Over the range of velocities used (1.8–6.8 m s⁻¹), the Rynchops bill cast experienced an approximately 14-fold increase in drag from 0.05–0.72 N (Figure 2A). The Tupuxuara model, immersed to the same depth (0.04 m), experienced drag that was nearly an order of magnitude higher at 0.30–5.18 N, with the data for Thalassodromeus of the same order (Figure 2B). The Rynchops data from the flume trials were modelled using an ordinary least-squares regression (OLS) of log-transformed drag on log-transformed velocity (0.286V^1.52, n = 17 flume runs, adjusted r² = 0.76, p < 0.001; Figure 2). To provide a conservative estimate of drag, the equation for the lower 75% confidence interval of the OLS was used to describe drag of the Rynchops bill (0.019V^1.26). Comparison of the thin plate approximation (Equations 1–6) of a Rynchops mandible with the lower 75% CI line of an OLS regression of drag on velocity for the physical model (Figure 2A) indicates that our hydrodynamic model captures the behaviour of the physical model appropriately and thus supports our use of this technique for the mandible of large pterosaurs. Drag measurements for the model Rynchops bill and both Tupuxuara sp. and the Thalassodromeus jaw fragment accord well with estimates derived from fluid dynamic principles to give conservative estimates of drag (Figure 2B).

Estimated flight costs including hydrodynamic drag were compared against an estimate of available metabolic power from Marden [29], using the hydrodynamic model for the pterosaurs to adjust immersion depth to 19%. Estimated minimum cost of flight (minimum power speed V_mp) and minimum cost of flight per metre travelled (maximum range speed V_mr) were 5.9 m s⁻¹ and 11.0 m s⁻¹ respectively for Rynchops (Figure 3A and 3B). Flight costs were considerably higher with V_mp at 3.9 m s⁻¹ and 4.3 m s⁻¹, and V_mr at 6.2 m s⁻¹ and 6.9 m s⁻¹ for Tupuxuara (Figure 3C and 3D) and Thalassodromeus (fragment; Figure 3E and 3F), respectively. The estimated foraging flight costs for powered flight (Figure 3) show both the reduced costs at intermediate speeds and the rapidly increasing costs at very low speed (resulting in higher stall speeds) associated with wing-in-ground effect flight [30]. When considered in relation to an estimate of maximal metabolic power available for powered flight based on body mass of flying vertebrates [29], Rynchops, despite high hydrodynamic drag costs (Table 1), is still capable of foraging for long periods with its bill submerged to a depth of 0.04 m. Neither of the pterosaurs appears able to meet the energetic costs of skimming. The fact that the observed flight speed of 10 m s⁻¹ for Rynchops [19,31] is closer to the theoretical V_mr suggests that the bird is attempting to reduce the cost of transport, and this finding supports the idea that foraging flight in this species maximises the distance travelled for a given energy input [31,32].

The added costs of flight due to hydrodynamic drag range from 20% of total costs in Rynchops, up to 68% in the low-mass estimate for Pteranodon (Table 1, Figure 4). Among the pterosaurs, even Tupuxuara, which seems potentially able to fly with the tip of its mandible immersed, uses half of the energy needed for flight on hydrodynamic drag. It is clear that in all species, hydrodynamic drag constitutes a major component of total flight costs. It is also clear that with hydrodynamic drag costs incorporated, the benefits of intermittently powered flight are negated for most species (Table 1).

Our results allow us to re-evaluate the foraging ecology of Rynchops. The hydrodynamic drag on the bill is considerably greater than that previously estimated by Withers and Timko [19]. In the context of our revised estimates of the costs involved in foraging flight, and contrary to the only other previous study [19], we found that hydrodynamic drag from the bill tip has significant implications for the energy budget of Rynchops. At the typical recorded foraging speed of R. niger (10 m s⁻¹ [19,31]), the drag extrapolated from our measurements for the model R. n. cinerascens (0.35 N) is at least three orders of magnitude greater than that previously estimated [19] (1 × 10⁻⁴ N). We attribute this disparity to inadequate consideration of dynamic scaling methods in [19] as accurate estimation of drag at the water surface requires test velocities close to those in real life to satisfy scaling of both the Reynolds number (Re) and Froude number (Fr) [33].

