Air speeds of migrating birds observed by ornithodolite and compared with predictions from flight theory.

We measured the air speeds of 31 bird species, for which we had body mass and wing measurements, migrating along the east coast of Sweden in autumn, using a Vectronix Vector 21 ornithodolite and a Gill WindSonic anemometer. We expected each species' average air speed to exceed its calculated minimum-power speed (Vmp), and to fall below its maximum-range speed (Vmr), but found some exceptions to both limits. To resolve these discrepancies, we first reduced the assumed induced power factor for all species from 1.2 to 0.9, attributing this to splayed and up-turned primary feathers, and then assigned body drag coefficients for different species down to 0.060 for small waders, and up to 0.12 for the mute swan, in the Reynolds number range 25 000-250 000. These results will be used to amend the default values in existing software that estimates fuel consumption in migration, energy heights on arrival and other aspects of flight performance, using classical aeronautical theory. The body drag coefficients are central to range calculations. Although they cannot be measured on dead bird bodies, they could be checked against wind tunnel measurements on living birds, using existing methods.

The concept of energy height in animal locomotion: separating mechanics from physiology.

The distance flown in gliding is proportional to the starting height, not to the starting potential energy, and it is independent of the body mass. By analogy, in powered flight, the quantity of stored fuel can be converted into a virtual "fuel energy height", defined as the height to which the fuel energy could lift the bird against gravity, if it were converted into work. This is a logarithmic function of the fuel fraction, not of the absolute amount of fuel, or of the body mass. It takes account of the strength of gravity, and of the efficiency with which fuel energy is converted into work. The "performance number" is the gradient on which a migrating bird comes "down" from its initial fuel energy height. It is mechanical (not physiological) in character, and corresponds to the lift:drag ratio in a fixed-wing aircraft. The concept of range as an initial energy height multiplied by a performance number can also be applied to swimming and running animals. Performance number, and also the related variable "cost of transport", are both independent of gravity in flying and running, but not in swimming. Migration by thermal soaring is analogous to powered flight with stopovers, except that the bird replenishes its potential energy by climbing in thermals, rather than replenishing fuel energy during stopovers. Rates of climb in thermals are typically higher than fuel energy rates of climb, but the available height band is two orders of magnitude smaller, and the intervals at which energy replenishment is needed are correspondingly shorter. Albatrosses replenish their kinetic energy by exploiting discontinuities in the wind flow over waves, requiring replenishment at intervals of tens of seconds, a further two orders of magnitude shorter than in thermal soaring. Fat energy height can be used as a measure of "condition", which is independent of the size or type of the animal. The fat energy height at which a migrant must arrive on the breeding grounds, in order to breed successfully, reflects the ecological characteristics of the habitat, not the size or character of the bird. Energy height expresses what an animal or machine can do with its stored energy, not the amount of energy.

Speeds and wingbeat frequencies of migrating birds compared with calculated benchmarks.

Sixteen species of birds passing Falsterbo in southwest Sweden during the autumn migration season were observed using short-range optical methods. Air speeds and wingbeat frequencies were measured, reduced to sea level, and compared with benchmark values computed by Flight.bas, a published flight performance program based on flight mechanics. The benchmark for air speed was the calculated sea-level value of the minimum power speed (V(mp)). The mean speeds of three raptor species that flew by flap-gliding were below V(mp), apparently because the flap-glide cycle involved slowing down below V(mp) when gliding and accelerating back up to V(mp) when flapping. The mean speeds of 11 species that flew by continuous flapping were between 0.82V(mp) and 1.27V(mp). Two passerine species that flew by bounding had mean speeds of 1.70V(mp) and 1.96V(mp), but these high mean speeds reflected their ability to fly faster against head winds. These results do not support predictions from optimal migration theory, which suggest that migrating birds 'should' fly faster, relative to V(mp). However, observations were restricted for technical reasons to birds flying below 200 m and may not represent birds that were seriously committed to long-distance migration. The benchmark wingbeat frequency (f(ref)) was derived from dimensional reasoning, not from statistical analysis of observations. Observed wingbeat frequencies ranged from 0.81f(ref) to 1.05f(ref), except in the two bounding species, whose wingbeat frequencies appeared anomalously high. However, the mechanics of bounding with a power fraction q imply that gravity during the flapping phase is increased by a factor 1/q, and when the value of gravity was so adjusted in the expression for f(ref), the wingbeat frequencies of the two bounding species were predicted correctly as a function of the power fraction. In small birds with more muscle power than is required to fly at speeds near V(mp), bounding is an effective method of adjusting the specific work in the muscle fibres, allowing conversion efficiency to be maximised over a wide range of speeds.

Uncertainty calculations for theoretical flight power curves.

