterior or posterior tip of the body were manually digitized either every 1 ms (for small spiders) or every 5 ms for the duration of a propulsion episode (from the beginning of the power stroke of legs III to the termination of the passive deceleration). We used these coordinates to analyze the displacement of the spider through time; acceleration of the spider was calculated as the slope of portions of the velocity versus time relationship, where each velocity was calculated discretely as the quotient of the change in displacement and the change in time increment (Fig. 3). Accelerations were calculated for the parts of the power and glide phases during which the change in velocity with time was approximately linear (Fig. 3). Newton’s second law (F=ma) allowed us to calculate the average net force exerted by the spider’s legs during the power phase and the average resistance exerted by the water during the glide phase.
Using the same experimental apparatus and analytical techniques, we measured the angular velocity of the propulsive legs II and III, relative to the longitudinal axis of the spider’s body, during the power stroke (Fig. 4). We used the angular velocity to calculate the velocity of the leg relative to the cephalothorax at known distances from the leg’s attachment to the spider; direct measurement of leg tip velocity was not possible during the power phase because, in many trials, the image of the leg tip was invisible in the refractive shadow of the leg-cum-dimple.
To measure forces on an isolated leg III in the horizontal plane, we used a horizontal balance to which a leg segment was attached (Fig. 5). At the core of the balance was an electronic clinometer (Applied Geomechanics Inc., model 900 Biaxial Clinometer) with a resolution of 0.01 ° (1.75×10−4 rad). The clinometer, attached to a pair of 25 cm tubular glass arms, could pivot around a horizontal axis that was perpendicular to the direction of water flow (‘pivot point II’ in Fig. 5). Because the leg segment was attached to a balance beam and not rigidly to the vertical arm of the balance (‘pivot point I’ in Fig. 5), the force pushing the leg segment down onto the surface of the water was always constant during a test. The force could be manipulated by adding weights to the leg end of the balance beam. The addition of weights to the weight post at the leg end of the balance beam caused an increase in the depth of the dimple, h, formed by the leg and its interaction with the water surface. We calculated h in still water by subtracting the height of the weight-post (measured using a micrometer caliper) with a test weight added from the height of the post with no weight added. The leg segment was judged to be at h=0 when, by close visual inspection in still water, the segment produced no visible dimple; note, however, that the irregularity of the leg surface (caused by leg hairs) meant that whenever the segment was in contact with the water there was some distortion of the water surface.
The distilled water or ethanol solution (see next section) moving past the leg segment contained no detectable flow disturbance, and so effectively replicated the still surface of a pond upon which a fisher spider would propel itself. This was accomplished because the water moving past the stationary leg was contained in a shallow, horizontally rotating cylinder and was moving with the same angular velocity as the cylinder itself. With respect to the container holding it, the water was not moving at all; but with respect to the leg segment on the surface, the water was moving at a velocity proportional to the angular velocity of the cylinder and the distance of the segment from the axis of rotation. The inside circumference of the cylinder (diameter 65 cm, depth 3 cm) was lined with a 1 cm thick layer of aquarium filter wool that effectively absorbed surface waves that reached the perimeter and therefore prevented the reflection of waves back into the experimental area. The cylinder rested on top of a turntable powered by a variable-speed d.c. motor (Bison Gear and Engineering Corporation, model 507-01-127). The only connection between the motor and the turntable was a pair of elastic bands, one to each of two speed-reduction pulleys, resulting in nearly vibration-free rotation of the cylinder and the fluid it contained. In practice, we made speed changes in small increments and collected force data only after the turntable had been at the new test speed for at least 1 min, allowing time for the water to attain the same angular velocity as the cylinder.
The output of the clinometer (in V) was digitized using an analog-to-digital (A/D) convertor (National Instruments Corporation, model NB-MIO-16L) driven by a custom-designed program (National Instruments software LabView 3) on a microcomputer (Apple Corporation, model Centris 650). The A/D converter also sensed voltage peaks in the output of an infrared tachometer mounted under the turntable. The software program converted the clinometer’s voltage into total or thrust force, Ft, and calculated the relative velocity of the water, U, past the leg segment, which was located approximately 24 cm from the center of the cylinder. For each experiment, consisting of a given leg length l, dimple depth h, and surface tension γ, scatter plots of force as a function of water velocity were fitted using power curves (CA-CricketGraph software III v. 1.5.3), since we anticipated that Ft would be proportional to U2 (see equations 2 and 3).