On the basis of the calculations provided by Withers and Timko [19], it has generally been assumed that the hydrodynamic costs of skimming are trivial compared to the aerodynamic costs of flight (e.g., [17,31–33]). Similar to our work, Withers and Timko [19] suggested that wave drag on the lower mandible could safely be ignored (see Materials and Methods), but they did not consider spray drag at all. Nonetheless, although we demonstrate that spray drag is less important than pressure drag, it is still an important component of the hydrodynamic drag costs. For the thin-plate approximation of the Rynchops bill, we find that as a percentage of total drag costs, the drag components contribute 79.1% (pressure), 13.5% (ventilation), and 7.4% (spray) at V_mp. However, whereas Withers and Timko [19] concluded that hydrodynamic costs for skimmers are less than 0.1% of the aerodynamic costs of flight, we argue that correct measurement of drag (matching Re and Fr) suggests that hydrodynamic costs are two orders of magnitude higher than that estimate (20% of aerodynamic costs). To this end, we suggest that the high energetic requirements of this foraging technique, despite considerable morphological specialisation, may explain its rarity in extant birds.

Hydrodynamic drag estimates for pterosaur mandibles are considerably higher than those of Rynchops, and in the context of aerodynamic estimates rule out skim-feeding for those animals heavier than 2 kg. Based on the flume results, our modelling appears to describe well the hydrodynamics of an immersed Rynchops bill tip and lends support to our use of the model for pterosaur jaws. In accordance with current thinking, powered flight seems an unlikely proposition for Quetzalcoatlus, even without the added drag from skim-feeding. Second, even with the conservative hydrodynamic drag estimates used, our calculations suggest that neither Thalassodromeus, Pteranodon, or Quetzalcoatlus would have been able to use skimming as a feeding method in continuous flight or with intermittent flapping, although Tupuxuara may have been able to meet skim-feeding energy requirements based on its proximity to its theoretical available metabolic power (Figure 4, broken red line). However, the estimated required power output is still around 10 W more than the other estimate of available power, based on empirical data from Marden [29]. This is a somewhat surprising finding, because Tupuxuara sp. was not a species we expected to be able to skim-feed, but the result most likely reflects our conservative modelling of drag. However, it does also present a possible explanation for evolution of the skimming habit. If, as some authors predict (e.g., [1,34]) pterosaurs had lower wing loading than extant birds, they were probably able to cope with some additional flight costs. If the ancestors of modern skimmers were either smaller or had proportionally larger wings, this may have given them the ability to tolerate the increased costs of skimming with a “normally” shaped bill and to take the step from dipping for prey to some kind of skimming behaviour. It also leaves the theoretical possibility that smaller (<2-m wingspan) pterosaurs such as Rhamphorhynchus may have been able to skim-feed occasionally without particularly specialised jaw morphology due to their lower flight costs (through lower wing loading [34]) than extant birds.

Our analyses assume air density at current levels, but this value may not have been constant over geological time [35,36]. Because we use gross estimates of available power, we are unable to estimate directly the implications for lift and drag of air with differing densities. However, if we assume a direct relationship between air density and lift, then over the relevant timescale for pterosaurs, the density minimum of the Triassic (87% of present [36]) and the density maximum of the early Tertiary (121% of present [36]) might be expected to increase or decrease the power requirements for flight by 13% and 21% respectively. Neither of these values is likely to change the overall conclusions presented here.

To date, analysis of potential skim-feeding adaptations in pterosaurs have been limited, typically focusing on single cranial or cervical features rather than considering the entire skull and neck. This suggests a misunderstanding of just how specialised skim-feeding is: modern skimmers require numerous adaptations across their cranial anatomy to facilitate efficient skimming and to resist forces generated during feeding [24,37]. Some 30 adaptations are seen in the skull and neck of Rynchops that cope with both hydrodynamic forces acting upon the skull whilst skimming and with the impact of capturing prey, as well as maintaining bill position within the water column, detecting prey, and regenerating abraded bills [24,37]. The most obvious of these adaptations include extreme lateral compression and pronounced horny extension of the mandibular symphysis, large jaw muscles, a reinforced quadrate-articular joint resistant to lateral disarticulation, developed medial bracing of the mandible, short mandibular rami, elongate mandibular symphysis, and a segregated, highly flexible neck [24,37]. Such adaptations are not seen in other birds and are thought to represent specialisations to the highly derived lifestyle of skimmers [24].

