Field estimates of body drag coefficient on the basis of dives in passerine birds

Abstract

During forward flight, a bird’s body generates drag that tends to decelerate its speed. By flapping its wings, or by converting potential energy into work if gliding, the bird produces both lift and thrust to balance the pull of gravity and drag. In flight mechanics, a dimensionless number, the body drag coefficient (CD,par), describes the magnitude of the drag caused by the body. The drag coefficient depends on the shape (or streamlining), the surface texture of the body and the Reynolds number. It is an important variable when using flight mechanical models to estimate the potential migratory flight range and characteristic flight speeds of birds. Previous wind tunnel measurements on dead, frozen bird bodies indicated that CD,par is 0.4 for small birds, while large birds should have lower values of approximately 0.2. More recent studies of a few birds flying in a wind tunnel suggested that previous values probably overestimated CD,par.
We measured maximum dive speeds of passerine birds during the spring migration across the western Mediterranean. When the birds reach their top speed, the pull of gravity should balance the drag of the body (and wings), giving us an opportunity to estimate CD,par. Our results indicate that CD,par decreases with increasing Reynolds number within the range 0.17–0.77, with a mean CD,par of 0.37 for small passerines. A somewhat lower mean value could not be excluded because diving birds may control their speed below the theoretical maximum. Our measurements therefore support the notion that 0.4 (the ‘old’ default value) is a realistic value of CD,par for small passerines.

Full text

The mechanical power required for flight in birds can be
calculated at different airspeeds according to a simple theory
(Pennycuick, 1975; Pennycuick, 1989). For cruising flapping
flight, three main power components sum to give the total
mechanical power: induced power arising from the rate of work
required to support the weight of the bird; profile power required
to overcome the drag of the flapping wings; and parasite power
required to overcome the drag of the body. Parasite drag depends
on both skin friction and the pressure drag caused by the body.
The skin friction drag component arises as a result of the body
surface roughness and the associated viscous shearing forces
tangential to the surface, while the parasite (or form) drag
results from the distribution of pressure normal to the surface
(e.g. McCormick, 1995). The different components of drag
responsible for any of the three power components can be
obtained from force measurements on mounted models in wind
tunnels (Rae and Pope, 1984). Using this method, the parasite
drag of bird bodies has been measured several times
(Pennycuick et al., 1988; Tucker, 1990). These estimates, using
dead, frozen bird bodies or models, have yielded body drag
coefficients within the range 0.14–0.4.
By convention, the drag coefficient multiplied by the body
frontal area yields the same drag as a flat plate with a fictitious
shape having a drag coefficient of unity and exerting the same
dynamic pressure in the airflow as the bird’s body. It was
observed that the frozen bird bodies caused turbulence in the
boundary layer (Pennycuick et al., 1988), and it has been
suggested that the body drag of a live, free-flying bird could
be lower than that of a mounted bird body. Tucker (Tucker,
1990) measured the drag of a wingless peregrine falcon (Falco
peregrinus) body in a wind tunnel and found a C
D,par
of 0.24,
while a smooth surface model gave a C
D,par
of 0.14.
Pennycuick et al. (Pennycuick et al., 1996) measured the
wingbeat frequency of a thrush nightingale Luscinia luscinia
and a teal Anas crecca and calculated that, to obtain a match
between the speed of minimum wingbeat frequency and the
aerodynamic model prediction of the minimum power speed
(V
mp
), C
D,par
would have to be 0.1 or even lower. When
calculating the mechanical power required for flight, for
example according to the programs of Pennycuick
(Pennycuick, 1989), a realistic value for C
D,par
has to be
assumed. Depending on the value used, derived properties such
1167
The Journal of Experimental Biology 204, 1167–1175 (2001)
Printed in Great Britain © The Company of Biologists Limited 2001
JEB3065
During forward flight, a bird’s body generates drag that
tends to decelerate its speed. By flapping its wings, or by
converting potential energy into work if gliding, the bird
produces both lift and thrust to balance the pull of gravity
and drag. In flight mechanics, a dimensionless number, the
body drag coefficient (C
D,par
), describes the magnitude of
the drag caused by the body. The drag coefficient depends
on the shape (or streamlining), the surface texture of the
body and the Reynolds number. It is an important variable
when using flight mechanical models to estimate the
potential migratory flight range and characteristic flight
speeds of birds. Previous wind tunnel measurements on
dead, frozen bird bodies indicated that C
D,par
is 0.4 for
small birds, while large birds should have lower values of
approximately 0.2. More recent studies of a few birds flying
in a wind tunnel suggested that previous values probably
overestimated C
D,par
.