The force transduction method outlined above measured forces at constant U and so is only appropriate in this study if the propulsive stroke of the live spider has a constant velocity. Because the angular velocity of a leg during most of its power stroke is constant (Fig. 4), the velocity of the part of the leg that is in contact with the water surface is also constant relative to the spider’s body. Although the spider’s body is being accelerated during the power stroke, its velocity is always much less than that of the leg tips (compare Figs 3 and 4). Thus, the velocity of the leg tips relative to the spider’s body approximates (with some exaggeration) the velocity of the leg tips relative to the water, and the velocity of the leg tips relative to the water is nearly constant. For these reasons, we can disregard the acceleration reaction and other unsteady mechanisms (Daniel, 1984).
Altering surface tension
In order to test the the predictions of the bow wave and surface tension hypotheses (see equations 2 and 4–7), we altered γ by adding ethanol to the distilled water. Ethanol in aqueous solution affects both γ and the kinematic viscosity of the solution: as the concentration of ethanol rises from 0 % (distilled water) to 9 %, γ falls from 0.072 N m−1 to 0.054 N m−1 and kinematic viscosity, ν, rises from 1.01×10−6 m2 s−1 to 1.59×10−6 m2 s−1 (Chemical Rubber Company, 1993). Note that the isolated leg experiments were performed at Reynolds numbers, Re, that ranged from approximately 30 to 1100 (for details, see below).
We measured horizontal forces, Ft, on an 18 mm (length) leg segment (see previous section) while increasing the concentration of ethanol in 1 % steps from 0 % (distilled water) to 9 %, yielding values of γ of 0.0716, 0.0696, 0.0676, 0.0657, 0.0637, 0.0617, 0.0597, 0.0578, 0.0558 and 0.0538 N m−1. We avoided using higher concentrations of ethanol because changes in the hydrophobicity of the leg segment’s surface might result, since segments held in 30 % ethanol for an hour lose their ability to form a dimple in the water surface, an indication that hydrophobicity has been lost. To keep experimental conditions as uniform as possible, we left the leg segment in place during changes in ethanol concentration. We made the changes by removing a small volume of the test solution and gradually replacing that volume with the same volume of a dilute ethanol solution the concentration of which had been calculated to increase the test solution’s concentration by 1 %. To determine whether significant alcohol evaporation would occur during the 2–3 h needed for a full ethanol series, we measured γ (using the capillary rise method) of a 10 % ethanol solution in the test chamber over 3 h. We could detect no change in surface tension, and concluded that evaporation of ethanol would not significantly affect our results. We used only a single length (18 mm) of a single leg because measurements of multiple leg lengths are not necessary to test the predictions of our hypotheses and because the use of a single leg for all trials eliminated the small variations in force that could have arisen from small variations in leg shape, surface contour and hairiness.
To test the predictions of equation 1, and hence the relative importance of Fw, we needed to determine at what relative water velocity waves first appeared and whether their appearance was abrupt or gradual as water flowed past an isolated leg segment. Note that we did not need to know the absolute magnitudes of the waves. To make these determinations, we used a 35 mm camera and an electronic flash to photograph (from above) the bottom of the rotating cylinder on which was cast the shadow of a 12 mm (length) leg segment as it rested (h=0.5 mm) on the distilled water surface. As waves appeared, the surface distortion caused refraction of the light from the electronic flash and that, in turn, caused wave shadows to appear on the bottom of the cylinder (Fig. 6). Twenty-six 35 mm photographs, taken at a variety of relative water velocities, were scanned (Epson ES-1200C) into a computer using software (NIH software Image v. 1.55f), and the pixel intensity of each image was measured along a standard transect running perpendicular to the wave crests and outside the leg’s shadow. By measuring the standard deviation of pixel intensity along the transect, we generated an index of the waviness of the water surface, not a measure of wave amplitude per se.
