O P T I M U M BIRD F L O C K SIZE IN F O R M A T I O N F L I G H T By M R I G E S H K S H A T R I Y ' 4 B.Sc, The University of British Columbia A THESIS S U B M I T T E D IN PAE.TIAL F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R OF ^ C | E N C E in T H E F A C U L T Y OF G R A D U A T E S T U I ^ g p (Department of Zoology) We accept this t!hesi$.: as. .cottfoj^ng to the required standard T H E U N I V E R S I T Y O f B K I T i p . C t l l p B J f © Mrigesh|.Is]iatriya^l9p In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Zoology The University of British Columbia Vancouver, Canada Date July, 11 1990 DE-6 (2/88) Abstract A theoretical model of flock size in migrating birds is developed. Although previous models of formation flight in birds show improved flight performance, they do not explain flock size variation across bird species or at different times of the year for a given bird species. This model captures some of the diversity in flock size observed in nature by incorporating energetic costs of flight and energy income from foraging. It turns out that within a myriad of possible flock sizes there - is one that is optimal for maximizing energetic efficiency (net energetic gain/energy expenditure) for a given maximum range speed, which minimizes flying cost per unit distance flown, and under certain migration conditions (i.e. flight distance and total time to complete the journey). Net energetic gain from foraging equals the rate of prey encountered times the time spent foraging. Energy expenditure from flying is determined from formation flight theory for a fixed wing aircraft. The benefit of formation flight, as derived from an approximation technique, is represented in close-form. This expression is a function of flock size and wing-tip spacing (WTS) and simplifies flight cost calculations. Under certain WTS, a good approximation to the induced drag for a member of a flock of size n is 1/nth of the induced drag of a single bird. In addition, optimum flight speed of a flock is (1/n) 1/ 4 of the optimum flight speed of a single individual. The approach taken here allows the prediction of flock size in migrating birds. Model results are discussed in relation to observation of flock size under various migration conditions. If migration is constrained by hours of daylight, seasional variation in flock size is expected if the start time of the north and southward migration are asymmetrical with respect to the summer solstice (June 21). Under certain conditions, such as long non-stop migration, solo flight is an optimum migratory strategy. ii Table Of Contents Abstract ii List of Figures v List of Tables . . ix List of Symbols x Acknowledgements xii Introduction 1 j Aerodynamics of Bird Flight 4 Solo Flight 4 Quasi-steady Assumption 4 Analysis of Steady Horizontal Flight 5 Formation Flight 7 Flow Field in the Neighborhood of an Airfoil 7 Upwash Velocity Calculations 18 Induced Drag Reduction 18 Approximation Formula 27 Model 45 Energy Expenditure . '. 45 Energy Gain 46 Ratio of Energy Gain to Energy Expenditure 47 Results 49 Conditions for Solo Flight . 49 iii Optimum Flock Size and Wing-tip Spacing 52 Optimum vs Maximum Flock Size 52 Effect of Umr on Optimum Flock Size 55 Seasonal Variation and Optimum Flock Size 58 Discussion 69 Aerodynamic Model 69 Optimum Flock Size Model . 73 Summary 80 References 81 Appendices . . 88 A. Limiting Case as n —• oo 88 B. Approximation Technique .89 C. Optimum Flock Size vs Wing-tip Spacing 91 D. Upper Bound for Maximum Flock Size 93 iv List of Figures Figure 1. Evolution of tip vortices behind a wing of finite span (based on Tokaty, 1971). See text for detail . 9 Figure 2. A unique circulatory flow (v — Y/2irr) superimposed (added) to the free stream flow (U) mimics the flow around a lifting wing 12 Figure 3. Circulation (r) superimposed on forward velocity (U) and downwash (wa) to produce lift (L) and induced drag (DT) respectively (Houghton & Brock, 1960) 15 Figure 4. Wing with rolled-up vortex sheet (left) and its equivalent horseshoe vortex: bound vortex (|||) and tip vortex (=) 17 Figure 5. Vertical flow in the vicinity of a lifting wing, moving in the positive x direction. The wing tips are at points x = 0 and y = ±1 20 Figure 6. Distribution of n = 5 birds and their horseshoe vortex on the x-y plane. Starting from the left, each bird is numbered from 1 to n. The distance, as projected on the y-axis, between bird number i and j is Syij = \i — j\(2b + s), where b is the semi-wing span and s is the distance between adjacent birds perpendicular to the flight path. Each bird is of span 26, represented by a bound vortex of length 2a, where a = 7r/46 26 Figure 7. Ratio of induced drag in formation to solo flight, as a function of flock size, n, and various wing-tip spacing, s, as a percent of the semi-wing span (solid). Curves are shown for s/b = +100, +50, +10, 0, -10, -15, and -20%. For the special case when the wing-tip spacing is such that the tip vortices of the two adjacent birds overlap (s = 2(a — b)) the ratio is given by the dash line. . . . . . . . 29 Figure 8. Percent contribution to the total induced drag reduction of the left most bird (Bird 1) by the neighboring birds numbered 2 through 15 (bottom x-axis), flying in abreast formation with a wing-tip spacing equal to zero (s = 0). The top v axis is the distance of each bird as measured from Bird 1. Each bird is assumed to be of unit semi-span (6 = 1). 32 Figure 9. Pictorial representation of the approximation technique for n = 6 birds, solid plus one diagonal filled wing. (A & C) The left and right most birds in the vee formation ([///|) are assumed to be flanked on the opposite leg by an infinite number of birds (1 | . . . ). (B) Each bird inside a flock, for example 1/ / /|, is assumed to be flanked by an infinite number of birds on both legs of the vee (••••)• • •' • 35 Figure 10. Effect of flock size and spacing ratio, s/b, on realtive error as result of the approximation technique. Curves are shown for s/b = -20, -15, -10, 0, +10, +50, and +100% . 38 Figure 11. Induced drag in formation flight, -Dj(n), relative to that in solo flight, Di, for n = 2,3,9,25 and infinite number of birds 40 Figure 12. Air flow pattern, as represented by a horseshoe vortex, of two wings each of semi-span 6, located at wing-tip spacing s = 2(a—b). Due to the positioning, tip vortex of the lead bird cancels the tip vortex of the second bird (A). The resulting flow, as shown by the horseshoe vortex in (B), is equivalent to a flow created by a single wing of span 2b 43 Figure 13. Condition for solo flight depends on the value of d, Umr and T. For a given Umr, a solid line divides the values of d and T pairs that predict solo flight (below the line) from d and T pairs that predict optimum flock size greater than one (above the line). To illustrate, say Umr = 1/2, (T, d) pairs that fall in the grey region correspond to solo flight while pairs that fall in the white region correspond to formation flight 51 Figure 14. Optimum flock size is plotted against T/ta. Curves correspond to wing-tip spacing, expressed by a non-dimensional ratio, s/b (+100, +50, +20, 0, vi -10, -20%). Smallest flock size occurs at wing-tip spacing s = 2(a — b) (dash line), flock configuration for maximum induced drag reduction 54 Figure 15. Normalized relative to a solo flyer, equation 59 is plotted against different flock sizes, n. The benefit of flying in a flock of size n = n* is identical to that of flying solo: R(l) = R(nm). In this example, a member of a flock of size n = nopt is 60% more efficient than a solo flyer 57 Figure 16. Effect of T, d and Umr on optimum flock size. Each panel, from left to right, is for a fixed T, 6, 12, and 18 hrs respectively, and is a plot of flock size vs distance, d (km) for Umr = 18, 36 and 54 km/hr ( 1 - 3 respectively). Solid horizontal line corresponds to nopt = 1, solo flight 62 Figure 17. Contour of day length (hrs) with time of year and with latitude (solid lines). The dashed zigzag lines represent movement, from 10°N to 50°N, of two hypothetical migratory bird species. By flying equidistant number of days before and after the summer solstice (top axis), species A (short dash) will experience the same hour of day length on both the north and southward migration; consequently, no seasonal variation in flock size is to be expected. H the two phases occur in an asymmetrical pattern, as for species B (long dash), the average day length during the northward migration will be larger than the day length during the southward migration; as a result, seasonal variation in flock size is to be expected. The degree of variation in flock size is plotted in figure 18 65 Figure 18. For migration from 10°N to 50° N, contours of the ratio R, are plotted against tf, and tr. The ratio Ra for bird species A and B is determined by the value of (tb,tr) pair which is (30,30) and (30,120) days respectively (see Figure 17). Ratio Rs < 1 corresponds to flock sizes being smaller during the southward migration compared to that during the northward migration. A ratio of one implies that flock sizes during the north and south migration are identical 68 Figure 19. Percent reduction in optimum flight speed relative to a single bird ([Umr — Umr(n)\IUmr) at different flock size, n, and spacing index, s/b (s is WTS, vii b is semi-wing span). The solid line corresponds to a WTS when trailing vorticies of adjacent bird cancel each other out 72 Figure 20. Predicted optimum flock size during the spring and autumn migration for varying distances between stop-over sites (top axis) or equivalentley the nubmer of stop-overs required to complete the 855 km journey (bottom axis). See text for detail 78 viii L i s t of T a b l e s Table 1. Relation of maximum range speed to body mass, after Greenwalt (1962, 1975) and Rayner (1979). Optimum flock size is shown divided by [T/d]4. (All quantities in MKS units.) 60 ix List of Symbols s wing tip spacing b semi wingspan A aspect ratio = (2b)2 /S parasite (body) drag coefficient Cr)(Para) profile (body) drag coefficient S area of the wings Ap frontally projected area of the body TO mass g gravitational acceleration (9.8m/s2) p air density U forward velocity Umr speed for maximum range Dp parasite (body) + profile (wing) drag Di induced drag L lift force kp a constant associated with parasite and profile drag ki a constant associated with induced drag n number of birds in a flock D total drag force acting on a bird Di(n) flock average induced drag x Umr(n) maximum range speed for flock of size n Efiy energetic cost of flying Eout energetic cost of flying in formation flight Ein net energy gain from foraging R ratio of energy gain to energy expenditure A number of prey (energy units) per unit time r circulation strength w vertical velocity component d flight distance ta airborn time tp time spent in a patch T total time available = tp + ta fiopt optimum flock size number ro* maximum flock size number tj Julian date (0 — Jan. 1, 1 = Jan. 2, ...) S(tj) solar declination as a function of Julian date H day length, hours between sunrise and sunset t latitude l\, £2 beginning and ending lattitudes of the migration tf>, tr beginning and returning time of the migration R8 ratio of expected flock size variation with season xi Acknowledgements I would like thank Dr. R. W. Blake for his support. Special thanks to Rebbeca Kolotylo and horacio de la cueva for allowing access to their large collection of scientific papers. I would also like to thank the BDC staff for computing assistance. xii Optimum Bird Flock Size in Formation Flight Introduction A great many birds migrate in flocks. The most characteristic migratory formation is a reversed \ / , apex in the direction of flight. "The birds are usually ordered in swept lines and they keep so small a span wise distance