We measured maximum dive speeds of passerine
birds during the spring migration across the western
Mediterranean. When the birds reach their top speed, the
pull of gravity should balance the drag of the body (and
wings), giving us an opportunity to estimate C
D,par
. Our
results indicate that C
D,par
decreases with increasing
Reynolds number within the range 0.17–0.77, with a mean
C
D,par
of 0.37 for small passerines. A somewhat lower mean
value could not be excluded because diving birds may
control their speed below the theoretical maximum. Our
measurements therefore support the notion that 0.4 (the
‘old’ default value) is a realistic value of C
D,par
for small
passerines.
Key words: aerodynamics, bird, drag, flight, body drag coefficient,
passerine, diving speed, migration.
Summary
Introduction
FIELD ESTIMATES OF BODY DRAG COEFFICIENT ON THE BASIS OF DIVES IN
PASSERINE BIRDS
ANDERS HEDENSTRÖM
1,
* AND FELIX LIECHTI
2
1
Department of Animal Ecology, Ecology Building, SE-223 62 Lund, Sweden and
2
Swiss Ornithological Institute,
CH-6204 Sempach, Switzerland
*e-mail: anders.hedenstrom@zooekol.lu.se
Accepted 14 December 2000; published on WWW 26 February 2001

1168
as predicted speeds for minimum power and maximum range
will be affected. On the basis of wingbeat frequency
measurements, Pennycuick now recommends that a C
D,par
value of 0.1 should be used as a default value with his programs
for birds with streamlined bodies (Pennycuick, 1999).
Estimating C
D,par
is notoriously difficult, and current values
should be considered as provisional until more direct
measurements of the mechanical power of birds become
available.
We have used a different approach to estimate C
D,par
in
small passerines. During migration, some birds occasionally
terminate their migratory flights by diving steeply towards the
ground, where they land. We obtained measurements of such
dives during routine tracking of migratory birds by means of
tracking radar. The tracks of a few of these birds were
measured for such a long time that the terminal speed could be
estimated with some confidence. The underlying assumption is
that, when the speed reaches a constant value, then the drag
will balance the pull of gravity.
Theory
When gliding or diving at constant speed along a path
inclined at an angle θ to the horizontal, the forces acting on a
bird will be at equilibrium, i.e. there will be no net forces
causing any acceleration. At any glide angle different from a
completely vertical dive, some lift must be generated, which
means that the wings have to be opened to some extent. In a
completely vertical dive, the wings are typically folded against
the body. By measuring the airspeed (V) and the angle of the
flight path with respect to the horizontal during a steady dive,
the lift L and drag D forces can be calculated on the basis of
airspeed and sinking speed (V
s
) as follows:
L = mg[1 − (V
s
/V)
2
]
1/2
, (1)
and
where m is body mass and g is the acceleration due to gravity.
Equation 2 gives the total drag acting on the bird, which can
be written as the sum of three aerodynamic components as:
D = D
ind
+ D
pro
+ D
par
, (3)
where the components are the induced, profile and parasite
drag, respectively. We neglected the possible effect on drag of
interference between the wings and body. Induced drag
represents the cost of generating lift, profile drag is the drag of
the wings and parasite drag is due to skin friction and the form
drag of the body. Induced drag is:
where k is the induced drag factor (the efficiency of the wing
in generating lift; ideally k=1, for a wing with perfect elliptical
lift distribution), L is lift (equation 1), ρ is air density, b is
wingspan and V is airspeed along the glide/dive path. Profile
drag is:
D
pro
= GρV
2
S
w
C
D,pro
, (5)
where S
w
is the wing area and C
D,pro
is a dimensionless drag
coefficient. Parasite drag is:
D
par
= GρV
2
S
b
C
D,par
, (6)
where S
b
is the body frontal area of the bird and C
D,par
is the
body drag coefficient, i.e. the parameter that we ultimately
wish to estimate. C
D,par
is a function of Reynolds number Re
(see below). By substituting equations 2 and 4–6 into
equation 3 and rearranging, the body drag coefficient can be
calculated as:
In a vertical dive with completely folded wings, V
s
=V, S
w
=0
and L=0, and hence the body drag coefficient becomes
C
D,par
=2mg/(ρS
b
V
2
), which is simply equation 6 rearranged;
the only aerodynamic force arises from the drag of the body.
To estimate C
D,par
for birds diving vertically with folded wings
would be desirable since this involves a minimum number of
assumptions. However, most birds tracked by radar showed
inclined dives, so equation 7 has to be used. The bird tail
(Thomas, 1993) and/or body can generate some lift, as
indicated from measurements of zebra finches Taenopygia
guttata (Tobalske et al., 1999), but there is no established way
of estimating the lift and the associated induced drag for the
body and tail. We will therefore assume that the wings are
responsible for the lift causing the inclined dive angles. By
assuming realistic values for the variables of equation 7 and
measuring the terminal airspeed of gliding dives, we were able
to estimate C
D,par
.
