Dynamic soaring is a small-scale flight manoeuvre which is the basis for the extreme flight performance of albatrosses and other large seabirds to travel huge distances in sustained non-flapping flight. As experimental data with sufficient resolution of th ese small-scale movements are not available, knowledge is lacking about dynamic soaring and the physical mechanism of the energy gain of the bird from the wind. With new in-house developments of GPS logging units for recording raw phase observations and of a dedicated mathematical method for postprocessing these measurements, it was possible to determine the small-scale flight manoeuvre with the required high precision. Experimental results from tracking 16 wandering albatrosses (Diomedea exulans) in the southern Indian Ocean show the characteristic pattern of dynamic soaring. This pattern consists of four flight phases comprising a windward climb, an upper curve, a leeward descent and a lower curve, which are continually repeated. It is shown that the primary energy gain from the shear wind is attained in the upper curve where the bird changes the flight direction from windward to leeward. As a result, the upper curve is the characteristic flight phase of dynamic soaring for achieving the energy gain necessary for sustained non-flapping flight.
Albatrosses excel in an extreme travelling performance by covering huge distances in their foraging trips with sustained non-flapping flight. Flight recordings show distances of thousands of kilometres as well as flights around the world in 46 days (Jouventin and Weimerskirch, 1990; Bonadonna et al., 2005; Croxall et al., 2005). Furthermore, albatrosses are able to fly persistently at high speed by using favourable winds (Catry et al., 2004).
The reason for this unique performance capability is a flight mode termed dynamic soaring. With dynamic soaring, the birds achieve an energy gain from the shear wind above the ocean surface, enabling sustained non-flapping flight. As a result, the birds can fly at virtually no cost when compared with flapping flight (Weimerskirch et al., 2000). By applying dynamic soaring (Sachs, 2005; Sachs et al., 2012), the birds gain access to an unlimited external energy source in terms of the shear wind above the sea surface. The unique advantage of having an unlimited energy source is due to the fact that there are permanently strong winds in the areas in which albatrosses live (Suryan et al., 2008). These brief considerations show that dynamic soaring is fundamental for the extreme travelling performance of albatrosses, enabling their unique way of flying and living.
The long distance flight of albatrosses, constituting a large-scale movement of the order of hundreds to thousands of kilometres, has been experimentally investigated at great length and is well documented (Jouventin and Weimerskirch, 1990; Weimerskirch et al., 2000; Weimerskirch et al., 2002; Catry et al., 2004; Bonadonna et al., 2005; Croxall et al., 2005). By contrast, the dynamic soaring flight mode of albatrosses and other large seabirds has not been experimentally investigated. This is because dynamic soaring is a small-scale movement of the order of tens to hundreds of metres (Sachs, 2005).
A primary goal of this paper is to advance the knowledge in the field of dynamic soaring in albatrosses and their unique flight method of gaining energy from the wind for flying without flapping. The current state of knowledge manifests in a variety of theories and explanations for the small-scale dynamic soaring flight of albatrosses. There is a theory termed wind-gradient soaring (Lighthill, 1975; Norberg, 1990; Spedding, 1992; Tickell, 2000; Dhawan, 2002; Lindhe Norberg, 2004; Azuma, 2006; Denny, 2009); according to this theory, soaring is continually possible using the wind gradient in the shear layer above the sea surface. Another theory termed gust soaring relates to discontinuities in the wind flow (Pennycuick, 2002; Pennycuick, 2008; Suryan et al., 2008; Langelaan, 2008; Langelaan and Bramesfeld, 2008); according to this theory, energy pulses are obtained from flight through the separated air flow region behind wave crests. Furthermore, wave soaring and wave lift are regarded as a technique to obtain energy for flying (Berger and Göhde, 1965; Wilson, 1975; Pennycuick, 1982; Sheng et al., 2005; Richardson, 2011); here, updrafts at waves are supposed to be usable for soaring. Another point relates to the aerodynamic ground effect (Blake, 1983; Hainsworth, 1988; Norberg, 1990; Rayner, 1991); this effect yields a decrease of the drag when flying close to the water surface so that an energetic advantage is possible at low levels.
To sum up, current theories and explanations are differing and show various findings and conclusions. As a result, there is a lack of both knowledge of dynamic soaring and clarity about this flight mode, particularly with regard to the magnitude of the achievable energy gain and the physical transfer mechanism of energy from the wind to the bird. This is due to the fact that there are so far no experimental data of the small-scale movements of albatrosses so that a full understanding of dynamic soaring could not be achieved.
