Swimming gaits, passive drag and buoyancy of diving sperm whales Physeter macrocephalus
1 Sea Mammal Research Unit, University of St Andrews, Fife, KY16 8LB,
Scotland
2 Biology Department, Woods Hole Oceanographic Institution, Woods Hole, MA
02543, USA
3 Department of Applied Ocean Physics and Engineering, Woods Hole
Oceanographic Institution, Woods Hole, MA 02543, USA
* Author for correspondence at address 1 (e-mail: pm29{at}st-and.ac.uk)
Accepted 22 March 2004
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Summary |
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Key words: drag, buoyancy, sperm whale, Physeter macrocephalus, swimming gait, diving
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Introduction |
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Hydrodynamic drag and basal metabolic rate are key determinants of optimal
swimming speed to minimize cost-of-transportation
(Fish and Hui, 1991;
Williams et al., 1993
). Low
drag has obvious benefits for migrating animals, which may travel large
distances without access to food (Sumich,
1983
), and also for diving mammals, such as the sperm whale, which
must travel between an oxygen source at the surface and food supplies at
depth. A theoretical drag coefficient, based on a series of flat plates in a
turbulent regime, was calculated as 0.0026 for the fin whale Balaenoptera
physalus (Bose and Lien,
1989
). Based on thrusting efficiency, Fish
(1998
) calculated the drag
coefficient of fast-swimming killer whales Orcinus orca to be 0.0029.
A drag coefficient of 0.0056 was calculated from glides for the Steller sea
lion Eumetopais jubatus with Reynold's number of
5x106 (Stelle et
al., 2000
).
Buoyancy forces, which can be stronger than drag forces, arise from gases
carried by a diving animal and differences in density between non-gaseous
animal tissues and the surrounding medium
(Lovvorn and Jones, 1991;
Beck et al., 2000
). While
buoyancy forces may have only a minor influence on horizontally transiting or
migrating animals (Ogilvy and DuBois,
1982
; Stelle et al.,
2000
), they add (or subtract) directly, to drag forces during
vertical diving (Skrovan et al.,
1999
). While drag forces always oppose the direction of movement
and increase with speed, buoyancy acts vertically and is not affected by
speed. Buoyancy due to air carried by a diving animal is strongly affected by
depth, with rapid changes in buoyancy near the surface as hydrostatic pressure
reduces air volume. Buoyancy from tissue density is not much affected by
pressure as both seawater and animal tissue have low compressibility
(Skrovan et al., 1999
). With
the exception of polar waters, seawater temperature decreases (and density
increases) with depth, which provides positive buoyancy to a diving animal as
it moves into colder, deeper water (Clarke,
1970
). Such temperature gradients may reduce external body
temperatures during long dives in cold, deep water, particularly if blood flow
to extremities is restricted
(Schmidt-Nielsen, 1997
).
Tissue buoyancy also varies seasonally, as the amount of relatively light
lipids carried by marine mammals changes between periods of feeding and
fasting (Webb et al., 1998
;
Beck et al., 2000
;
Biuw et al., 2003
).
Marine mammals employ diverse swimming styles referred to as `gaits' (i.e.
steady fluking, gliding, stroke-and-glide, porpoising) that appear to result
in lower transportation costs when used in the appropriate conditions
(Williams et al., 2000).
Differences in buoyancy forces, both within dives and across animals,
particularly influence use of glides during ascent and descent. Diving
bottlenose dolphins glide more during descent after air volumes have collapsed
due to hydrostatic pressure (Skrovan et
al., 1999
). Prolonged glides were made during descent by leaner
Weddell seals Leptonychotes weddellii, while fatter seals employed
stroke- and-glide swimming (Sato et al.,
2003
). Gliding during descent appears to reduce oxygen
consumption, resulting in longer dives and more efficient foraging
(Williams et al., 2000
).
Broadly speaking, species with negative buoyancy appear to glide more during
descent (e.g. phocid seals, balaenopteridae;
Williams et al., 2000
), while
positively buoyant species glide more during ascent (e.g. balaenids;
Nowacek et al., 2001
). The
vertical velocity of elephant seals Mirounga leonina, whose buoyancy
was altered experimentally, changed during descent, but ascent velocity
appeared to be unaffected (Webb et al.,
1998
). Similarly, leaner grey seals Halichoerus grypus in
the post-moult period had faster descent velocities, but also, seemingly
paradoxically, faster ascent velocities
(Beck et al., 2000
). These
different influences of buoyancy on descent and ascent velocities may reflect
the fact that these animals glide during descent, but actively swim during
ascent (Webb et al., 1998
;
Williams et al., 2000
).
Sperm whales are accomplished divers, making dives greater than 1000 m
depth and 1 h in duration (Watkins et al.,
2002). Their ability to accomplish these dives successfully
depends critically on the drag and buoyancy forces acting on them, and the
speed and style of swimming employed to overcome these forces and successfully
capture prey. There has been interest in the buoyancy of sperm whales at depth
(see Whitehead, 2003
), based
upon the premise that sperm whales would benefit from obtaining neutral
buoyancy at depth (Clarke 1970
,
1978c
). Clarke
(1978c
) analyzed factors
affecting the relative density of a diving sperm whale and the seawater
medium, and determined that if sperm whale tissue (without air) is neutrally
buoyant at the surface, a whale would have a positive buoyancy of
100650 N over much of its dive. To achieve neutral buoyancy, Clarke
(1970
,
1978c
) proposed that sperm
whales might increase tissue density by cooling oils in their massive
spermaceti organ. Clarke (1970
)
further proposed that sperm whales might heat their oils actively during
ascent to gain positive buoyancy. While this idea has been debated
(Ridgway, 1971
;
Norris and Harvey, 1972
;
Cranford, 1999
;
Madsen et al., 2002
;
Whitehead, 2003
), no data have
been published to date reporting measurements of buoyancy or swimming patterns
from diving sperm whales.