Calculating the drag
The total lift and drag were estimated for each bird track
using equations 1 and 2. From the lift, we calculated the wing
area needed to generate the lift from the relationship
L=GρV
2
S
w
C
L
, assuming a lift coefficient, C
L
, of 0.5 (see
Pennycuick, 1968). We investigated the effect of varying the
values of the lift coefficient by ±50% to investigate the
sensitivity of the results to assumptions. When the wings are
flexed, the mean chord remains almost constant (Rosén and
Hedenström, 2001), and this was used to calculate an effective
wingspan associated with the wing area required. Here, we
assumed that only the wings generate any useful lift force. The
mean chord of the wing c was calculated as c=S
max
/b
max
, where
S
max
is the wing area and b
max
is the span when the wings are
in the fully outstretched position. For the profile drag
coefficient, we assumed C
D,pro
=0.014 (Pennycuick, 1989), but
we also investigated the effects of varying C
D,pro
by ±50%
around this default value. Pennycuick et al. (Pennycuick et al.,
1992) found a mean value of 0.02 for the wing of a Harris’
hawk (Parabuteo unicinctus), but also lower values in the
neighbourhood of 0.014. The body frontal area, which is the
(7)
C
D,par
=
−
S
w
C
D,pro
−
.
4kL
2
πρ
2
b
2
V
4
1
S
b






2mgV
s
ρV
3
(4)
D
ind
=
,
2kL
2
πρb
2
V
2
(2)
D =,
mgV
s
V
A. HEDENSTRÖM AND F. LIECHTI

1169Body drag coefficient of birds
reference area when calculating the parasite drag, was
calculated on the basis of body mass m as S
b
=0.00813m
0.666
(Pennycuick, 1989). When calculating the drag components,
we used the calculated air density for the relevant mid-
point altitude of each dive (US standard atmosphere; Lide,
1997).
Reynolds number
The Reynolds number Re is a dimensionless index
describing the relative importance of inertial and viscous
forces; it is calculated as Re=Vl/ν, where V is airspeed, l is a
characteristic length and ν is the kinematic viscosity of air,
defined as dynamic viscosity divided by density (see Batchelor,
1967; for examples of drag and Re relevant to biology, see
Vogel, 1994). We calculated Re using the diameter of the body
frontal area as the reference length. Dynamic viscosity µ (Pas)
was estimated for the relevant dive midpoint altitude for each
track according to the equation µ=1.79×10
−5
−(3.32×10
−10
)Z,
where Z is altitude (data from Lide, 1997). Air density
estimated for the relevant altitude was used when calculating
the kinematic viscosity ν.
Materials and methods
Study sites
Bird migration was studied by tracking radar during spring
1997 in an area of the Western Mediterranean (from 19 March
to 26 May). One site was situated on Mallorca, the largest of
the Balearic islands, close to the southern tip of the island
approximately 2km from the coast (3°4′E, 39°18′N; 10m
above sea level). A second observation site was on the southern
coast of Spain, approximately 25km east of Malaga, 100m
inland from the east–west oriented coastline (4°8′W, 36°44′N,
20m above sea level). Birds arriving on Mallorca had flown at
least 300km across the open sea, whereas from the north
African coast to Malaga they had covered approximately
170km.
Data collection and analysis
At both sites, flight paths of individual birds were recorded
by an X-band tracking radar of the type ‘Superfledermaus’
(for details of the characteristics of this type of radar, see
Bruderer, 1997). The fluctuation of the echo-signature,
representing the wingbeat pattern of the bird, was
continuously digitised, whereas the spatial position of the
tracked birds (X, Y and Z coordinates) was automatically
saved every second. Wingbeat frequency was analysed by
selecting parts of a flight path with a low noise level and a
clear wingbeat pattern and then determining the wingbeat
frequency using Fast Fourier transformation analysis. For this
study, we selected tracks consisting of two obviously
different flight conditions. In one part (generally at the
beginning of the track), wingbeat frequency had to be clearly
identifiable, whereas in the other part sinking rate had to be
greater than 20ms
−1
over a period of at least 20s. From these
segments of rapid descent, we selected the parts with a
maximum sinking rate over at least 4s to represent maximum
diving speeds. In total, 16 tracks from Malaga and 23 from
Mallorca were selected for analysis (Table 1). All dives were
tracked during the day.
To match a specific wingbeat frequency to a species, we
used the abundance of migratory species monitored at least
every second day in the area surrounding the radar stations. In
addition, almost all targets were observed and classified
visually by a telescope mounted parallel to the radar antenna.