Flow pattern disturbance
One indicator of the presence of sub-surface drag force (equation 3) is the disturbance of the flow pattern around the sub-surface structure (Vogel, 1994). Specifically, by disturbance we mean evidence that the momentum of the fluid perpendicular to the flow has been altered, which indicates, in the case of our isolated leg experiments, that the leg-cum-dimple is removing momentum from the sub-surface flow (Fig. 2B). The level of disturbance was measured as the amount of change in a pattern of light reflected by the surface of the water (Fig. 7). First, we mounted a card perpendicular to the distilled water surface in the turntable, positioned so that the reflection (on the water surface) of the horizontal lines on the card could be photographed by a 35 mm camera (as in Fig. 1). The card, the camera’s film plane and the water that had just passed under and around the leg-cum-dimple were parallel to one another. As a result, distortions in the water surface caused by sub-surface flow disturbance were visible as distortions in the reflections of the horizontal lines. Photographic images were digitized in the same manner as were those for the wave index measurements (see previous section) and were then analysed using NIH software (Image v. 1.55f) as described in Fig. 7.
where y is Ft, c is the regression constant, x is U, and b is the regression exponent. We chose this relationship based on the a priori expectation that Ft would be an exponential function of U, with b having values between 1 and 2 (Vogel, 1994). Note that b is also the logarithmic slope when equation 9 is log-transformed. For each experimental trial, b then becomes the single response variable that summarizes the relationship between Ft and U.
To test for the homogeniety of slopes between treatments, the first stage in ANCOVA, the variance in b was tested in a two-way analysis of variance (ANOVA), with h and γ as the categorical independent variables, with 2 and 9 degrees of freedom (d.f.), respectively. Planned contrasts were made between categories within each independent variable. For h, we investigated whether the mean values of b differed between 0 and 1 mm, and between 1 and 2 mm. For γ, we investigated whether the mean values of b differed between neighboring values of γ and between the middle and end values (SAS Institute, v. 6.11).
Modeling drag from first principles
In addition to the experimental procedures described above, we also modified equation 3 to calculate drag of the leg-cum-dimple, reasoning that if drag were the primary propulsive force, approximations of Fd from first principles should approximate Ft on the isolated leg (see equation 8). The key to accurate drag estimates is an accurate measurement of the coefficient of drag, Cd, which varies as a function of the shape and Reynolds number, Re, of the object (Vogel, 1994). For simple objects, such as flat plates and circular cylinders, the relationship between Cd and Re is well known.
During the acceleration phases of surface locomotion by D. triton, the leg velocities and whole-body forces, derived from videographic measurements (Fig. 3) made on 19 spiders of different sizes, indicate that the spiders produced net leg tip velocities with means ranging from 0.16 m s−1 to 0.62 m s−1 (Fig. 8A) and mean forces in the range 0.008–0.327 mN during acceleration (Fig. 8B). In the acceleration phase, the forces were generated by the backwards sweep of the third and second pairs of legs (Fig. 3) but, because acceleration was approximately constant while legs III, then III plus II, and then II alone were providing thrust (Fig. 3), we could treat the system as if only one pair of legs was used throughout. Thus, the thrust produced per leg varied between 0.004 and 0.164 mN (the range given above divided by two). Because this forward force had to be the difference between the forward force generated by the legs and the backward resistance of the water on non-propulsive parts of the spider, the estimate of the force generated by each leg must be an underestimate.
During the trials, dimples of leg III were as deep as 3.5 mm, and the largest spiders often had more than 18 mm of each propulsive leg in contact with the water surface (although only the tip of the leg would have been at the maximum depth). Thus, the thrust estimates, Ft, based on data from an isolated leg segment of 18 mm length (see next section), underestimate the actual thrust that the legs of a spider should be able to generate.
Isolated leg experiments
At experimental values of U below 0.21 m s−1, a bow wave, or a wave of any kind, was not detected, as indicated by the wave index (Fig. 9). At U of 0.21 m s−1, waves appeared abruptly and, with increasing U, increased rapidly in amplitude and frequency. This result supports the prediction (Denny, 1993) that bow waves should not be available as a source of resistance to the leg at low U (equation 1). Because Fw cannot be operating at U<0.20 m s−1, we restricted our experimental manipulations of h and γ to this lower range of U.
