Gait switches in deep-diving beaked whales: biomechanical strategies for long-duration dives

Locomotor efficiency is critical for any active foraging animal but is especially so for animals that must travel long distances between resources (van Ginneken et al., 2005; de Knegt et al., 2007), such as breath-hold divers who access oxygen at the sea surface and food at depth (Kramer, 1988). For these animals, the reduced cost of transport increases the time and energy available for foraging (Williams, 1996). Numerous anatomical, behavioural and physiological factors influence the cost of transport for swimming animals including body shape (Williams, 1999), stroke frequency (Mori et al., 2010), speed (Aguilar de Soto et al., 2008), net buoyancy (Miller et al., 2012) and muscle type (Rome, 1992). These factors can vary over long time scales, e.g. as a result of ontogenetic (Noren et al., 2006) or body condition changes (Biuw et al., 2003; Mori et al., 2010), but some factors also vary within individual dives such as buoyancy forces, which change with depth (Lovvorn and Jones, 1991).

Terrestrial animals modulate their locomotor power and speed by changing the rate and amplitude of propulsive motions, but also by changing their gait, i.e. the pattern of movements. Aquatic animals also exhibit gait changes: fish switch from pectoral to caudal locomotion (Webb, 1994a,,b) and squid from undulatory to jet propulsion (Webber and O'Dor, 1991; Svendsen et al., 2010). Most marine mammals use only a caudal propulsor while swimming, but they use distinct stroking rhythms (i.e. continuous stroking, gliding, stroke-and-glide), analogous to gaits in terrestrial animals, to meet changing power requirements (Williams et al., 2000; Cook et al., 2010). Continuous stroking is used when high thrust levels are required; for example, to counter strong positive buoyancy from air in the lungs when the animal dives away from the surface (Miller et al., 2004). Gliding is only possible when buoyancy forces are small or act in the direction of movement (Skrovan et al., 1999; Miller et al., 2004). At other times, diving animals often use a stroke-and-glide gait. Despite the elevated drag associated with reacceleration between glides, this interrupted swimming has been estimated to save 15–50% of locomotor cost compared with steady swimming in fish (Weihs, 1974; Fish et al., 1991) and around 28% in Weddell seals (Williams et al., 2000).

In addition to using different swimming styles, aquatic animals modulate the amplitude and rate of strokes to change power output (Skrovan et al., 1999). Variations in stroke frequency and amplitude compensating for dynamics in buoyancy and drag have been recorded in turtles, diving birds and marine mammals (Skrovan et al., 1999; Wilson and Liebsch, 2003; Hays et al., 2007). Faster and larger strokes are also used to produce bursts of speed for pursuit (Wilson and Liebsch, 2003; Aguilar de Soto et al., 2008) or escape, at least in fish (Blake, 2004). However, the usual stroking frequency in steady swimming aquatic animals follows a mass^−0.3 law over more than four orders of magnitude of body mass whereas usual swimming speed remains relatively constant at about 1.5 m s⁻¹ (Sato et al., 2007). It has been suggested that there is an optimal size-related stroke frequency during diving that minimizes the cost of transport in seabirds (Mori et al., 2010). There may also be an optimal stroke amplitude because high-amplitude strokes produce more thrust (Fish et al., 2014), but also increase induced drag due to the greater flow disturbance of the extended propulsor (Fish et al., 1988).

For fish, the ability to swim with different gaits and stroke rates is closely connected with distinct types of muscle fibres optimized to different activity levels. Two fibre types are found in the musculature of fish (Jayne and Lauder, 1994): slow-twitch oxidative fibres (red muscle or Type I) are used for slow steady swimming, while fast-twitch fibres (white muscle or Type II) are recruited to power brief bursts of fast swimming during predator avoidance or prey capture (Alexander and Goldspink, 1977). Type I and II muscle fibres are also present in the swimming muscles of marine mammals but less is known about how these are employed (Kanatous et al., 2008; Velten et al., 2013). It is generally assumed that diving mammals minimize their reliance on anaerobic metabolism by choosing dive durations in keeping with their activity levels (Butler and Jones, 1997). The longest dive that an animal can make while relying primarily on oxygen stored in the blood and muscle is named the Aerobic Dive Limit (ADL) (Kooyman and Ponganis, 1998) or diving lactate threshold (Butler and Jones, 1997). The ADL for animals with the same oxygen store per unit mass should increase with body size (Martin and Smith, 1999; Halsey et al., 2006) due to lower mass-specific metabolic rates (White et al., 2009). However, 25,000 kg sperm whales and 1000 kg beaked whales perform dives of comparable duration (30–50 min) and depth [600–1200 m (Tyack et al., 2006; Watwood et al., 2006)], while more active 2500 kg pilot whales dive to similar depths but for substantially shorter intervals [15–20 min (Aguilar de Soto et al., 2008)], raising the question of how the smaller beaked whales can dive for such long durations.

Beaked whales, a speciose family of deep diving toothed whales, have several anatomical structures including unnotched lunate flukes, a slender peduncle and small pectoral flippers that fit into depressions in the body wall (True, 1910; Mead, 1989) which seem to promote efficient swimming (Fish and Rohr, 1999). Like other deep-diving marine mammals, beaked whales also have physiological adaptations that prolong their ADL: high lipid volume density and myoglobin concentrations increase oxygen stores while extremely low mitochondrial volume densities and large fibre diameters reduce the metabolic rate (Velten et al., 2013; Davis, 2014). Nonetheless, it is likely that the two smallest ziphiid species studied to date, Cuvier's and Blainville's beaked whales, routinely dive beyond their ADL, powering some portion of locomotion in deep dives anaerobically (Tyack et al., 2006; Velten et al., 2013). Despite this, both species make distinctive and enigmatic low-pitch ascents from deep dives. These ascents, which appear to be free of foraging, often take 2- to 3-times longer than descents (Tyack et al., 2006), suggesting a conservative slow swimming strategy. But rather than having the expected high percentage of slow-twitch oxidative fibres (Kanatous et al., 2002; Williams et al., 2011), the locomotor muscles of Blainville's beaked whales comprise about 80% fast-twitch fibres similar to those of sprinting athletes (Velten et al., 2013). The unusual combination of long dives and a predominance of anaerobic muscle, at least in the one species evaluated, makes beaked whales excellent candidates to explore how swimming gaits change in response to the progressive depletion of oxygen stores through the course of deep dives.

