The energetics of ‘airtime’: estimating swim power from breaching behaviour in fishes and cetaceans

The maximum speeds an animal can achieve are rarely displayed in the laboratory, or indeed in the field; wild animals are only occasionally observed moving flat out (Lutcavage et al., 2000; Wilson et al., 2015). Even during predator–prey interactions, maximum speeds are rarely exhibited (e.g. Husak et al., 2006). Measures of maximum power in animals are therefore difficult to obtain, and methods that encourage maximum physical effort from subject animals can be ethically questionable. This is unfortunate, because an understanding of maximum speed and power provides insights into the morphological and physiological capabilities and limitations that have evolved in species usually highly adapted to the environments they inhabit.

Possibly, however, there is a natural behaviour exhibited by a diversity of aquatic animals that is not only sometimes undertaken with maximum speed and power, but is also easy to record – jumping clear of the water, otherwise known as breaching. In some species at least, locomotion speed is documented to be greater during breaching than at any other time (Johnston et al., 2018; Watanabe et al., 2013). There is only a small amount of published data quantifying the nature of breaches in jumping animals (Tanaka et al., 2019), but many breaches have been documented on video, which are freely available on the worldwide web. An initial estimate of an animal's breaching speed can often be obtained from video recordings simply by timing the duration that the animal is above the water surface, coupled with breaching angle from horizontal (Johnston et al., 2018). From breaching speed, the mechanical power needed to achieve that breach can be estimated with knowledge of the drag coefficient and hydrodynamic efficiency. In turn, the drag coefficient can be estimated using semi-empirical methods (Iosilevskii and Papastamatiou, 2016); the hydrodynamic propulsion efficiency (the ratio of the mechanical power to the power supplied by the muscles) can be estimated from swimming gait. We present and interpret the analysis of the speed and power of breaching by fish and cetaceans ranging in length from 20 cm to 14 m.

Data on breaching speed and breaching angle from horizontal were collected for 14 species of fishes and cetaceans spanning (tip of snout to fork of tail) lengths from 0.2 to 14 m (Table 1). Breaching data were obtained predominantly from videos available on www.youtube.com. Only those segments of video that represented the entirety of clearly discernible jumps shown at full speed were analysed. The time that the animal was out of the water τ during a breach was estimated from video footage, following a refined approach to that taken by Johnston et al. (2018) for basking and white sharks; that study also validated the approach with direct measures of speed obtained from an animal-attached data logger. The time was measured from the moment the snout of the animal broke the water surface until the animal's estimated centre of mass reached the same height above the water on descent that it was during ascent at the point that the body just cleared the water. The angle of the breach from horizontal, γ, was estimated visually at the same point (only breaches close to vertical were included in analysis). Justification of this approach is in Appendix 1. The length of the animal was bracketed ±30% around the typical length associated with the particular species according to specialist taxonomy websites Wikipedia and FishBase, or, when possible, the typical length determined from direct observations (Parsons et al., 2016).

Estimation errors in Eqns 1 and 5 are assessed in Appendix 3. To minimize these errors, only steep jumps (where γ exceeded 70 deg) were included. Under this restriction, the errors in the breaching speed v₀ are estimated at about 10%, whereas the errors in the mass-specific power (P/m) can possibly reach 30%.

Velocities of all breaches for each species for which the breaching angle exceeded 70 deg, and the mass-specific mechanical power deemed needed to achieve these, are plotted against body length in Fig. 1. Maximum breaching speed has an upper bound of about 11 m s⁻¹, while maximum mass-specific power has an upper bound of about 50 W kg⁻¹. These limits coincide at about 2 m body length. Breaching velocity increases up to 2 m body length with larger animals not exhibiting a systematic relationship between length and velocity; mass-specific power is approximately constant up to 2 m body length, at around 50 W kg⁻¹, and is variously lower at greater body sizes. While sample size varies considerably between species, there is no substantive regression between the mean breaching speed of the top three fastest breaches and sample size (Spearman's rho: 0.141; P=0.645). The breaching velocity of common bottlenose dolphins has been determined by a different method using a high-speed underwater camera (Rohr et al., 2002), which returned a range of maximum speeds similar to those we report, providing further validation for our method. Based on our calculations, as species get larger up to 2 m in length, maximum breaching velocity exhibited increases, to a highest breaching velocity of nearly 11 m s⁻¹ (achieved by the common bottlenose dolphin; Fig. 1A).

So, there is some suggestion from the breaching data that a number of particularly large animals do not exhibit higher breaching speeds than do 2 m long species. The maximum swimming speed of an animal is limited either by its maximal thrust or its maximal power (Iosilevskii and Papastamatiou, 2016). Thus, maximum swimming speed is the lower of the theoretical speeds at which hydrodynamic drag equals maximal thrust, and at which rate of work done by the animal on the water (loosely, the product of drag and speed) equals maximal power. Maximal thrust is proportional to the cross section area of the animal's locomotion muscles, and hence scales with the length of the animal squared. Maximal power is proportional to the volume (mass) of those muscles, and hence scales with the length of the animal to the third power. Because hydrodynamic drag is proportional to the product of the swimming speed squared and length of the animal squared, while maximal thrust is proportional to animal length squared, if the maximal speed is limited by muscle thrust, maximal speed should be independent of length. If, on the other hand, the maximal speed of an animal is limited by power, it should scale with length to the power (1/3). Our data suggests that for animals smaller than 2 m in length, the breaching speed increases with length to the power (1/3), implying that it is limited by the mass-specific (volume-specific) power of the locomotion muscles. For larger animals, the breaching speed remains practically independent of length, implying that it is limited by the alternative possibility – the thrust it can generate per unit cross section area of its muscles.