Our goal here is to describe the swimming behaviour of sperm whales, and to
relate their behaviour to the drag and buoyancy forces acting on them. We
detail the movements of diving sperm whales using a 500 g solid-state archival
tag containing a depth sensor, an audio-band acoustic recorder, and 3-axis
accelerometers and magnetometers. When sampled at high rates this sensor suite
can be used to observe fine-scale details of the tagged whale's behaviour,
including its 3-D orientation, fluke strokes and vertical velocity
(Johnson and Tyack, 2003). We
analyze data from this instrument to describe the swimming gaits employed by
sperm whales during deep dives and shallow surface dives. By fitting measured
acceleration during glides to a model of drag and buoyancy forces, we obtain
the first field estimates of drag and both air- and tissue-induced buoyancy
forces that act on diving sperm whales.
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Materials and methods |
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Tag design and attachment
We recorded the diving and swimming behaviour of sperm whales Physeter
macrocephalus L. using a high-resolution digital recording tag `Dtag'
deployed on sperm whales using suction cups. The Dtag sensors include a
hydrophone, a depth sensor, a temperature sensor, and 3-axis accelerometers
and magnetometers (Johnson and Tyack,
2003).
Sperm whales were located at sea either by visual observers on the flying
bridge or acoustically using a towed hydrophone array. Once sperm whales had
been located, visual and acoustic teams made initial observations from the R/V
before a small rigid-hull inflatable boat was launched to attach tags to the
sperm whales. Diving whales were tracked from the tag-boat using a
custom-built directional hydrophone, and were approached upon surfacing. We
approached whales at slow speeds, typically from directly behind, with the tag
mounted on the end of a cantilevered 12 m-carbon pole
(Moore et al., 2001). We
sought to place the tag high on the animal, and most placements were just
forward of the dorsal hump. Tag attachments were recorded using digital video
and the response of the animal was carefully assessed.
Once the tag was applied to the animal, the team on the tag boat inspected
the position of the tag on the animal, measured the whale's heading, and
attempted to take identification photographs. The tagged whale was identified
and followed via a VHF signal from the tag along with visual and
acoustic tracking (Lerczak and Hobbs,
1998; Zimmer et al.,
2003
). Once the tag detached from the animal, it was retrieved
from the sea surface by tracking the VHF signal. CTD casts were often made
from the research vessel near the location where tags were recovered. Suction
cups were inspected for the presence of sloughed skin, which was preserved in
DMSO and subsequently genetically analyzed to determine the sex of the tagged
animal (Berube and Palsbøll,
1996
). The magnetometers on the tag were calibrated immediately on
retrieval to account for the remanent magnetic field held by steel and nickel
components in the tag. A least-squares fitting method was used to reduce
temperature- and pressure-related offsets in the 3-axis accelerometer and
magnetometer signals (Johnson and Tyack,
2003
).
Length estimation/allometry
Estimation of drag and buoyancy parameters described below requires values
for each whale's length, mass and surface area. To obtain these values for
tagged whales, observers on the tagging boat made calibrated video recordings
at measured ranges from tagged whales resting at the surface (`logging'). We
positioned the small boat directly broadside of the logging whale at >100 m
range and took frame images of the whale using a calibrated Canon GL1 digital
video recorder at a fixed zoom setting (full zoom). The range to the logging
whale was measured using a Bushnell Yardage Pro 1000 range finder (accuracy of
±1 m) and recorded by voice on the audio channel of the video recorder.
To ensure that the logging whale was oriented broadside to the camera, we took
multiple measurements as we moved from slightly behind to slightly ahead of
broadside of the logging animal. In several cases we were able to confirm the
broadside orientation of the whale by inspection of the tail flukes when the
whale dived.
Digital images were transferred from the camera to computer. The number of pixels from the blow-hole to the dorsal fin was measured, and this measurement was converted to centimeters based upon the range to the whale and the camera calibration factor. The camera was calibrated at the full zoom setting using a measuring tape at measured ranges from 16 to 180 m to determine the conversion factor of pixels to degrees. In addition, we measured the length of several known-length targets at sea, with an error of less than 1.0%.
Having obtained the distance from the blow-hole to the dorsal hump, this
distance was converted to animal length, mass and surface area based upon
published analyses of sperm whale allometry. Total animal length was estimated
using Gordon's equation (Gordon,
1990) with a small correction of half of the dorsal fin width
(Fujino, 1956
) as Gordon's
equation was based upon the rear end of the dorsal while we measured the tip
of the dorsal. Gordon's equation for length
(length=0.3875+1.679xdb0.015xdb2),
where db is the distance from the dorsal to blowhole) was based upon
188 male and female sperm whales and was a good fit for animals of size
ranging from 2 to 17 m in length (see fig.
3 in Gordon,
1990
). This technique was used to estimate the length of all of
the whales except sw250, for which we estimated length based upon the position
of tag-placement and the length of the tagging pole from a digital video
recording.
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Tag data analysis
Data downloaded from the Dtag were analyzed to obtain high-resolution depth
and 3-dimensional orientation values. Raw sensor data were initially filtered
and down-sampled by eight times (four times for sw250) to obtain a common
effective sampling rate of 5.88 Hz across all tag deployments. Pressure
readings on the depth sensor were converted to meters using calibrated values,
and magnitude readings from the 3-axis accelerometers and magnetometers were
converted to pitch, roll and heading of the tag in the earth frame, following
the technique described in detail in Johnson and Tyack
(2003). We derived the
3-dimensional orientation of the whale in the earth frame by correcting for
the orientation of the tag on the whale. This correction was estimated based
upon visual inspection of the position of the tag on the whale, and then
refined using measured values of the heading of the whale at the surface. The
criteria for an accurate tag-frame to whale-frame conversion were: (1) whale
pitch and roll should equal zero when the whale was resting at the surface;
(2) whale headings measured by the tag should match those measured visually;
(3) the rapid change in pitch upon diving should not correspond with a change
in roll; and (4) the tag-frame to whale-frame conversion should match the
visually inspected orientation of the tag on the whale.