A prerequisite for understanding the biomechanical strategies of swimming animals is the quantification of stroke rate, amplitude and surge acceleration under varying load conditions. Underwater cameras, accelerometer and gyroscope tags have been used to study stroking in cetaceans, pinnipeds, turtles and diving birds (Williams et al., 2000; Wilson and Liebsch, 2003; Hays et al., 2007; Aguilar de Soto et al., 2008; Fossette et al., 2010). However, all three approaches have limitations. Cameras only capture the displacement of the propulsor, not the resulting body accelerations (Davis et al., 2001), while accelerometers are sensitive to both the orientation and the specific acceleration (i.e. the surge and heave acceleration) of the body (Fish et al., 2003). As stroking generates both specific acceleration and body rotations at the stroking rate, these two movement components cannot be quantified separately without additional sensors (Sato et al., 2007; Simon et al., 2012). Gyroscopes can provide independent information about orientation, enabling estimation of the specific acceleration when combined with magnetometers and accelerometers, but the power consumption and drift of miniature gyroscopes (Noda et al., 2012) make them unattractive for animal biologging devices.

Here, we investigate if and how beaked whales adapt their swimming gates throughout deep dives as oxygen stores are consumed. For this, we adapt a method developed for human body tracking (Bonnet and Héliot, 2007) to separate body rotations and accelerations during stroking movements. The method requires two high-rate on-animal sensors that are already widely used in tags: magnetometers and accelerometers. The new approach allows us to quantify simultaneously the rate, relative amplitude and surge acceleration of fluke-strokes. We use the method to examine the swimming movements of three ziphiids: Blainville's beaked whale (Mesoplodon densirostris Blainville 1817), Cuvier's beaked whale (Ziphius cavirostris Cuvier 1823) and the northern bottlenose whale (Hyperoodon ampullatus Forster 1770), and show that all three species produce distinctive, fast, high-thrust fluke-strokes intermixed with normal strokes during ascents from deep dives. The additional acceleration generated by fast strokes does not seem to lead to shorter ascents (i.e. with higher pitch or speed) but results in brief glides that may enable muscle recovery while reducing the average drag, possibly improving overall energetic efficiency when compared with continuous stroking.

A total of 118, 96 and 16 h of tag recordings from 10 Mesoplodon densirostris (Md; 7 individual whales), 9 Ziphius cavirostris (Zc) and 2 Hyperoodon ampullatus (Ha) were analysed containing 49, 43 and 6 deep dives, respectively. Unresolved tag moves on the body of the whale led to nine dives being removed from analysis. At least two entire deep dives were recorded from each whale (Table 1). As described elsewhere (Hooker and Baird, 1999; Baird et al., 2006; Tyack et al., 2006; Arranz et al., 2011), these beaked whales typically perform long and deep foraging dives (Md: means of 49 min and 844 m; Zc: 59 min and 1044 m; Ha: 49 min and 1572 m in the data analysed here). For Md and Zc, ascents from deep dives were about twice as long in duration as descents and were performed at proportionally lower vertical speeds and absolute pitch angles (Table 1). Because of the typically low pitch angle, forward speed could not be estimated accurately throughout ascents but in the few intervals with higher pitch angle, the forward speed was about the same in ascents and descents, suggesting that the long ascents are a result of the low pitch angle rather than slow swimming (Tyack et al., 2006). The situation is different for Ha: the pitch angle of descents and ascents is similar for this species but swimming is slower during ascent, leading to a smaller but still significant difference in duration.

Tagged whales stroked continuously in the initial part of descents, presumably to overcome the buoyancy of gases carried from the surface (Fig. 1) (Miller et al., 2004; Simon et al., 2009,, 2012). Stroking was continuous until a mean depth of 128 m (Md), 197 m (Zc) and 95 m (Ha). The remainder of the descents were performed with prolonged glides and occasional bursts of stroking or stroke-and-glide. On ascents, terminal glides began at depths of 30 m (Md), 34 m (Zc) and 36 m (Ha). Below this, the gait varied from continuous stroking to stroke-and-glide with some strokes having distinctly larger peak high-pass filtered dorso-ventral acceleration, ã_zt (Fig. 1). These high acceleration strokes were typically performed singly and followed by a short glide. However, there were occasional bursts of up to 3, 14 and 2 consecutive of these high-acceleration strokes in Md, Zc and Ha, respectively. The consistent presence of these high acceleration strokes during ascents but not descents for all three species of beaked whales (Fig. 2) led us to investigate their occurrence and characteristics. Because the accelerometer output is affected by both body rotations and specific acceleration, the increased high-frequency dorso-ventral acceleration in these strokes could result from a larger stroking amplitude or a faster stroke, or both.

To examine separately the contribution of specific acceleration and body rotation in Md and Zc, we first used the magnetometer method to estimate body rotations, r_t, and then combined this with the accelerometer data to estimate the heave and surge accelerations. As asymmetric strokes were often encountered in these signals, we analysed half-strokes defined as the intervals between successive zero-crossings in the estimated body rotations ⁠. This signal, being insensitive to acceleration, is relatively smooth (Fig. 3) and so half-strokes can be reliably detected and measured.

Each half-stroke was parameterized by the RMS value of the rotation, and the surge, heave and ã_zt over the interval. Plots of these parameters as a function of half-stroke duration (Fig. 4) show two clearly separated clusters, indicating that the observed gait variations during the ascent correspond to distinct stroke types rather than a continuum of strokes with varying parameters. These plots indicate that the high acceleration strokes noted in Fig. 2 also have consistently shorter duration and we refer to these short, strong strokes as ‘type-B’ strokes here to distinguish them from the more numerous ‘normal strokes’. As the RMS heave acceleration typically showed the clearest bimodality, we used a threshold in this parameter to distinguish normal and B-strokes. To account for different tag placements, the threshold was selected separately for each animal from a histogram of RMS heave acceleration (Table 2). While normal up- and down-strokes had similar accelerations and body rotations, up and down B-strokes, were often asymmetric (Fig. 4). We report the average over all half-strokes when comparing parameters between normal and B-strokes.