The horizontal force, Ft, generated by an isolated leg segment (18 mm in length) and its associated dimple, as it moved across the water surface varied from 0 to 0.6 mN at relative leg velocities, U, of up to 0.2 m s−1 (Fig. 10). The relationship between Ft and U for each experiment (constant h and γ) produced a range of power curves with exponents (b in equation 9) varying from 1.601 to 2.135. This range of exponents includes the value of 2 predicted from the bow wave (equation 2) and drag (equation 3) resistance hypotheses. Note that the surface tension resistance hypothesis (equations 4–7) also predicts an increase in Ft with increasing U (see Table 1), although it is unclear what the exact relationship should be, since Fs depends on the exact shape of the dimple during the power stroke, and we did not measure this feature.
As revealed by ANOVA (Table 2), changes in both dimple depth, h, and surface tension, γ, significantly altered b. A comparison of the F-values of the two factors reveals that much more of the variance in b is explained by the variance in h than that in γ. Statistical comparisons (Table 3) indicate that with increasing dimple depth, h, from 1 to 2.0 mm, the power exponent, b, of Ft increases significantly (Fig. 11). This result is consistent only with the predictions of the drag resistance hypothesis (equation 3; see also Table 1). Statistical comparisons using ANCOVA (Table 3) also show that with increasing surface tension, γ, from 0.054 to 0.064 N m−1, the power exponent, b, of Ft increases significantly (Fig. 12), as predicted by both the bow wave (equation 2) and surface tension resistance hypotheses (equations 4–7; see also Table 1). None of the simple predictions of the three theories predicted the decrease in b as γ increases from 0.064 to 0.072 N m−1.
The surface of the water downstream from a leg segment was undisturbed, as measured by the flow disturbance index, when U=0, but disturbance of the water surface increased steadily as U increased (Fig. 13). This result supports the hypothesis that drag is operating at all values of U, since the disturbance is an indicator that momentum is being removed from the water by the leg (Fig. 2B). The smooth increase in the index at U from 0 to 0.33 m s−1 indicates that there is a continuous transfer of momentum from 0.07 to 0.33 m s−1 and that no abrupt alteration in the transfer rate occurs at the velocity at which surface waves begin to form (0.21 m s−1, Fig. 9).
At in vivo values of U (0.075–0.20 m s−1) and with h=0 and 1 mm, the model of the isolated leg-cum-dimple as a circular cylinder is an excellent predictor of the measured values of Ft, as indicated (a) by a ratio of modeled Fd to measured Ft of 0.80–0.98 (Fig. 14A) and (b) by the estimated values of Fs as the difference between measured Ft and modeled Fd approached 0 mN (Fig. 14B). The circular cylinder model is less accurate when h=2, yielding Fd/Ft ratios of 1.22–1.31 and Fs values of −0.01 to −0.15 mN.
Experiments on the isolated legs of D. triton demonstrate, for the first time, that drag is the dominant force operating to permit fisher spiders, and quite possibly most other surface-dwelling arthropods, to move horizontally across the water surface. In living spiders, the legs and the dimples they create move backwards at nearly constant velocity, accelerating the body forward in response. The net propulsive force experienced by the spider’s body during surface locomotion can be sufficiently explained by the drag forces on the isolated legs, which were measured at in vivo flow velocities. Other forces acting on the legs – those due to surface tension and bow wave pressure – appear to be negligible at conditions most closely resembling those seen during non-galloping water surface locomotion in live spiders.