The comparison of theoretical and experimental estimates of the mechanical power requirement for flight is currently impossible owing to the absence of complete experimental data based on mechanical power, as opposed to measurements of metabolic rates. Nevertheless, comparisons of measured and predicted characteristic speeds, and inferred power curves are frequently made, despite the total absence of uncertainty estimates of the theoretically predicted quantities. Here the method for correct calculation of uncertainty estimates in mechanical power models is outlined in detail, and analytical and numerical results are derived for realistic examples. The sensitivity of the calculated variations in power requirement varies greatly among the independent variables, and the practical and theoretical consequences of this variation are discussed. Pending the arrival of appropriate experimental measurements, it is now possible, in principle, to make quantitative comparisons with theoretical predictions.

Horizontal flight of a swallow (Hirundo rustica) observed in a wind tunnel, with a new method for directly measuring mechanical power.

A swallow flying in the Lund wind tunnel was observed from the side and from behind, by two synchronised high-speed video cameras. The side-view camera provided a record of the vertical position of a white mark, applied to the feathers behind and below the eye, from which the vertical acceleration was obtained. The rear-view camera provided measurements of the mean angle of the left and right humeri above horizontal. From these data, the force acting on the body, the moment applied by each pectoralis muscle to the humerus and the rotation of the humerus were estimated and used to analyse the time course of a number of variables, including the work done by the muscles in each wing beat. The average mechanical power turned out to be more than that predicted on the basis of current estimates of body drag coefficient and profile power ratio, possibly because the bird was not flying steadily in a minimum-drag configuration. We hope to develop the method further by correlating the mechanical measurements with observations of the vortex wake and to apply it to birds that have been conditioned to hold a constant position in the test section.

A new low-turbulence wind tunnel for bird flight experiments at Lund University, Sweden

A new wind tunnel for experiments on bird flight was completed at Lund University, Sweden, in September 1994. It is a closed-circuit design, with a settling section containing five screens and a contraction ratio of 12.25. The test section is octagonal, 1.20 m wide by 1.08 m high. The first 1.2 m of its length is enclosed by acrylic walls, and the last 0.5 m is open, giving unrestricted access. Experiments can be carried out in both the open and closed parts, and comparison between them can potentially be used to measure the lift effect correction. The fan is driven by an a.c. motor with a variable-frequency power supply, allowing the wind speed to be varied continuously from 0 to 38 m s-1. The whole machine can be tilted to give up to 8 ° descent and 6 ° climb. A pitot-static survey in the test section showed that the air speed was within ±1.3 % of the mean at 116 out of 119 sample points, exceeding this deviation at only three points at the edges. A hot-wire anemometer survey showed that the turbulence level in the closed part of the test section was below 0.04 % of the wind speed throughout most of the closed part of the test section, rising to approximately 0.06 % in the middle of the open part. No residual rotation from the fan could be detected in the test section. No decrease in wind speed was detectable beyond 3 cm from the side walls of the closed part, and turbulence was minimal beyond 10 cm from the walls. The installation of a safety net at the entrance to the test section increased the turbulence level by a factor of at least 30, to 1.2 % longitudinally and 1.0 % transversely.

Actual and 'optimum' flight speeds: field data reassessed

Previously published field observations of the air speeds of 36 species of birds, all observed by the same method (ornithodolite), were compared with estimates of the corresponding minimum power speeds, calculated with a default body drag coefficient of 0.1. This value, which was derived from recent wind tunnel studies, represents a downward revision from default values previously used and leads, in turn, to an upward revision of estimated minimum power speeds. The mean observed air speeds are now distributed around the minimum power speed, rather than in between the speeds for minimum power and maximum range, as they were before. Although the field data do not represent migration, examination of the marginal effects of small changes of speed, on power and lift:drag ratio, indicates that flying at the maximum range speed on migration may not represent an 'optimal' or even a practical strategy and that cruising speeds may be limited by the muscle power available or by aerobic capacity. Caution in constructing 'optimisation' theories is indicated.

Stress and strain in the flight muscles as constraints on the evolution of flying animals.

The minimum mechanical power needed for an animal to maintain level flight can be estimated, as can the wingbeat frequency, from measurements of the animal's mass, wing span and wing area, and of the strength of gravity and the air density. Dividing the power by the frequency gives the work done per cycle, and dividing this by the muscle mass gives the specific work, meaning the work done in each contraction by unit mass of muscle. This in turn is the product of the average stress during shortening and the strain, divided by the muscle density. The minimum specific work for level flight is strongly size dependent. To account for even minimum performance in the largest species known to be capable of prolonged, aerobic flight (whooper swan), the specific work of the myofibrils needs to be 57 J kg-1, which could be achieved, for example, by a stress of 240 kN m-2 combined with a strain of 0.25. The upper limits of stress and strain for sustained exercise are not known, but are not likely to be much higher than these figures. Much larger birds, such as the Miocene fossil Argentavis, would require improbably high values of stress and strain for level flight, unless the air density were much higher in Miocene times than at present, and/or the strength of gravity were much less. Birds of small and medium size have more than the minimum amount of muscle required for level flight. This opens a wide range of possibilities for different species to be specialised for different types of activity. The potential diversity for evolution in large species is less than for medium-sized or small ones, and dwindles to zero above a body mass of about 14 kg. There is also a strong positive trend in the aerobic scope, from about 3 in small, long-winged passerines such as swallows, to 47 in the whooper swan.