Because the tags used on Hyperoodon lacked a high-rate magnetometer, a different method was needed to distinguish strokes in this species. Half-strokes were identified from zero-crossings in ã_zt. Although wide hysteretic thresholds were used to avoid detecting multiple zero-crossings due to transient accelerations, the resulting half-stroke durations are more variable for this species than for Md and Zc indicating that there are likely to be frequent errors in stroke detection. Nonetheless, reasonably distinct clusters corresponding to the two stroke types during ascents appear in plots of RMS ã_zt versus duration and an ã_zt threshold was used to separate normal and B-strokes (Table 2).

The three species spend more time gliding over the 100 to 400 m depth range in descents versus ascents (Md, mean of 5.6 vs 2.2 min desc/asc; Zc, 9.1 vs 3.3; Ha, 6.8 vs 4.5; see Table 1 for s.d.). Counting both stroke types, Md and Ha produced about 2- to 3-times as many half-strokes to ascend over this depth range than to descend (Md, mean of 0.23 vs 0.57 strokes/vertical metre desc/asc; Ha, 0.11 vs 0.21). For Ziphius, the stroke counts were more balanced (0.31 vs 0.27 desc/asc) with some individuals stroking more in descents than in ascents. For the shallow ascents of Md and Zc, more strokes are expected to cover the 300 m depth range. But even considering this, there are more ascent than descent strokes per metre travelled for Md (mean of 0.20 versus 0.30 desc/asc) whereas for Zc, the opposite is true (mean of 0.29 versus 0.17 desc/asc). The mean duration of normal half-strokes was 1.1 and 1.4 s for Md and Zc, respectively, implying stroking rates in normal continuous swimming of 0.44 (Md) and 0.36 (Zc) Hz. B half-strokes identified during the ascent phase of deep dives had a mean duration of 76% (averaging up- and down-strokes) of normal half-strokes in Md and Zc, implying a 30% increase in the instantaneous stroking rate. Body rotation was also larger for B-strokes by about 52% and the high-frequency dorso-ventral acceleration as well as the estimated surge and heave accelerations were 2.5- to 4-times larger in B-strokes compared with normal strokes (Table 2). These ratios were surprisingly stable across individuals and between the two species. For Hyperoodon, normal half-stroke duration was 1.6 s, giving a stroking rate of 0.32 Hz (Table 1). For Ha, the duration of B half-strokes was 70% that of normal strokes while the dorso-ventral acceleration was 2.5-times higher in B-strokes.

Given the 4-fold increase in estimated surge acceleration for B-strokes in Md and Zc, there should be a corresponding increase in forward speed during these strokes. Unfortunately, the ascent pitch angle of these species was usually too low to allow forward speed estimation from vertical speed. However, forward speed could be estimated during some brief episodes of steep ascent in one Zc (Fig. 5). For this animal, there was a significant difference in the speed increment over two consecutive B-strokes compared with two normal strokes (t-test, P<0.001). Consistent with the increased thrust, each burst of two B-strokes produced a speed increment of 0.5±0.1 m s⁻¹, accelerating the animal to speeds of 1.7–2 m s⁻¹, whereas pairs of normal strokes produced no speed increment (0.0±0.10 m s⁻¹). Thus in Fig. 5, the long bout of normal strokes only produced sufficient thrust to maintain a steady forward speed of 1.3 m s⁻¹.

While normal strokes were present in all descents and ascents of deep and shallow dives, gait switches to and from B-strokes occurred almost exclusively in deep-dive ascents. B-strokes were detected in 42 of 49 deep-dive ascents of Mesoplodon, 37 of 43 for Ziphius and 5 of 6 for Hyperoodon, and comprised an average of 30% of the ascent strokes in Md and Ha, and almost 50% in Zc (Table 1). B-strokes were also identified in the ascent phase of 5 out of 104 shallow dives performed right after a deep dive. Mesoplodon foraging dives without B-strokes (N=7) were performed by two different whales and were amongst the shortest deep dives recorded (37.7±7.3 min cf. overall mean of 49±5 min; means±s.d.). The six Ziphius deep dives without B-strokes were performed by three different whales and were also relatively short (43.3±5.9 min cf. overall mean of 59±12 min; means±s.d.). The only Hyperoodon deep dive without B-strokes, however, was not short (52 min cf. 49±4 min; mean±s.d.).

To explore the possibility that B-strokes were associated with longer, faster or more vertical swimming, we used a generalized linear model relating stroke presence to time and vertical speed (see Appendix). Because time into dive appeared to have a significant effect on the presence of B-strokes in all three species, we compared the actual occurrence of B-strokes in ascents to a model which allocated the same number of B-strokes at random throughout the ascent. B-strokes began during the ascent at mean times of 34 min (Md), 44 min (Zc) and 35 min (Ha) well after the mean start of ascent times of 30 min (Md), 38 min (Zc) and 31 min (Ha). In each species, B-strokes appeared significantly later than expected by the random model. Although B-strokes tended to co-occur with higher vertical speed, this factor explained little of the variability in stroking. Thus, the 4-fold increase in surge acceleration during B-strokes does not translate consistently into a commensurate increase in ascent speed. This seems to be a consequence of the frequent co-occurrence of B-strokes with glides. Taken as a unit, B-strokes and glides involve similar average activity as continuous stroking (Table 2; see Appendix).

Although distinct gait switches were not evident in acceleration signals during descents, 0–16% of B-strokes detected using the heave acceleration threshold occurred during deep-dive descents in Md and Zc. About 66% of these occurred in the first 100 m of the descent where high accelerations are needed to overcome buoyancy due to air in the lungs (Miller et al., 2004). The duration and rotation magnitude of B-strokes were similar for descents and ascents, but the heave and surge accelerations were smaller (∼50–80% of their ascent values, inferred from Table 2).