Fig. 1A also includes data for burst swimming fish during containment in several different swimming apparatuses reported across multiple studies (taken from Table 4 in Videler and Wardle, 1991). In all cases, except for small mackerel, the maximum speeds observed in the lab for burst swimming fish are much lower than breaching speeds we calculated for similarly sized species, suggesting that those burst swimming fish were not using their maximal power. Castro-Santos et al. (2012) have developed a flume larger than used in previous studies of fast-swimming fish, which appears to elicit close to maximal swimming speeds in multiple relatively small species. Trout Salvelinus fontinalis and Salmo trutta of 0.145 m length volitionally swum against a fixed flow at up to about 4 m s⁻¹, while herring Alosa aestivalis (0.22 m) reached 4.5 m s⁻¹. Similar feats were observed in comparably sized barbels Luciobarbus comizo and nase Pseudochondrostoma duriense (Sanz-Ronda et al., 2015). These speeds are remarkably similar to the highest speeds at which we would predict species of this size to breach based on our data (Fig. 1A). Owing to size constraints, however, very few fish greater than 1 m in length have been swum in the laboratory (although see Sepulveda et al., 2007), particularly at higher speeds.

Small cetaceans can be trained to swim fast in captivity, and the fastest swimming speed reported for dolphins under such conditions (11 m s⁻¹; Lang and Pryor, 1966) matches the speed exhibited by dolphins during their fastest breaches calculated in the present study. Fish (1998) recorded captive orca swimming up to 7.9 m s⁻¹, a speed that does not match the fastest breaching speed we calculated of 9 m s⁻¹. Extensive tabulations of cetacean swimming speeds are provided in Fish and Rohr (1999).

Maximum speed capabilities of larger fishes can be potentially recorded in the field, although sometimes, tagged individuals do not exhibit such behaviour. For example, tagged blue marlin Makaira nigricans, at least 1 m long, were recorded swimming no faster than 2.25 m s⁻¹ during 165 h of continuous tracking (Block et al., 1992). This may indicate that they did not hunt while tagged. In contrast, however, 1.5 m long sailfish Istiophorus platypterus hunting sardines exhibited maximum speeds of 8.8 m s⁻¹ (mean of top 3 fastest bursts; Marras et al., 2015; P. Domenici, pers. comm.). This speed is very close to the maximum breach speed observed for animals of a similar length in our dataset, and might suggest that during such hunts the sailfish are swimming close to their maximum speed. Devil rays Mobula tarapacana, about 3 m long (S. Thorrold, pers. comm.), reach speeds of up to 6 m s⁻¹ during descents into the water column (Thorrold et al., 2014), while short-finned pilot whales Globicephala melas, 4 m in length, were recorded at mean maximum sprint speeds of 6 m s⁻¹ (Aguilar Soto et al., 2008). These swimming speeds are commensurate with breaching speeds of similarly sized animals (Fig. 1A). During burst swimming, tagged sperm whales Physeter microcephalus, between 6 and 10 m in length, were never observed swimming faster than 8 m s⁻¹ (Aoki et al., 2012); this top speed is similar to that exhibited during breaches by the similarly sized orca and humpback whale (Fig. 1A). Spinner dolphins responding to an approaching ship reached swimming speeds up to 4.8 m s⁻¹ (Au and Perryman, 1982) – considerably slower than the single fastest breach we recorded. Humpback whales have not been recorded in the wild swimming as fast as their most powerful breaches (e.g. 4.1 m s⁻¹; Williamson, 1972).

Given that breaching speeds represented in our data set match the speeds of animals swimming in flumes designed to elicit maximum effort, and also match if not surpass the maximum swimming speeds of animals observed in the wild, the locomotion athleticism of fish and cetaceans during their fastest breaches may represent maximum capability. We conservatively estimate that the maximum mass-specific mechanical power exhibited by breaching species is about 35–50 W kg⁻¹ of body mass (Fig. 1B), though our calculations have assumed fin retraction, neutral buoyancy and that the animal is no longer accelerating just prior to breaching. Interestingly, this power production was attained by species across an order of magnitude in size from 20 cm to 2 m body length (very approximately 100 g to 100 kg) and as such may represent a power ceiling in general, based on an isometric relationship between maximum power and body mass within this range.