To address the aims of this paper, in-flight measurements of the small-scale movements in free-flying birds were performed so that experimental verification of dynamic soaring in albatrosses could be accomplished. The experimental verification was achieved with new in-house developments of appropriate GPS logger hardware and of a novel mathematical method for computing the albatrosses' flight path and speed with the high precision required for this small-scale flight mode.
MATERIALS AND METHODS
New in-house developments were accomplished in order to achieve the required high precision in determining the small-scale movements of dynamic soaring in albatrosses (Traugott et al., 2008a). One pertains to the hardware, yielding a miniaturized GPS logging unit for recording raw L1 phase measurements at high sampling rate. The other in-house development is a new mathematical method for determining the flight path and speed with high precision. The new in-house developments were tested for correct functioning prior to being used for the albatross flight measurements. These trials included test runs with a car as well as flight tests with several aeroplanes (Traugott et al., 2008a; Traugott et al., 2008b).
New logger hardware
Three different types of GPS data loggers were developed for the albatross flight measurements. The core of each device was a passive 25×25 mm patch antenna plugged to the single frequency GPS module LEA-4T (size 17×22.4 mm, mass 2.1 g; u-blox, Thalwil, Switzerland). This module is capable of calculating position and velocity online with a sampling rate of 4 Hz. When logging the data to the on-chip 8 MB flash memory or to an external memory this yields a data stream of 0.8 MB h−1. As an additional feature, the LEA-4T module makes the GPS raw data available with a sampling rate of up to 10 Hz (~9.6 MB h−1). These data are the basis for online positioning but can also be re-processed back for precision and sampling rate augmentation. Because of known issues when forcing the module to output both the online solution and the raw data at maximum rates, the online solution rate was limited to 1 Hz during the measurement campaign (210 kB h−1). The average power consumption of the loggers was given by 40–60 mA and 2.9–3.7 V. Energy was provided by two or three 3.6 V primary lithium–thionyl chloride cells (Saft LS 14500 and LS 14500 C).
Two types of loggers featured an external 2 GB memory card. Equipped with three batteries, these devices were capable of logging the raw GPS data throughout a period of up to 6 days. Their form factor was 89×55×22 mm and 108×55×19 mm at a total mass of 103 and 93 g.
The third type of logger additionally featured a 3-axes MEMS accelerometer. No external memory was provided but all data were logged to the on-chip 8 MB flash which yielded possible recording intervals to about 40 min. For bridging the delay between mounting a logger on a bird at the nesting site and the bird finally cruising over open waters, these loggers implemented a reliable sleep/wake-up logic triggering high rate recording only when the bird had left a predefined area and was exceeding a given speed threshold.
In Fig. 1A, a miniaturized GPS data logger is shown. The complete logging unit includes the logger, the batteries, the wiring and the casing. The GPS logging unit was flight tested prior to being used for the albatross flight measurements. One of the aircraft used in the flight test programme is shown in Fig. 1B. This test vehicle is a research aircraft of the Institute of Flight System Mechanics of the Technische Universität München. The goal of the trials was to test the GPS data logger hardware as well as the new mathematical method for precise position determination. As no experience was on hand with regard to the required high position precision and high sampling rate, the development of the system involved significant test efforts. An issue of proper functioning for the planned albatross flight measurements was GPS signal shadowing at large bank angles, like the values occurring in dynamic-soaring-type flight manoeuvres of albatrosses. Therefore, the flight tests comprised, among others, dynamic manoeuvres with high bank angles to simulate those of dynamic soaring in albatrosses.
New mathematical method
A new mathematical method was developed to achieve relative position precision in the low decimetre range depending on environmental conditions. The high precision is obtained by forming single differences between raw L1 phase measurements taken by the moving receiver at two moments in time. Neither a second, nearby base receiver nor any (static) initialization procedures as commonly used in geodetic applications exploiting the same type of precise measurements are required by this differential GPS (D-GPS) approach. Details of the method are given elsewhere (Traugott et al., 2008a; Traugott et al., 2008b).