Natural selection on breath-hold divers should favour adaptations that reduce the cost of transport and thereby increase the time and energy available for foraging at depth (Williams, 1996); one such adaptation would be to choose gaits that are efficient given the changing physiological and mechanical constraints throughout a dive (Alexander, 2003; Mori et al., 2010). Here, we examined the swimming gaits employed by three beaked whale species with similar morphology and diving patterns, but a 7:1 range of adult body masses (Silva and Downing, 1995; Allen et al., 2011). All three species produced a previously un-described stroke-and-glide gait involving two distinct types of strokes that occurred primarily during ascents from deep dives: the normal locomotory stroke and a faster, more energetic stroke, which we term B-stroke (Figs 2 and 3). Type-B strokes had 2.5-times more dorso-ventral high-frequency acceleration than normal strokes (Table 2) and so were readily detected in on-animal accelerometer data (Fig. 2). But because accelerometers are influenced by both body rotations and specific acceleration, the increased acceleration could have resulted from either a larger stroke amplitude or a faster stroke, or both. To distinguish between these possible drivers of the acceleration signal, we combined high-resolution accelerometer and magnetometer signals to separately quantify body rotations and specific accelerations during stroking (Bonnet and Héliot, 2007). Using this method we confirmed that B-strokes are distinct from normal strokes in terms of duration and body rotation, as well as surge and heave acceleration. We also found that B-strokes often precede short glides which are presumably enabled by the increased thrust of this stroke.

Because of the difficulty of encountering and tagging beaked whales, the sample size of individual animals is relatively small. However, the consistent presence of B-strokes in all tagged animals from three species suggests that this locomotion pattern may be a common component of ziphiid diving behaviour. In the following we first discuss the magnetometer method and its limitations and then consider what role the observed gait switching may play in the diving strategy of beaked whales.

Magnetometers are commonly employed in tags to estimate heading and are generally sampled at a low rate to minimize power consumption. Adapting the method of Bonnet and Héliot (2007), we demonstrated that high sampling rate on-animal magnetometer data can also be used to determine body rotations during stroking. Triaxial magnetometers are a vector sensor like accelerometers and so can be used to deduce two components of the instantaneous orientation of an animal but, unlike accelerometers, they are not sensitive to specific acceleration. This means that for relatively simple motions like swimming, in which the rotations largely occur in a single plane, the instantaneous orientation of the animal can be estimated solely from the magnetometer. By combining these body rotation estimates with simultaneously sampled high-resolution accelerometer data, the orientation and specific acceleration components in the accelerometer can be separated, enabling a detailed analysis of stroking styles. Although several simplifying assumptions are required for this method, the single plane movement model appears to be a good approximation for ascent and descent swimming in Md and Zc (see Appendix). An additional advantage of using magnetometers to study stroking is that stroke duration can be measured more precisely with this sensor than with an accelerometer. This is because the body rotations measured by a magnetometer vary relatively smoothly because of the rotational inertia, i.e. the resistance to angular acceleration, of the animal. In comparison, the ã_zt accelerometer signal can have substantial high-frequency transients, making it more difficult to identify individual strokes reliably (Fig. 3).

The method developed here can be modified for other fauna provided that the locomotion oscillation occurs mostly in one plane (e.g. for seals and caudally propelled fish, it would be a yaw rotation rather than a pitch). There are, however, some limitations to the method. Because of the assumption that low-frequency components in the accelerometer are entirely due to orientation changes, only the dynamic forces during stroking can be estimated. Thus slowly changing forces such as mean drag, lift and buoyancy cannot be estimated. Secondly, as with accelerometers, body rotations will not be observed by a magnetometer if the animal is swimming such that the axis of rotation is parallel with the field vector. These body orientations were rare in our data (<3.3% of Md and Zc analysed dive time) but could be more prevalent at low latitudes where the magnetic field vector is nearly horizontal. Finally, a limitation with all tagging methods is that measurements are only made at one location on the animal, making it difficult to predict how other parts of the body are moving. Stroke duration and therefore stroking rate are not affected by tag location, but the magnitude of rotations, specific accelerations and the mechanical events analysed as up- or down-strokes are. To overcome this sensitivity, we calculated the ratio of body rotations and specific acceleration between normal and B-strokes, in effect normalizing these parameters of B-strokes to the values in normal strokes. These ratios appear to be fairly stable across individuals and/or tag locations within each species but are even broadly similar across Md and Zc (Table 2). This suggests that the proportional increase in acceleration and rotation due to the gait switch may be a fundamental aspect of the movement that is dictated more by anatomical structures than behavioural decisions.

The magnetometer method provides detailed information about the two different stroke types produced by beaked whales during deep-dive ascents. For the two species (Md and Zc) that could be analysed with this method, type-B half-strokes were some 76% of the duration of normal strokes and had 52% greater rotation. Assuming that body displacement is proportional to rotation, the heave acceleration should therefore increase 2.6-times (i.e. displacement×rate²=1.52/0.76²) under a sinusoidal movement model. Actual RMS heave accelerations estimated using the magnetometer method were a mean of 3.1 (Md)- and 3.5 (Zc)-times larger in B-strokes compared with normal strokes (Table 2), indicating that B-strokes are consistently more energetic than expected by simply scaling normal strokes. Surge accelerations are also greatly increased in B-strokes compared with normal strokes (mean 4-fold increase for Md and Zc) and the waveforms for both accelerations (Fig. 6C) show short high-magnitude transients in B-strokes that are largely absent in the more sinusoidal signals of normal strokes.

Thus, B-strokes not only utilize increased speed and amplitude of propulsor motion but are also performed more abruptly than in normal swimming, leading to the production of brief but powerful bursts of thrust. These bursts translated directly into increased forward speed in the one Zc that ascended steeply enough to allow reliable forward speed estimation (Fig. 5). While continuous normal strokes generated sufficient thrust to maintain a speed of some 1.3 m s⁻¹, pairs of B-strokes rapidly accelerated the animal to 1.7 m s⁻¹. However, over longer time intervals, B-strokes did not lead to substantially increased forward speed, as evidenced by the weak explanatory power of a model matching speed to presence of B-strokes in 10 s (Md) or 15 s (Zc and Ha) intervals. This is because individual short bursts of B-strokes were typically followed by glides with duration equivalent to 2–3.5 normal strokes as in Fig. 5. In comparison, normal strokes generally occurred in prolonged bouts and so were seldom followed by a glide.

