Faculty of Aerospace Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel

Abstract

Background

Drafting in cetaceans is defined as the transfer of forces between individuals without actual physical contact between them. This behavior has long been surmised to explain how young dolphin calves keep up with their rapidly moving mothers. It has recently been observed that a significant number of calves become permanently separated from their mothers during chases by tuna vessels. A study of the hydrodynamics of drafting, initiated in the hope of understanding the mechanisms causing the separation of mothers and calves during fishing-related activities, is reported here.

Results

Quantitative results are shown for the forces and moments around a pair of unequally sized dolphin-like slender bodies. These include two major effects. First, the so-called Bernoulli suction, which stems from the fact that the local pressure drops in areas of high speed, results in an attractive force between mother and calf. Second is the displacement effect, in which the motion of the mother causes the water in front to move forwards and radially outwards, and water behind the body to move forwards to replace the animal's mass. Thus, the calf can gain a 'free ride' in the forward-moving areas. Utilizing these effects, the neonate can gain up to 90% of the thrust needed to move alongside the mother at speeds of up to 2.4 m/sec. A comparison with observations of eastern spinner dolphins (

Conclusions

A theoretical analysis, backed by observations of free-swimming dolphin schools, indicates that hydrodynamic interactions with mothers play an important role in enabling dolphin calves to keep up with rapidly moving adult school members.

Background

The problem of separation of mother-calf pairs in chase situations has become a serious concern in fishing-related cetacean mortality, in particular in the eastern tropical Pacific Ocean where tuna are fished with a purse-seine method, in which schools of dolphins are encircled with a fishing net to capture the tuna concentrated below

The hydrodynamics of the drafting situation is extremely complex, as it deals with unsteady motions of two flexible bodies of different size, moving, while changing shape, at varying speeds and distances from the water surface and from each other, and periodically piercing the surface. In addition, there are several different preferred drafting positions for the calf

Older calves are seen more often in the 'infant position', which involves swimming under the mother's tail section with the neonate's head (or melon) lightly touching the mother's abdomen. Once they are several months old, calves swim in the echelon position about 40% of the time and swim in the infant position about 30% of the time. Gubbins

There is very little quantitative information on drafting in dolphins, and much of the extant data is qualitative (for example, 'close proximity' is not reported in actual distances in the different positions). Such data as do exist will be briefly reviewed here, and mentioned again when specifically used in the calculations later in this article.

The experimental comparisons used here are based on data for eastern spinner dolphins,

Aerial photographs of swimming dolphins

Aerial photographs of swimming dolphins. **(a) **An actual leaping sequence; **(b) **several mother-calf pairs swimming at high speed. In (a), the calf performs a bad leap, resulting in a large splash (frame 4), slowing it down and losing the close connection required for drafting. The data in (b) are the basis for several of the entries in Table

In some aerial records of mother-calf pairs moving at high speed, one can observe the calf moving from one side to the other obliquely behind the mother. This motion may be due to the bias in yaw that the calf experiences when moving on one side, and an attempt to 'even' this out, by periodically changing sides.

The first step in the analysis is to try to extract the dominant effects of the postulated hydrodynamic interaction and to build a simplified model, which will be able to give quantitative predictions for the major parameters, without losing relevance. This model can then serve as a building block for further, more complicated descriptions. This procedure is complicated, however, by the fact that empirical data are scarce and partial with large inherent errors. Such a model should be simple enough to be solvable by semi-analytic methods before delving into full numerical analysis, the accuracy of which will be compromised by the large scatter in experimental input data.