We acknowledge that pterosaurs may have possessed skimming adaptations distinct from those of Rynchops, but the structures of pterosaur and bird skulls are not so different that some functional convergence should be expected if, indeed, pterosaurs did skim. However, few of the adaptations expected are seen in these postulated skimming taxa. For instance, pterosaur skulls are often elongated, but not one pterosaur yet found shows the degree of lateral compression seen in the mandible of Rynchops. Relatively pronounced lateral compression of the mandibular symphysis is seen in Pteranodon, Rhamphorhynchus, and Thalassodromeus, with tapered dorsal and ventral symphyseal surfaces also seen in the latter taxa [2,26,38], but none exhibit the extreme condition of this feature seen in Rynchops (Figure 1).

The rarity of soft tissue preservation limits understanding of pterosaur jaw soft tissues, and it remains possible that some postulated pterosaur skim-feeders could have possessed horny mandibular sheaths similar to those of Rynchops to extend the jaw and pronounce lateral compression. This structure appears vitally important for skimming Rynchops, because it extends the feeding depth and provides an expendable jaw tip that can be replaced if damaged accidentally on submerged obstacles or grounded on substrata [24]. A review of Rynchops literature suggests that such accidents may be relatively common [17]. The rhampotheca of Rynchops is highly vascular and grows continuously, presumably to regenerate the abraded or broken bill tips [24]. Such an adaptation, therefore, seems crucial to skim-feeding. However, no known pterosaurs show a mandibular sheath comparable to that of Rynchops. Rhamphorhynchus does show a soft-tissue mandibular extension, but it is considerably shorter than that of skimmers and shows dorsal curvature [38] instead of ventral as seen in skimmers.

Several pterosaurs are skimmer-like in possessing short mandibular rami (e.g., Pteranodon and Quetzalcoatlus, though not Thalassodromeus), but their rami differ from Rynchops in being relatively slender. This suggests that pterosaurs had comparatively little mandibular anchorage for their jaw adductor musculature, which is known to be large in Rynchops. This increased muscle mass in Rynchops is critical for stabilising the mandible and counteracting depressive forces acting on the mandible whilst skim-feeding [24]. Skimming pterosaurs would almost certainly require similarly developed musculature to forage successfully.

Rynchops possesses a reinforced jaw joint supported by bony bracing at the quadrate and the basitemporal plate [24,37]. Several other bird groups demonstrate a similar condition, but none show the development of this bracing to the level seen in Rynchops [37]. Reinforcement of the jaw is thought to be necessary to withstand the impact phase of skim-feeding [24], and no pterosaur yet known shows jaw bracing of this kind. The jaw articulator facets of Rhamphorhynchus and Tupuxuara are deep (DM Unwin, personal communication) and a small laterally bracing “peg” is known in Quetzalcoatlus [13], but despite this, pterosaur jaw joints are considerably less robust than those of Rynchops.

The neck of Rynchops is also subjected to large stresses during skim-feeding [24], including caudoventral forces on the skull and neck that are countered by powerful neck muscles anchored to a flexible, robust cervical series. The cervical vertebrae are segmented into articulatory groups and highly sculpted to situate enlarged neck musculature [24], with the anterior vertebrae particularly robust. Such adaptations are needed when an impact occurs during skimming, as these force the head and anterior cervical segments to rotate caudoventrally. Strong, flexible necks appear critical to absorbing impact forces while skim-feeding.

Amongst pterosaurs, the cervicals of Rynchops are most distinct from those of azhdarchids. These pterosaurs bear slim, elongate cervicals with broad, interlocking zygopophyses and reduced or absent neural spines and transverse processes [39]. Many authors have noted the inflexibility of the azhdarchid cervical series (e.g., [27,40,41]), and this would appear to be a severe handicap to a skim-feeder. So great are these distinctions in cervical vertebrae between azhdarchids and Rynchops that we echo the suggestion of Ősi et al. [27] that neck osteology alone casts serious doubt on the skimming potential of azhdarchids. The cervical vertebrae of Rhamphorhynchus, Pteranodon, and tupuxuarids compare more favourably with those of skimmers in being relatively robust and bearing developed processes. The biomechanics of pterosaur necks are still largely uninvestigated, but we note that neither Rhamphorhynchus nor Pteranodon show the distinct segmentation and size increase of the anterior cervical series seen in Rynchops (Figures 27 and 28 of [24]). The cervical series of tupuxuarids are among the most robust and flexible of all pterosaurs (DM Unwin, personal communication), but we also note that many feeding strategies require flexible necks, not just skimming. Hence, these features of tupuxuarid anatomy do not necessarily advocate skim-feeding habits in these forms.