Averaging over the full gait unit, B-strokes and glides produced about as much force per unit time as continuous normal stroking (VeDBA in Table 2). Thus, the extra thrust of B-strokes facilitates a lower stroking duty cycle in ascents rather than faster swimming. Although by far the majority of B-strokes occurred in ascents from deep dives, a small proportion of descents strokes were also categorized as B-strokes using a heave acceleration threshold. These descent B-strokes did not occur with glides, but most occurred in the first 100 m of descent where continuous stroking may be needed to counter positive buoyancy from air in the lungs. This energetic stroking style thus appears to be utilized in two ways: (1) as a low-duty cycle alternative to normal stroking, and (2) as a high-thrust gait to overcome strong forces. A third use may be to accelerate animals when escaping predators or pursuing prey, although this behaviour was not observed in the data analysed here. But while the biomechanical advantage of a high acceleration stroke in initial descents, prey chase and predator avoidance is evident, the benefit of using this gait in ascents is less clear. Nonetheless, the B-stroke-and-glide gait powers some 53% of vertical travel in ascents of all three species, making it an important component of transport. Preferential use of this gait would be understandable if it is energetically more efficient than continuous normal stroking but, in which case, why would it not be used for most or all transport locomotion? Two different modelling approaches show a weak but significant trend for this gait to be used later in dives, suggesting that it replaces continuous stroking as oxygen stores dwindle rather than offering an efficient alternative throughout dives. This is supported by the observation that the few ascents performed without B-strokes by Md and Zc are among the shortest dives in our data. To explore this further, we compare the diving behaviour of the three beaked whale species.

As body size increases, oxygen stores increase relative to metabolic rate, other factors being equal (Martin and Smith, 1999; Halsey et al., 2006). Although the metabolic cost of diving depends upon a number of factors in addition to body mass, there should be sizeable differences in the diving capabilities of the beaked whale species studied here. The overall diving pattern of the three species is broadly similar, comprising long, deep foraging dives interspersed with sequences of shallow dives with no apparent foraging (Hooker and Baird, 1999; Tyack et al., 2006). However, the durations of deep dives do not show the expected variation with body mass (Table 1). Hyperoodon with 7-times the mass of Mesoplodon performed similar duration dives (mean of 49 min) in our data albeit to almost twice the depth (Table 1). Ziphius made consistently longer dives (mean of 59 min) than recorded for Md and Ha, but to intermediate depths of about 1000 m, although dives to 3000 m have been reported for this species (Schorr et al., 2014). Despite these disproportionate dive durations, all three species perform B-strokes in most (84 of 98) deep-dive ascents. The swimming speed and distance covered in dives also contribute to the metabolic cost. Mesoplodon densirostris, the smallest of the three species, ascend at one half the vertical speed of descents, primarily because of a halving of the pitch angle. Neglecting lift, these long shallow ascents cover roughly 75% more distance over a 100–400 m depth range than descents but require 2.5-times as many strokes (Table 1), indicating a body density greater than water. Ziphius cavirostris also ascend at a low pitch angle but use about the same number of strokes to ascend as to descend, suggesting a more neutral body density. Hyperoodon ampullatus descend and ascend at high pitch angles in the limited data available, but ascents take 20% longer and require twice as many strokes as descents over the 100–400 m depth range, suggesting a dense negatively buoyant body (Table 1). This simple analysis indicates two overlapping distinctions between these ziphiid species: body density and ascent angle. Low-pitch ascents reduce the thrust required per stroke to counter negative buoyancy (Gleiss et al., 2011b), permitting a lower propulsive power albeit for longer duration, and this may be the strategy adopted by Md. In comparison, the greater oxygen stores in dense-bodied Ha appear to be expended in powerful steep ascents that reduce transport-time and may enable access to deeper resources. The more neutral body of Zc compared with Md may lower the overall cost of transport (Sato et al., 2013) in this species, enabling, in concert with the larger body size, the longer and deeper foraging dives. However, the shallow ascents in Zc are less easy to explain biomechanically and this behaviour in both Md and Zc may actually be a predator avoidance tactic in which animals invest energy in ascents to move away from the last point at which they vocalize (Tyack et al., 2006; Aguilar de Soto et al., 2012). If so, this crypsis must be obtained at the cost of reduced foraging time. In practice, a number of evolutionary forces may have contributed to the diverse diving behaviour of these species. However, irrespective of the specific drivers, all three species have substantial propulsive needs in the last part of deep dives: to power long shallow ascents in Md and Zc, and to drive steep, fast ascents in Ha. These calls on metabolic resources when oxygen stores are nearly depleted coincide with the appearance of B-strokes and led us to question what benefits this specialized high acceleration gait may offer.

Although stroke-and-glide gaits have been reported for a number of marine mammals, this is the first observation of a distinct stroke type within such a gait. This may be partly a consequence of sensor sampling rate as the wide bandwidth accelerometers (e.g. >10-times the stroking rate) required to reliably detect B-strokes have only been used relatively recently. However, high acceleration strokes have not been reported for other deep-diving toothed whales tagged with the same devices as used here (Miller et al., 2004; Aguilar de Soto et al., 2008), suggesting that the routine production of B-switches may be peculiar to beaked whales. If so, it is surprising to find such energetic stroking in animals that rely on long, deep foraging dives and would therefore be expected to adopt a conservative swimming style.

The high acceleration strokes described here have some 76% of the duration and involve a 52% larger body rotation than normal strokes implying a faster and larger amplitude movement of the flukes. To achieve this, the rate of muscle contraction must be doubled (i.e. 1.52/0.76) in B-strokes compared with normal strokes if the relationship between fluke extension and body rotation at the tag location is the same for both stroke types. This, in turn, implies a substantial increase in the metabolic cost of B-strokes, which may be further exacerbated by a larger active drag due to the increased propulsor amplitude (Weihs, 1974). However, the increased thrust of B-strokes also enables short glides resulting in some 70% reduction in stroking duty cycle while maintaining roughly the same average speed in ascents. The lower stroke rate and reduced drag in glides will offset the cost of B-strokes, but it seems unlikely that B-strokes are more efficient than continuous normal stroking, at least when muscles are well oxygenated, given the near absence of this gait early in dives and even throughout some short deep dives by Md and Zc.