Obviously the model needs to be accurate and detailed enough to give useful results. Lang

Results and discussion

The modeling process is started by looking at drafting in water far enough from the surface to neglect surface wave (Froude number) effects. Viscosity (boundary layer) effects are left out at this stage, allowing the use of the linear, potential flow model. This will allow superposition of solutions, as mentioned above. Effects of viscosity will be included where required at a later stage (see below). This is the equivalent of using the Kutta-Joukowski condition in airfoil theory

Next, assume that both mother and calf are moving without changing body shape - that is, with a fixed (rigid) body shape (no tail oscillations). On the basis of observations on several dolphin species, this shape is taken to be an oblate ellipsoidal shape of aspect ratio of about 6 (see Figure

Schematic description of **(a) **a mother-calf pair of dolphins, and **(b) **two ellipsoids modeling them

Schematic description of **(a) **a mother-calf pair of dolphins, and **(b) **two ellipsoids modeling them.

Thus the drag on a swimming dolphin will be estimated as that of a 6:1 ellipsoid moving in the direction of its long axis, multiplied by 3, while a coasting dolphin will have the drag of a 6:1 ellipsoid. The drag on streamlined bodies at zero angle of attack (measured between the direction of motion and the animal's longitudinal axis) is well known

The basic premise here is that drafting is advantageous as a result of the mother producing a flow-field that has areas of forward-moving water, resulting from a non-uniform pressure field. When the calf positions itself in these areas, it needs to produce less thrust, as the relative velocity it experiences is lower than the absolute speed of motion, and the energy required is roughly proportional to the relative velocity cubed. This is the same principle I and others used in developing models for fish schooling

A series of cases simplified sufficiently to allow semi-analytical solutions (that is, solutions that do not require numerical analysis of the differential equations, but use computations to obtain numerical values of the solution functions) are now analyzed. The flow around ellipsoidal shapes is calculated. As mentioned above, these closely approximate dolphin shapes, when excluding fins. First a single ellipsoid is analyzed, and then two ellipsoidal shapes of equal or different sizes in close proximity.

Motion of a single ellipsoid

The first model developed here is of a single ellipsoid moving in still waters. The flow field obtained is an accurate representation of the force on each point in the flow field and can be seen as the flow field experienced by a body much smaller than the ellipsoid itself. Thus, such a calculation is a good approximation for the positioning of pilot fish in the vicinity of sharks, but less so for dolphin calves, which at birth are already about one half the length, and over one tenth of the mass, of an adult

This model is only an approximation to the flow pressure distribution on a large body, showing the generally advantageous areas for calf positioning. Figure

A snapshot description of the flow around a single ellipsoid moving from right to left at speed

A snapshot description of the flow around a single ellipsoid moving from right to left at speed

But the positions identified here need to be carefully scrutinized to make sure that they are relevant when the simplifications in the model are taken into account. Thus, some of the 'best' positions predicted by this model are irrelevant. These are as follows: first, a position just ahead of the mother's nose, being 'pushed' forward; this position will not be adopted for several reasons, including that the pushing motions are unstable

Thus, the actual best positions will be obliquely in front, and obliquely behind the mother's center-line. The forward position is less practical, for the reasons explained above, and so is used only in the first few hours after birth. The remaining preferred zone is obliquely behind the mother's equator, which is more reminiscent of the 'infant' position, to which we will return later. All positions, except directly in front of or behind the mother's center-line, experience lateral velocity components, which need to be compensated for by a lateral force if the calf is to swim in a straight line. Thus, an optimal trade-off between forward velocity contribution and loss due to sideways compensation can be found.

Two slender ellipsoidal shapes moving in proximity

Here, the analysis is based on the studies by Tuck and Newman

Planar coordinate systems for a mother-calf pair of ellipsoidal shapes

Planar coordinate systems for a mother-calf pair of ellipsoidal shapes. _{1}, _{2 }and _{1}, _{2 }are calf and mother lengths and speeds, respectively. The instantaneous longitudinal and lateral distances between the centers of mass are

As mentioned in the additional data file, each of the bodies can be defined by a distribution of doublets along the longitudinal axis

where we take

where _{i}_{i}_{i }is the maximum area of each at the equator. Without loss of generality, the calf is assumed to be non-moving and the mother moving at _{2 }relative to the calf.