that the wing-tips of two adjoining birds lie about one behind the other" (Hummel, 1983). This mode of formation flight is typical of geese, ducks, pelicans, cranes, and shore-birds (Dorst, 1962). Formation flight, however, is not common to all migratory species. Some birds, such as the cuckoos, nitejars, orioles, numerous birds of prey, journey alone (Dorst, 1962). The reason why some species of birds adopt formation flight and others do not is not clear. Furthermore, the advantage of flying in this fashion is also unclear. Numerous explanations have been offered, however. Fransiket (1951) purposed that formation flight allows for good optical contact and thus decreased risk of collision. It may, on the other hand, function to decrease risk of predation, a function similar to that suggested for schooling in fish (Vine, 1971; Weihs, 1973). Equally, formation flight may allow older birds with previous migratory experience to act as navigators for the inexperienced birds of the flock (Dorst, 1962). It has been suggested that formation flight may assist navigation by averaging the direction preference of individual birds (Heppner, 1974) although contradictory information was reported by Keeton (1970). Another possibility is that formation flight, as with airplanes, may provide aerodynamic benefit over solo flight. Early aviators noted that patrol leaders burned significantly more fuel than the other planes in the formation (Dorst, 1962; Houghton & Brock, 1960). While many explanations have been put forward for formation flight in birds, the aerodynamic hypothesis allows for quantifiable energetic savings. Weiselsberger (1914) was the first to give the correct description of the power reduction for formation flight. Flight tests with formations of two airplanes show that the power reduction was in close agreement with that predicted by aerodynamic theory (Hummel & Beukenberg, 1989). The aerodynamic rationale for formation flight 1 Optimum Bird Flock Size in Formation Flight is that while each wing creates lift by generating downward momentum within its span, an upwash is created beyond the wing. In a vee formation, each bird flys in an upwash field generated by all other wings in the formation. This is equivalent to flying in an upcurrent and consequently each member of the flock experiences a reduction in flight power. The magnitude of the power reduction depends on wing tip spacing, and on the number of birds in the flock. In a quantitative analysis, Lissaman and Schollenberger (1970) plotted the re-duction in power of formation flight of 3, 9, 25 and an infinite number of birds showing that as the flock size increases flight power requirement decreases. Conse-quently, their results imply that in order to minimize flying cost birds should fly in a formation of infinite size! Although this is a meaningless statement, the corollary which follows is that birds should fly in the largest flock possible. In particular, two or more flocks should not fly beside each other but rather join to form a single large flock. However, numerous flocks of Snow Geese (Anser caerulescens) were observed flying beside each other at the Riefel Island Bird Sanctuary. While individuals may sometimes change flock arrangment in mid air, flocks do not merge to form a single large flock. More importantly, Lissaman and Schollenberger (1970) results do not explain avain flock size varation observed across species. For example, some species of birds such as the common heron that nest in large colonies have the opportunity to form large flocks often migrate in groups of only a few birds (Dorst, 1962). On the other hand, some species of birds migrate in large flocks, even if they are nor-mally solitary (McFarland, 1987). In addition, some migrants, such as wild ducks and geese travel in small numbers. Other species such as Short-eared owl (Asio flammeus) and Scops Owls (Ofus scops) often fly in large numbers. Black Storks 1 migrate in flocks of sixty to seventy, whereas there are often fifty to 100 Brant geese (Branta bernicla) and 500 to 1,000 common scoters (Oidemia nigra) in a flock (Dorst, 1962). The aim of this study is to show that there is an optimal flock size for birds during migration. This is done by quantifying the flight energetics of formation 2 Optimum Bird Flock Size in Formation Flight flight and foraging energetics. Taking the energy expenditure of formation flight and energy gain of foraging together, the model shows that optimum flock number is a function of morphological parameters of the bird, flight distance, and total time available for migration. Since optimum flock size is a function of the three variables mentioned above, it is also possible to predict, among other things, under what conditions solo flight is more advantageous than formation flight. 3 Optimum Bird Flock Size in Formation Flight Aerodynamics of B i rd Flight Solo Flight Quasi-steady Assumptions The aerodynamics of napping bird flight can be based on the quasi-steady approach: that the instantaneous aerodynamic forces on a flapping wing are those that a wing would experience in steady motion at the same instantaneous speed and angel of attack (Ellington, 1984). In other words, it is assumed that averaged over the wing-beat cycle, the aerodynamic forces are of the same magnitude as that of a fixed wing. The aerodynamic forces may be subdivided into four components, profile, parasite and induced drag, plus the drag created by the unsteady motion of the wing. The profile and parasite drag is determined by the wetted surface area of the wing and by the degree of streamlining of the body, respectively. Collectively called the profile drag, it is proportional to the velocity square. Induced drag is associated with generation of lift forces and is inversely proportional to the square of the velocity. During optimal cruising flight speed, profile and induced drag are of equal magnitude (Pennycuick, 1972). However, the fourth term, related to flapping motion, is of low magnitude if the reduced frequency, ratio of tip flapping speed to the flight speed, is low (Lissaman & Shollenberger, 1970). Spedding (1987) studied the structure of a wake behind a kestrel (Falco tinnun-culas) in flapping flight using stereophotogrammetery of multiple flash photographs of the motion of small soap-covered helium bubbles. His findings showed that at moderate flight speeds the rate of momentum generated in the wake approximated that of the quasi- steady state model. Spedding, concludes that "on occasion where an estimate of the induced power requirement at moderate flight speeds is required as a component in the calculation of the energy budget of a flying bird, the added complexity of a more rigorous and complete aerodynamic model is not justified by the small improvement in accuracy in estimating one component of the total aerody-namic power requirement." In the case of a kestrel of mass m = 0.12 kg the reduced 4 Optimum Bird Flock Size in Formation Flight frequency was calculated to be « 1. From allometric equations given by Pennycuick (1972) and Rayner (1979a) it can be shown that reduced fequency oc l / m 1 / 6 . In other words reduced frequecny decreases with mass of the bird. If the wing beat kinematics of the kestrel are taken to be representive of birds of similar or lower reduced frquency (i.e. greater mass) then the quasi-steady model can be applied to estimate bird flight energetics. However, in the flight of small passerines where reduced frequency is relatively high, unsteady aerodynamic effects due to flapping become important (Kokshaysky, 1979; Ellington, 1984). As a first approximation to the energy budget of a flying bird the cyclic variation in flapping flight is assumed to be of second order and therefore is ignored. In short, the average aerodynamic drag experienced by a bird during flapping flight is assumed to be equivalent to drag on a bird with fixed wings in the horizontal position. The following analysis applies to fixed wing flight on the basis of classical aerodynamic theory (Pennycuick, 1969,1972,1978; Rayner 1979a). Analysis of Steady Horizontal Flight Aerodynamic drag in steady horizontal flight may be expressed in the form D = Dp + Di (1) where Dp is the sum of the parasite and profile drag and DT is the induced drag. The parasite and profile drag are given by Dp = ^pSCD(pro)U2 + ^pApCD(para)U2 (2) where p is the air density, U is the flight speed, S is the wing area and Ap is the frontal projected area of the body; Cr)(pr0) is the drag coefficient of the body and Cr)(pa.ra) x s the drag coefficient of the wings. The induced drag may be written as Optimum Bird Flock Size in Formation Flight where L is the lift force generated by the wings; b is the semi-wing span. The total * drag on a bird flying at a constant speed U is D = \PSCD{pro)U2 + \ P A p C D { p a r a ) U 2 + 2 ^ 7 2 " ( 4 ) For the case of horizontal flight the lift force balances the weight of the bird, in other words L = mg (5) i where m is the mass of the animal and g is the gravitational acceleration (9.8 m/s2). Grouping parameters, equation (4) can be written as D = kpU2 + k{U~2 (6) where H = ^pSCD(Pro) + ^pApCoipara) and k = 2^52 ( 7 ) The total aerodynamic force acting on a bird flying at constant speed U is given by equation (6). The energy required to fly a distance d is Efiy =D-d. (8) Flight energy in relation to forward speed follows a U-shaped curve. At higher velocities the parasite and profile drag become the dominant terms; while at lower velocities the induced drag dominantes. As a result there is a speed at which energy expenditure for flight is at a minimum. Specifically, the cruising speed, which minimizes flying cost per unit distance travelled, is calculated by differentiating (Efiy/d), and setting the first derivative equal to zero: ± EQL = 0 = 2kpU - 2kiU-3. (9) du d Solving the above equation for U gives r f c i 1 / 4 Umr = T~ (10) . P . Optimum Bird Flock Size in Formation Flight In migratory flight, cruising speed, which minimizes energy consumption per unit distance flown, allows migrants to cover the maximum range on a given energy reserve. Speeds from radar studies, collected by Alerstam (1976), show that mean air speed (flight speed corrected for influence of wind) of migrants is generally close to the predicted cruising speed, Umr- The maximum range speed should be used whenever the longest distance should be covered on a given amount of energy. Formation Flight Flow Field in the Neighborhood of an Airfoil In order to quantify the savings obtained in formation flight, it is necessary to model the flow close to a airfoil. The most common approach used is the horseshoe vortex model. A vortex is a "core" of rotating fluid, around which air flows in concentric circles (Houghton &; Brock, 1960). In the model, the airfoil is replaced by a bound vortex, and the trailing wake is replaced by two tip vortices. Under the right conditions, tip vortices can be seen behind an airfoil. The formation of these trailing vortices can be explained as follows. The pressure on the upper surface of a wing is smaller than that on the lower surface of a lifting wing. A pressure gradient exists between the upper and lower surfaces which equalizes beyond the wing tips. The tendency of the air is to equalize any pressure difference; as a result, air "leaks" to the top from the bottom of the wing, from the region of high pressure to the region of low pressure. This pressure equalization at the wing tips causes an inward deflection of the streamlines above the wing and an outward deflection below the wing (Figure 1). The difference in spanwise velocity will cause air to roll up in into a number of small vortices (a vortex sheet), distributed along the wing span. A short distance downstream the vortices roll up and combine into two distinct cylindrical vortices. These two vortices, referred to as tip vortices, rotate in opposite directions and form 7 Optimum Bird Flock Size in Formation Flight Figure 1. Evolution of tip vortices behind a wing of finite span (based on Tokaty, 1971). See text for detail. 