Almost without exception, pterosaur anatomy appears poorly adapted for skim-feeding. Comparisons between pterosaurs and Rynchops reveal little of the convergence expected between animals postulated to have such similar, specialist lifestyles, with the relatively wide mandibular symphyses, apparent absences of elongate, abradable mandibular sheaths (at least observable in Rhamphorhynchus), and lack of a suitably reinforced jaw joint particularly pertinent arguments against pterosaur skimming. We note that Dsungaripterus, a pterosaur specialized for crushing hard-shelled foods [16], shares an upturned, dorsally tapered mandibular symphysis with many of the postulated skimming pterosaurs discussed here. Upturned jaws may provide increased grip on food through the larger occlusion surface with the bolus along the lower jaw. Straight jaws, by contrast, can only hold an object at a single point on each jaw. Dorsally tapered symphyses may also allow for greater application of pressure on food items and do not necessarily indicate skimming habits.

The idea that pterosaurs were skimmers appears to have originated through limited morphological comparisons with modern forms (e.g., [2]) and inadequately investigated functional inferences (e.g., [15]). Both modelling the energy requirements of skimming pterosaurs and analysing their osteology casts serious doubt on the ability of pterosaurs to habitually skim-feed. Although our physical modelling suggests that small pterosaurs may have been energetically capable of skimming, there is no anatomical evidence to assume that Rhamphorhynchus or any other small pterosaurs were skimmers. From this, we stress the difficulty of using limited morphological convergence to interpret the ecology of extinct forms: comprehensive analysis of this nature can be insightful but should ideally be supported with additional biomechanical data.

Model bills were cast in epoxy resin (R. n. cinerascens, BMNH 1914.12.2.9), or modelled with dense, varnished, polyethylene foam (Tupuxuara sp., [IMCF 1052] and Thalassodromeus fragment [DGM 1476-M]) from specimens and data [28]. Model dimensions are given in Table 2. These physical models were instrumented with strain gauges connected to an amplifier, with the output signals recorded on a computer. Models were suspended from a trolley towed above a water flume (0.5 m width × 12.0 m length, 0.25-m water depth, 8.0-m trolley travel) at different speeds selected from the known range for Rynchops and determined by speed settings for the towing motor. Velocity data were extracted from video footage of the trials (Sony DCR HC1000, 25 frames per s) and bending moments present at the strain gauges were used to calculate the drag force acting on the models.

In physical terms, the laterally compressed mandible of Rynchops acts as a surface-piercing strut, and as such, the drag that it experiences is potentially comprised of five additive components: friction drag, pressure drag, wave drag, spray drag, and drag due to ventilation [42]. Of these five components, the latter three are specific to surface-piercing struts and are the result of energy losses from work performed in displacing water vertically against the force of gravity [42].

Drag coefficients representing pressure and friction drag vary with Reynolds number (Re). The distal section of the mandible of Rynchops is similar to a reversed streamlined airfoil (e.g., NACA 0012), for which data collected at Re ≥ 10⁶_, suggests coefficients of drag twice that of a typical streamlined airfoil oriented with its blunt leading edge facing the direction of movement [33,42]. Such sections are apparently adapted to reduce drag components associated with the air–water interface [33,43]. We have no reason to believe that the coefficient of drag for such a reversed airfoil will vary with Re in the same way as a conventionally oriented airfoil will, especially at Re < 10⁶, a range that has received comparatively less attention from fluid dynamicists. Although we know almost nothing about the behaviour of reversed airfoils, Hoerner [42] suggests that the coefficient of drag for sharp-nosed foil sections (i.e., thickness ratios less that 15%) at Re < 10⁶ may behave in a manner similar to that of a thin plate in laminar flow. Composite simulation and experimental data from work on turbine blades supports this theory by suggesting the drag coefficient of a reversed NACA 0012 airfoil is approximately constant over the range 10⁴ ≤ Re ≤ 10⁷ [44]. We thus describe the Re dependence of Cd_pro (coefficient of profile drag, including pressure and friction components) of the tip of a Rynchops mandible by using the theoretical drag coefficient for a thin plate in laminar flow corrected for a drag coefficient based on frontal area where 1.328/Re^1/2 is the coefficient of skin friction drag for laminar flow C_fL [42]. The ratio c/t_mid is the inverse of the thickness ratio t_mid/c where t_mid is the thickness of the plate at the midpoint between its tip and the water surface and c is the chord of the plate at the water surface. We calculate profile drag as where ρ is the density of water, V the velocity of the mandible, and A_bill is the projected frontal area of the bill, from its tip to the water interface.