After some rather complicated algebraic development (see the additional data file), one can finally obtain expressions for the forces by substituting equation (1) into equations (A-8) and (A-9) in the additional data file. The force in the longitudinal direction on the calf (that is, the force pushing the calf forward) is:

and the lateral (side) force (including the Bernoulli effect mentioned by Kelly

where

Some results of these calculations are presented in non-dimensional form, in Figure

The forces on the calf calculated from equations (2) and (3)

The forces on the calf calculated from equations (2) and (3). Definitions of the parameters appear in Figure **(a) **The non-dimensional peak longitudinal force _{max }(thrust) on the calf as a function of the normalized lateral distance _{1 }from the mother, for different mother/calf size ratios (as indicated by the numbered arrows). The dashed red line indicates closest probable proximity. The ratios relevant here are from 1 (fully grown calf - the solid blue line) to 2 (neonate). **(b) **The non-dimensional peak lateral force on the calf, for different mother/calf size ratios, as a function of the normalized lateral distance _{1 }from the mother. The peak lateral force is obtained when the centers of mass of mother and calf are on a line perpendicular to the long axis (**(c) **The variation of forces and moments as one animal is placed at different normalized longitudinal positions relative to the other in the fore-aft direction. Positive values on the horizontal axis indicate that the mother's center is ahead of the calf. The lateral distance is one quarter of the calf's length. The curves marked

These results are now applied to the dolphin-drafting situation. The mother and calf are in the same horizontal plane, but the results are applicable also for depth differences, as the assumption is that both are approximated by bodies of revolution, so that all that is required is that the plane including the two center-lines is defined as the horizontal.

Figure _{2}/_{1 }is 2 at birth, so that the top line is applicable. Both mother and calf are approximately 6:1 ellipsoids, so that the minimal distance in terms of calf length is _{1 }= 0.16, with a value of 0.2-0.3 probably best to avoid collisions, beyond the first few hours after birth. The non-dimensional force can be seen to be about 3.3 at _{1 }= 0.3 for newborn calves, going down to about 1.3 for almost fully grown calves (when _{2}/_{1 }≈ 1). The dotted red line in Figure

Figure

Figure

Force calculations

To find the actual forces in specific cases, we need to define a normalizing coefficient based on specific data, to make the remaining terms in the integral dimension-free. This coefficient is obtained by taking all constants in equations (2) to (4) out of the integration. The coefficient is thus

Recalling that we assumed 6:1 ellipsoidal bodies of revolution for both mother and calf with no allometric changes during growth, the ratios _{i}/_{i}^{2 }can be calculated once and for all as

From

^{2}*0.95^{2}*0.0218^{2 }= 2.54 (7)

A reasonable minimal distance between mother and calf center-lines is the sum of half the mother's thickest section plus half the calf's thickest section. This is 1.9/12 + 0.95/12 = 0.24 m for the neonate, and 2*1.9/12 = 0.32 m for a fully grown calf. The spacing parameter is therefore at best _{1 }= 0.25 for neonates and _{1 }= 0.17 for fully-grown calves.

The maximal forward force on the calf can now be obtained from Figure _{1 }= 0.25 and length ratio _{2}/_{1 }= 2, in Figure

These values can now be compared to viscous drag on the calf, recalling that the drag force is defined as ^{2}_{D' }where _{D }is the drag coefficient. At speeds of 2.4 m/sec the drag coefficient based on wetted area for a 6:1 ellipsoid is approximately 0.003 ^{2}, so that the drag is about 12 N for the stretched body and about 36 N for the swimming calf. Comparing these drag estimates to the forward and lateral forces found previously, it is seen that the drafting forward force is close to 90% of the total drag force (that is, 10.6/12 for a coasting, stretched-straight calf) and the Bernoulli suction is much larger, but in a different direction. Thus, even when considering the enhanced drag when performing swimming motions, we see that the mother can provide a large proportion of the force required for a neonate. These numbers are reduced for larger calves, but this is again reasonable, as the larger calves are both more powerful and more adept at swimming. The cost to the mother is increased by the presence of the calf, obviously, as the curve for thrust (