8 <0 Optimum Bird Flock Size in Formation Flight just inboard of the wing-tips. The distance between the two tip vortices is 2a where a « jb and b is the semi-wing span (Higdon & Corrsin, 1978). Whereas the tip vortices can be easily verified visually, the bound vortex is a useful theoretical abstraction to simulate accurately all the properties of a real airfoil. In general, lift produced by a wing is explained from the fact that airflow is accelerated over the top surface of a wing while that flowing below is decelerated relative to the free stream. The difference in speed, by Bernoulli's theorem, results in reduced pressure above the wing and increased pressure below it, and this pressure difference is what supports the weight of the wing. The difference in speed above and below the airfoil can be resolved into a circulatory flow, centered at the wing, superimposed on the free-stream. Circulatory flow is created by replacing the airfoil by a bound vortex, which causes air to flow in concentric circles outside its core. The pictorial equation of figure 2, shows that a free stream flow plus the flow due to the bound vortex creates a flow identical to one at an airfoil. The bound vortex can be considered as a rigid rotating body (rotating cylinder) (Prandtl & Tietjens, 1957). The strength of the vortex, T (m2/s), is defined as r = 27rftr2, (11) where £1 is the angular velocity and r is the radius of the core (Prantdl Sc Tietjens, 1957). Replacing the airfoil by a vortex, which acts like a rotating cylindrical body, not only describes the flow at an airfoil but also allows the calculation of the lift force and the induced drag created by the airfoil. A "sideways" force associated with a spinning object moving through the air has been recognized since ancient times (Houghton& Brock, 1960). It is usually referred to as the Magnus effect. In the case of the horseshoe vortex model, the spinning cylinder, or the bound vortex, will create a lift force perpendicular to the free-stream velocity. The lift force created per unit span, Ii, is Ii = pTU. (12) 10 Optimum Bird Flock Size in Formation Flight Figure 2. A unique circulatory flow (v = r/27rr) superimposed (added) to the free stream flow (U) mimics the flow around a lifting wing. 11 Optimum Bird Flock Size in Formation Flight This expression is the algebraic form of the Kutta-Zhukovsky Theorem, and is valid for any system where circulation is superimposed on a uniform velocity (Houghton & Brock, 1960). In order to create the same lift as an airfoil of span 26, the bound vortex has to be a length 2a, corresponding to the distance between tip vortices. Thus the total lift of an airfoil of span 26 is L = 2apYU. (1.3) It can be shown that the downward velocity (downwash) created by an airfoil iss a constant along the span and is given by wd = L (14) (Houghton & Brock, 1960). The result of applying the Kutta-Zhukovsky theorem to the downwash velocity, wd, is a force in the opposite direction of the motion iof the airfoil and is referred to as induced drag, DT, (Figure 3) Dj = 2apTwd. (15) The induce drag can also be written by substituting wd from equation (14) to give 1 Dr = 1PT*. (16) A more common form of the induced drag equation (Equation 3) can be derived by substituting in the above equation for T from equation (13) to give „ 7T pirUb/2\ 2pitbW 1 2 (17) In summary, then, it is possible to represent a wing by a vortex system forming three sides of a rectangle (Figure 4): along the y axis a vortex filament of strength T of length 2a represent the wing, and along the x axis two trailing vortices of the same strength as the bound vortex represent the trailing wake. The system of vortices can also be looked upon as a single long vortex bend in the shape of horseshoe. 13 Optimum Bird Flock Size in Formation Flight Figure 3. Circulation (r) superimposed on forward velocity (17) and downwash (wd) to produce lift (L) and induced drag (Di) re-spectively (Houghton & Brock, 1960). 14 15 Optimum Bird Flock Size in Formation Flight Figure 4. Wing with rolled-up vortex sheet (left) and its equiva-lent horseshoe vortex: bound vortex (|||) and tip vortex (=). 16 17 Optimum Bird Flock Size in Formation Flight Upwash Velocity Calculations The vector velocity field induced by a line vortex in a shape of a horseshoe and strength Y is calculatd by applying the Biot-Savart law (McCormick,1979). The i. vertical velocity component at a distance Sx, Sy away from center of the wing is /x x \ r 1 / Sy + a Sy-a wiox.oy) = —— — < —. . v »' A<n8x\ y/Sx2 + (Sy + a)2 y/Sx2 + (Sy - a)2 r i f ! - * \ (is) 4TT (Sy - a) [ y/8x2 + (Sy - a)2 r I J\ Sx 4TT (Sy + a) [ /^cSx2 + (5y + a)2 J (Milne-Thomoson, 1958; Lugt, 1983). Figure 5 shows the magnitude of the vertical velocity near a moving airfoil, with its center at the origin, and its tip at y = ± 1 . Updraft occurs in front and also in the outboard region of the tip vorticies, located on the line y = ± 7 r / 4 . The downwash occurs behind the airfoil and between the two tip vortices. While the downwash is constant behind the airfoil, the upwash takes its maximum value at the tip vortex and then rapidly diminishes with increasing distance from the wing. , Induced Drag Reduction ~-\ Flying performance of a bird is influenced by the upwash created by other birds in the flock. The upwash brings about a decrease in the total induced drag of the flock. The extent of the reduction in induced drag depends on the wing-tip spacing and the number of birds in the flock. A general account of the influence of two birds flying in formation is given and then expanded to n birds. An isolated wing produces a vertical velocity component, Wd, that is due to the wing's own vortex system. The upwash, w, is the vertical velocity induced by the vortex system of the second wing. Flying in the influence of each other results in a smaller downwash, w<* + w (the sign of the downwash is always positive and 18 Optimum Bird Flock Size in Formation Flight Figure 5. Vertical flow in the vicinity of a lifting wing, moving in the positive x direction. The wing tips are at points x = 0 and y = ±1 . 19 20 Optimum Bird Flock Size in Formation Flight the sign of the up wash is negative). The induced drag of the first wing, by analogy with equation (15), is D / i = 2apT(wd + w) (19) which can be written as D n = 2apTwd + 2apTw (20) D i 1 = D I 1 1 + D I 1 2 . (21) The component D m is called the "self-induced drag" and D m is known as the "mutually induced drag"(Reid, 1932). In other words, D m denotes the self-induced drag of wing 1, while D m is the mutually induced drag experienced by wing 1 as a result of the vertical velocities induced by wing 2. The same argument can be made with the influence of the second wing on the first. The induced drag on the second wing is D n = D m + D I 2 1 . (22) The total induced drag of a flock containing 2 birds is 2?/(2) = D m + D m + D m + D I 2 1 . (23) In general, the total induced drag of a flock with n birds (wings) is equal to the sum of the self-induced drags of its component wings plus as many mutually drags as there are permutations of its wings in pairs (Reid, 1932). Mathematically, the total induced drag of a flock of n birds is n n i = i j=i The mutually induced drag for a pair of wings depends on the relative position of each wing. The greater the distance between the wings the smaller is the mutual induced drag. More accurately, the second term of equation (20) should be written as Dm = 2apVw(Sx,Sy), (25) 21 Optimum Bird Flock Size in Formation Flight where wing 2, whose vortex influences the flow at wing l,is located at the point (Sx,dy). Produced by wing 2, the upwash, w(8x,dy), varies along the span of wing 1. Taking the integral mean value of the upwash over the span a of wing 1 gives -D/12 = 2apTw (26) (Hummel, 1983). I r6y+a w(8x,8y) — — / w(8x,T])dr]. 2 a JSy-a (27) The average upwash velocity induced by a wing at a point (8x,dy) is 'Sy+a 'After substituting the result of the above integration into equation (26), the induced drag on a wing centered at (8x,8y) (wing 1) due to the presence of a vortex system at the origin (wing 2) is given by PT2 DI12 = 4.TT ^{ yj8x2 +(ri- a)2 - ^/8x2 + („ + a)2 -{ l°g(>7 - a) +log + < \og(rj + a) +log 8x + y/8x2 + (T / - a)2 (r}-a) 8x + y/8x2 +(n + a)2 (v + o) (28) rr=6y+a rf=6y — a (Moran, 1984). In the case of calculating the total induced drag of a flock, as opposed to the induced drag of each bird with respect to position in a flock, Munks Stagger theorem is applied to simplify the calculations. The theorem states that "a collection of lifting surfaces (birds) may be translated in the streamwise direction (x-axis) without affecting the total induced drag of the system (flock) as long as the circulation of every wing (or lift) is unchanged" (Higdon and Corrsin, 1978). For example, a flock in line abreast formation will have the same total induced drag as one in V formation. Although, staggering in the x direction causes a redistribution of the induced drag for each bird, the total of the induced drag of the flock does not change. The consequence of this theorem is that the total induced drag of a flock depends on the scatter along the y-axis and not on the depth distribution 22 Optimum Bird Flock Size in Formation Flight along the x-axis. Since the calculation of the total induced drag is independent of the x-axis, 8x in equation (28) is set to zero, which is equivalent to saying that the flying formation is unstaggered. The mutually induced drag can now be simplified to £>/12 = log | rj + a 77 — a r)=Sy+a r)=6y — a (29) which further reduces to Dm = ^—log 47T i - i £ by (30) In the case of unstaggered flight formation, it can be shown that the mutually induced drag terms between a pair of birds is equal. In other words, Dn2 = DT 21 (31) (Reid, 1932; Prandtl & Tietjens, 1957). Substituting Dm = fpT2 (equation (16)), into equation (30) gives Dm = 2Dn 7T" -log dy (32) the mutually induced drag of a bird that is Sy apart from the second bird. In general, the distribution of n birds along the y-axis is a function of the wing-tip spacing s and the semi-wing span b (Figure 6). If the left most bird in a flock is numbered 1, and the next bird numbered 2 and so on to n, the distance between bird i and j is given by Sya = \i - j\(2b + 3). (33) Mutually induced drag of bird i due to presence of bird j can be written as 2n Dnj = —T— log T*1 1- 2a | » - j | (26 + *) (34) 23 Optimum Bird Flock Size in Formation Flight By assuming that all birds are identical, in that they have the same wing span, aspect ratio, and weight, the problem of calculating the total induced drag (Equa-tion (24)) reduces considerably. For instance, all terms such as Dm, D /22 • • • Dinn are equal and can be replaced by n-Dm. Furthermore, the mutually induced drag terms in general can be written as DHJ = Diji (35) as a result of equation (31). Equivalently, equation (24) can now be written as n—1 n 1 £>/(n) = nDni + 2 ^ X Dlii' i=i j=i+i Substituting equation (34) for DHJ, equation (36) can be written as (36) Dl(n) = nDm + ^ l o g »=1 j=t-|-l 1-2a \i-j\(2b + s) 2n (37) for n birds in a flock. In a flock of size n, the induced drag on a single bird, Di(n), is equal to the total induced drag of the flock divided by its size. That is Djin) Dj(n) = n (38) Substituting equation (37) for Dj(ra) in equation (38), and noting that Dm and Di are notationaly equivalent, the above equation can be written as 2 i 7r z n —' —' 1- 2a i -j\(2b + s) (39) Solo term Flock—associated terms Essentially, the above equation has the following meaning. The first term, Di, represents the induced drag experienced if a bird was to fly solo. The second term is the flock associated reduction in induced drag as a result of flying in the upwash 24 Optimum Bird Flock Size in Formation Flight Figure 6. Distribution of n = 5 birds and their horseshoe vortex on the x-y plane. Starting from the left, each bird is numbered from 1 to n. The distance, as projected on the i/-axis, between bird number i and j is Sytj = \i — j\(2b + s), where b is the semi-wing span and s is the distance between adjacent birds perpendicular to the flight path. Each bird is of span 2b, represented by a bound vortex of length 2a, where a = 7r/46. 