We now turn to the spray and wave drag components associated with the breaking of the water surface by the mandible. Both wave and spray drag depend on the Froude number (Fr), which is a measure of the ratio of inertial and gravitational forces and is used to compare the wave-making resistance between bodies of various sizes and shapes. Froude number, based on the chord of the mandible, is given by where, as before, V is the speed through the water, g is the acceleration due to gravity (9.81 m s⁻²), and c is the chord of the mandible at the water surface. Spray drag effects are unfortunately poorly described at 1 ≤ Fr < 3 [42]. However, we do know that the coefficient of spray drag (Cd_spray, based on spray drag divided by mandible thickness-squared [42]) is relatively constant and of the order 0.24 for Fr ≥ 3 for semi-streamlined double arc sections with forebody thickness ratios (t/x) ≤ 0.4 (where x is forebody length, from leading edge to point of maximum thickness on a double arc or aerofoil section; Figure 25, section 10–13 of [42]. Both the mandible of Rynchops and the NACA 0012 airfoil fall into this category, and even at 1 m s⁻¹ for the Rynchops bill, Fr = 3.5. For struts at Fr < 3, the velocities involved are small and we assume that spray drag is negligible. The spray drag is thus given by

Wave drag represents the kinetic energy transferred from an object travelling at the water surface as water is accelerated upward [42]. Wave drag increases to a maximum at Fr ≈ 0.5 [42], but then decreases steadily to zero near Fr = 3. Thus for the mandible of Rynchops, wave drag is likely to be zero at velocities near 1 m s⁻¹, and we consider its effects to be negligible in comparison to spray drag down to 0.5 m s⁻¹ and therefore assume that it can be ignored.

The ventilation drag component is caused mainly by the presence of an area of negative pressure behind a moving object that draws air downwards and increases its effective chord [42]. Drag due to ventilation is very much dependent on the shape of the strut, being greater in those with flat trailing edges, but in general, the coefficient for drag due to ventilation can be given by where d is the depth to which the tip of the mandible penetrates the water [42]. We therefore estimated the drag due to ventilation as

Finally, the three relevant drag components (D_pro, D_spray, D_vent) were summed to give an estimate of the total drag (D_tot) acting on the immersed tip of the mandible. We also used drag components calculated for a NACA 0012 aerofoil oriented with a blunt leading edge as a comparison to the flat plate approximation of a mandible tip, because its profile is more similar to that of the tips of the pterosaur mandibles. This implementation of the model differed from that given above, only in that Cd_pro was calculated (after [42]) as with C_fL from Equation (1), and C_fT, the coefficient of skin friction drag for turbulent flow, well described by the “Schoenherr line”, an empirically derived equation to describe skin friction. The coefficient of skin friction drag for turbulent flow can be approximated within 2% by (Equation 26, section 2–5 of [42]). Throughout the modelling, the mandible was assumed to be angled at 45 °, and this inclination was accounted for by adjusting the drag function by a factor of 1.415 following the results of Withers and Timko [19].

Whereas Rynchops was treated as having the distal 31% of the mandible immersed, physical models for Tupuxuara and the Thalassodromeus fragment were, for logistic reasons, immersed to the same absolute depth (0.04 m) as the Rynchops cast. To account for the discrepancy, we modelled the pterosaur bills using the hydrodynamic model with an immersion depth of 19% of its length. This immersion was chosen as it represents the posterior region of the mandibular symphysis in the Thalassodromeus jaw and thus the point after which sudden lateral expansion of the mandible occurs.