Drag is the primary propulsive force
The bow wave force, Fw, was eliminated as a candidate for the primary source of resistance since waves were not detected behind or in front of the isolated leg at flow velocities, U, of 0.20 m s−1 or below (Fig. 9). Since the leg tips of live spiders at times move at U<0.20 (Fig. 8) and the more proximal portions of the legs move still more slowly, some other resistance mechanism must be operating at low speeds. Most of the resistance is caused by drag, Fd, and not surface tension force, Fs, as indicated by several lines of evidence. First, Ft varied in proportion to U to the power 1.601–2.135 (Fig. 10), all values of b (see equation 9) within the range seen in a variety of biological systems (Vogel, 1994). The value of b alone, however, does not rule out the possibility that surface tension is also providing resistance, since Fs may also increase with the same proportionality to U (Table 1). More convincing is the fact that b increases significantly as the dimple depth, h, increases from 1 to 2 mm; this was the predicted relationship for Fd but not for Fs (Table 1). Furthermore, the significant changes in b are due mainly to h and not to γ (compare the F values in Table 2). At the same time, γ may have an effect on Ft (to be discussed in the next section) as shown by significant changes in b with increasing γ (Fig. 12; Table 3).
Second, a circumstantial piece of evidence supports the primacy of a drag-based mechanism. Flow behind the leg-cum-dimple was disturbed (Fig. 13), relative to still water, at all flow velocities measured (U>0.07 m s−1). Because flow disturbance is evidence that momentum is being transferred from the water to the leg in the sub-surface flow (see Fig. 2B), this photographic assay strongly implicates drag as the source of that flow disturbance.
Third, the modeling of the leg-cum-dimple as a circular cylinder oriented perpendicular to flow (equations 13–15) generates values of Fd that account for 75–98 % of the measured Ft when h=0 or 1 mm and U’2!0.075 m s−1 (Fig. 14A). This low value for U was half of the lowest in vivo value for the leg tip (Fig. 8); parts of the leg in contact with the water, but proximal to the tip, will move more slowly. Note that if only leg tip velocities are used, with a corresponding minimum in vivo leg tip velocity of 0.16 m s−1, the drag model is more accurate, accounting for at least 85 % of the measured Ft (Fig. 14A). When the entire range of U is considered (0<U:s;0.20 m s−1), Fs never exceeds values of approximately 0.03 mN when h=1 mm (Fig. 14B). This simple drag model is less accurate when h=2 mm; with that configuration, Fd overestimates Ft by 20–30 % (Fig. 14A).
We do not understand how Fs could be negative when h=2 mm (Fig. 14B). Precise measurements of in vivo h and dimple shape (ra and rp, αa and αp in equations 4–7) might offer an explanation. A negative Fs would actually pull – not resist – the leg, and the dimple geometry required to explain this is αa>αp (see Fig. 2C). But this configuration seems unlikely given that the dimple’s leading edge cannot travel as a wave and move ahead of the leg, decreasing αp, since waves are not produced at the speeds we are considering (U<0.20 m s−1). Instead, αa would have to increase, which could happen if a deep dimple has more vertical walls as the curvature of the surface moves beyond that which can be supported by surface tension (see Denny, 1993). While theoretically possible, such drastic changes in dimple geometry have yet to be visualized. Clearly, dynamic three-dimensional measurements of dimple structure are warranted.
The shape of the dimple may also explain an unpredicted result – the decrease in b with increasing γ approaching that of fresh water (Fig. 12). Recall that our prediction of an increase in Fs (Table 1) was predicated on the assumption that the trailing edge of the dimple would reconfigure such that αa<αp (Fig. 2C). The rate of increase in Ft with respect to U would decrease, as indicated by a drop in b (Fig. 12), if αa increased, decreasing Fs at any given U (assuming a constant Fd), if an increase in γ ‘stiffened’ the dimple wall.
Propulsion in living spiders
A drag-based mechanism operating at leg speeds, U, below cmin (see equation 1) solves Denny’s paradox (1993), which states that small or slow-moving surface-dwelling arthropods should not be able to propel themselves horizontally, since their legs would never travel at velocities faster than the 0.23 m s−1 needed to create a bow wave and the resulting propulsive force. No such speed limit is imposed on a drag-based mechanism. Thus, the early instars of fisher spiders and water striders are free to locomote across the water surface using legs that never exceed 0.23 m s−1.