25 t I I I M t l H I I I I M I I I I H I I l t l l l M l l l l l l l t l l l l l l l l l l l l l t l l l l l l l l l l l l l l l l l l l l l t t l l l l l l l l l l l l l U l l l l f l l l l H I H MNMMWniMMIIIIIIIII • • • l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l t l l l l l t l l l l l t l M l l l l l l l l l l l l l l I I I M I I I P I I I I I I I I I I I t l l l l l l l l l l l l l l t l l l l l l l l l I t t t f l l iMMMtl f l imi lUt lMiMimHt lMIMMItMIMMmil l iMIIMMI IIIMMMII MmiMI MH I I tt l l l l l l l l l l l l l l lMH I I I I I I I MMMI I I t lMll l l l l t t t l l l l lMI 26 Optimum Bird Flock Size in Formation Flight field created by the other n - 1 birds. Hock-associated reduction is negative. As a result, the induced drag for a member of the flock is less than what would be experienced if flying solo. Only if the formation is such that each bird flys in the flight path of other (in the downwash region, see Figure 5) results in the induced drag being higher than that for a single bird. The extent of reduction of induced drag by formation flight is dependent on wing-tip spacing, s, the distance between wing-tips of adjacent birds perpendicular to the flight path and flock size, n. Since the region of the upwash occurs inboard of the actual wing-tips, certain overlap of the wing-tips may occur (Figure 5). In this case the wing-tip spacing, s, is negative. The ratio Di(n)/Di is used to assess the benefit of formation flight vs solo flight. A small ratio implies large savings in formation flight. When the ratio is equal to one-both formation and solo flight, in terms of induced drag, are equivalent. As the wing-tip spacing as percent of the semi-wing span increases for a fixed flock size, say 20 birds, the ratio approaches one (Figure 7). Thus a flock with very large wing-tip spacing benefits less than a flock that flys in tighter formation. The benefit of formation flight increases with flock size; however, there is a limiting value as n —• oo (Lissaman & Schollenberger, 1976). The induced drag of a single bird in a flock as n becomes-indefinitely large is given by Um 73/(n) = Di + ^ log (see appendix A for derivation). Approximation Formula In its current form, equation (39) is computationally intensive and mathemat-ically awkward. To calculate the induced drag of a bird in a flock of size n requires at least n 2/2 operations. For instance, a flock of size n = 100 requires at least 104/2 operations to calculate the induced drag. There are two main advantages of the approximation formula. Firstly, it is computationally less time-consuming, since sin(2a7r/(26 + s)) 2an/(2b + s) (40) 27 Optimum Bird Flock Size in Formation Flight Figure 7. Ratio of induced drag in formation to solo flight, as a function of flock size, n, and various wing-tip spacing, s, as a percent of the semi-wing span (solid). Curves are shown for s/b = +100, +50, +10, 0, -10, -15, and -20%. For the special case when the wing-tip spacing is such that the tip vortices of the two adjacent birds overlap (s = 2(a — b)) the ratio is given by the dash line. 28 Optimum Bird Flock Size in Formation Flight there are considerably less operations. Secondly, it provides a basis for generaliza-tion. The goal of the approximation technique is to eliminate the summation series in equation (39) and replace it with an analytic function. There are basically two ways to reduce the number of operations in a formula containing a summation series. The first method is to truncate the series at some arbitrary term and discard the rest of the terms in the series. As a result of this method, the calculated value from the truncated series underestimates the true value. The second method is to assume that there is an indefinite number of terms in the series. In other words, the original series is assumed to be an infinite series. This method often leads to representation of the infinite series by an analytic function. The degree of overestimation depends on the contribution of the extra terms to the sum. As the distance between a pair of birds gets large, the corresponding flock-associated term, in equation (39), gets small. To illustrate, in a flock of 15 birds, the percent contribution by the other birds to the total induced drag reduction of the left most bird (Bird 1) is shown in figure 8. In the case of flying abreast, with a wing-tip spacing of zero (s = 0) and semi-wing span of one (b = 1), the closest bird to Bird 1 contributes 65% of the entire induced drag reduction. For other birds, percent contribution drops off as the distance from them to Bird 1 increases. Incidentally, the smallest contribution to the entire induced drag reduction is by the 15th bird, furthest from Bird 1. It follows, then, that the addition of extra birds right of the 15th bird, in this case, will not greatly affect the total drag reduction of Bird 1. The basis of the approximation technique is to assume that in a flock of size n there is an indefinite number of birds creating an upwash field when in reality there are only n — 1 birds. Figure 9 gives a pictorial representation of the approximation technique. Specifically, in a formation of n birds, the left and right most birds are assumed to be flanked to either starboard or port respectively, by an infinite 30 Optimum Bird Flock Size in Formation Flight Figure 8. Percent contribution to the total induced drag reduction of the left most bird (Bird 1) by the neighboring birds numbered 2 through 15 (bottom x-axis), flying in abreast formation with a wing-tip spacing equal to zero (s = 0). The top axis is the distance of each bird as measured from Bird 1. Each bird is assumed to be of unit semi-span (6 = 1). 31 01 09 09 Ov 0E 02 01 0 (%) uoipnpey 6BJQ paonpui ui uoiinqujuoQ 32 Optimum Bird Flock Size in Formation Flight number of birds (Figure 9(A & C ) ) . However, the remaining birds, each having adjacent birds on either side, are assumed to be flanked on the left by an infinite number of birds and on the right by an infinite number of birds (Figure 9(B)). Following the above assumptions, the induced drag reduction on a bird in a flock of size n can be approximated by ir* n (see appendix B for detail). In the above form, the estimation of induced drag on a bird in formation flight is greatly simplified. To test the accuracy of this method, the true value as calculated from equation (39), is compared to the value from the above approximation equation. The degree of accuracy is measured by the relative error, defined as (42) (Taylor, 1982). The relative error incurred by this approximation technique is plotted in figure 10. Evidently, the approximation formula is quite accurate for a specific range of wing-tip spacing and of flock size. Relative error is the largest for formations with small wing-tip spacing relative to the semi-wing span (s/b). In such tight formation, the up'draft created by each bird contributes a significant amount of reduction in induced drag on any other bird in a flock. Not surprisingly, in a case where each bird has a significant effect, the introduction of an indefinite number of birds to a flock results in large overestimation of the drag reduction; consequently, the relative error for this case is large. The approximation formula for wing-tip spacing relative to semi-wing span of —20% guarantees the relative error to be less than 23%. However, as the wing-tip spacing increases, the relative error drops quickly. Figure 10, also, shows that the accuracy of the approximation formula increases with flock size. For example, the relative error drops from 12 to 5% for a flock size of 3 and 100 birds, respectively, at zero wing-tip spacing J AS the sin(2a7r/(26 + s)) 2a?r/(26 + s) (41) Relative Error = true — approximated true 33 Optimum Bird Flock Size in Formation Flight Figure 9. Pictorial representation of the approximation technique for n = 6 birds, solid plus one diagonal filled wing. (A & C) The left and right most birds in the vee formation ( 1 / / /]) are assumed to be flanked on the opposite leg by an infinite number of birds (| |...). (B) Each bird inside a flock, for example 1/ / /[, is as-sumed to be flanked by an infinite number of birds on both legs of the vee (1 | . . . ) . 34 Optimum Bird Flock Size in Formation Flight flock size increases the assumption of indefinite number* of birds in a flock becomes truer; correspondingly, the relative error becomes smaller. Another test for validity is to compare the calculated result of Lissaman and Schollenberger (1970, Figure 2) with the above approximation formula. In this case, the same order of magnitude of drag reduction turns out as calculated by Lissaman and Schollenberger. In other respects the behaviour of the estimate formula is similar to the family of curves they obtain. In particular, significant drag saving occurs at small wing-tip spacing; and the magnitude of the saving for a formation of 25 or more birds is nearly the same as for an infinite number of birds (Figure 11)-It should be remarked that equation (39) and consequently equation (41) has a singularity at wing-tip spacing s = 2(a — 6), where a.= ^6. In other words, the value of the function becomes infinite at the point s = 2(a — b). Physically, this wing-tip spacing corresponds to when the tip vortices of adjacent birds overlap. Of interest is the total induced drag of a flock in such a flight formation. Before generalizing for n birds, an account of the influence of two birds flying in formation for this case is given. Figure 12 shows the air flow due to the interaction between the two wings at such a spacing. As done previously, each wing in the flock is replaced by an equivalent horseshoe vortex. For a single wing of semi-span b, the bound vortex is of length 2a; and the distance between the two counter rotating tip vortices is 2a (Figure 12 (A)). Due to its position, the lead bird's tip vortex, rotating clockwise, cancels the tip vortex, rotating anti-clockwise, of the second bird. The resulting flow is another larger horseshoe vortex. The bound vortex is now of length 4a and the distance between the tip vortices is also 4a. Equivalently, the resulting vortex system can be generated by a single wing of semi-span 26 (Figure 12 (B)). This follows from the relation that a = ^6. If a, as in this case, is doubled, then 6 is increased by the same amount. Thus the air flow of two birds, each of semi-span 6, in this formation is equivalent to the flow produced by a single wing of semi-span 36 Optimum Bird Flock Size in Formation Flight Figure 10. Effect of flock size and spacing ratio, s/b, on realtive error as result of the approximation technique. Curves are shown for s/b = -20, -15, -10, 0, +10, +50, and +100%. 37 8C Relative Error (%) 0 5 10 15 Optimum Bird Flock Size in Formation Flight Figure 11. Induced drag in formation flight, Di(n), relative to that in solo flight, Z?j, for n — 2,3,9,25 and infinite number of birds. 