In Fig. 2, the basic concept of the approach for coping with the position determination task is graphically presented: in this exemplary scenario, a starting point is specified for an arbitrary time tb1 at the beginning of a flight manoeuvre of interest. The corresponding position of the vehicle is not known exactly but is estimated by techniques such as code-based single point positioning, a technique yielding robust, absolute position information limited to metre-level accuracy. Therefore, the start position is biased from the true location by Δ. The subsequent trajectory is now determined by time-differential processing relative to this starting point. In other terms, the base vectors pointing from the starting epoch to the current position are determined exactly. Hence, all fixes are afflicted by the same bias Δ. Phase measurements are sensible to signal shadowing – a short upside-down interlude can cause signal obstruction and prevent further processing. A new base at tb2 can be imported from the single point solution right after the manoeuvre (no re-initialization) and processing can be continued relative to the new base. Such an event causes a gap in the resulting trajectory. In the case shown in Fig. 2, the solution fails again between the base epoch tb2 and the current time tj. However, this time there are enough healthy satellites observed at tj−1 and tj to calculate the baseline between these two points. A base hand-over preventing a gap in the solution can be realized and processing is hereupon continued to tb3. A detailed description is given elsewhere (Traugott et al., 2008a; Traugott et al., 2008b).
The new mathematical method yields a position precision in the low decimetre region (Traugott et al., 2008a) so that an improvement in precision by a factor of 10 when compared with the state of the art is achieved. This holds for position precision in the longitudinal and lateral directions as well as in the vertical direction. In addition to this precision improvement, the position data are recorded at a sampling rate of 10 Hz, which is also significantly higher than the state of the art as regards comparable miniaturized devices.
Wind information was obtained using SeaWinds on QuikSCAT Level 3 Daily, Gridded Ocean Wind Vectors (JPL SeaWinds Project; winds.jpl.nasa.gov). The data are sampled on an ~0.25°×0.25° global grid twice a day (equal to 28 km in north–south direction and 18 km in east–west direction at 49° south). The data provide local wind velocity vectors at a reference altitude of 10 m with an accuracy of 2 m s−1 (or 10% for velocities above 20 m s−1) and the wind direction within ±20°. Additional information is provided elsewhere (Perry, 2001).
For calculating the wind at the respective trajectory point, an in-house computational procedure was developed using one-dimensional linear interpolation in the time domain and bivariate Akima interpolation in the position domain (Müller, 2009).
The field work took place during a research stay of 3 months close to the Albatross colony at Cap Ratmanoff, Kerguelen Archipelago, southern Indian Ocean, on wandering albatrosses (Diomedea exulans Linnaeus 1758).
The miniaturized GPS logging units were taped to the back feathers of the birds using TESA tape according to the procedure suggested by Wilson et al. (Wilson et al., 1997). The mass of the miniaturized GPS logging units, which was 107 g in the heaviest version, represented about 1.0–1.3% of the mass of the birds, thus being less than 3% of the birds' mass as recommended by Phillips et al. (Phillips et al., 2003). The GPS units were recovered at the end of a foraging trip. Twenty GPS units were deployed, of which 16 provided high-quality flight data of the bird's trajectories. Two GPS units were damaged by saltwater, two others provided data from sitting birds that delayed their departure.
RESULTS AND DISCUSSION
Large-scale and small-scale movements
Reference is first made to the large-scale movement in terms of a foraging trip of an albatross to show a complete data recording of high-precision tracking at 10 Hz. In Fig. 3, the ground track of the flight that begins and ends at the Kerguelen Archipelago is presented. The overall duration of the trip was 3.2 days and its length was 1120 km.
A closer examination of the large-scale movement reveals that there are individual cycles constituting the bird's small-scale movements which are continually repeated. While the large-scale movement appears as a steady-state cruise-type motion horizontal to the Earth's surface (Fig. 3), the small-scale movements are of a pronounced three-dimensional and highly dynamic nature, yielding repetitive cycles (Fig. 4). They show distinct motions in the longitudinal, lateral and vertical directions. There are four flight phases, which are the characteristic elements of each cycle, denoted by numbers 1–4 at the first cycle: (1) windward climb; (2) upper curve from windward to leeward flight direction; (3) leeward descent; and (4) lower curve from leeward to windward flight direction. A flight cycle is the basic constituent of dynamic soaring.