Unfortunately, exact species identification was not possible
because of the large distances (at least several hundred metres),
but all tracks included in this study refer to passerine birds. The
observers detected no signs of flapping wings when the birds
were diving, indicating that diving birds held their wings
motionless and did not supply any additional power. No signs
that the legs were held in an outstretched position, causing
extra drag, were observed, although this could have been
difficult to detect.
To calculate air density, air pressure and temperature at the
bird’s flight altitude, data were taken from the radio-sonde
measurements at Gibraltar (for the Malaga site) and Palma de
Mallorca (12h UTC). Wind profiles were collected every 4h
at the study sites by tracking helium-filled balloons by radar
up to 4000m above sea level. To calculate horizontal air
speed and diving angle, the flight path of the balloon between
the height at the start and end of the dive was subtracted
from the bird’s flight path. Calculations were performed
incorporating the wind data from the measurements before
and after the track of the dive. According to these two
measurements, diving angle for an individual bird differed
maximally by 12° (mean 3.8°) and horizontal air speed by
7ms
−1
(2.4ms
−1
). In seven cases, data from only one wind
measurement were available.
Body mass and wing morphology
It was impossible to obtain data on body mass and wing
morphology for those individuals tracked by radar. Information
was therefore obtained from Pennycuick (Pennycuick, 1999)
and from our own unpublished database for the species that
were likely candidates on the basis of the wingbeat signatures
obtained prior to the commencement of dives. Species and
morphological data used for the calculations are shown in
Table 2.
Results
General patterns of dives
Information for each tracking is given in Table 1. All tracks
included in this analysis refer to rapid descents from cruising
altitudes during migration of up to 3700m above sea level to
relatively low altitudes. Most descents probably refer to the
termination of migratory flights, although a few may include
birds that continued the flight at a lower altitude. A few
examples of dive patterns are shown in Fig. 1, in which altitude
is plotted with respect to time. For example, bird 267
descended from almost 3500m to approximately 1200m in

1170
80s, but the descent was interrupted a few times during
this period (Fig. 1A). The mean rate of descent (29.9ms
−1
)
was therefore lower than the instantaneous maximum speed
(52.3ms
−1
) achieved over at least 4s. This pattern of
‘stepwise’ descent was also evident in other birds, e.g. 232 and
352 (Fig. 1C,D), while others showed rather continuous and
uninterrupted descents. Gliding/diving angles varied from 50°
to a near vertical dive of 83.5°. There was no significant
relationship between maximum speed along the flight path and
dive angle (analysis of covariance, ANCOVA, F
1,25
=0.44,
P=0.51; Fig. 2). Species category had no significant effect on
speed (F
12,25
=0.87,P=0.58), while the effect of body mass was
marginally significant (F
1,25
=4.18, P=0.051). The maximum
speeds ranged from 23.8 to 53.7ms
−1
with a mean value of
37.5±6.9ms
−1
(mean ± S.D., N=39) There was a positive
relationship between the height at the middle of the dive
A. HEDENSTRÖM AND F. LIECHTI
Table 1. Data for 39 tracks of diving birds used to estimate body drag coefficients
ρ Z
1
Z
2
∆tV
s
θ V
Species No. (kgm
−3
) (m) (m) (s) (ms
−1
) (degrees) (ms
−1
) C
D,par
Wheatear 378 1.06 1051 894 4 39.3 83.3 39.5 0.41
Wheatear 268 1.16 716 265 15 30.1 69.8 32.0 0.54
Meadow pipit 212 1.19 258 141 4 29.3 61.7 33.2 0.42
Meadow pipit 342 1.15 425 330 4 23.8 52.4 30.0 0.50
Nightingale 283 1.16 619 368 8 31.4 60.4 36.1 0.37
Nightingale 319 1.14 942 686 9 28.4 67.9 30.7 0.56