Tag records were divided into five exclusive intervals: surface time, descent phase of dive, bottom period of dives, ascent portion of dives and shallow dives. The beginning of a descent was the time the whale left the surface, while the end was the time when whale pitch first exceeded 0° (i.e. when it was no longer oriented downward). Conversely, an ascent was defined to start at the last point in time when an animal's pitch was downward (<0°) and ended when the whale reached the surface. Dives were considered to be `shallow' if the whale did not exceed 350 m depth. For ascent and descents of deep dives, we calculated the mean pitch and vertical velocity. These values were averaged for each whale, and the mean values were averaged across whales.
We measured the percentage of time each whale was actively thrusting with
its flukes (`fluking') during descent and ascent periods. Fluking can be
detected clearly on the tag record as oscillations in the pitch record or in
the raw accelerometer signals themselves with a period ranging from 68
s (Nowacek et al., 2001;
Johnson and Tyack, 2003
). This
method of observing active fluking can be confirmed by listening to low
frequency flow noise over the tag, which also oscillates with fluking. We used
acceleration values in the whale's Z-axis (dorsalventral),
based on the correction of tag-placement described above
(Fig. 1B) to quantify when the
whale was fluking versus gliding. First we calculated the rate of
change in Z-axis accelerations or `jerk' by taking the difference in
successive accelerometer values. To reduce noise from non-fluking
perturbations to pitch, we band-pass filtered jerk to periods of 410 s.
Filtered jerk was squared, averaged and square-rooted over the period of a
typical fluke stroke (5.4 s or 32 samples) to obtain root mean-square jerk as
a metric of fluking-energy (Fig.
1C). Most ascents had a long terminal glide, and a threshold of
3x the fluking energy during these glides was used as a threshold to
identify gliding periods in the dive records. The accuracy of this threshold
was checked visually. Individual whales oscillated their flukes at a
consistent frequency, so gliding time is inversely proportional to mean
overall thrusting rate, as defined by Sato et al.
(2003
).
Drag and buoyancy forces during glides
In order to describe the forces acting on diving sperm whales, we developed
a model of drag and buoyancy based upon the anatomy of sperm whales. We
explored how well this model fit accelerations during glide periods using a
modified version of the method described by Bilo and Nachtigall
(1980) including equations from
Skrovan et al. (1999
) and Sato
et al. (2002
). The assumption
in the method described by Bilo and Nachtigall
(1980
) is that animal
acceleration during a glide is solely a function of drag forces; any buoyancy
effects are ignored. For a whale ascending at a steep pitch, however, vertical
buoyancy forces contribute to the forces affecting caudalrostral
acceleration during a glide as a function of sin(pitch)
(Fig. 2;
Sato et al., 2002
). Buoyancy
is equal to the mass of the fluid displaced by a body minus the mass of the
body, and is considered positive when the mass of the displaced fluid exceeds
that of the body. Dividing the body into air and non-air tissue portions, this
can be expressed as:
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![]() | (2) |
![]() | (3) |
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The influence of glide depth
The depth of a glide affects many of the terms in
Equation 3. Seawater density
w is the only depth influence on term 1 and can be calculated
for any depth using CTD profiles near tag locations to obtain salinity and
temperature (Fofonoff and Millard,
1983
; Morgan,
1994
). Air volume at depth is equal to air volume at the surface
(diving lung volume) divided by (1+0.1d) according to Boyle's Law,
where d is depth in meters. Conversely, the density of air
air increases with (1+0.1d). The term
(
w
air) changes only slightly with
respect to depth as water is highly incompressible and
air is
small compared to
w. In the Ligurian Sea,
(
w
air) is (1027.34.8) or 1022.5
kg m-3 at the shallowest analyzed glide depth of 38 m and
(1030.576.7) or 953.8 kg m-3 at the deepest glide depth of
757 m, a decrease of only 6.5%.
With full depth effects, term 2 of
Equation 3 can be written as:
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The density of animal tissue in term 3 is a function of both pressure and
temperature. Most animal tissues have the same compressibility as seawater
(Skrovan et al., 1999;
Beck et al., 2000
). While
certain tissues such as blubber are somewhat more compressible than water,
others such as bone are less compressible
(Clarke, 1978c
). Therefore, we
can neglect the pressure influence of animal depth on tissue buoyancy as
compression equally increases the densities of tissue and displaced
seawater.
While we can assume that the whale is at ambient pressure at depth, we
cannot assume that it is at ambient temperature, and the influence of seawater
temperature changes with depth may be substantial if whale temperature is
constant (Clarke, 1970). The
colder and denser water that the whale displaces at depth creates a positive
buoyancy force relative to when the whale is at the surface. The force is
equal to the temperature-induced density change of the displaced seawater. One
way to visualize this is to imagine that the whale is equivalent to an
insulated bag of seawater, which can be maintained at a different temperature
from the surrounding seawater. The difference in density of the warm bag of
water from that of the displaced fluid is defined as
w(T), which can be determined from CTD data.
For example, in the Mediterranean Sea, seawater is roughly 20° near the
surface and decreases to 13° at 1000 m. At 1000 m depth, the densities of
seawater at 13°C and 20°C are 1034 kg m-3 and 1032 kg
m-3, respectively. Thus, the value for
w(T) is 2 kg m-3, which is the
effect of the temperature difference at 1000 m depth at this location. Note
that the
w(T) term is quite small compared to
the actual density.
If the density of tissue is assumed to remain constant with respect to
temperature, but varies equally with the surrounding seawater with respect to
pressure, the depth-dependence of term 3 can be written as:
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Rewriting Equation 3 in matrix
form with full depth-dependencies obtains the following model of glide forces:
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Model fit and estimation of drag and buoyancy parameters
Our goals are to use experimental measurements of animal depth D,
pitch p, speed v, acceleration a, mass m
and area A, to estimate values for the unknown coefficients
Cd, Vair(0)/mw and
w/
tissue1 and to determine how
accurately the model fits the observed data.