Whereas data from the model jaw of Tupuxuara sp. could be used to estimate foraging flight costs, the jaw fragment attributed to Thalassodromeus [28] was considerably thicker than is suggested by Kellner and Campos [2]. In the absence of mandible dimensions other than length and chord, the dimensions of the Thalassodromeus fragment were used for both Thalassodromeus jaw models, as well as for those of Pteranodon and Quetzalcoatlus (Table 2).

To model flight costs, we estimated the body mass (M) and wing area (A_wing) of pterosaur specimens in this study using a regression formula based on data on mass and wingspan (W) for the following 13 species of pterosaur for which wing area estimates were also available [34]: Campylognathoides zitteli, Dorygnathus banthensis, Dsungaripterus weii, Eudimorphodon ranzi, Nyctosaurus gracilis, Pteranodon sp., Pterodactylus antiquus, P. elegans, P. kochi, P. micronyx, Rhamphorhynchus intermedius, R. muensteri, and Scaphognathus crassirostris. In addition, we used data on wingspan and area for Tapejara wellnhoferi, Anhanguera piscator, and Quetzalcoatlus northropi from Chaterjee and Templin [1]. The dataset of Hazelhurst and Rayner [34] provides three estimates of wing area based on the postulated posterior attachment point of the wing membrane, whereas that of Chatterjee and Templin [1] provided only one, which fitted best when placed with Hazelhurst and Rayner's type III area (ankle attachment). OLS regressions of log(M) on log(W) and log(A_wing) on log(W) all gave good fits (M = 0.229W^2.215, n = 13, p < 0.0001, adjusted r² = 0.978; A_wing1 = 0.0668W^1.854, n = 13, p < 0.0001, adjusted r² = 0.992; A_wing2 = 0.0886W^1.778, n = 10, p < 0.0001, adjusted r² = 0.990; A_wing3 = 0.138W^1.692, n = 15, p < 0.0001, adjusted r² = 0.925). The estimated masses are given in Table 2.

Wingspan estimates for Quetzalcoatlus [1], the Thalassodromeus holotype [2], and Tupuxuara sp. (Witton and Martill, personal observation) were used to estimate other parameters for these species using the regressions generated from the Hazelhurst and Rayner data. The regression method produced estimates of mass for larger species that fell near the lowest published estimates, so to account for the variation in current mass estimates for large pterosaurs, we also used two published masses for Pteranodon and Quetzalcoatlus from the upper end of accepted estimates [45,46]. Wingspan for the Thalassodromeus specimen of Veldmeijer et al. [28] was estimated from the holotype [2] by scaling to the height of the mandibular symphysis.

We estimated the metabolic power (P_met) required for both flapping and flap-gliding flight using the methods of Pennycuick ([47] and summarised by [48]) and Norberg [49], respectively. There are three components to the power requirements for level flight: parasite power, induced power, and profile power. Following Ward et al. [50], the parasite power was estimated as where ρ_air is the density of air (≈1.23 kg m⁻³), S is the projected frontal area of the animal's body (calculated as 8.13×10⁻³M^0.666, after [47], but see also [51,52]), and Cd_body is an estimate of the coefficient of drag of the body. We use Cd_body = 0.25 in accordance with the revised estimates of Pennycuick et al. [53] for species with a prominent head and enlarged bill. The second term on the right hand side of Equation (9) represents the costs associated with ploughing the tip of the mandible through the water as a function of velocity.

The induced power was estimated as where k is an induced velocity scaling factor related to the degree of efficiency with which flapping wings generate lift, and set to 1.2 throughout, based on aeronautical arguments as in the majority of avian flight studies [53]. From Equation (10) it is possible to find the unique value of V that minimises the sum of P_par and P_ind. The final component of power output, profile power P_pro, is estimated as 1.2 times this minimal sum of P_par and P_ind as given by Equations (9) and (10). From the above, the total mechanical power expended in flight is simply the sum P_par + P_ind + P_pro. Although estimation of profile power in this way is common, it is too simple to be fully realistic because it does not allow P_pro to vary with flight speed. However, when dealing with extinct taxa for which wing kinematics are not known, it is the only available option.

In the above calculations, we assumed that the costs of ploughing the tip of the mandible through the water and of flight were not separable, and should not be simply summed or ignored as they have been in previous studies where mandible drag was considered negligible [19,31,32]. We therefore included mandible drag in the flight power model as additional parasite drag as recommended by Ward et al. [50] for drag due to respirometry masks in flight studies to take account of its effects on performance in addition to the direct drag costs.