Characteristics of dynamic soaring
Altitude, speed and total energy
A more detailed examination of dynamic soaring cycles yields the behaviour and magnitude of quantities relevant for this flight mode. With knowledge of these features, further characterization of dynamic soaring is possible. For this purpose, two cases were selected from the experimental data obtained in the in-flight measurements. These cases refer to the altitude interval that the bird traverses during dynamic soaring cycles, yielding cycles with a large or a small altitude interval (Figs 5 and 6, where Fig. 5 is the case already shown in Fig. 4). Selection of the altitude interval – instead of another quantity – is based on its importance for energy gain. This is because the greatest wind speed that the bird encounters in a dynamic soaring cycle is determinative for the achievable energy gain. The greatest wind speed is at the top of the altitude interval because the wind speed increases with altitude (Stull, 2003), i.e. with distance from the sea surface, where it is practically zero. With the altitude intervals selected for Figs 5 and 6, a wide range is covered, yielding representative results for dynamic soaring in albatrosses.
The altitude region extends from zero to about 15 m in the larger altitude interval case, and to less than 9 m in the small altitude interval one. Altitude and inertial speed (i.e. the speed related to the Earth as an inertial reference system) show a cyclic behaviour such that they are continually repeated, with the speed lagging behind the altitude. The speed range run through during a cycle is greater in the larger altitude interval case, while the highest speed level, which is close to 30 m s−1, is the same in the two cases. The duration of a cycle is of the order of 10 s, with somewhat greater values in the large altitude interval case and smaller values in the other.
There are characteristic properties concerning the energy gain in terms of the energy extraction from the shear wind and its transfer to the bird. The main point is the section of the dynamic soaring cycle where the energy gain is achieved. This is graphically illustrated in Figs 5 and 6 (indicated by the grey shading). The total energy begins to increase during the climb. At the top of the trajectory, the energy increase, being in full progress, continues. It comes to a stop in the course of the descent, after the altitude has already decreased. At this point, the total energy of the bird is at its maximum. There are large energy gains in all cycles, reaching as high as 300% of the value at beginning of a cycle (Fig. 5, first and third cycle). The way in which the energy gain is achieved holds for all cycles shown in Figs 5 and 6. Thus, it applies to the large altitude interval case as well as to the small interval case.
The energy diagrams in Figs 5 and 6 also show the kinetic and the potential energy. A comparison of the individual energy curves reveals that the increase in the total energy during the energy gain phase consists primarily of an increase in the kinetic energy, whereas the potential energy increase is significantly smaller. Thus, the total energy gain from the shear wind is mainly a kinetic energy gain. This means that, with regard to the motion of the bird, there is an increase in favour of speed when compared with altitude.
Upper curve: trajectory section of energy gain
For verification of dynamic soaring, the trajectory section in which the energy gain is achieved is determinative. This is made recognisable in Figs 7 and 8, with colour coding used to show the relationship between total energy and trajectory. The central issue is the trajectory section associated with the energy gain. The energy gain is achieved in the curve where the flight direction changes from windward to leeward, as indicated by the colour change from blue to red. Reference to Figs 5 and 6 reveals that this flight direction change occurs in the upper altitude region of each cycle, around the top of the trajectory. Thus, the curve in question where the change of flight direction from windward to leeward takes place is in the upper altitude region, yielding the upper curve of the dynamic soaring cycle. The trajectory position at which the maximum total energy is reached is at the end of the upper curve. This holds for all cycles, in the large as well as in the small altitude interval cases. As a result, the upper curve can be qualified as the characteristic flight phase of dynamic soaring for achieving an energy gain.
The lower curve is the trajectory section where a total energy loss occurs (with red to blue change, Figs 7 and 8). This also holds for all cycles, in both the large as well as small altitude interval cases. As the wind speed is low here (Stull, 2003), the energy loss is not as large as it would be at a higher altitude. This is the reason why the curve where the flight direction changes from leeward to windward is in the lower altitude region, close to the sea surface.
Dynamic soaring cycles on a large-scale representation
In Figs 4, 5, 6, 7 and 8, the decisive motion quantities and the energy balance essential for dynamic soaring as well as their relationship to the flight trajectory are presented so that the physical mechanism of the energy gain from the wind can be shown and identified as a unique characteristic of this flight mode. Supplementary to the small-scale movement representation, trajectory sections of greater length are depicted in Fig. 9 to illustrate how dynamic soaring manifests on a larger scale. Three wind cases are selected because of the essential significance of the wind for dynamic soaring, i.e. downwind flights at low and at high wind speed as well as upwind flight. Basically, all three trajectory sections show the typical pattern of dynamic soaring, which consists entirely of winding and curving segments.