While we cannot fully validate our model estimating power, we can compare the resultant values with those in the literature obtained by other means. The fastest breaching species in our dataset were dolphins; our maximum power estimates for breaching dolphins are similar to the maximum fluke-beat-averaged value of 48 W kg⁻¹ for the Pacific white-sided dolphin Lagenorhynchus obliquidens swimming at 7.4 m s⁻¹ calculated by Tanaka et al. (2019), the common bottlenose dolphin while tail standing (62 W kg⁻¹; Isogai, 2014) and porpoises Stenella attenuata encouraged to swim maximally fast along a 25 m course (50 W kg⁻¹; Lang and Pryor, 1966). Because the mass of the locomotor muscles is approximately half of the body mass (fao.org/3/T0219E/T0219E01.htm), and, at a given instant, only half of them are propelling the animal during the tail beat cycle, muscle power output at our estimated maximum is 140–200 W kg⁻¹. We therefore propose that these values represent an approximate maximum attainable power output by fish and cetaceans. And this is supported by the observation that 200 W kg⁻¹ muscle during fast flights in Phyllostomus bats (Thomas, 1975; Weis-Fogh and Alexander, 1977) and 214 W kg⁻¹ of muscle exhibited by small lizards during vigorous movement (Curtin et al. 2005) are the highest reported power output values we have found in the literature for any species within the size range of the breaching species represented in the current study (some higher values have been recorded for individual muscles; Table 1 in Josephson, 1993). Moreover, power output measurements for human participants asked to apply maximum effort at best match these values. For example, high-level rugby players produced a mean peak power of 66.6 W kg⁻¹ body mass during standing jumps (N. Tillin et al., 2013); assuming 25–30 kg of leg muscle mass (Tillin, pers. comm.), their leg muscles were providing around 200 W kg⁻¹ of power.

Valid estimates of maximum power are not only insightful physiologically, but in turn, they elucidate an animal's behavioural limitations. In the case of breaching, for example, calculations of the necessary dimensions of dam spillways to enable fish to pass up them (Baigún et al., 2012; Beach, 1984) will greatly benefit from an understanding of breach velocity.

The fish, however, is not a point mass, and it likely gains energy (on the account of reduced wet area) between the moment when its nose (snout) pierces the water surface and the moment when its tail leaves the water. Consequently, the speed that the fish reaches immediately before piercing the water surface, v₀, is probably smaller than v′₀. Moreover, between the time the centre-of-mass leaves the water and the time the tail leaves the water, the motion of the fish is still assisted by buoyancy, and during that time, the fish decelerates at less than the acceleration of gravity. It increases the actual airtime τ as compared with the airtime τ′ it would have had if it were a point mass leaving the water at v′₀. The aim of this appendix is to estimate the relations between τ and τ′, and between v₀ and v′₀. To remain concise, the analysis will be based on the following six assumptions: (1) the fish is neutrally buoyant; (2) it has reached a constant speed before piercing the water surface; (3) its thrust (T) remains constant until the tail clears the water; (4) its drag (D) is proportional to the wetted area and to the swimming speed squared; (5) its body is symmetrical nose to tail; (6) its fins (other than caudal) are contracted. This list pertains to Appendix 2 as well.

Recalling that the accuracy of capturing the flight time is a few percentage at best (one frame count at 30 frames s⁻¹ for flight time of the order of 1 s), we seek a simplified (approximate) variant of the above formulae.

follows Eqn A61 by Eqn A28. The two correcting factors are the energy added due to reduced drag – this is the factor (1 − Ā/4) in the first term – and trajectory angle correction, manifested in the last term.

Having a multitude of non-retractable fins, sharks probably represent the most hydrodynamically ‘dirty’ of the fusiform fish. At zero lift, their drag coefficient exceeds the estimate of Eqn A71 by about 30% (Iosilevskii and Papastamatiou, 2016).

Based on Eqn A75, a 30% uncertainty in Ā has practically no effect on v₀ (less than 1%). Based on Eqn A76, a 30% uncertainty in length has no effect on v₀ if the breach is near vertical (say, above 75 deg), but can lead to a large uncertainty if the animal is large (e.g. 5 m) and breaches at a shallow angle (Fig. A5A). Based on Eqn 77, a 3% uncertainty in the airtime yields a 3% to 6% uncertainty in v₀ (see Fig. A5C). Based on Eqn A78, a quarter-radian uncertainty in the breaching angle has no effect on v₀ when the animal breaches almost vertically, but can render the estimate of v₀ unreliable when the breaching angle becomes less than about 60 deg (see Fig. A5E).

Being dependent on v₀³, the uncertainties in P/m are naturally larger than those in v₀. In particular, uncertainties of 30% in length and in Ā are reflected in comparable uncertainties in P/m – see Eqns A79, A80, A75 and A76. A 3% uncertainty in the airtime yields 9–18% uncertainty in P/m – see Eqns A81, A77 and Fig. A5D. Perhaps the biggest uncertainty is associated with the breaching angle – see Eqns A82, A78 and Fig. A5F; it can be reduced by considering only those cases where the breaching angle exceeds 70–75 deg.

Andreas Fahlman, Chris Lawson, Yuuki Watanabe, Theodore Castro-Santos, Paolo Domenici, Simon Thorrold, Neale Tillin and in particular Roger Seymour, provided insightful thoughts and information.