Wingbeat frequency of birds in steady cruising flight: new data and improved predictions

Wingbeat frequencies of 15 species of birds, observed in the field in level, cruising flight were compared with predicted frequencies, calculated according to the formula derived from an earlier sample of 32 species. All of the data were collected by the author, using the same methods throughout. The new observations were predicted well for species with low wingbeat frequencies, but were underestimated at the higher frequencies. The following revised proportionality gave the best fit of the wingbeat frequency (f) to the combined data set of 47 species: where m is the body mass, g is the acceleration due to gravity, b is the wingspan, S is the wing area, I is the wing moment of inertia, and is the air density. As measurements of I were not available for most species, its exponent was combined with those of m and b, by assuming that Imb2. The following equation was fitted to the data on this basis: These formulae are dimensionally correct, according to the rules derived in the earlier paper, and the equation is also numerically correct as it stands, without requiring a multiplication factor. For allometric comparisons between geometrically similar species, where body mass and wing measurements vary together (including wing moment of inertia), the expected relationship is f m-1/6. If the mass alone varies, owing to feeding or consumption of fuel, while the wing measurements and other variables remain unchanged, the expected relationship is f m1/2. These relationships apply to any dimensionally correct formula and are not affected by adjusting the coefficients within the dimensional constraints.

Wingbeat frequency and the body drag anomaly: wind-tunnel observations on a thrush nightingale (Luscinia luscinia) and a teal (Anas crecca)

A teal (Anas crecca) and a thrush nightingale (Luscinia luscinia) were trained to fly in the Lund wind tunnel for periods of up to 3 and 16 h respectively. Both birds flew in steady flapping flight, with such regularity that their wingbeat frequencies could be determined by viewing them through a shutter stroboscope. When flying at a constant air speed, the teal's wingbeat frequency varied with the 0.364 power of the body mass and the thrush nightingale's varied with the 0.430 power. Both exponents differed from zero, but neither differed from the predicted value (0.5) at the 1 % level of significance. The teal continued to flap steadily as the tunnel tilt angle was varied from -1 ° (climb) to +6 ° (descent), while the wingbeat frequency declined progressively by about 11 %. In both birds, the plot of wingbeat frequency against air speed in level flight was U-shaped, with small but statistically significant curvature. We identified the minima of these curves with the minimum power speed (Vmp) and found that the values predicted for Vmp, using previously published default values for the required variables, were only about two-thirds of the observed minimum-frequency speeds. The discrepancy could be resolved if the body drag coefficients (CDb) of both birds were near 0.08, rather than near 0.40 as previously assumed. The previously published high values for body drag coefficients were derived from wind-tunnel measurements on frozen bird bodies, from which the wings had been removed, and had long been regarded as anomalous, as values below 0.01 are given in the engineering literature for streamlined bodies. We suggest that birds of any size that have well-streamlined bodies can achieve minimum body drag coefficients of around 0.05 if the feet can be fully retracted under the flank feathers. In such birds, field observations of flight speeds may need to be reinterpreted in the light of higher estimates of Vmp. Estimates of the effective lift:drag ratio and range can also be revised upwards. Birds that have large feet or trailing legs may have higher body drag coefficients. The original estimates of around CDb=0.4 could be correct for species, such as pelicans and large herons, that also have prominent heads. We see no evidence for any progressive reduction of body drag coefficient in the Reynolds number range covered by our experiments, that is 21 600­215 000 on the basis of body cross-sectional diameter.

Units of measurement for fractal extent, applied to the coastal distribution of bald eagle nests in the Aleutian Islands, Alaska.

Attempts to measure the nesting density or territory size of bald eagles led to a fundamental difficulty, inherent in all such measurements on organisms which are distributed along an irregular boundary, such as a coastline. The "length" of such a boundary is not a meaningful measure, and neither can a meaningful "area" be associated with each nest. Mandelbrot's (1983) fractal geometry applies to the problem, but has not previously supplied practical units of measurement for fractals such as coastlines or rugged surfaces. A practical method is given for measuring the "extent" of such fractals, introducing a unit of variable dimension, the "metron", which includes the existing SI units of length, area and volume as special cases. A linear measure, the "spacing" allows densities on fractals of different dimensions to be compared directly. The method is applied to the distribution of bald eagle nests along the coastlines of two islands in the Aleutians, and an extension of the method to handle distributions on mountainsides and island surfaces is indicated.

Elastic energy storage in primary feather shafts.

It is proposed that the kinetic energy of a pigeon's wing, in hovering or slow forward flight, is transferred to the air at the end of the downstroke by a mechanism involving temporary storage of additional energy in bent primary feather shafts. Estimates of the amounts of energy which can be stored and recovered in this way are compared with the requirements of the theory. The hypothesis is not rejected, as far as present evidence goes. If is is correct, high-velocity pulses of calculable magnitude should be detectable in the wake.