We only conducted this detailed analysis for sperm whales from which we had recorded at least two `steep' ascents, because this analysis requires multiple glides by whales at high pitch across a range of depths and speeds. `Steep' ascents are defined as ascents in which the whale maintained a pitch of 60° or higher up to a depth of 50 m and maintained a fairly consistent vertical velocity (i.e. no pauses during ascent). In ascents classified as `non-steep', whales reduced their pitch at various times throughout the ascent, presumably to translate their position horizontally during ascent.
Animal speed through the water v, depth D and pitch
p were taken as the mean value during each glide period measured
(Fig. 1). We calculated speed
through the water as: D/sinp, based on the assumption
that lift forces are minimal and the whale moves in a caudalrostral
direction through the water (Fig.
2). The minimum pitch angle for which we attempted to calculate
speed through the water was 50°, for which the correction is 30.5%. At the
average pitch of 79.6°, the correction is only 1.7%. To increase the range
of glide velocities, we divided most glides in half into two sub-glides except
the long terminal ascent glide, which we divided into multiple sub-glides
depending on its duration (Fig.
1). Acceleration during each sub-glide was measured directly using
a linear regression of speed versus time. We found the change in
speed versus time to be quite linear in the sperm whale glides, so it
was not necessary to use inverse speed as suggested by Bilo and Nachtigall
(1980
). For each sub-glide we
obtained a Reynold's number (Re) using the animal's length estimate and the
mean speed during the sub-glide (Vogel,
1981
). The value for me was set to 1.06 based
on the measure for a prolate spheroid of fineness ratio 5.0
(Skrovan et al., 1999
).
The linear coefficients Cd,
Vair(0)/mtissue and
(w/
tissue1) were then estimated using
linear least-squares fitting (Strang,
1991
) for each whale. Linear least-squares estimation of the
unknown terms in Equation 6 is
equivalent to fitting the observed acceleration data to a 3-term regression
model with no constant (Zar,
1984
). The slopes of the three terms in the regression are the
estimates for Cd and
Vair(0)/mtissue, and
w/
tissue1. The statistics of the model
fit were obtained by fitting all sub-glides from each whale to a 3-term linear
regression model with no constant in Systat
(Zar, 1984
), using a
P-value of 0.01. For each whale, all three coefficients were first
estimated using all ascent sub-glides from all dives. Because
Vair(0) can vary between dives, the Cd
and
w/
tissue1 coefficients were
constrained by the overall estimates, and
Vair(0)/mtissue was then estimated
separately for each dive. An estimate was made of the drag coefficient
Cd for each sub-glide by subtracting the effect of the
buoyancy terms, and Cd was compared to the Reynold's
number (Re) of each sub-glide.
The fit of the model (r2) to the observed acceleration data was calculated for each whale and the model was used to predict acceleration during descent glides made by the same whale. A check against descent glides is important because drag and tissue-buoyancy may be inter-correlated when only ascent glides are used to build the model, and the range of speeds observed is limited. There is less concern for inter-correlation with air-buoyancy because glides were measured over a large depth range. The direction of the drag force is reversed during descent, so testing descent predictions provides a strong test of the overall accuracy of the model.
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Results |
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Swimming gaits/gliding rates
In order to describe fluking patterns during descent, ascent and shallow
dives for the same individual, we limited our analyses to the 23 whales from
which we recorded at least one complete deep dive (Tables
1,
2). Descent was marked by a
steep initial descent, with small oscillations in downward pitch, steady
changes in roll and active clicking (Fig.
1; see also Zimmer et al.,
2003). Whales fluked actively during descent and maintained a
fairly constant speed through the water (Figs
1,
3). Ascents generally had an
early phase of low-pitch ascent with steady fluking, followed by a later phase
of steeper ascent with significant gliding and few changes in roll (Figs
1,
3).
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Whales descended at a mean (± S.D.) vertical velocity of 1.15±0.14 m s-1 at a mean pitch of53.3±6.3° (Table 2). Vertical velocity during ascent averaged 1.33±0.16 m s-1 at a mean pitch of 56.6±10.4°. Vertical velocity was generally quite constant during steady fluking, but oscillated during stroke- and-glide swimming (Figs 1, 3). The magnitude of the pitch angle did not differ between ascent and descent (paired t22=1.84, P=0.08), but vertical velocity was higher during ascent than descent (paired t22=5.20, P<0.001). Correcting ascent/descent velocity for animal pitch (Fig. 2) obtains a mean descent speed through water of 1.45±0.19 m s-1, which was significantly lower than the pitch-corrected mean ascent speed of 1.63±0.22 m s-1 (paired t22=4.04, P<0.001).
All sperm whales exhibited strong fluking at the beginning of descent and gliding during the terminal phase of the ascent for all dives (Fig. 3), probably reflecting the buoyancy effect of air carried from the surface. Overall, mean gliding time was much lower during descent (5.3±6.3%) than ascent (36.0±16.4%) phases (paired t22=6.8, P<0.001). The most common swimming gait observed was steady fluking during descent, and stroke-and-glide during ascent (Figs 1, 3A,B). This pattern was observed during all dives for 12 individuals and during at least a subset of dives for another 10 whales (Table 2).
Inter-individual variability of percentage time gliding (with coefficients of variation of 119% for descent and 44% for ascent) was greater than for all other measures of dive behaviour. Twenty whales glided more during ascent than descent, while three individuals glided somewhat more during descent than ascent (Table 2). Across these 23 whales, there was a significant negative relationship between the percentages of time gliding on ascent versus descent (Spearman rank r=0.59, P<0.0001), consistent with the expectation that tissue buoyancy should affect gliding behaviour inversely on ascent versus descent.