Information about muscle type and myoglobin concentrations are missing for most ziphiids but a recent anatomical study of Mesoplodon densirostris revealed an unexpectedly high percentage, about 80%, of fast-twitch fibres in locomotor muscles (Velten et al., 2013). Although these fibres may power bursts of high-speed swimming, e.g. when escaping predators or chasing prey, the authors propose that the primary function of Type II fibres during dives is not to power locomotion directly, but to provide a large, metabolically inexpensive oxygen store within the locomotor muscle, supplying oxygen to Type I fibres during routine swimming. But if swimming in foraging dives only used Type I fibres, then there is no reason to expect the strongly bimodal stroking reported here. Rather, stroke rate and amplitude could be adjusted continuously within the capabilities of these fibres to meet locomotion requirements (Johnson et al., 1994).

If the fibre composition reported for Md does have a connection with the B-stroke-and-glide gait reported here, then a possible explanation is that Type II muscle fibres are recruited to supplement propulsion when oxygen stores in Type I fibres are low, and the resulting faster contraction rates of Type II fibres produce B-strokes. Similar fast-twitch recruitment when muscles are fatigued has been observed in exercising humans (Houtman et al., 2003). This hypothesis is supported by the strong bimodality in the stroke types and the disproportionately high accelerations generated in B-strokes, both of which suggest that these high-energy strokes are produced in a way that is biomechanically distinct from normal strokes. If this interpretation is correct, the short glides accompanying B-strokes may serve to maintain an average speed close to the minimum cost of transport while also reducing drag (Fish et al., 1988). However, the connection between muscle recruitment and movement is generally complicated (Jayne and Lauder, 1994; Flammang and Lauder, 2008) and it is unlikely that there are simple one-to-one relationships between muscle types and specific movement modes. Moreover, our biomechanical interpretation of fibre function and the physiological function inferred by Velten et al. (2013) are not mutually exclusive: the large resource of Type II fibres in Blainville's beaked whales could provide both an oxygen store during most of the dive and an anaerobic power source when oxygen stores dwindle. But if B-strokes do result from recruitment of fast-twitch fibres, then our data indicate that at least three species of beaked whales rely on anaerobic metabolism to power swimming in most deep-dive ascents. This strategy may enable both a long foraging phase and a slow ascent, as in Mesoplodon and Ziphius, or a fast powerful ascent for Hyperoodon. In either case, recruitment of anaerobic muscles must necessitate a post-dive recovery interval to process lactate, and this may help explain the prolonged periods between deep dives reported for the beaked whale species studied here (Hooker and Baird, 1999; Tyack et al., 2006). Thus, this strategy comes at the cost of a lower time available for foraging: although many other factors influence diving rates, the proportion of time spent actively searching for prey on a daily basis by beaked whales [17–25% (Tyack et al., 2006; Arranz et al., 2011)] is well below that of sperm whales [53% (Watwood et al., 2006)].

Recruitment of different fibre types to produce diverse swimming gaits has been well documented in fish (Jayne and Lauder, 1994), with anaerobic pathways generally being reserved for predator avoidance or prey capture rather than routine transport (Alexander and Goldspink, 1977). Though we did not measure muscle fibre contractions directly, the results presented here are consistent with a process whereby anaerobic fibres are employed for routine locomotion by some extreme diving mammals to prolong foraging at depth. The observation that marine mammals can utilize strokes with substantially different mechanical output and energetic cost has an important bearing on methods used to assess locomotor cost. If animals produce a diversity of strokes, the number of strokes alone may be insufficient to characterize the cost of transport (Williams et al., 2004) and a method that takes into account either the duration of individual strokes or the acceleration [e.g. Overall Dynamic Body Acceleration (ODBA) (Wilson et al., 2006)] may be more accurate.

Tagging was conducted on three beaked whale species: 10 Md were tagged from 2003–2010 off the island of El Hierro in the Canary Islands, Spain; 9 Zc were tagged from 2003–2008 in the Ligurian Sea, Italy; 2 Ha were tagged in 2011 in the Gully, Canada. Some Md were tagged unintentionally more than once due to the difficulty in recognizing individuals at the moment of tagging, resulting in tag deployments on seven different whales (Table 2). Mesoplodon and Ziphius were tagged with DTag sound and movement recording tags following the methods described by Madsen et al. (2005). The tag dimensions were 20×10×3 cm giving a flow cross-section ≪1% of the frontal area of Md and Zc. Tags were attached with four 6-cm-diameter suction cups, minimizing the relative movement of the sensors with respect to the whale. The tag included sensors for pressure, triaxial acceleration and triaxial magnetism. All sensor channels were anti-aliased filtered with a single pole filter at 5 Hz before sampling at 50 Hz. Data were then decimated in post-processing to a 25 Hz sampling rate using identical low-pass filters (10 Hz cut-off frequency) on each sensor channel. Hyperoodon were tagged with single suction cup Little Leonardo tags (W2000-3MPD3GT, 17.5 cm long×3.0 cm diameter) following the data collection methods described in Aoki et al. (2012). These tags recorded pressure, swim speed, temperature and three-axis magnetic field at a sampling rate of 1 Hz and three-axis accelerometer data at 32 Hz. No anti-alias filter was present in the Little Leonardo tags.

Beaked whales perform long, deep foraging dives interspersed with protracted intervals of shorter shallow dives (Hooker and Baird, 1999), which do not appear to include foraging, at least in Md and Zc (Tyack et al., 2006). As a result, the dive-depth histogram for each species is bimodal with clearly separated shallow and deep dives. Using these separations, we defined deep dives as those with maximum depth greater than 350 m, 600 m and 800 m for Md, Zc and Ha, respectively.

The accelerometer and magnetometer signals recorded by the DTags and Little Leonardo tags were corrected for the orientation of the tag on the whale, which was estimated at each surfacing from the stereotypical movements during respiration (Zimmer et al., 2005). Suction cup tags occasionally move on the animal; if a tag was found to have moved during a dive and the time of the move could not be determined from the data, that dive was excluded from further analysis.