Next, the effect of increased lateral distance between the center-lines of mother and calf is assessed. This is obtained from Figure _{2}/_{1}. As mentioned above, a reasonable minimal distance between mother and calf center-lines is the sum of half the mother's thickest section plus half the calf's thickest section. Table

The forward force (in N) on a neonate, as a function of lateral distance from the mother

_{1}

Ordinate

Force = Ordinate*2.54

Percentage of the maximal force possible for

0

0.25

4.16

10.6

100%

10

0.36

2.60

6.6

62%

20

0.46

1.78

4.52

43%

30

0.57

1.23^{†}

3.3

31%

The value of ^{†}The value is beyond the scope of Figure

As shown above, the mother can provide close to 90% of the thrust needed for the calf to move at 2.4 m/sec when the mother and coasting calf move side by side, almost touching. Table

A similar calculation for full-grown calves, where _{1 }= 1.9 m, and _{2}/_{1 }= 1, appears in Table _{1 }= 0.317/1.9 = 0.167. The factor

The forward force (in N) on a fully grown calf, as a function of lateral distance from the mother

_{1}

Ordinate

Force = Ordinate*10.16

Percentage of the maximal force possible for

0

0.167

2.93

29.8

100%

10

0.22

2.09

21.2

71%

20

0.27

1.58

16.0

54%

30

0.32

1.20

12.2

41%

The value of

Tables

The peak side force (in N) on a neonate, as a function of lateral distance from the mother

_{1}

Ordinate

Force = Ordinate*10.16

Percentage of the maximal force possible for

0

0.25

12.1

30.7

100%

10

0.36

7.50

19.1

62%

20

0.46

5.20

13.2

43%

30

0.57

3.61^{†}

9.17

30%

The value of ^{†}The value is beyond the scope of Figure

Similar results can be obtained for the side force and yawing moments. We only show the side force on the neonate, for which the Bernoulli effect is most important. This appears in Table

The rapid decrease of the transmitted forces with lateral distance is a clear indication that forced 'running', as in chases by fishing vessels, can easily cause loss of the mother-calf connection. Moving at high speeds will require strenuous, large-amplitude motions by both mother and calf, so that in order not to interfere with each other they would have to enlarge the lateral distance, from almost touching (

Further hydrodynamic effects

In order to obtain an exact mathematical solution to the drafting problem, I had to make some simplifying assumptions: first, the propulsive motions were not accounted for; second, no free surface effects were considered (water of infinite depth); third, inviscid flow was assumed; and fourth, uniform velocity (no jumps) was assumed. The effects of relaxing these assumptions are now examined, to see what effect such relaxation has on the results presented above. Obviously, only rough estimates of these additional, complicated effects can be made.

Effects of propulsive motions

The propulsive motions of the mother and calf are now considered. These can be described as a vertical oscillation of the body and caudal flukes, with amplitude minimal at the shoulders, and growing as one moves rearwards. For example, Romanenko

where _{T }is the maximal vertical excursion of the fluke and _{n }= _{max}sin(2π_{max }is obtained from (8),

In the burst-and-coast mode, the animal accelerates during the burst and decelerates during the coast. Thus, a calf using this mode of energy saving would appear to move relative to the mother. This would appear as forward motion during the burst, and slipping backwards during the coast. It might be difficult to observe this behavior, as the effectiveness of the burst-and-coast rises (more energy is saved) when the bursts are short and the velocity does not change appreciably ^{3}/3.2^{3 }= 12.1) by the calf to keep up with the mother when the swimming speed doubles from 3.2 m/sec (fast cruising) to 6.4 m/sec (escape speeds).