The forces experienced by an isolated leg, which are indicative of its propulsive capabilities, are only sufficient to explain surface propulsion in a living spider if the following criteria are met. First, we must rule out propulsive contributions from other structures or mechanisms. Second, we must demonstrate that the isolated legs were tested at physiological conditions. Third, the forces on an isolated leg must account for approximately half (one of four propulsive legs, usually working in pairs) of the total net force acting on the spider’s body. Fourth, the proposed mechanism must be able to account for variation in locomotor speeds seen with changes in size or behavior.
First, the evidence that legs II and III alone are generating thrust comes from the fact that during locomotion they are the only appendages that are in motion relative to the body. A possible propulsive mechanism that would not require leg rowing might include an asymmetric change in γ by the secretion of a surfactant (Denny, 1993). We know of no evidence for this mechanism in spiders. Second, the physiological relevance of these experiments is demonstrated by the fact that we used the isolated leg segment at physiological values of h (see Results) and of U (Fig. 8A).
Third, the maximal thrust force, Ft, measured on the isolated leg was approximately 0.6 mN (Fig. 10), a value within the range of mean body forces measured in live spiders (Fig. 8B). These single leg values are also within the range of horizontal forces measured in single legs of the dead-head cockroach, B. discoidalis (Full et al. 1995; Full and Ahn, 1995); when adjusted for body mass (kg), one of our spiders, of 0.7 g mass, producing 0.6 mN force with one leg at a leg velocity of 0.2 m s−1, would generate a mass-specific single leg force of 0.86 N kg−1, a value close to the horizontal force of a 3 g B. discoidalis (1.7 N kg−1) during running.
Finally, we can suggest simple ways in which a spider might alter locomotor speeds. First, it can increase the depths of dimples by shifting its weight distribution, which permits larger forces to be applied by the legs to the surface. Second, a spider can alter the angular velocity of its limbs, thus changing the fore-to-aft velocity of the legs and hence drag. Third, it can alter the water-contact length of each leg, thereby altering surface area and drag. By varying the speed and size of the leg-cum-dimple, a spider may also be able to compensate for changes in surface tension that are expected in nature (Denny, 1993). We are pursuing these sources of variability because it is clear that the physics of locomotion at the water surface can only be part of the story for surface-dwelling animals – behavior, especially changes in posture and weight distribution, is likely to be as crucial in this form of locomotion by very small organisms as it is in aerial dispersal by ballooning in spiders (Suter, 1992). We are also pursuing the allometry of the propulsive and glide phases of surface locomotion by measuring the forces acting on first- and later-instar water striders and adult female fisher spiders (a 7000-fold range of masses).
Multiple mechanisms can operate
Finally, it should be noted that in spite of the predominance of drag, the three hypothetical sources of resistance to the motion of a spider’s leg at the water surface (Fig. 2; equation 8) are not mutually exclusive. Under some conditions, in fact, all must contribute to the resistance against which a spider or other surface-dwelling arthropod must push when supported by the surface tension of water. For example, when the legs form dimples in the water surface and are moving at velocities greater than 0.23 m s−1 relative to the water surface, a bow wave will be produced, the moving leg-cum-dimple will encounter some drag, and the dimple will be distorted (Fig. 2). We can also predict the relative importance of the three kinds of resistance under certain conditions: at leg tip velocities less than 0.23 m s−1, bow waves should be unimportant; at leg tip velocities greater than 0.23 m s−1 and at dimple depths very close to zero, dimple distortion and drag should be minimized and the relative contribution of bow wave resistance should be maximal; and at dimple depths between 1 and 3.5 mm and velocities below 0.23 m s−1, drag should predominate. The data presented in this study make possible not only an evaluation of those predictions, but also the conclusion that a single physical process, drag, is ascendant under most conditions.
Mathematical symbols used in the text
We thank Edgar Leighton at the University of Mississippi for providing us with the subjects of this study, Erin Murphy for some of the data collection, Pam Murphy for some of the spider illustrations, and two anonymous reviewers for their careful and very constructive comments on an earlier version of the manuscript. The study was supported in part by funds provided by Vassar College through both the Undergraduate Research Summer Institute and the Class of ‘42 Faculty Research Fund. J.H.L. was funded by the Office of Naval Research (Grant N00014-97-1-0292).