Cetaceans propel themselves with dorso-ventral oscillations of their flukes which generate surge and heave accelerations (i.e. specific acceleration in the longitudinal, and dorso-ventral axes, respectively), as well as body rotations (Fish et al., 2003). All of these movements appear as cyclical variations at the stroking rate in accelerometer signals recorded on a swimming whale and are overlaid on the constantly changing postural orientation as the animal manoeuvres (Johnson and Tyack, 2003; Sato et al., 2007). In steady swimming, an animal's overall body posture varies slowly in comparison with the stroking rate and so this component can be removed by filtering (Sato et al., 2003). To design this filter, we calculated power spectra (FFT length 512, 50% overlap, 25 Hz sampling rate) of the longitudinal and dorso-ventral accelerometer signals during descents and ascents to determine the dominant stroke frequency sensu Sato et al. (2007), f_r, for each animal. A symmetric Finite Impulse Response (FIR) low-pass filter, L, was then designed with a cut-off frequency of 0.4 f_r to separate stroking and postural signals. This filter was applied to each axis of the accelerometer and the filtered data were shortened by removing samples equivalent to half the filter length from the start of each signal, effectively cancelling the group delay of the filter (Stein, 2000). The measured and low-pass filtered accelerometer signals at time t are referred to as A_t and ⁠, respectively, with each being a three-element vector with axes, x (longitudinal), y (lateral) and z (dorso-ventral).

The U-shaped deep dives of beaked whales (Baird et al., 2008) naturally divide into three phases: descent, bottom and ascent. To distinguish dive phases, the pitch angle was computed from the low-pass-filtered acceleration using with negative pitch angles indicating descent. Following Miller et al. (2004), we defined descents as the interval from the start of the dive until the pitch first exceeds 0 deg. Ascents were taken as beginning at the last time in the dive with a sustained downward pitch (<0 deg). The reliability of this method was checked visually and corrected in several cases in which a brief descent occurred in the middle of the ascent or vice versa. Rapid manoeuvring during foraging in the bottom phase makes it difficult to detect individual fluke-strokes and so movements were not analysed in this phase.

DTags do not have a speed sensor but assuming that animals move in a caudo–rostral direction, forward speed, u(t), can be estimated by dividing vertical speed (i.e. the differential of depth) by the sine of the instantaneous pitch angle [i.e. (Miller et al., 2004]. The resulting speed estimate is noisy because of specific acceleration corrupting the pitch estimate and because of noise in the depth sensor amplified by the differentiation, and so requires low-pass filtering. As we are interested in the speed increment over individual fluke strokes, we computed the speed estimate at 25 Hz and then low-pass filtered this with a 0.6 Hz cut-off frequency. To minimize errors, this speed estimate was only used when the average absolute pitch angle was more than 30 deg (Simon et al., 2012). Data analyses were performed in Matlab 7.0 (MathWorks, Natick, USA) and statistical tests were made in R (R Development Core Team, 2009).

Two methods were used to detect and quantify fluke-strokes, leveraging the different sensor suites in the Little Leonardo and DTag tags. The first method requires high sampling rate (i.e. ≫twice the stroking rate) accelerometer data which was recorded by both tags. In this method, stroking rate variations were derived by subtracting the low-pass-filtered acceleration from the measured acceleration (Sato et al., 2004). The resulting high-pass-filtered acceleration, ⁠, contains both rotation and specific acceleration and so cannot be used to quantify either of these movements but it does provide an overall estimate of the magnitude of accelerative signals during stroking. We used the dorso-ventral component of Ã_t (i.e. ã_zt) as a proxy for stroking motions that could be computed for all tags.

The second method requires synchronous accelerometer and magnetometer data at high sampling rates so could only be applied to DTag recordings. Magnetometers are sensitive to the instantaneous orientation of the animal but not to its acceleration, providing an opportunity to separate rotation and specific acceleration. As only two orientation angles can be deduced from this triaxial vector sensor, the magnetometer enables precise tracking of body rotations in planar movements independent of acceleration (Bonnet and Héliot, 2007). Here, we use this method to track orientation during stroking. In this method (see Appendix), estimated body rotations (⁠⁠), derived from the high-pass-filtered magnetometer signals (⁠⁠), are used to predict the dynamic orientation component in the high-pass-filtered accelerometer signals and so arrive at an estimate of the specific acceleration (⁠⁠).

The accelerometer and magnetometer methods described above provide scalar signals, high-pass-filtered dorso-ventral acceleration (ã_zt) and estimated body rotations (⁠⁠), respectively, which can be used to detect intervals with and without stroking. An exploratory examination of these signals suggested that asymmetric and incomplete strokes were common, leading us to analyse stroking on the basis of half-strokes. These are defined as the interval between successive zero-crossings in the estimated body rotations for Zc and Md, and ã_zt for Ha. We used log-frequency plots (Sibly et al., 1990) of these signals to pick suitable hysteretic thresholds, J, for detecting zero-crossings. Stroking was considered to occur whenever there were zero crossings in which the signal passed from below −J to above +J or vice versa within 2/f_r seconds, where f_r is the dominant stroking rate for each animal. Up-strokes were defined as the period between a positive and a negative zero-crossing and down-strokes from negative to a positive zero-crossing. Intervals >2/f_r between zero crossings were defined as glides. The accuracy of this stroke and glide detection method was checked visually, relying on the usually evident differences in stroking and gliding in accelerometer signals. The automatic method was accurate with the exception that partial fluke strokes at the start and end of fluking bouts were often not detected, neither as strokes nor glides, on account of having only one valid zero-crossing. These movements appear to generate little acceleration and may primarily serve to position the flukes for the next gait. Thus, the exclusion of these transitional movements from the analysis has little impact on the results. The lack of a concluding stroke in some dives impedes the method to detect the terminal glide in ascents and these glides were added manually.

Stroke parameters were only calculated for half-strokes of Md and Zc in which the single plane rotation model explained more than 80% of the variance in the magnetometer signal (see Appendix). For these half-strokes, the duration and root mean square (RMS) magnitude of the stroking signal were calculated along with the RMS of the estimated heave and surge accelerations.