The downwind flight cases at low and high wind speed are similar with regard to the extension of the curved segments in the longitudinal and lateral directions. Furthermore, the overall flight directions in both cases appear as a straight movement on a large-scale basis. Compared with this, the upwind case shows significant differences. Here, the extension of the curved segments in the longitudinal and lateral directions is much more pronounced. The overall flight direction is not so straight, being more of a meandering type. The individual segments are rather irregular, particularly in comparison with the downwind cases.
Comparison with current theories and explanations for dynamic soaring
Theory of wind-gradient soaring
Theory of gust soaring
The theory of gust soaring is concerned with discontinuities in the wind flow (Pennycuick, 2002; Pennycuick, 2008; Suryan et al., 2008; Langelaan, 2008; Langelaan and Bramesfeld, 2008). According to this theory, there is an alternative flight mode by which pulses of energy are obtained from a flight through the separated air flow region behind wave crests. It implies that this happens at low level.
Investigation of the energy behaviour (Figs 5 and 6) shows that all total energy curves are continuous and smooth. This means with regard to the theory of gust soaring that there are no discontinuities and no energy pulses, but there is a continuous extraction of energy from the shear wind to the bird. Furthermore, the energy gain is not attained at low level. Rather, it is achieved in the upper altitude region, around the top of the trajectory.
This conclusion is further confirmed by data from the flight of an albatross over flat land. In Fig. 10, the trajectory obtained from in-flight measurement of an albatross flying over land is presented. The flight over land consists completely of dynamic soaring cycles, without any straight trajectory portion in between. All cycles show the characteristic pattern of dynamic soaring. The elevation of the terrain in the area of the dynamic soaring trajectory over land is presented in Fig. 11, revealing how flat this area is. Because of the flatness of the land, separated air flow is not possible. As a result, there cannot be an effect such as energy pulses obtained from flight through a separated air flow region. Rather, the energy gain is achieved in a continuous and smooth manner in the upper curve of dynamic soaring.
Wave soaring and wave lift
Wave soaring and wave lift are considered to be a possible source of energy gain (Berger and Göhde, 1965; Wilson, 1975; Pennycuick, 1982; Sheng et al., 2005; Richardson, 2011). By flying along the flanks of waves, the birds can make use of rising air currents directly related to the waves. This would imply flight at low altitude, close to the waves.
The dynamic soaring cycles over flat land, as shown in Figs 10 and 11, provide evidence for another energy transfer mechanism. As the ground is flat there are no geometric forms similar to waves. Thus, no rising air currents exist that may be used for achieving an energy gain. Instead, there is only horizontal wind. This means that the energy is extracted from horizontally moving air rather than from rising air currents.
Furthermore, wave soaring and wave lift would show flight segments close to the water surface where altitude and speed are constant or slowly changing. However, this is not the case, as may be seen in Figs 4, 5, 6 and 7. Rather, the energy gain is achieved in the upper altitude region where the effect of waves on air currents can be assumed to be negligible, if not zero. As a result, this effect plays no role in energy gain.
Aerodynamic ground effect
The aerodynamic ground effect yields a decrease of the drag when a bird is close to the water surface (Blake, 1983; Hainsworth, 1988; Norberg, 1990; Rayner, 1991). It rapidly reduces with the distance from the ground. The aerodynamic ground effect has an influence only during flight at low level. It yields a decrease of the induced drag factor by about 10% at a distance of a wing semispan from the ground, with less of a decrease for larger distances (Rayner, 1991). Flight involving a high bank angle, as is the case in the lower curve of dynamic soaring of albatrosses, suggests that the effectiveness of the aerodynamic ground effect would be reduced.
A main point in this context is that the aerodynamic ground effect cannot increase the total energy because it merely reduces the aerodynamic drag so that there is still a dissipative effect of the drag. As a result, the aerodynamic ground effect plays no role in the energy gain.
Energy gain mechanism in dynamic soaring: propulsive force due to the wind
A deeper insight into the physical mechanism underlying the energy gain in dynamic soaring is possible with an analysis of the force effecting this gain. There is a propulsive force that acts at the bird and yields an increase of the total energy. The generation of this force is shown in Fig. 12.
Fig. 12A shows the lift vector, as seen from behind in direction of the airspeed vector. The lift vector can be decomposed into two components: and . The component is in the plane of the speed vectors, while the component is perpendicular to that plane. This means that is doing work and, thus, exerts an effect on the total energy. By contrast, is doing no work so it has no effect on the total energy.