Nightingale 320 1.11 732 607 5 25.0 61.1 28.6 0.63
Barn swallow 267 1.00 2930 2722 4 52.0 83.5 52.3 0.23
Barn swallow 231 1.07 1301 97 32 37.6 74.7 39.0 0.37
Robin 304 1.09 1354 1216 4 34.5 63.8 38.4 0.35
Yellow wagtail 336 1.04 2891 2728 4 40.8 64.0 45.3 0.26
Yellow wagtail 217 1.17 981 821 4 40.0 78.5 40.8 0.31
Yellow wagtail 236 0.99 2347 2035 8 39.0 75.8 40.2 0.38
Yellow wagtail 352 1.11 3060 2910 4 37.5 66.6 40.9 0.31
Yellow wagtail 242 1.15 1865 1644 6 36.8 58.2 43.3 0.24
Yellow wagtail 250 1.09 1764 1547 6 36.2 65.6 39.7 0.33
Yellow wagtail 221 1.16 1510 1375 4 33.8 75.8 34.8 0.43
Yellow wagtail 250 1.08 1987 1793 6 32.3 66.3 35.3 0.42
Yellow wagtail 276 1.12 654 425 8 28.6 72.8 30.0 0.59
Yellow wagtail 336 1.10 1026 927 4 24.8 50.9 31.9 0.43
Spotted flycatcher 297 1.05 3500 3042 10 45.8 59.6 53.1 0.17
Spotted flycatcher 312 1.18 629 547 4 20.5 59.5 23.8 0.77
Redstart 279 1.13 1540 1358 5 36.4 58.9 42.5 0.25
Redstart 289 1.12 1882 1742 4 35.0 55.7 42.4 0.24
Redstart 347 1.12 1409 1279 4 32.5 70.9 34.4 0.42
Reed warbler 232 1.18 691 561 4 32.5 77.2 33.3 0.41
Pied flycatcher 290 0.89 2498 2311 4 46.8 60.6 53.7 0.19
Pied flycatcher 335 1.18 689 545 4 36.0 66.4 39.3 0.28
Pied flycatcher 314 1.05 2558 2273 8 35.6 56.2 42.9 0.24
Pied flycatcher 314 1.16 542 400 4 35.5 72.0 37.3 0.32
Pied flycatcher 340 1.17 809 466 10 34.3 71.3 36.2 0.34
Pied flycatcher 281 1.15 322 211 4 27.8 74.3 28.8 0.55
Serin 282 1.15 224 87 4 34.3 54.0 42.3 0.21
Willow warbler 215 1.06 1791 1572 6 36.5 54.6 44.8 0.19
Willow warbler 209 1.11 2448 2303 4 36.3 71.3 38.3 0.29
Willow warbler 325 1.15 914 796 4 29.5 63.7 32.9 0.35
Willow warbler 229 1.18 427 315 4 28.0 57.8 33.1 0.32
Willow warbler 325 1.16 525 422 4 25.8 50.1 33.6 0.29
Goldcrest 332 1.13 931 826 4 26.3 75.1 27.2 0.49
No. indicates the identification number for the track, ρ is estimated air density at the midpoint altitude [(Z
1
+Z
2
)/2], Z
1
is the altitude at whic
h
the dive starts, Z
2
is the altitude at which the dive segment stops both measured as metres above ground level, ∆t is the duration of the dive o
f
maximum speed, V
s
is vertical sinking speed, θ is dive angle with respect to the horizontal, V is dive speed along the dive path and C
D,par
is the
estimated body drag coefficient.
Species refer to the ‘best guess’ according to wingbeat signature (see Materials and methods).
Scientific names are given in Table 2.

1171Body drag coefficient of birds
and the speed (ANCOVA, F
1,25
=43.2, P<0.001). A simple
regression indicated that dive speed increases by
approximately 5.7ms
−1
per 1000m altitude.
Estimating the body drag
The calculated body drag coefficients (C
D,par
) for all trackings
of diving birds are given in Table 1. The mean estimated C
D,par
was 0.37±0.13 (mean ± S.D., N=39), ranging from 0.17 to 0.77.
We changed the assumed value for C
L
(0.5) by ±50%, but this
had a negligible effect (±0.3%) on the estimated mean C
D,par
.
The assumed value for C
D,pro
(0.014) was also changed by
±50%, which again had a very small effect (±0.15%) on the
estimated C
D,par
. We will therefore use the baseline assumptions
for further analyses in this paper. The estimated C
D,par
showed
a positive relationship with body mass, but the relationship was
not statistically significant (ANCOVA, F
1,25
=3.60, P=0.069).
There was no significant relationship between C
D,par
and diving
angle (ANCOVA, F
1,25
=1.50, P=0.23). Our estimates of C
D,par
did, however show a negative relationship with the Reynolds
number (Fig. 3; ANCOVA, F
1,25
=329.3, P<0.001). The lowest
values of C
D,par
of approximately 0.2 were obtained in the range
of Re from 57000 to 84000 (Fig. 3). The linear regression
equation between C
D,par
and Re was C
D,par
=0.82−(7.5×10
−6
)Re;
when two data points where C
D,par
>0.6 were excluded, the
regression was C
D,par
=0.70−(5.8×10
−6
)Re.