Inter-individual variability of swimming gaits corresponded to that in percentage time gliding. Four sperm whales showed evidence of stronger than average positive buoyancy by fluking steadily during descent and making at least one prolonged glide over 350 m during an ascent (e.g. Fig. 3C). Conversely, some other individuals showed evidence of less positive buoyancy. As noted above, three whales glided more during descent than ascent. Also, five sperm whales did predominantly stroke-and- glide swimming during at least one descent, and two of these also fluked steadily during at least one ascent (Table 2).
A total of 59 shallow dives were recorded from 13 of the 23 whales. These dives had a mean duration of 11.96 min(S.D.=6.5; min., 2.82; max., 31.47) and a mean maximum depth of 16.3 m (S.D.=12.0; min., 3.6; max., 64.23). The predominant swimming gait during 30 of these dives was stroke-and-glide, while steady fluking was observed in 26 dives. During three shallow dives, whale sw254a (which made one prolonged glide from a deep dive; Table 2) maintained a steady depth less than 20 m with its head oriented toward the surface for 14.1, 11.3 and 11.1 min. During these shallow dives, there was no change in depth, there was no flow noise audible acoustically, and no fluking was apparent in the accelerometer records until the whale moved to the surface. The sound of bubbles from released air was audible during these dives, with a near simultaneous slowing of ascent by the whale. We conclude that the animal was resting during these intervals and modulated its air content to achieve neutral buoyancy.
Data usable for drag and buoyancy model
We were able to obtain video-photogrammetry measurements from eleven
whales, of which eight had at least two complete dives
(Table 1). Measured lengths
ranged from 8.513.4 m. All tagged whales in the Ligurian sea were
confirmed or probably male and exceeded 12 m in length. Whales in the Gulf of
Mexico were generally less than 10 m in length except for sw208b, which was
12.4 m and confirmed male. Three of the smaller Gulf whales were confirmed to
be female, while the others were either adult females or adolescent whales of
unknown sex.
Five of the eight measured whales (sw250, sw265, sw275b, sw191b and sw209c)
had at least two steep ascents and thereby met the criteria for detailed
analysis of drag and buoyancy parameters
(Table 1). All five whales
utilized the most commonly observed swimming gait of steady fluking during
descent (gliding 0.66.4%) and stroke-and-glide swimming during ascent
(gliding 43.052.0%; Table
2). These five whales were either isolated animals or in a widely
spaced social aggregation associated with males
(Lettevall et al., 2002).
From these five whales, we recorded a total of 20 steep ascents, and extracted a total of 382 sub-glides for analysis. The sub-glides had a mean (± S.D.) duration of 10.2±4.5 s and pitch of +79.6±7.0° covering a depth range of 38757 m.Speed through the water during glides ranged from 1.1 to 2.2 m s-1, with an overall mean of 1.5±0.2 m s-1. Accelerations ranged from0.023 m s-2 to +0.012 m s-2, and were strongly affected by both animal depth and speed through the water (Fig. 4).
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Evaluation of the drag and buoyancy models
Using Equation 6 above, we fit
the observed accelerations to the model of glide forces, obtaining estimates
for drag coefficient Cd, air carried to depth
[Vair(0)/mtissue] and animal density
tissue. The model fit measured accelerations with an
r2 of 99.199.8% for each whale, average 99.6%. All
three terms were statistically significant at P<0.01 for all
whales. Error residuals versus depth were quite flat
(F1,380=0.02, P=0.96), suggesting that the model
adequately accounted for depth-specific effects acting on acceleration.
Because gliding during descent was quite rare, we identified a total of only seven descent glides by four of the five animals that were both of sufficient duration and steep enough pitch for acceleration to be reliably measured. We compared these measured descent accelerations to predicted values derived from ascents based on Equation 6. The slope of predicted versus observed glide accelerations was 0.80 (t5=4.04, P<0.01), and not significantly different than 1.0, which is the expected value for a model fit (Fig. 5). The mean difference between observed descent-glide accelerations and predicted values was0.0052 m s-2.
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Thus, the model of glide forces had a statistically strong fit to ascent glide accelerations (r2 = 99.199.8%), and the model based upon ascent data correlated with descent glide accelerations with a small constant offset. Based on the mass of the different whales, this offset is consistent with a decrease in whale density of 0.064% during the surfacing interval between an ascent and the subsequent descent.
Drag and buoyancy parameters
Treating individual whales as the unit of analysis, the mean estimate
(± S.D.) for the drag coefficient Cd was
0.00306±0.00015 (Table
3). The estimate of the drag coefficient was close to the
predicted value for a completely turbulent spindle with a fineness ratio of
5.5 (Fig. 6). Airvolume carried
by the whale at the surface
(Vair(0)/mtissue), was estimated as
26.4±3.9 l kg-3, and air volume carried by whales did not
vary much between dives. Whale tissue density tissue during
ascents was estimated at 1030.0±0.8 kg m-3, which is
0.31%±0.07% more dense than seawater at the 40 m reference depth.
Correcting for the offset between predicted and observed descent glides
accelerations (Fig. 5), the
density during descent would be slightly lower, 1029.3 kg m-3, if
the offset is due to warming at the surface (see Discussion).
|
|
To describe the drag and buoyancy forces acting on a diving sperm whale, we
used the coefficients to predict drag and buoyancy forces during ascents and
descents across a typical diving depth range for a 12.5 m,
25x103 kg whale traveling at 1.5 m s-1 in the
Ligurian Sea (Fig. 7). Air
buoyancy is quite strong near the surface (>5000 N), and decreases quickly
with depth. The absolute value of the air buoyancy force becomes smaller than
drag at roughly 170 m. Buoyancy imparted by sperm whale tissue is strongly
affected by the increase in seawater density at depth, changing from710
N near the surface to180 N at depth. Combining air and tissue buoyancy
after descent, the animal is within 20 N of neutral buoyancy at 800 m
depth. After cooling during bottom time, which increases tissue density by the
0.064% difference between observed and predicted accelerations during descent
glides, the depth of neutral buoyancy becomes
250 m
(Fig. 7). Drag forces are
predicted to be 333334 N across all depths.