From an analysis of Fpropulsive using data from the albatrosses' in-flight measurements, the results presented in Fig. 13 are obtained. These results basically show: (1) Fpropulsive is the force that generates the total energy gain; this takes place in the upper curve; and (2) the effect of Fpropulsive is very strong.
Positive Fpropulsive values mean that energy is extracted from the shear wind and transferred to the bird, yielding a total energy gain, while negative Fpropulsive values cause an energy loss. The range of positive Fpropulsive values is associated with the upper curve, as indicated in Fig. 13 by the grey shading. The fact that Fpropulsive reaches its greatest level around the middle of the upper curve means that it is most effective in the higher altitude region of the dynamic soaring cycle. This again verifies that the upper curve is the characteristic flight phase of dynamic soaring for achieving an energy gain.
The total energy management in the dynamic soaring cycle is dominated by Fpropulsive. This is confirmed by a comparison with the contribution of the drag, which is the only other force doing work and influencing the total energy. The analysis of the force relationship using data from the albatrosses' in-flight measurements is also concerned with the drag, yielding an estimation of its component DcosαW, which is the drag component effective in doing work (Fig. 12B). In Fig. 13, the component DcosαW is also shown (the negative of DcosαW is selected because of its dissipative effect concerning the energy). Comparison with Fpropulsive reveals that DcosαW is substantially smaller. This particularly holds for the upper curve, where DcosαW is practically negligible. Thus, the total energy behaviour is influenced by the drag to a very small extent only, but dominated by Fpropulsive.
A comparison with the force characteristics in the lower curve further deepens the insight into the energy management during dynamic soaring. In Fig. 14, the lift vector in the lower curve is presented, as seen from above on the plane of the speed vectors. The lift component sinαW is now acting in the opposite direction to the inertial speed vector . Thus, it yields a decrease of the total energy. However, there is an effect that reduces the size of sinαW. This is basically due to the wind speed being small at the altitude of the lower curve. As a result, the angle αW is also small, yielding a reduction of sinαW. In Fig. 14, it is assumed for comparison purposes that the same size applies for and for the wind direction as in Fig. 12B, while the wind speed VW is smaller because of the lower altitude. Comparison of Fig. 12B and Fig. 14 reveals that sinαW differs considerably in the two cases, showing a significantly smaller value in the lower curve. This is an indication that the energy loss in the lower curve is smaller than the energy gain in the upper curve so that there is a net energy gain for the complete dynamic soaring cycle with regard to the effect of Fpropulsive. The described force difference is confirmed by the time history of Fpropulsive depicted in Fig. 13, where the level of the negative values is smaller than that of the positive values.
Evidence of non-flapping in dynamic soaring
The results presented in Fig. 13 provide evidence that there is no wing flapping in dynamic soaring. Rather, the bird is in a gliding condition where the wings are held motionless and in a fixed position. Evidence for non-flapping is substantiated in the following.
A propulsive force this high cannot be generated by wing flapping. This is because for large birds like wandering albatrosses with a mass of about 10 kg the maximum propulsive force possible with wing flapping is smaller by an order of magnitude.
The existence of a propulsive force this high (Fpropulsive,max≈1.2mg as opposed to Fflapping,max≈0.06mg) is evidence of the fact that dynamic soaring is performed without flapping the wings. Instead, the wings are kept in a fixed position, and hence in the position of soaring.
The authors thank J. Mardon for his help in the preliminary trials of the study, and Giacomo Dell'Omo, Wolfgang Heidrich, Franz Kümmeth and Alexei L. Vyssotski for contributing to technical tools as well as for conducting performance tests and optimizing the equipment.
The authors are grateful to the Institut Polaire Français Paul Emile Victor, which supported this work (IPEV, Program No. 354) – the work was performed according to guidelines established by IPEV and CNRS for the Ethical Treatment of Animals. The authors are also grateful for the support provided by National Science Foundation International Research Fellowship for A.P.N. (no. 0700939).
LIST OF SYMBOLS AND ABBREVIATIONS
propulsive force due to flapping
acceleration due to gravity
lift vector component in speed vector plane
lift vector component perpendicular to speed vector plane
wing reference area
arbitrary time base (1–3)
speed at the minimum drag-to-lift ratio, (CD/CL)min
speed at minimum power required
No competing interests declared.