Bird 267 is of particular interest; it dived almost vertically
in a stepwise manner (θ=83.5°) and reached a maximum speed
V of 52.3ms
−1
(Fig. 1A). This bird showed the wingbeat
pattern typical of the barn swallow Hirundo rustica, which
makes this observation especially valuable. The nearly vertical
dive means that the wings were almost completely folded and,
hence, the estimated C
D,par
of 0.23 is probably quite a reliable
estimate for this bird. However, two other birds with near-
vertical dives (θ=83.3° and θ=77.2°) both showed an
estimated C
D,par
of 0.41, which makes us suspect that they
might have been braking using partially opened wings or by
holding their legs/feet outstretched, although this could not be
seen by the observer (see below). These two birds (378 and
232; see Table 1) reached maximum speeds of only 39.5ms
−1
and 33.3ms
−1
, respectively.
Table 2. Body mass, body frontal area and wing morphology used to estimate the aerodynamic properties of the birds tracked by
radar
mS
b
b
max
S
max
Chord
Species (kg) (cm
2
)
a
(m) (m
2
) (m)
Wheatear
Oenanthe oenanthe L. 0.0232 6.6 0.264 0.01366 0.052
Meadow pipit
Anthus pratensis L. 0.0199 6.0 0.273 0.0143 0.052
Nightingale
Luscinia megarhynchos Brehm 0.0197 6.0 0.221 0.01059 0.048
Barn swallow
Hirundo rustica L. 0.0182 5.6 0.328 0.01446 0.044
Robin
Erithacus rubecula L. 0.0182 4.6 0.224 0.01026 0.046
Yellow wagtail
Motacilla flava L. 0.0176 5.5 0.248 0.01051 0.042
Spotted flycatcher
Muscicapa striata Pallas 0.0153 5.0 0.262 0.01209 0.046
Redstart
Phoenicurus phoenicurus L. 0.015 5.0 0.2 0.01006 0.050
Reed warbler
Acrocephalus scirpaceus Hermann 0.0123 3.9 0.2 0.00779 0.039
Pied flycatcher
Ficedula hypoleuca Pallas 0.012 4.4 0.2 0.00873 0.044
Serin
Serinus serinus L. 0.0114 3.9 0.214 0.00828 0.039
Willow warbler
Phylloscopus trochilus L. 0.0087 4.0 0.194 0.00768 0.040
Goldcrest
Regulus regulus L. 0.0054 3.5 0.146 0.00504 0.035
a
Calculated as S
b
=0.00813m
0.666
(Pennycuick, 1989).
m, body mass; S
b
, body frontal area; S
max
, maximum wing area; b
max
, maximum wingspan; C, mean wing chord.

1172
Discussion
Diving speeds
We have presented radar tracks of migrating passerine
birds commencing rapid descents from their cruising altitude.
The birds were tracked during their spring migration, and the
dives probably represent the termination of migration flights
across the Mediterranean Sea. The bird with the steepest dive
angle (83.5°) reached a maximum speed of 52.3ms
−1
, which
is near the maximum speed recorded at a lower dive angle for
another bird (53.7ms
−1
at dive angle 60.6°; bird 290). It
seems that small birds can reach very high diving speeds,
such as the maximum speeds recorded (>50ms
−1
), but that
many birds control the speed of their dives by adjusting the
positions of their wings and perhaps their legs to increase
drag. This is supported by the strong correlation between the
height at the midpoint of the dive and the dive speed, which
might be caused by greater speed control at lower altitudes
where the birds might head for a specific site to land. Any
characteristic flight speed is expected to increase by
approximately 5% per 1000m increase in altitude because
air density decreases with altitude. This effect would cause
a reduction in the mean dive speed (37.5ms
−1
) of
approximately 1.9ms
−1
per 1000m reduction in altitude,
which is less than we observed (5.7ms
−1
per 1000m change
in altitude). Nevertheless, we believe that the maximum
speeds achieved will represent situations in which the
minimum possible drag will be exhibited. The mean and
maximum diving speeds recorded are comparable with
measurements of maximum diving speeds in larger bird
species. For example, large falcons, famous for their stoops
when attacking aerial prey, have been recorded as achieving
top speeds in the range 39–58ms
−1
(Alerstam, 1987; Peter
and Kestenholz, 1998; Tucker et al., 1998). Tucker et al.
(Tucker et al., 1998) even mention preliminary measurements
of speeds of 70ms
−1
in wild peregrines (Falco peregrinus)
A. HEDENSTRÖM AND F. LIECHTI
221
0
500
1000
1500
2000
0 50 100 150 20
0
267
0
1000
2000
3000
4000
0 50 100 150
232
0
500
1000
1500
0 100 200 300
Altitude (m)
352
0
1000
2000
3000
0 100 200 300
212
0
500
1000
1500
0 50 100 150 200
Time (s)
276
0
1000
2000
3000
4000
0 100 200 300
Time (s)
A
B
C
D
E
F
Fig. 1. Altitude versus time for passerine birds showing rapid
descents during migratory flights as recorded by radar. On the basis
of their wingbeat signature, the six tracks illustrated are referred to as
(A) barn swallow, (B) yellow wagtail, (C) reed warbler, (D) yellow
wagtail, (E) meadow pipit and (F) yellow wagtail. The numbers
shown in each panel refer to the track identification number also
given in Table 1.