39 Optimum Bird Flock Size in Formation Flight 2b. As a result, the induced drag equation of a single wing (Equation 3) can be used to calculate the total induced drag of a flock. In particular, substituting b for 2b and L for 2L in equation (3) gives the total induced drag of a flock containing two birds D ' < 2 ) " a J w " 5 ^ = D - ( 4 3 ) Accordingly, the induced drag for a single bird in this case is 5(2) = BL. (44) In other words, the induced drag of a bird in a flock of size 2 is half the induced drag if the bird was to fly solo (-Dj). In general, the total induced drag for n birds can be calculated by substituting the lift L to be the total lift of all n birds, that is n • L and by substituting the semi-wing span 6 to be the total semi-span of all n birds, that is n-6 into the induced drag equation for a single wing. Making the above substitutions into equation (3) gives the total induced drag of a flock of size n Dr(n) = DL (45) For any single bird in a flock of size n, the induced drag is Dl{n) = ^L. (46) n Thus in a formation when the tip vortices of adjacent birds overlap, the induced drag of a single bird in a flock of size n is 1/nth the induced drag if it was to fly solo. Of all possible wing-tip spacing, maximum reduction in induced drag, as demonstrated by the dashed line of figure 7, occurs at wing tip spacing s = 2(a — b), corresponding to when the tip vortices overlap. In summary, within a range of wing-tip spacings the approximation formula (Equation (41)) is quite an accurate measure of induced drag for a member of a flock. Due to the nature of the approximation equation, flight formations where 41 Optimum Bird Flock Size in Formation Flight Figure 12. Air flow pattern, as represented by a horseshoe vortex, of two wings each of semi-span b, located at wing-tip spacing s = 2(a — 6). Due to the positioning, tip vortex of the lead bird cancels the tip vortex of the second bird (A). The resulting flow, as shown by the horseshoe vortex in (B), is equivalent to a flow created by a single wing of span 26. 42 -e-6 r €7 43 Optimum Bird Flock Size in Formation Flight the wing-tip spacing s = 2(a — b) equation (46) should be used. Interestingly, both formulas show that the induced drag of a bird in a flock is inversely related to flock size. This result greatly simplifies flight energetic calculations of a bird in different sized flocks. 44 Optimum Bird Flock Size in Formation Flight Model In formulating an optimal behaviour model three details are often considered: (1) choosing a currency (2) choosing an appropriate cost-benefit function with the appropriate constraints and (3) solving for the optimum (Pyke et al, 1977; Schoener, 1971). The currency in this case is energy, the cost-benefit function is net energy gained per unit energy expended, and the optimum is the flock size which maximizes the cost-benefit function, subject to various constraints. Here, energy gain (Ein) and energy expenditure (Eout) are the result of foraging and flying, respectively.: Energy Expenditure (Hying cost) Analogous to solo flight cost equation 8, energy expenditure for a bird in formation flight is given by E O H t = (DP+DI(n))d.' (47) In formation flight, the sum of profile and parasite drag is the same as for a solo bird, D p , whereas the induced drag, Z)j(n), is reduced, and it is now a function of wing-tip spacing, and flock size. As shown previously, induced drag can be expressed by two approximation formulas, depending on the choice of wing-tip spacing. The result of substituting equation (41) for Di(n) is shown in appendix C. The case when the wing-tip spacing causes the tip vortices to overlap is considered here. In such formation Di(n) = Di/n; accordingly, the flight cost equals Eout = {DP + ^-)d. (48) n Substituting equation (2) and (3) for D p and D j respectively, and grouping terms gives the energy expenditure of a bird in a flock of size n flying at a speed U for a distance d E 0 U t = ( k p U 2 + - U - 2 ) d . (49) 45 Optimum Bird Flock Size in Formation Flight The maximum range speed, which minimizes flying cost per unit distance travelled, is given by solving the equation d_ du E, out 2kpU-^U-3 = 0. n (50) Solving for U gives the maximum range speed of the flock 1/4 U m r ( n ) Ajp Th (51) Equation (51) shows that with increasing flock size cruising speed of the flock decreases. Relative to the optimum speed of a solo flyer, Umr = (ki/kp)1/4, the maximum range speed of the flock decreases as ra-1/4. Due to a lower cruising speed, birds in a larger flock will take a longer time to travel the same distance than birds in a smaller flock. In effect, the airborne time ta during a migration leg of distance d is ta = d U m r ( n ) = d ki 1/4 n 1/4 (52) Finally, substituting equation (51) for U in equation (49) gives the flying cost of a bird in a flock of size n, travelling a distance d at the maximum range speed E0ut = 2 u k p • k{ —=. (53) Equation (53) shows that flying cost for an individual decreases as the number of birds increases in the formation. More specifically, this cost decreases as n - 1 / 2 . Energy Gain (feeding) It is assumed that birds do not feed during their journey and so any energy gain occurs only at a foraging site. The total energetic gain, Ein, will equal the rate of prey encountered times the time spent foraging. Assuming that an individual forager captures food according to a Poisson process with parameter A (number of 46 Optimum Bird Flock Size in Formation Flight prey (net energy equivalent) per unit time) (Mangel h Clark, 1988), then the net energy gain for a patch residence time, tp, is Fin — \ ' tjj (54) The total time spent flying and foraging, T, is given by 7 — "bp ~\~ ta, (55) Substituting for tp into equation (54) gives E i n = X(T-ta). (56) Equation (56) shows that as the airborne time increases the total energy gain decreases. Substituting for ta (Equation (52)) in equation (56) gives 1/4 E i n = X ( T - d n 1/4 (57) In effect, net energy gain for an individual decreases with increasing flock size. .Ratio of Energy Gain to Energy Expenditure Let the ratio of net energy gain/energy expenditure be Ein R = Eout (58) (Pyke etal, 1977; Schmid-Herhpel etal, 1985). Substituting equations (53) and (57), into the above ratio gives R(n) = ^\ \ (59) The above equation is maximized when dR(n)/dn = 0. So, R(n) is maximized when, 2dy/kT^k~ \2 \ T n - " 2 - -d 1/4 n - 1 / 4 = 0. (60) 47 Optimum Bird Flock Size in Formation Flight Solving the above equation gives nopt, a value of n that maximizes equation (59), nopt — 16 ki_ (61) Finally, substituting U m r = (ki/kp)1/* (Equation (10)) into the above equation gives r n 4 Ifi T (62) » - 1 6 TT* JLopt — umr where Umr is the maximum range speed for a solo bird. For completeness, it can be shown that equation (59) is a maximum at n = nopt, for at this value the second derivative of the function is negative; in particular, 5 < 0. (63) o?R(n) dn2 n=nopt 729 kj 1024 kj 48 Optimum Bird Flock Size in Formation Flight Results Conditions for Solo Flight The model makes predictions on the optimum number of birds in a flock. Provided that the value of d, Umr, and T are known a corresponding flock size can be calculated. Here, however, value of d, Umr and T that yield a flock size of one (solo flight) is examined. In other words, n o p t is set equal to one in equation (62): d = ^ U m r T . (64) The conditions for solo flight depends on the value d, Umr, T such that the above equality holds. For instance, in figure 13, for a given Umr, a solid line divides the value of d and T pairs that predict solo flight (below the line) from the pairs that predict optimum flock size greater than one (above the line). Furthermore, figure 13 shows that as the value of Umr increase the region for solo flight (below the line) also increases. Thus birds with large Umr are more likely to fly solo than birds with low Umr. Moreover, the condition for solo flight can also be expressed as a ratio of two time variables. Substituting d = Umr • ta into equation (64) gives which can be also expressed as T _ ta+tp _ 3 ta ~ ta ~ 2' Thus the condition for solo flight is when (66) In other words, under the conditions where time spent feeding is one half the flight time birds should fly solo. 49 Optimum Bird Flock Size in Formation Flight Figure 13. Condition for solo flight depends on the value of d, Umr and T. For a given Umr, a solid line divides the values of d and T pairs that predict solo flight (below the line) from d and T pairs that predict optimum flock size greater than one (above the line). To illustrate, say Umr = 1/2, (T, d) pairs that fall in the grey region correspond to solo flight while pairs that fall in the white region correspond to formation flight. 50 T3 51 Optimum Bird Flock Size in Formation Flight Optimum Flock Size and Wing-tip Spacing Birds in a flock have a choice of whether to fly in tight or loose formation. A tight formation is when the wing-tip spacing is small or, when the wing tips overlap, negative. As was shown in figure 7, maximum induced drag reduction occurs when the wing-tip spacing is such that the tip vortices of adjacent bird cancel each other out. The corresponding optimum flock size for such a spacing is given by equation (62). However, for other spacings, the optimum flock size is given by - lint - B m where both A and B are a function of wing tip spacing, s (see appendix C , Equations C3 & C4). As shown in figure 14, for the same value of T/ta, the smallest optimum flock size results when the wing-tip spacing gives the maximum drag reduction (dash line). As the wing-tip separation relative to the semi-wing span (s/b) increases the predicted number of birds in a flock also increases. To compensate for a lower induced drag reduction which occurs at large wing-tip spacing the number of birds in a flock is increased. Finally, figure 14 shows that for any wing-tip spacing, as the ratio of T / t a increases so does the predicted optimum flock size. Optimum vs Maximum Flock Size Although the model predicts an optimum number of birds in a flock, an ex-amination of the cost-benefit function (Equation (59)) shows that a flock may form which exceeds the optimum size (Sibly, 1983). Figure 15 shows that the curve spec-ifying individual efficiency, R, as function of flock size is concave downward and has a peak at n o p t , the size yielding maximum efficiency. Due to the downward concavity of the curve, individual efficiency decreases in the region where flock size is larger than nopt. Specifically, at a flock size n*, the curve is at the same level as that for a solo flyer. Accordingly, individual birds in a flock of size n* will expe-rience the same benefit as a solo bird. By forming flocks larger than n* results in 52 Optimum Bird Flock Size in Formation Flight Figure 14. Optimum flock size is plotted against T/ta. Curves correspond to wing-tip spacing, expressed by a non-dimensional ratio, s/b (+100, +50, +20, 0, -10, -20%). Smallest flock size occurs at wing-tip spacing s = 2(a — b) (dash line), flock configuration for maximum induced drag reduction. 