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Discussion |
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Our model of forces acting on sperm whales during glides
(Equation 6) explained over 99%
of the variability in measured accelerations during glides from five whales,
and all three terms of the model had statistically significant fits at
P<0.01 for all whales. Predictions of accelerations during descent
glides correlated with observed values, with a small offset of 0.0052 m
s-2 (Fig. 5). This
offset is consistent with a decrease in animal density of 0.064% while at the
surface, or a similar increase in drag during descent as opposed to ascent.
Sperm whales are made up of roughly one-third blubber, which is one-third
lipid. Blubber lipid contains 60% wax esters, similar in structure to that of
spermaceti oil (Lockyer,
1991). For a sperm whale's density to increase by 0.064% would
require the densities of these wax esters to increase by 0.9%. Spermaceti oil
decreases density by
0.5% per 1°C in the range 2832°C
(Clarke, 1978b
), which is the
temperature of blubber in fresh-killed sperm whales (see appendix 1 of
Clarke, 1978c
). Thus, even
neglecting changes in the other oils in blubber, a 2°C change in blubber
temperature could account for the offset between observed and predicted
accelerations during descent glides. A 2°C change could occur as blubber
cools at depth, and then warms when the whale is at the surface.
Thermal-imaging techniques (Westgate et
al., 2001
) would be useful to explore temperature variations in
the peripheral tissues of sperm whales, to test whether such warming does
occur at the surface.
Drag coefficient, air volume and tissue density
The mean (± S.D.) estimate for passive drag coefficient
Cd was 0.00306 (±0.00015) at an average Reynold's
number of 1.9x107 (±0.36x107).
Because air volume is expressed per unit mass, and mass was simplified out of
the tissue buoyancy term in the model, errors in the allometry analysis
directly influence only the estimate for the drag term. The influence of
errors on the drag estimate is somewhat reduced because animal area is in the
numerator while mass is in the denominator. We refit the accelerations with a
±10% change in length (and carried this through to mass and area),
which resulted in Cd changing by ±6%. Given the
potential for errors in the length estimate in either direction for any one
whale, the mean Cd value is the best estimate.
The drag coefficient estimates were close to those predicted for a
completely turbulent spindle of fineness ratio 5.5
(Fig. 6; see
Stelle et al., 2000). The
sperm whale has smaller control surfaces, and may therefore be more ideally
streamlined, than other odontocete cetaceans
(Fish, 1993
). At the high
Reynold's number experienced by these large animals, flow is likely to be
entirely turbulent (Vogel,
1981
). In a turbulent flow regime, adaptations in shape or skin
texture that delay separation of the turbulent boundary from the whale would
reduce drag. Interestingly, the region of the sperm whale posterior of the
dorsal fin is highly convoluted with a corrugated external surface (see fig.
18 in Berzin, 1972
), which may
be an adaptation to delay separation of the turbulent flow from the body.
Measurements of hydrodynamic flow over this type of structure could be made to
test this possibility.
Our results argue against Whitehead's conjecture
(Whitehead, 2003) that sperm
whales are less hydrodynamic than other large marine mammals. The largest
animal for which a drag coefficient has previously been calculated from
kinematic data is the killer whale, with an estimated drag coefficient of
0.0029 at Re of 3.7x107
(Fish, 1998
), very close to
our estimate for sperm whales. Based on a series of flat plates in turbulent
flow, a theoretical drag coefficient of 0.0026 was estimated for a fin whale
at a Reynold's number of 4.28x107
(Bose and Lien, 1989
). Drag
coefficients were calculated from glides for the Steller sea lion with an
estimated Cd of 0.0056 for Re of 5.52x106
(Stelle et al., 2000
). In the
slightly smaller California sea lion, Cd was estimated at
0.0039 at a Reynold's number of 2.9x106 where flow is thought
to be partly laminar (Feldkamp,
1987
). Thus, the marine mammal species for which drag has been
estimated or measured match each other fairly closely based upon the flow
regime at the appropriate Reynold's number
(Vogel, 1981
).
Estimated volumes of air carried to depth ranged between 21.9 and 32.6 l
10-3 kg, with an overall mean of 26.4 l 10-3 kg.
Variability across dives in estimate air carried to depth by each whale was
quite small, with a standard deviation of 1.1 l 10-3 kg or less for
each of the five animals (from Table
3). To our knowledge, no reliable measurements of lung volume have
been made for sperm whales. However, lung mass has been measured
(Omura, 1950) and is a similar
percentage of body mass in sperm whales as in the bottlenosed whale
Hyperoodon ampullatus where lung volume was measured
(Clarke 1978a
). Based upon
these weight measurements, lung volume of sperm whales is estimated at
17.823.5 l 10-3 kg, which is close to our estimate
(Clarke, 1978a
). A larger lung
volume is predicted from a general mammalian body size regression line that
includes smaller mammals (Kooyman,
1973
), but there are few data and significant variability for the
larger whales. In the absence of accurate measurements of the volume of the
sperm whale lung, the lung mass data summarized by Clarke
(1978a
) are the best
information currently available. Our estimate of total air carried from the
surface is roughly one-half the value of diving lung volume used by Kooyman
and Ponganis (1998
) to
estimate total oxygen stores in the sperm whale.
The most likely function for air carried to depth by sperm whales is sound
production at depth (Madsen et al.,
2002), as little gas exchange is likely to occur while sperm
whales are diving. While there may be gas exchange as long as the alveoli
contain air, the lungs of cetaceans appear to have adaptations for rapid
alveolar collapse upon diving (Kooyman,
1973
). Sperm whales also have large rigid trachea and bronchi that
support the collapse of alveoli (and cessation of gas exchange) upon diving
(Berzin, 1972
). If gases were
released from tissues such as the rete mirabile in large quantities
during ascent, we would expend to find negative residuals for the shallowest
glides where gas should be released most quickly from solution
(Kooyman, 1973
). However, the
residuals from our model were flat versus depth. We conclude that the
impact on buoyancy due to gas release from tissues during ascent is negligible
at depths greater than 38 m.