0
10
20
30
40
50
60
70
30 40 50 60 70 80 90 100
Dive angle (degrees)
Maximum speed (m s
-1
)
Fig. 2. Maximum airspeed along the flight path versus dive angle
(horizontal=0°) in 39 dive measurements of passerine birds tracked
by radar. The maximum speeds refer to segments of tracks at least 4s
long. Each track is represented by only one such segment (see
Table 1).
0
0.2
0.4
0.6
0.8
30 000 50 000 70 000 90 000
Reynolds number, Re
Body drag coefficient, C
D,par
Fig. 3. Estimated body drag coefficients (C
D,par
) versus Reynolds
number (Re) for 39 dive measurements of passerine birds tracked by
radar. Reynolds number was calculated using the diameter of the
body frontal area as the reference length. The regression equation
was C
D,par
=0.82−7.5×10
−6
Re (t=4.77, P<0.001).

1173Body drag coefficient of birds
that would represent the maximum speed ever measured for
a bird using reliable methods. Our data refer to passerines,
which are much smaller than the birds with the previously
reported top speeds. Even if only a few dives achieved speeds
in excess of 50ms
−1
, this indicates that small birds also have
the capacity to achieve very fast speeds when diving, and that
the size of the bird has little or no effect on the maximum
dive speed.
There is an expected size-dependent maximum glide/dive
speed (e.g. Andersson and Norberg, 1981), but the available
data lend little support to this prediction. We found no
statistically significant (P=0.051) relationship between
maximum speed and body mass within our data set of rather
limited size range. The lack of such a relationship may
not, however, be a critical rejection of the size-dependent
maximum diving speed since birds may control their speeds
below the theoretical maximum for other than aerodynamic
reasons. Tucker (Tucker, 1998) derived theoretical top speeds
for a falcon of 89–112ms
−1
with C
D,par
=0.18, and even up to
138–174ms
−1
for a C
D,par
of 0.07. Not even the peregrine
mentioned above achieved such high speeds.
Sources of error
Before discussing the body drag coefficients obtained, we
will briefly discuss some potential sources of error that could
have influenced the results. Identification of species was
indirect because it was based on wingbeat signatures obtained
from the radar echo signal. This method has been used before
(e.g. Bloch et al., 1981) and, even if the species was wrong,
the wingbeat frequency is strongly related to the size of
the bird (Pennycuick, 1996). Recently, the radar wingbeat
signatures of identified swallows and house martins (Delichon
urbica) were found to agree with those obtained from the same
species observed in a wind tunnel (L. Bruderer, personal
communication). Track 267 was positively identified as a barn
swallow, and that bird achieved the second highest diving
speed recorded in an almost vertical dive.
Horizontal winds will affect the horizontal airspeed derived
from the trackings and, hence, the estimated dive angles with
respect to the air. We calculated the horizontal airspeeds on the
basis of the mean wind speed measured before and after each
bird track registration, separated by 4h. We also calculated the
horizontal airspeed on the basis of the first and second wind
measurements separately, but the differences were quite small.
Vertical winds were not measured, and rising thermals or
sinking air could have affected the estimated vertical speeds.
However, even in the tropics, where thermals are strong, they
are of the order of 2–5ms
−1
(Pennycuick, 1998), which is
small compared with the speeds measured for the birds when
diving. In conclusion, there are a few potential sources of error,
but they are not expected to be systematic and will not,
therefore, influence the general conclusions of this study to any
significant degree.
Values of C
D,par
Our results gave values of C
D,par
between 0.17 and 0.77,
with a mean of 0.37. The mean value is close to the ‘old’
default value for small passerines suggested by Pennycuick
(Pennycuick, 1989). However, we suspect that some birds
controlled their speeds below their potential maximum by
increasing drag, resulting in some of the quite large values
(>0.4). The gyrfalcon (Falco rusticolus) studied by Tucker et
al. (Tucker et al., 1998) controlled its speed by changing the
angle of attack of its cupped wings and by lowering its tarsi
and feet from their normal position tucked up under the tail.
Therefore, we think that some of the larger values (0.6–0.7)
might be unrealistic for bird bodies, but that our mean estimate
of C
D,par
=0.37 could be typical for passerines at cruising
speeds. This is near the ‘old’ default value of 0.4, while our
lowest values are close to the value of 0.24 obtained for frozen
bodies of large birds measured in a wind tunnel (Pennycuick
et al., 1988).