53 54 Optimum Bird Flock Size in Formation Flight benefits smaller than that of flying solo. Thus, it follows that the maximum flock size is n*. Mathematically, the maximum flock size, n*, satisfies R(l)=R(n*). (69) Grouping terms (see Appendix D), the following relation can be derived between nopt and nm (70) 16 nJ - 1 While not directly solving for n», it is shown in appendix D that the upper bound of n* is be given by n» < ^-rnopt « 5 • nopt. (71) lb Thus the maximum flock size is no greater than five times the predicted optimum flock size. When a flock size exceeds 5'nopt birds, the benefit of solo flight outweigh that of group flight. Due to decreasing benefit, as the number of birds increases beyond the optimum size, a large flock may become unstable. For example, a flock will continue to attract individuals until the benefit of group flight equals that of solo flight. For this reason no other bird will join a flock that contains the maximum number of birds, n*. However, compared to a large single flock, a greater benefit can be achieved by breaking up in to smaller flocks, each of size approximately equal to nopt. Hence, breaking up of a flock is highly likely once its size has surpassed the optimum number. Effect of Umr on Optimum Flock Size Estimates can be made of flight speed for a particular animal from equation (10), provided that at least its weight and wing span are known. Substituting morphological relationships for wing characteristics given by Greenwalt (1962) for three bird groups : passerforms, shorebirds, and ducks into equation (10) a mass specific Umr can be determined for each group. Although this division is based 55 Optimum Bird Flock Size in Formation Flight Figure 15. Normalized relative to a solo flyer, equation 59 is plot-ted against different flock sizes, n. The benefit of flying in a flock of size n = n* is identical to that of flying solo: R(l) = -R(n*). In this example, a member of a flock of size n = n o p t is 60% more efficient than a solo flyer. 56 00 Flock Size (n) Optimum Bird Flock Size in Formation Flight largely on disc loading, and not on taxonomy (Rayner, 1979), maximum range speed as a function to body mass for the three groups and all birds combined is given in table 1. Substituting mass specific Umr into equation (62) results in a mass specific optimum flock size. The corresponding flock size for the three groups and across all groups is given in column two of table 1. Since members in a group fall within a small range of mass, for example, the majority of ducks fall into a narrow range of about lkg, Umr does not vary as greatly within a group as across a group. Accordingly, if birds within a group migrate similar distances then a small variation in flock size is to be expected. The interspecific equation for velocity (table 1) shows that a large variation in flock size is expected when comparing across a group then within a group. Although morphological relationships give a general trend, a more accurate measure of flock size can only be made when the values of T, Umr and d are known for a specific species. Flock size for various combinations of T, Umr and d, are plotted in figure 16. Each panel, from left to right, is for a fixed T, 6, 12, and 18 hrs respectively, and is a plot of flock size vs distance, d (km), for three migration speeds. These panel show certain trends. First, for any fixed Umr and d, flock size increases with increasing T. Second, for any given T and d, flock size increases with increasing Umr. Finally, the critical distance at which to fly solo, irrespective of T, increases with higher cruising speed of the animal. Seasonal Variation and Optimum Flock Size Total time available, T, is an important variable in determining optimum flock size. On any leg of the migration, T equals the sum of the airborne time, t„ and the foraging time, tp. Day length, in some cases, may constrain the total time available to complete these two activities. Since day length changes with time of year, the optimum flock size at the time of the northward migration may differ from the size at the time of the return journey, on the southward migration. In other words, due 58 Optimum Bird Flock Size in Formation Flight Table 1. Relation of maximum range speed to body mass, after Greenwalt (1962, 1975) and Rayner (1979). Optimum flock size is shown divided by [T/d]4. (All quantities in MKS units.) 59 Maximum Optimum flock range speed size/[T/d]4 Interspecific rule 10.04M 0 2 4 2.00 x K F M 0 - 9 6 Intraspecific rule passiforms 7.75 M 0 1 1 0.71 x 1 0 3 M 0 4 4 shorebirds 8.98 M 0 0 2 1.28 x l O 3 / ^ 0 0 8 ducks 12.43M 0 0 1 4.71 x lCPAf 0 0 4 60 Optimum Bird Flock Size in Formation Flight Figure 16. Effect of T, d and Umr on optimum flock size. Each panel, from left to right, is for a fixed T, 6, 12, and 18 hrs respec-tively, and is a plot of flock size vs distance, d (km) for Umr = 18, 36 and 54 km/hr (1-3 respectively). Solid horizontal line corre-sponds to nopt = 1, solo flight. 61 Optimum Bird Flock Size in Formation Flight to seasonal varation in day length there will be a corresponiding change in flock size if T equals the number of hours between sunrise and sunset. Hours of daylight depends on the solar declination and latitude of the observer (Griffiths, 1976). Solar declination, as a first approximation, is given by where tj is the Julian date (Jan. 1 = 1,... Dec. 31 = 365) in the year (Oke, 1987). Day length, hours between sunrise and sunset, is given by o H(£, tj) = — arccos(- tan(*) tan(S(tj))), (73) lo where £ is latitude, in degrees, and H is day length, in hours (Griffiths, 1976). Figure 17 illustrates the variation in day length with time of year and with latitude. From the beginning of the year, day length progressively gets longer, reaching 12-hours at the vernal equinox (March 21), and continually increases until the summer solstice (June 21), where it then starts to decrease, reaching again a 12-hour length at the autumn solstice (September 23). Over a year, this pattern of day length change is symmetrical about the summer solstice. In other words, days equidistant from the summer solstice have the same number of daylight hours, irrespective of latitude. Now that T can be quantified by equation (73) it is interesting to study how it affects nopt. Let the beginning of the northward migration be at time tb, number of days before the summer solstice and let the return journey be at time t r, number of days after the summer solstice. To a first approximation, and disregarding changes in hour length during the time of flight, the average number of hours of daylight from the start of the migration at latitude £\ at time tj, to the end of the migration at latitude £2 can be calculated by R^ = i^rT1J^mtj)di (74) (Greenspan and Benney, 1973). Substituting tb and tr for tj in the above equation, gives the expected day length, H(tb) and H(tr), respectively, and can be assigned tio 63 Optimum Bird Flock Size in Formation Flight Figure 17. Contour of day length (hrs) with time of year and with latitude (solid lines). The dashed zigzag lines represent move-ment, from 10° N to 50° N, of two hypothetical migratory bird species. By flying equidistant number of days before and after the summer solstice (top axis), species A (short dash) will experience the same hour of day length on both the north and southward mi-gration; consequently, no seasonal variation in flock size is to be ex-pected. If the two phases occur in an asymmetrical pattern, as for species B (long dash), the average day length during the northward migration will be larger than the day length during the southward migration; as a result, seasonal variation in flock size is to be ex-pected. The degree of variation in flock size is plotted in figure 18. 64 09 Optimum Bird Flock Size in Formation Flight variable T in equation (62). Depending on the values of H(tb) and H(tr), optimum flock size for a given bird species may differ at time tb from that at time tr. Let Rs be the ratio of flock size at the the time of the northward migration to flock size at the time of the return journey. That is 16 Trt H ( * r ) 4 P _ 81 d* _ B(ir) 16^4 H(n)* [H(tb)\ -, 4 (75) Using equation (74), the above equation can be written as r rt: H(£,tr)di &H(t,th)dl (76) The degree of seasonal variation in flock size can be implied from the above ratio. Contours of R8 are plotted against tb and tr in figure 18 for a bird species that migrates from t\ = 10° N to fe = 50° N. When the start and the return time is equidistant with respect to the summer solstice, there is no seasonal variation in flock size because the average number of day light hours is the same on both leg of the journey. In this case the ratio, Ra, equals one. Birds of similar migratory range, but of asymmetrical start and return time with respect of the summer solstice, however, should exhibit seasonal variation in flock size. The degree of this variation is expressed by the value of R„. For example, if the start time is 30 days before and the return time is 120 days after the summer solstice then the flock size on the return journey from latitude 50° N to 10° N will be 40% smaller than that during the northward migration from 10° N to 50° N (point B, figure 18). In conclusion, figure 18 shows that variation in flock size depends on the time of the north and southward migration with respect to the summer solstoice. For bird species that migrate in a fashion where the north and southward migration times are symmetrical about the summer solstice no seasonal variation in flock size is expected. It is shown that asymmetrical time of the north and southward migration relative to the summer solstice is what causes variation in flock size. The degree of variation in flock size at these two times can be calculated from equation (76). 66 Optimum Bird Flock Size in Formation Flight Figure 18. For migration from 10° N to 50°N, contours of the ra-tio Rs are plotted against tb arid tr. The ratio Ra for bird species A and B is determined by the value of (tb,tr) pair which is (30,30) and (30,120) days respectively (see Figure 17). Ratio Rs < 1 corre-sponds to flock sizes being smaller during the southward migration compared to that during the northward migration. A ratio of one implies that flock sizes during the north and south migration are identical. 67 68 Optimum Bird Flock Size in Formation Flight Discussion Aerodynamic Model The aerodynamic model of formation flight makes certain predictions on ge-ometry, optimum speed and energy expenditure of a flock. In terms of geometry, formation close to a crescent rather than a vee shape allows for equal distribution of induced drag reduction. In such a case, the crescent shape has approximately a formation angle (angle measured between the two legs of the formation) of 109 ° (Lassaman & Schollenberger, 1970). However, in most observed flocks the formation angle is much smaller than 109 ° , and the formation is in a distinct vee shape (Timothy et al. 1976; Gould & Heppner, 1974). This finding would imply that induced drag reduction is not equally distributed among members of a flock. Calculations by Hummel (1983) show that among all other birds in a formation the lead bird experiences the lowest induced drag reduction, while retaining a flying cost lower than a solo flyer. Taking turns leading during flight may be one way to assure that work is equally distributed within all members. No study as of yet has been done on the dynamics of birds within a flock although there are many accounts of birds changing lead position during flight. While distribution of induced drag reduction is influenced by formation angle the magnitude of this reduction is dependent on wing tip spacing (WTS). Within limits, tighter the formation higher the reduction in induced drag. By flying in a vee formation, certain overlap in WTS (s < 0) can occur which would be impossible, for example, in abreast formation. The limit of the overlap is determined by the position of the vortex filament shed from the bird ahead. A wing of semi-span ft sheds two trailing vortex a distance a = from the center of the wing (Houghton & Brock, 1970; Higdon & Corrsin, 1978). As shown in figure 7 the maximum induced drag reduction occurs when the trailing vortex of adjacent birds cancel each other out, corresponding to a WTS s = 2(a — b). Observed WTS of Canada Geese (Branta canadensis) in two studies report a median WTS of —0.198 m (range —1.28 69 Optimum Bird Flock Size in Formation Flight to 2.89 m) and —0.337 m (range —1.2 to 1.41m) (Hainswdr^ tli, 1987; Hainsworth in press). Since in both studies it was assumed the birds were of constant length and of semi-wing span of 6 = 0.75 m, the corresponding predicted WTS is calculated to be s = —0.36m. Although there is