We estimated the total power consumption of flight by using the aerodynamic equations to calculate the mechanical component of power output during flight (P_mech) and obtain an estimate of P_met from this by: where flight muscle efficiency (E_FM) is defined as P_mech/metabolic power consumed by the flight muscles, and P_BMR is basal metabolism ([47,49] and see [50] for a discussion of the errors associated with this estimate]. The constant value of 1.1 results from the assumption that the extra costs of respiration and circulation during flight each contribute 5% to P_met [47]. A recent study examining E_FM in starlings (Sturnus vulgaris) suggested values of around 0.18 for birds in the size range of starlings (approximately 0.1 kg [50]). However, in the absence of other data, and given the known increase in efficiency with size, we used the commonly used value of 0.23 (e.g., [47]). We used an allometric equation for non-passerines (P_BMR = 3.8 M^0.72, [54]) to estimate P_BMR from mass.

As a further step, a method for calculating the cost of flying close to the ground (wing-in-ground effect) is presented by Rayner [30]. From published photographs of foraging skimmers (e.g., [17]) we estimated that when held parallel to the water's surface, the skimmer's wing is some 10 cm above the water. This corresponds to the height estimated from considering the geometry of the bird in flight, i.e., (L − d) cos(45°), where L is mandible length and d is immersion depth. From this we calculated a relative height as β = h/b, where h is the height of the wing from the water surface and b the wing semispan. Next, the aspect ratio of the wings r_W is calculated as 4b²/S, and from this, the minimum drag-to-weight ratio,

Two further coefficients denote the reduction in induced drag due to the ground effect (σ), and a circulation factor due to ground effect (τ). These coefficients are graphed as functions of β in Figure 2 of Rayner [30].

From Equation (11), we identify V_mp, the velocity that minimises energetic costs (away from the ground effect). We can then calculate the costs of flying at any speed V correcting for the effects of flying near the ground P_ge according to where D_ge is the total drag of the animal corrected for ground effect, and δ is the relative drag, where v is the scaled velocity V/V_mp. To calculate D_ge, we first calculate γ, the relative circulation [30]:

Then Flap-gliding or intermittently powered flight is an energy-saving behaviour and is frequently seen in Rynchops [19,31]. Flap-gliding is characterised by the use of powered flight to increase height before gliding using open wings. During the gliding phase, the bird supports its weight with its wings but does no mechanical work; it loses height as potential energy is converted into work against drag. The resulting undulating flight path was approximated by the following method, based on that of Norberg [49]. The assumption was made that the minimum power velocity for powered flight (V_mp) was equal to both the glide (V_glide) and climb (V_climb) velocities. From this it follows that the vertical sinking speed of the animal at a glide angle ϕ is

The backward drag component D in this situation is approximated by

The proportion of time spent climbing from the water surface to elevation h is given by where φ is the slide angle. The mechanical power required to climb using powered flight is given by

To produce an estimate of flight costs during flap-gliding, climb angle φ was set at 45 ° and ϕ was adjusted to minimise the cost of flight at V_climb (i.e., minimise the product α P_climb), with the limitation that ϕ ranged from 1 ° to 30 ° (corresponding to reasonable glide angles for extant volant animals and human gliders [33]). The total average power required for undulating flight is then simply given by the sum of metabolic power required to climb, power required for forward flight with the mandible in the water, and metabolic costs of basal metabolic rate during gliding

It should also be noted that this method does not account for changing drag costs associated with variable mandible submersion, drag of the mandible while gliding, or changes in flight costs due to impacts of elevation on wing-in-ground effect. It does, however, give a rough estimate of the flight costs during flap-gliding for comparison with powered level flight.

To evaluate the sensitivity of our aerodynamic model, we varied the values of k and Cd_body to simulate high- and low-power cases for Rynchops, Tupuxuara, and Thalassodromeus (holotype) following [55] and [56]. For the low-power case, k was decreased from 1.2 to 1.0 and Cd_body was reduced by 60% to 0.16. In the high-power case, k was increased to 1.4 while Cd_body was increased by 60% to 0.40. These values allowed us to span the range of current estimates for these two parameters and so bracket the power outputs for a given flight speed (see shaded regions in Figure 3).