The model estimated that the density of the non-air portion of all five
sperm whales was 1030 kg m-3, or 0.31% denser than seawater at the
reference depth of 40 m. Sperm whales killed in whaling operations are
typically positively buoyant, but they are reported to sink in rare cases when
air is lost due to shooting or a wound
(Clarke, 1978c). Although
traditional whalers preferred sperm whales because they would float upon
death, Berzin (1972
) reported
that whalers pumped air into sperm whales to keep them from sinking and noted:
`At present sperm whales are considered to sink after death, and in fact,
some of them do, primarily young animals, that have a smaller amount of
fat.' Clarke (1978c
)
reports a case in which a sinking whale was easily recovered on the line,
suggesting only slight negative buoyancy. It therefore appears that at least
some sperm whales are negatively buoyant when they have lost their air, in
agreement with our findings.
Predicted forces and fluking patterns
It is important to note that all five whales for which we estimated drag
coefficients, air volume carried to depth, and tissue density used the
most-commonly observed pattern of fluking: steady fluking on descent and
stroke-and-glide swimming on ascent. Gliding rates of these five whales
(descent: 1.4%, ascent: 39.9%) were close to the overall average of all 23
sperm whales of 5.3 and 37.7%, respectively
(Table 2). In contrast to these
five loosely aggregated animals (mostly male) that primarily ascended without
pauses at steep pitch, animals in larger social groups were more likely to
show non-steep ascents with shallow ascent angles and more pauses. Several of
these animals were confirmed female and were observed at the surface
interacting in social groups containing small adults with a calf. These social
whales appear to make use of their ascents from depth to translate their
position horizontally as well as vertically, likely to maintain contact with
preferred associates (Whitehead,
2003).
Based on the parameters estimated in the model (Equation 6), we are able to predict the forces acting on a `typical' sperm whale during ascent and descent (Fig. 7) and relate these predictions to observed fluking patterns (Table 2, Fig. 3). For `typical', we predicted forces for a 12.5 m sperm whale (mass 25x103 kg) with a surface area of 58.1 m2 traveling at 1.5 m s-1. Air buoyancy forces are small (+55 to +200 N) from 1000 m to 300 m. Air buoyancy then increases more rapidly to +550 N at 100 m and very quickly in the shallowest waters to +3000 N at 10 m. The buoyancy force imparted by the tissues of the sperm whale is strongly affected by the sharp decrease in density of the seawater at depth. In the Mediterranean Sea, tissue density forces for our hypothetical 12.5 m sperm whale are710 N near the surface and decrease to180 N at depth due to the increase in density of the displaced seawater. Note there can be a rapid change in buoyancy force as the animal passes through the thermocline (Fig. 7).
Across the depth range of a diver, the most apparent feature of these summed forces is strong positive buoyancy in the top 100 m of the water column. For an ascending whale, positive buoyancy exceeds drag and tissue buoyancy forces at roughly 80 m depth. This prediction closely matches our observations that all whales fluked strongly during the initial descent but glided during the terminal portion of the ascent. Many whales, in fact, began terminal glides at greater depths, initially decelerating and later accelerating (Fig. 3A,C).
Based on our model of forces acting on sperm whales, forces acting against descending whales are much stronger than those acting against the motion of ascending whales, at least in the top 300400 m of the water column (Fig. 7). This overall summary of predicted forces links strongly with our finding that whales glide significantly more during ascent than descent, even though ascent speeds were higher (Table 2). The strong relationship between predicted forces and swimming gaits in our study supports the hypothesis that buoyancy affects behavioural swimming decisions in diving animals.
While there is a strong overall link between swimming behaviour and predicted forces in our study, we did find evidence of stronger positive buoyancy in four animals that made prolonged glides during ascent from depths as deep as 600 m (Table 2; Fig. 3). While no animals produced prolonged glides during descent, five whales did predominantly stroke-and-glide swimming during at least one descent and three animals (sw248a, sw249a and sw254b) glided more on descent than ascent. This suggests that some whales differ in buoyancy (both positive and negative) from the overall average observed in our 23 animals. This interpretation is most strongly supported by the finding that whales that glided more during ascent glided less during descent, and vice versa.
In fact, we expect significant variability in tissue buoyancy across
individuals depending on the thickness and lipid content of their blubber
(Beck et al., 2000;
Biuw et al., 2003
). Controlling
for both animal size and sampling location on the body, the thickness of sperm
whale blubber is highly variable across individuals
(Lockyer, 1991
), even those
from the same social grouping (Evans et
al., 2003
). The buoyant middle-ventral blubber region is
919 cm thick in whales of 10 m length and is composed of 2558%
lipids, with typical ranges of 30% over much of the sperm whale body
(Fig. 5 in
Lockyer, 1991
). Lipid content
appears to vary independently of blubber thickness, which may further increase
inter-individual variability in tissue density
(Evans et al., 2003
). Based
upon lipid density of 850 kg m-3 and 10% lipid content of sperm
whale tissue, a 5 cm increase (decrease) in blubber thickness will change
overall tissue density to 0.16% less (0.20% more) dense than seawater. For a
25x103 kg animal, this would result in tissue-buoyancy forces
changing by ±500 N. Adding 500 N to the model of predicted forces
(Fig. 7) would result in
buoyancy exceeding drag over most of the ascent depth range (creating the
potential for prolonged glides during ascent), while subtracting 500 N results
in negative buoyancy exceeding drag once whales reach depths greater than
250 m (supporting more gliding in descent).