a large range of observed WTS, the latter study is in close agreement to the predicted WTS for maximum induced drag reduction. Large variation in observed WTS may be due to several factors such as local turbulence caused by change in wind direction or due to unpredictable moves by birds ahead (Hainsworth, 1988). While accounting for the dynamic nature of WTS between birds (see Figure 10, Hainsworth, 1988) the degree of overlap suggests that a relatively high reduction in induced drag can occur. In order to achieve the best flight performance, all individuals of a flock must travel at an optimum speed. This speed, minimizing flying cost per unit distance flown, is a function of two variables, flock size, n and WTS, s. In general, optimum speed of a flock, L7mr(n), is slower than that of a single bird, Umr (Figure 19). Specifically, equation 51 and C6 show that the optimum speed increases to the level of a single bird with increasing WTS or with decreasing flock size. In other words, the aerodynamic model predicts a negative correlation between speed and flock size for a given WTS. However, in three studies, two with Oystercatchers (Haematopus ostralegus) (Preuss, 1960; Noer, 1979) and one with Arctic Terns (Sterna paraiisata) (Alerstam, 1985), no significant correlation between speed and flock size was found. This apparent discrepancy between model and observation can be explained by the fact that all three studies failed to measure WTS, which affects optimum speed. For example, a 17% reduction in relative speed to a single bird is predicted for a flock containing 40 birds and 4 birds, at spacing s/b = 0 and s/b = —20% respectively. Not surprisingly then, no significant correlation can be established without knowledge of WTS. On the whole, any study to investigate the dependence of speed on flock size must measure WTS and other variables, such as wind speed and direction, which also effects optimum speed (Pennycuick, 1972). 70 Optimum Bird Flock Size in Formation Flight Figure 19. Percent reduction in optimum flight speed relative to a single bird ([Umr — (7mr(ro)]/i7mr) at different flock size, n, and spacing index, s/b (s is WTS, b is semi-wing span). The solid line corresponds to a WTS when trailing vorticies of adjacent bird cancel each other out. 71 72 Optimum Bird Flock Size in Formation Flight Finally, the aerodynamic model predicts that formation flight be more ad-vantageous than solitary flight in terms of energy expenditure. Observation on white pelicans (Pelecanus erythrorhynchos), commuting between nesting and for-aging area, show lower flying cost for different types of formations if wingbeat is considered as an indicator of flying cost (O'Malley & Evans, 1982). The number of wingbeats per hour, calculated from wing beat frequency (beats per minute) and percent time flapping, was lowest in vee formation and greatest in single birds. It was found that birds in a vee formation had an average 6042 wingbeats/h whereas single birds flew with an average 7169 wingbeats/h (table 5; O'Malley & Evans, 1982) suggesting indirectly that formation flight provides aerodynamic advantage over solitary flight. Optimum Flock Size Model Verification of the model requires comparison with appropriate data. Four i: variables, WTS (s), cruising speed (Umr), flight distance (d), and cumulative feeding and flying time (T) are interrelated components of this model. Presently, such data is sparse and where it exists it is often incomplete to validate this model. For example, many studies, while reporting flock size, are ambiguous about the behaviour of the flock. As is already known, there are many other types of behaviour that flocking is associated with, such as colonial nesting, communal roosting and anti-predator avoidance (McFarland, 1987), in addition to formation flight during migration. Clearly, a system to classify flock behaviour should be adapted to include behaviour of aerial flocks. For instance, formation flight is not restricted only to migration but may occur during local daily movements between nesting and foraging site, as in the case of Pelicans. Thus group flight may be associated with many types of behaviour. Here, model predicts size of migratory flock since non-migratory flock may travel at a speed different than maximum range speed (Pyke, 1981; Norberg 1981). Because literature information pertaining to this model is often incomplete, certain assumptions have to be made in order to determine optimum flock size of 73 Optimum Bird Flock Size in Formation Flight migrating birds. Estimates of optimum flock sizes compared to what is observed in the field for cranes and Canada geese are considered next. Although taking advantage of thermal soaring when meteorological conditions allow for it, common cranes' (Grus grus) primary mode of migration is powered flight, either in V or J flock formation (Pennycuick et al. 1979). In this species observed flock size range f^rom 2-31 birds during part of spring migration from Rugen (54°30'N, 13°30'E), in the southern Baltic Sea, to lake Hornborgasjon (57°30'N, 14°02'E), in Sweden, a flight distance of nearly 400 km (Pennycuick et al. ,1979). K flying and feeding is restricted between sunrise and sunset, the amount of time to complete these two activities during the April migration is approximately 13 hrs (Figure 17). Calculated maximum range speed for this species using a representative weight of 5.5 kg (Pennycuick et al. , 1979) is 55km/hr (Table 1). Using field data to estimate flight distance (d = 400 km), general mathematical expressions to estimate both flight speed (Umr = 53km/hr) and day length (T == 13 hrs), and condition for maximum induced drag reduction to estimate WTS (s = —0.436), gives an optimum number of 1.72 birds/flock (Equation (62)), a value near the lower range of observed size. Not all birds, however, make the journey in one day (Alerstam, 1974). Thus, if the journey is broken in two phases, with a single stopover 200 km away, the optimum number in this case (d = 200 km) is 27.6 birds/flock. Likewise, using field data to estimate flight distance and speed allows calcu-lations of optimum flock size in Whooping cranes (Grus americana) . By track-ing daily movements of transmitter-tagged cranes along the Central flyway from Wood Buffalo National Park (58°30'N, 112°00'W), in Alberta to Aransas National Wildlife Refuge (26°30'N, 97°10'W), on the Texas Coast, the average distance be-tween stopover sites was estimated to equal 325 km (Emanuel, 1982). From table 1, flight speed for this species was calculated at 58km/hr, using an average male weight of 7.3 kg (Johnsgard, 1983). During fall migration period, between mid-October and early November, day length approximately equalled 10 hrs. Setting 74 Optimum Bird Flock Size in Formation Flight d = 325 km, Umr = 58km/hr and T = 10 hrs in equation (62), gives an optimum number of 2 birds/flock; Central fly way fall migration records report flock size ranging from 1 - 9 birds, with an average of 2.6 birds/flock (Table 2, Johnsgard, 1983). Finally, migration timing and distance data on a population of Canada Geese (Branta canadensis), which nests at Marshy Point Goose Sanctuary (50°32'N, 98°07'W), Manitoba and migrate to Silver Lake (44°00'N, 92°20'W) in Rochester, Minnesota (Wege & Raveling, 1983), allow the estimation of seasonal variation in flock size and the determination of conditions for solo and group flight. The data shows that transmitter-tagged geese took on an average 8.3 and 2.3 days during the spring (8 - 15 April) and autumn (28 Oct. - 2 Nov.) migrations respectively to complete the 855 km journey between Lake Rochester and Marshy Point Goose Sanctuary. On the average birds flew 371 km, with some flying non-stop during the autumn migration period. In contrast, all the tagged geese interrupted their spring migration. In fact because stopover areas in the spring were pastures and gain fields, many birds broke the migration to renew fat reserves while advancing northward (Elkins, 1988; Wege & Raveling, 1983). As a result, map locations of stopover sites were used to estimate flight distance (Fig. 2, Wege & Raveling, 1983). On the average birds flew 170 km during the spring migration period, about 50% less than on their southward migration. Hours of daylight also differed during the two migration periods and was equal to 14 and 11 hrs at the time of the north and southward migrations respectively. Figure 20 shows predicted flock size during the spring and autumn migration for varying distance between stopover sites. For this case, the figure shows two trends: firstly, that there is a seasonal variation in flock size, with flocks being largest in the spring and secondly, that flock size should decrease as the distance between stopover sites increase, with the possibility of flying solo. Data on flock size, while lacking in Wege & Raveling (1983) study, is varied. For example, data collected in central Illinois report flock size ranging from 23 - 300 birds, with an average of 96 birds/flock (Bellrose, 1980). Other studies 75 Optimum Bird Flock Size in Formation Flight report for this species an average size of 36.08 (Wing, 1941), 27.02 (Heppiier, 1974) and 5 birds/flock (Elder & Elder, 1949). On the whole, it is predicted that Canada Geese may migrate in flocks that are highly variable in size, determined, among other factors, by distance between stopover sites and time of year. It is recognized that a number of assumptions have been made in determining optimum flock size. Nevertheless, the estimates of flock size agree with what is observed in the field, and supply an understanding, if tentative, basis for interpreting flock size in migrating birds. Not all birds migrate in flocks. Condition for solo flight admittedly depends on the value of Umr, T and d. However, since flock size approaches a value equal to one with increasing flight distance, this distance largely determines solo flight condition for the case of long non-stop migration. In such cases flight distance can be thousands of kilometres, often over uninhabitable regions such as oceans, deserts, and glaciers and flight time may range from 20 to 100 hours, depending on the energy reserve of the migrant (Alerstam, 1976). To illustrate, geese travel long distances non-stop, such as the lesser snow geese (Amer caerulescens) in North America which fly from James Bay to the Gulf of Mexico, a distance of about 2700 km in 60 h and the Brant geese ( Branta bernicla) which fly about 4000 km across the Pacific Ocean from the Alaska Peninsula to South California coast (Ogilive, 1978). Other bird species such as the European cuckoo fly over 3000 km from southern Europe to equatorial Africa migrate singly or in small flocks (Wyllie, 1981). Likewise, travelling singly or in a small group, Pacific Golden Plover fly approximately 4000 km over open ocean from Alaska to Hawaii (Johnston & McFarlane, 1967). On the whole, for the case of long non-stop migration, the model predicts solo flight as a possible migration strategy. The approach taken here allows the determation of aerial flock sizes in migrat-ing birds. However, the model can accommodate more detailed information on a variety of issues. Specifically, observation of flocks sizes for different bird species, 76 Optimum Bird Flock Size in Formation Flight Figure 20. Predicted optimum flock size during the spring and autumn migration for varying distances between stop-over sites (top axis) or equivalentley the nubmer of stop-overs required to complete the 855 km journey (bottom axis). See text for detail. 