To provide a simple comparison of swimming effort during descents and ascents, and therefore help predict whether whales are denser or less dense than water, the total number of half-strokes between 100 and 400 m in each phase was calculated. This depth range excludes depths near the surface in which air in the lungs will contribute more strongly to buoyancy and includes all but 4 deep dives of Md for which the descent phase does not reach the 400 m.

where sin(r_t) in Eqn A8 has been replaced with its estimate from Eqn A7 to compute Ŝt. The components of Ŝt in the longitudinal, lateral and dorso-ventral axes are termed surge, sway and heave accelerations, respectively.

The swimming model used to generate assumes that the body rotates around the transverse axis, i.e. a pitch rotation. To assess the quality of fit of this movement model for the magnetometer estimator in Eqn A7, we computed the R² statistic, i.e. one minus the relative squared residuals ⁠, for each half-stroke. The model provided an acceptable fit (R²>0.8) to the magnetometer data in 86% (Md) and 92% (Zc) of half-strokes giving an overall R² of 0.93 (Md and Zc) for the analysed half-strokes. About 12% of half-strokes were rejected due to an R²<0.8. These likely coincided with transient fast manoeuvres for which the spectral separation assumption breaks down. Although stroking may also involve some rolling or yawing rotations, the high R² values suggest that the simple pitch model is a good approximation for ascent and descent swimming in Md and Zc.

The R² was also computed for the accelerometer estimator of Eqn A9, ignoring specific acceleration, as  ⁠. This was not done to assess quality of fit but rather to give a sense for how well a rotation-only model fits the observed accelerometer signals. The much lower R² values of 0.65 (Md and Zc) indicates, as expected, that body rotation explains a smaller part of the accelerometer signals.

To explore the possibility that B-strokes were associated with longer, faster or more vertical swimming we used a generalized linear model (GLM) with a logit link function and two explanatory variables: time into the dive (min) and vertical speed (m s⁻¹); as pitch and vertical speed are highly correlated, pitch angle (deg) was not included in the model. A mixed effect model was used for Md and Zc with individual as a random effect while a fixed effect model was used for Ha with individual as a fixed factor due to the small sample size. The presence of one or more B-half-strokes in each 10 s (Md) or 15 s (Zc and Ha) interval in the ascent was used as a dichotomous response variable. These analysis intervals were chosen to encompass the typical stroke-and-glide gait duration for each species. Models were fit using the lmer and glm functions in R and goodness-of-fit assessed following Nakagawa and Schielzeth (2013).

Ascent models showed significant positive relationships between occurrence of B-strokes and both vertical speed (odds increase by 17, 13 and 31% for Md, Zc and Ha, respectively, per metre) and time into dive (odds increase by 11, 13 and 20% for Md, Zc and Ha, respectively, per minute). However, these models explain relatively little variance with marginal R² (i.e. the variance fraction explained by the fixed effects) of 0.22 (Md), 0.37 (Zc) and 0.50 (Ha). For Md and Zc, the random effect increased the conditional R² to 0.45 and 0.51, respectively, indicating considerable individual variability. The weak association of B-strokes with vertical speed may be a consequence of the frequent co-occurrence of B-strokes with glides within the 10 s or 15 s analysis windows. Despite there being 1.2- to 13-times more normal strokes than B-strokes, a high percentage of ascent glides were performed after a B-stroke (85% in Md, 63% in Zc and 67% in Ha). These glides lasted about 2- to 3.5-times longer than a normal stroke and so substantially reduced the overall acceleration of the stroke-and-glide gait. To quantify this, we calculated a relative measure of activity, similar to the Vectorial Dynamic Body Acceleration (VeDBA; Gleiss et al., 2011a) during the same ascent intervals. This measure was calculated as the sum of the vector magnitude of Ã_t for Ha, and ⁠, for Md and Zc. Intervals with B-strokes had values of this metric that were 1.1- to 1.5-times greater than intervals with normal stroking (Table 2) with the exception of two Zc for which the B-stroke value was twice the usual stroke value because of occasional bursts of up to 10–14 consecutive B-strokes.

Based on the significant relationship between B-strokes and the elapsed time in a dive, we evaluated when B-strokes began to be produced in dives. Excluding the few B-strokes in the initial part of descents, the onset time of B-strokes was calculated as the elapsed time from the beginning of each deep dive until the first minute with at least the median ascent rate of B-strokes for each whale. The onset times and the corresponding depths were averaged for each whale, and then animals were pooled within species. The bottom phase of dives could not be analysed and so these onset times will be biased if B-strokes were used in some dives before the start of the ascent. It is, in fact, difficult to assess how these onset times compare with those that would be obtained by chance if B-strokes occurred at random in ascents because of two forms of censoring: left-censoring occurs when B-strokes are found immediately at the start of an ascent whereas right-censoring occurs when no type-B strokes are performed in an ascent. A non-parametric maximum likelihood estimator (NPMLE) allowing interval-censored data was used to estimate type-B stroke onset time for Mesoplodon and Ziphius (see supplementary material Fig. S1), but not for Hyperoodon, because of the small sample size. The analysis was carried out in R using the surv and survfit functions in R package ‘survival’ (Therneau, 2012). Taking into account the interval-censored data, B-strokes appeared significantly later than expected by chance (i.e. the raw data confidence intervals for onset time were to the right of the randomized curve). For Md and Zc, B-strokes appeared more than 4 and 6 min later than expected by chance, respectively, in 50% of dives.

We thank A. Crespo, A. Fais, A. Schiavi, A. Escánez, C. Reyes, J. Marrero, K. Gikopoulou, M. Bayona, M. Moral, M. Tobeña, P. Arranz, P. Madsen, A. Bocconcelli, A. Sturlese, G. Mojoli, M. Ballardini, T. Pusser, K. Aoki, M. Bivins, R. Antunes, T. Narazaki, V. Deecke and many others for their help in the field in the Canary Islands, Ligurian Sea and The Gully. Special thanks to Mick Wu and Iosu Paradinas Aranjuelo for his statistical support. Thanks also to Mike Fedak, Frants Jensen, Peter T. Madsen, Rene Swift, Ann Pabst, Brandy P. Velten, Peter L. Tyack, Hans M. Nielsen, Paul Ponganis, Gerald Kooyman and the anonymous reviewers for insightful comments which have greatly improved the paper. This paper is dedicated to Guido Mojoli whose enthusiasm kept us going.