Shallow dives
We analyzed a total of 59 shallow dives to less than 65 m that lasted
between 2.8 and 31.5 min. The whales swam using steady fluking or
stroke-and-glide gaits in all but three of these shallow dives. While we
cannot speculate why sperm whales used different swimming gaits in this
context, it does suggest that choice of swimming gait may be an important
behavioural option in shallow traveling dives. The efficiency with which sperm
whales make use of glides appears to be relevant both for deep diving and
surface travel contexts.
Based upon predictions from the glide-forces model (Fig. 7), we expect shallow-divers to be positively buoyant. At the mean shallow dive depth of 16 m, a 25x103 kg whale should have a positive buoyancy of more than 2000 N if it dived with a lung-full of air. While such a force could undoubtedly be overcome through hydrodynamic control during active swimming, this would entail a significant cost in swimming efficiency for the whale. Also, for three dives, one whale hung motionless in the water at less than 20 m depth for more than 10 min. This whale (sw254a) also made one prolonged glide during an ascent, suggesting that it is even more positively buoyant than the `typical' whale modeled in Fig. 7. If 2000 N buoyancy were the only force acting on this inactive whale, it would move 20 m to the surface in less than 30 s.
Instead, we observed a clear example of buoyancy regulation by this sperm
whale during shallow resting dives. After the whale had reached its maximum
depth and was slowly drifting back to the surface, it released a quantity of
air, which produced clearly audible bubble sounds on the tag. The ascent rate
of the drifting whale slowed rapidly upon the release of the air.
Occasionally, a second burst of bubbles was heard. While this behaviour was
only observed in animal sw254a in the data-set reported here, it has been
subsequently observed in at least two more whales tagged in the Gulf of Mexico
in 2003 (P. Miller, unpublished observation). In addition to releasing
bubbles, sperm whales could also regulate the volume of air they inhale before
diving, as appears to occur in diving penguins
(Sato et al., 2002).
Spermaceti function for buoyancy control?
It is clear from this study that buoyancy is an important factor in the
ecology of sperm whales, and that sperm whales actively regulate their
buoyancy, at least during shallow resting dives. Clarke
(1970,
1978c
) proposed that a
function of the large spermaceti organ could be to regulate the buoyancy of
the sperm whale at depth. The density of the spermaceti oils increase rapidly
upon cooling, which could act to counter positive buoyancy that Clarke
predicted for sperm whales. Lockyer
(1991
) suggested similar
possible functions for the wax-rich lipids in the outer blubber layer of sperm
whales. Several objections have been raised to this proposed function for the
spermaceti organ (see review in Cranford,
1999
), but few directly relevant measurements have been made. The
most direct test of this hypothesis would be to measure the internal
temperature of the spermaceti and other organs during dives. While this is
clearly a difficult task with current technologies, indirect measures of
temperature via sound-speed in spermaceti suggest that no significant
cooling takes place during dives (Madsen
et al., 2002
).
The model of glide forces that we fit to ascent glide accelerations
slightly underestimated the amount by which whales slow down during descent
glides. This is consistent with the notion that cooling of some tissues (and
increased density) may occur at depth. The scale of cooling that we observe,
however, is quite small and can be fully explained by passive cooling of
blubber alone. Note also that warming at the surface (and therefore lower
tissue density) increases the cost of descent, while cooling at depth (which
leads to higher tissue density) increases the cost of ascent. Because the
blubber of sperm whales is well vascularized, heat transfer from the blood to
the blubber is possible (Berzin,
1972). Clarke
(1970
) further suggested that
warming of spermaceti oils (or blubber) through such a mechanism could
increase buoyancy during ascent. To explore this possibility, we fit a second
ascent model, which assumed that the whale decreased its tissue density in
equilibrium with seawater temperature. This second model fit the observed
acceleration data significantly worse than the glide-forces model presented in
this paper and predictions of descent glide accelerations did not correlate
with observed values. While we cannot rule out any warming of tissues during
ascent, the glide acceleration data recorded from these five animals more
strongly supports a fixed-temperature regime at least to 40 m depth, with
possible warming only nearer or at the surface.
In addition, there are two aspects of our study that appear to contradict
some of the important assumptions behind Clarke's hypothesis that cooling at
depth may be functional for a sperm whale. First, the model fit to the glide
accelerations suggests that the five sperm whales in our study are
not neutrally buoyant (sans gases) at the surface, as
predicted by Clarke (1970,
1978c
), but have considerable
negative tissue buoyancy. The sum of negative tissue and positive air buoyancy
forces becomes very small when the whale is at depth
(Fig. 7). The high levels of
inter-individual variability in descent and ascent gliding patterns
(Fig. 3) suggest that different
sperm whales have varying buoyancy depending on their body condition. Fatter
and more buoyant whales are the only group that would appear to benefit from
extra cooling of a large organ such as the spermaceti. It seems unlikely that
the massive spermaceti organ would evolve through natural selection for such a
marginal benefit to those individuals that are already the most fit.
A second assumption of Clarke's hypothesis is that sperm whales have a need
to remain motionless at depth to wait for prey based on reports that whales
would resurface close to the location where they dived (Clarke,
1970,
1978c
;
Lockyer, 1991
). Instead, our
tag records show clearly that sperm whales actively swim during the bottom
phase of their dives. This fact is borne out by steady depth-excursion
throughout the bottom phase of the dive, the constant presence of acoustic
flow noise, and through analysis of the accelerometer signals (see also
Whitehead, 2003
). Whales in
different geographic locations may use different hunting strategies depending
on prey type, but none of the 23 sperm whales in this study were sit-and- wait
predators, all appeared to pursue prey. Therefore, any need to control
buoyancy hydrostatically is reduced because the whale can overcome these
forces hydrodynamically.
While more work is needed to quantify spermaceti or blubber cooling during deep dives of sperm whales, our research suggests that a special adaptation supporting additional cooling for a buoyancy function would not be particularly beneficial to sperm whales.
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Acknowledgments |
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