Tucker (Tucker, 1990) took great care when measuring a
frozen peregrine body and obtained a C
D,par
of 0.24; he also
prepared a smooth surface model of the peregrine that gave a
C
D,par
of 0.14. Our lowest values are within this range,
although our measurements refer to passerines. Pennycuick et
al. (Pennycuick et al., 1996) arrived at even lower values for a
thrush nightingale and a teal, using the speed of minimum
wingbeat frequency and the calculated speed of minimum
mechanical power. To get the two speeds to match, the
calculated mechanical power curve was shifted by reducing
C
D,par
to 0.07. In a wind tunnel study of the mechanical power
of a swallow (Hirundo rustica), Pennycuick et al. (Pennycuick
et al., 2000) derived a novel method for estimating the
mechanical power required to fly on the basis of the vertical
accelerations of the body and the wingbeat kinematics during
the course of a wingbeat. They did not, however, find close
agreement between the speeds of minimum wingbeat
frequency and estimated minimum power, which may question
the assumption that these minima should coincide. Be that as
it may, on the basis of our data, we may conclude that C
D,par
for passerines used for calculating flight performance in birds
(for example, by using the programs published by Pennycuick,
1989), should be of the order of 0.4 (the ‘old’ default value).
We found no values as low as 0.1, the ‘new’ default value
recommended by Pennycuick (Pennycuick, 1999), which is
approximately half our lowest value (0.2; Fig. 3). One
advantage of the present data is that they were obtained using
a different method from those of previous studies, and yet we
achieved surprisingly realistic values.
The negative correlation between C
D,par
and Re shown in
Fig. 3 illustrates a potentially interesting feature of the drag of
bird bodies. At some critical value of Re, there is usually a
transition from a laminar to a turbulent boundary layer, which
is associated with a reduction in C
D,par
. In a circular cylinder,
this transition occurs at approximately Re=300000 (Anderson,
1991), but this point can be reduced to approximately
Re=50000 by mounting a thin wire just in front of the leading
edge on model aircraft wings (Simons, 1994). Diving passerine
birds may be operating in the zone of Re where turbulence in
the boundary layer can be induced and used to reduce C
D,par

1174
(Pennycuick, 1989), although it remains to be demonstrated
that this is the mechanism for real bird bodies. Passerine
birds flying at typical cruising speeds (10–20ms
−1
) operate at
Re≈30000, which is clearly below the range of Re of the diving
birds (see Fig. 3). The lowest values of C
D,par
were found at
Re⭓57000, again suggesting that the ‘old’ default value of 0.4
could be typical of passerine birds in cruising flight.
The aerodynamic performance of passerine birds is certainly
impressive, allowing them to migrate long distances, such
as across the Sahara and the Mediterranean Sea, without
refuelling. Changing C
D,par
to much lower values, as suggested
by recent studies, will have quite dramatic consequences for
our interpretation of flight performance (Pennycuick et al.,
1996). For example, the lift to drag ratio will increase if C
D,par
is reduced and, hence, the potential flight range calculated on
the basis of flight mechanical theory will increase. Also,
characteristic flight speeds, such as those associated with
minimum power, maximum range and maximum overall
migration speeds (sensu Hedenström and Alerstam, 1995),
will increase if C
D,par
is reduced. Because of the difficulties
involved in challenging birds to minimise drag when diving in
the field and methodological difficulties when studying birds
in wind tunnels, we predict that the last word concerning body
drag coefficients in birds has not yet been written.
List of symbols
b wingspan
b
max
maximum wingspan
c mean chord of wing
C
D,par
parasite drag coefficient
C
D,pro
profile drag coefficient
C
L
lift coefficient
D drag
D
ind
induced drag
D
par
parasite drag
D
pro
profile drag
g acceleration due to gravity
k induced drag factor
l length, diameter of body
L lift
m body mass
Re Reynolds number
S
b
body frontal area
S
max
maximum wing area
S
w
wing area
V airspeed
V
mp
minimum power speed
V
s
sinking speed
X, Y, Z, spatial coordinates
Z
1
start altitude of dive measurement
Z
2
stop altitude of dive measurement
∆t time interval
θ dive angle with respect to horizontal
µ dynamic viscosity
π ratio of circumference to diameter of a circle
ρ air density
ν kinematic viscosity
We are grateful to Dieter Peter and Herbert Stark for
collecting a large part of the diving data on migrating birds, and
T. Steuri for maintenance of the radar and the development of
the recording equipment. B. Bruderer directed the whole project
in the Western Mediterranean. We are grateful to Lukas Jenni,
Geoff Spedding and two anonymous referees for constructive
comments on the manuscript. Financial support was obtained
from the Swiss National Science Foundation (No. 31-432
42.95), the Silva Casa Foundation and the Swedish Natural
Science Research Council.
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