77 Optimum Bird Flock Size in Formation Flight and under different migration conditions would be rewarding. Further detailed work on seasonal time budget of migrating birds and their flight speed is still needed. Eventually, this would provide a basis of for comparative study of flock sizes for different migrating species and for different times of the year. In addition, several aspects of the aerodynamic model such as the optimal adjustment of air speed for different flock size and WTS still needs to be explored. 79 Optimum Bird Flock Size in Formation Flight Summary (i) The aerodynamic of formation flight theory for a fixed wing aircraft has been applied to avian formation flight. Two approximation techniques are derived which greatly simplify induced drag calculations for a single bird flying in a group of n birds at wing-tip spacing s. The approximation formula guarantees the relative error to be less than 23% for spacing index greater than s/b = —20%. Relative error decreases with increasing n and s. (ii) The results show that a striking saving in induced drag can occur during formation flight. Typically, a group of 25 birds in tip-to-tip formation (i.e. s = 0) has approximately 50% reduction in induced drag than a lone bird. The greatest reduction in induced drag occurs in formations where tip vortices of adjacent birds overlap. In such formations, induced drag for a member of a flock is 1/nth the induced drag of a single bird. (iii) Maximum range speed of a flock decreases with flock size. For a group of 25 birds in tip-to-tip formation there is approximately a 15% reduction in optimal speed relative to a single bird. In formations where tip vortices of adjacent birds overlap, maximum range speed of a flock is reduced by a factor of (1/n)1/ 4. (iv) The optimum flock size model is based on the energetic cost of flight and energy income from foraging. Optimum size of a flock is a function of four variables: Umr, T and d. 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S C H M I D - H E M P E L , P., K A C E L N I K , A. A N D H O U S T O N ( 1 9 8 5 ) Honeybees maximize efficiency by not filling their crop. Behav. Ecol. Sociobiol 17: 61 - 66. S C H M I D T — K O E N I G , K . ( 1 9 7 9 ) Avian Orientation and navigation. Academic Press: London. S C H O E N E R , T .W. ( 1 9 7 1 ) Theory of feeding strategies. Ann. Rev. Ecol. Syst. 11: 369 - 404. SlBLEY, R . M . ( 1 9 8 3 ) Optimal group size is unstable. Animal Behaviour 31: 947 - 948. S P E D D I N G , G . R . , R A Y N E R , J .M.V. , A N D C.J . P E N N Y C U I C K ( 1 9 8 4 ) Momen-tum and energy in the wake of a pigeon (Columba livia) in slow flight. J. Exp. Biol. I l l : 81 - 102. S P E D D I N G , G . R . ( 1 9 8 7 ) The wake of a kestrel (Falco tinnunculus) in flapping flight. / . Exp. Biol. 127: 59 - 78. T A Y L O R , J ( 1 9 8 2 ) An introduction to error analysis. University Science Books: Mill Valley, California. T A L A Y , T. ( 1 9 7 5 ) Introduction to the aerodynamics of flight. 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Universe Books: New York. 88 Optimum Bird Flock Size in Formation Flight Appendix A Limiting Case as n —» oo Induced drag of a bird in a flock of size n is n—1 n D!{n) = DT + -J2 E DW (A1) 1=1 J=l+1 Substituting equation (34) for DHJ, the second term of the above equation can be written as n—1 n n-1 n E E % = * E E 1o« t'=l j=i+l i=l j ' = t + l 2a \i - j\(2b + s) (A2) Applying the logarithm law, ]og(«i)+log(u2)H hlog(u n ) =log(«iw2 • • • « n ) n r n J^log[u;] = log JJ t t ; i = l i = l (A3) to the inner summation, gives n—l n n-1 E E D"i i=l j=i+l Let k = \i — j\ then i=l n .j=i+l 2a \i-j\(2b + s) (AA) n-1 2Dj ^ i=l n fc=i i - ( *» \k(2b + s) (A5) As n —• oo, the upper hmit of product term changes from n to oo, that is n-1 i = l n ,fc=i I - 2a k(2b + s) Given the identity, sin(a;) n k=i L (A6) (A7) 89 Optimum Bird Flock Size in Formation Flight (Hyslop, 1954) the infinite product term can be written as sin(2a7r/(2& + s)) n ifc=i I X \ 1 - -r \irk J J 2air/(2b + s) where, in this case, x 2cnr 2b+ s (A8) (A9) Now, the double summation can be expressed as 'sin(2a7r/(26 + s)) n-1 n _ n n - 1 E E i=l j=i+l t=l 2air/(2b + s) (AIO) Since the terms in the summation on the r.h.s. are independent of the index i, the preceding equation can be written as n—1 n 2D/ (n - l)log sin(2a7r/(2& + s)) 2air/(2b + s) (AU) i=l j=i+l Substituting the double summation of equation A l with the above result gives 'sin(2a7r/(26 + *)) = , x „ 4Dj (n - 1) , D/(n) = + ^ — ± log 2air/(2b + s) (A12) Consequently, as n —• oo, (n — l)/n—>1. Thus ADT lim D^n) = £>/ + — - i log sin(2a?r/(26 + *)) 2air/(2b + s) (A13) Appendix B Approximation Technique The goal of the approximation technique is to represent the total induced drag of a flock of size n (equation Bl) in a close form; in other words, by an analytic function. Figure 9 gives a pictorial representation of the approximation technique. Specifically, in a formation of n birds, the left and right most birds are assumed to be flanked on either the starboard or port, respectively, by an infinite number of birds. The remaining birds, having neighbors on both sides, are assumed to be 90 Optimum Bird Flock Size in Formation Flight flanked on both the port and starboard by infinite number of birds. Following the above assumptions, the total induced drag, n—1 n D(n) =nDI + 2j2 X Dli'> i=l j=i+l (Bl) can be decomposed into n —1 oo D(n) « nDi + 2 X j + 2 X X ^ j=2 i=2 j=i+l (52) Calculation of the first and second summation series gives the total mutual induced drag for the two end birds, and for the remaining n — 2 birds within the flock, respectively. Substituting equation (34), for DHJ the first summation series can be written as X ^ = ^ X l o s 1 -2a \l-j\(2b + s) 3=2 j=2 with some manipulation (see Appendix A) it can be expressed as l^Dnj = — log j=2 n U=l \-( 2a \k(2b + s) Using the identity in Appendix A, the above equation reduces to J'=2 sin(2a7r/(26 + s)) 2air/(2b + s) (B3) (BA) (B5) In a similar fashion, the second summation series j'=2 j=i+l i=l n fc=i 2a k(2b + s) can be reduce to (see Appendix A) n— 1 oo i=2 j=i+l n-1 i=2 sin(2a7r/(26 + 5 ) ) 2a7r/(2b + s) (B6) (B7) 91 Optimum Bird Flock Size in Formation Flight Since each term in the above summation is independent of the index i , the above equation simplefi.es to n— 1 oo E E 0 , y = ^ - 2 ) i o g i=2 sn(2<Mr/(26 + s)) 2an/(2b + s) (BB) Substituting the first and second summation series of eqaution B2 by the above results (equations B5 and B8 respectively) gives Dx(n) « n£> 7 + — i l o g 7T Z sin(2a7r/(26 + *)) 2a7r/(2fe + s) • ^ ( n - 2 ) l o g I T ' an(2a?r/(26 + s)) 2a7r/(2b + s) (B9) Grouping terms, gives an approximation function to the total induced drag of a flock of size n (equation B l ) ADi Di(n) w n D j -\ — (n - l ) log It* sin(2a7r/(26 + s)) 2air/(2b + s) Accordingly, the average induced drag, Z)j = Di(n)/n, is "sin(2a7r/(26 + *)) n 2air/(2b + s) (BIO) (511) Appendix C Optimum Flock Size as a Function of Wing-tip Spacing Energy expenditure for a bird in formation flight is given by EOUT = (DP + Dj(n))d, (CI) where in this case and 5/(n) = Dr(A + B/n) A = l + — l o g 7T^ B = - ~ log 7T^ sin(/?) P sin(/3) (C2) (C3) (CA) 92 with 8 = 2air/(2b + s). Thus Optimum Bird Flock Size in Formation Flight Eout = {DP + D^A + B/n))d. (CA) Substituting equation (2) and (3) for Dp and Di respectively, and grouping terms gives Eout = (kpU2 +ki(A + B/n)U~2)d. (Ch) The optimum cruising speed is given by / f c \ 1 / 4 U m r ( n ) = [ ^ - j (A + B/n)1'*. (C6) Airborne time to fly a distance d is Finally, the flying cost at U = Umr(n) is Eout = 2dy/k~k~p^A + B/n. (C8) Ein is given by Ein = \(T-ta). (C9) Substituting equation C7 for ta in the above equation gives Ein = \ ( r - d(^j ' (A + S / n ) 1 / 4 ) (CIO) The ratio = |^ = VJi- - — L . (Cll) Eout 2dyfk^y/A + B/n V ; Differentiating the above ratio with respect to n gives dR(n) B\(2ky\3/\An + Bf'^T - 3 ^ / 4 r c ) d n ~ 8dJbf / 4JbJ / 2n 5 / 4(An + S ) V 4 93 = 0. (C12) Optimum Bird Flock Size in Formation Flight Finally, solving for n gives the optimum flock size as a function of wing-tip spacing, B nopt(s) = 81 kp 16 TT (C13) since both A and B are a function of s (Equations C3 & C4). Finally, subsituting equation (62) in the above equation gives nopt(s) — B l/riopt - A Appendix D Upper Bound for Maximum Flock Size Let the maximum flock size n* be the flock size at which a member of a flock has the same benefit as a solo flyer. That is R(l) = R(n*), X ( T ~ d [ f c ] ( T - d [ £ ] V 4 nV^ (DI) 2d ^/ kp • rtjj 2d yj kp • k{ Solving for n*, Vn7= 1 + d kp 1 1/4 3/4 71*' — d kp 1/4 Given that Umr = (ki/kp)1/* the above equation can written as d n 3 / 4 d 1 -y/n7 = 1 + T U„ T U„ Collecting terms gives n 1/2 d The r.h.s of the above equation, in terms nopt is 3 ! / 4 _ T — Tl. 2 o p t ~ d (D2) (D3) (DA) (Db) 94 Optimum Bird Flock Size in Formation Flight Thus equation D4 can be written as n 3 / 4 - 1 n 1/2 _ 0 I/4 — 7inopt • {DG) Solving for n o p t gives 16 81 3/4 1/2 , nJ - 1 (D7) While it is not possible to solve for n* in the above equation, an upper bound for this variable can be calculated since the denominator, n J 2 — 1, is less than n J 2 16 n o p t > -7Z 3/ 4 — 1 ~\ 4 1/2 (D8) Furthermore, n^ 4 4= < n^/4 it follows that n o p t > 16 81 1/4 n 1/2 ^ 1 6 r 1/4 < 18 K (D9) Finally, the upper bound for n* can be written as 81 • nopt > (D10) 95
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Optimum bird flock size in formation flight Kshatriya, Mrigesh 1990
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Title | Optimum bird flock size in formation flight |
Creator |
Kshatriya, Mrigesh |
Publisher | University of British Columbia |
Date Issued | 1990 |
Description | A theoretical model of flock size in migrating birds is developed. Although previous models of formation flight in birds show improved flight performance, they do not explain flock size variation across bird species or at different times of the year for a given bird species. This model captures some of the diversity in flock size observed in nature by incorporating energetic costs of flight and energy income from foraging. It turns out that within a myriad of possible flock sizes there - is one that is optimal for maximizing energetic efficiency (net energetic gain/energy expenditure) for a given maximum range speed, which minimizes flying cost per unit distance flown, and under certain migration conditions (i.e. flight distance and total time to complete the journey). Net energetic gain from foraging equals the rate of prey encountered times the time spent foraging. Energy expenditure from flying is determined from formation flight theory for a fixed wing aircraft. The benefit of formation flight, as derived from an approximation technique, is represented in close-form. This expression is a function of flock size and wing-tip spacing (WTS) and simplifies flight cost calculations. Under certain WTS, a good approximation to the induced drag for a member of a flock of size n is 1/nth of the induced drag of a single bird. In addition, optimum flight speed of a flock is (1/n)⅟₄ of the optimum flight speed of a single individual. The approach taken here allows the prediction of flock size in migrating birds. Model results are discussed in relation to observation of flock size under various migration conditions. If migration is constrained by hours of daylight, seasional variation in flock size is expected if the start time of the north and southward migration are asymmetrical with respect to the summer solstice (June 21). Under certain conditions, such as long non-stop migration, solo flight is an optimum migratory strategy. |
Subject |
Birds -- Flight |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-11-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0098469 |
URI | http://hdl.handle.net/2429/30004 |
Degree |
Master of Science - MSc |
Program |
Zoology |
Affiliation |
Science, Faculty of Zoology, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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