It may seem that using burst-and-coast could be counter-productive, as the calf moves away from the rather narrow range of beneficial positions. This loss can be minimized if the calf starts the burst and accelerates when it is at the optimum position for maximum forward force, at about 65% of the mother's length where the acceleration is easiest (Figure

The propulsive motions cause a periodically varying pressure field, which affects the results shown previously in two additional ways. First, the fact that the body, and especially the caudal flukes, produce a backward-moving wake makes the zone directly behind the mother highly undesirable, as moving in that area means moving against a backward-flowing current (as in schooling

Exclusion zones for the calf due to increased energy expenditure

Exclusion zones for the calf due to increased energy expenditure. Exclusion zones are bounded by dashed lines. The **(a) **side and **(b) **top views show the zone of exclusion around the tail; **(c) **the front view shows the preferred angular sector for calf placement.

In the analysis for non-oscillating ellipsoids presented in the previous section, the effects are axially symmetric, such that any angular displacement between the line connecting the center-lines of mother and calf and the horizon gives the same result. It is clear, however, that moving closely above or directly below the mother is more difficult, because of the vertical body oscillations (and, in addition, the surface effects if the calf were to be above the mother). As a result, we see that the calf is limited to zones between approximately 45° above and below the mother's center-line. The next question addressed here is what are the preferred positions for the calf, in the vertical plane, relative to the mother.

Logvinovich showed (as cited in

where _{8 }is the undisturbed pressure, _{n }is the vertical excursion velocity. _{n }can be related to the vertical excursion of the dolphin's body as

where ^{2}

Looking at cases where the mother helps the calf, and not

In the study of fish schooling mentioned previously

Recently some studies have appeared

A rough estimate for a neonate with 16 cm diameter is presented next. The detached wake of the swimming mother has a total circumference of roughly π_{m }+ 2*0.2_{m }where the subscript _{m }= 32 cm and _{m }= 1.9 m. The second term is due to the oscillations. So, the calf will pass through 16/(π*32 + 2*0.2*190) = 0.0625; the gain in thrust due to the shed boundary layer can therefore be, at best, 6.2% of the drag on the mother's fore-body (or about 3% of the total drag on the mother, which is approximately 12% of the neonate's drag at the same speed). It is important to mention here that this gain can be obtained even when the mother is coasting, so one can predict that suckling calves will preferentially draft during coasting.

Free surface effects

The main influence of the free surface of the water is to increase the energy required to move at a given speed as a result of the energy wasted on lifting the free surface. This can be roughly modeled by an increase in drag coefficient, by a factor of up to 5, depending on the ratio of depth to body hydraulic diameter, and on the swimming speed (Froude number); further details may be found in

Another free-surface effect stems from the fact that the interaction is negligible when in air, so that breaching effectively breaks up the drafting interaction. This effect is not too harmful for juveniles, as the ballistic motion the mother and calf perform means that if they leave the water together, and return together, they will be able to re-establish drafting. But infants, and especially neonates, who are less adept at porpoising, may either breach or return at non-optimal penetration angles (see analysis below, and Figure

The effects of non-optimal porpoising leaps by a mother-calf pair

The effects of non-optimal porpoising leaps by a mother-calf pair. **(a) **Optimal for distance (45° water exit and entrance) and minimal splash, with longitudinal penetration; and **(b) **non-optimal.

Viscous flow

Viscous flow theory is required to estimate the original drag force on the animal, before calculating the interactive corrections. However, as we are interested in the mother-calf interactions here, we do not need this type of calculation. Furthermore, as the Reynolds numbers are large (^{6})), the boundary layer approximation is sufficiently accurate. This means that only thin layers of fluid are affected by viscosity. The thickness of these layers is not more than 1-3% of the body radius, so that the body may be assumed to be that much thicker and to move in inviscid fluid (displacement thickness model). At distances of 25-50% of body radius, where the calf may be found, the effects are therefore negligible, to the level of accuracy of the present discussion.

Synchronization of jumping

At high swimming speeds, dolphins usually resort to porpoising

Thus, if the mother-calf pair exits the water at the same speed and angle as each other, they will land in the same relative positions as when leaving. On exit, leaving at the wrong angle can reduce the distance crossed in the air; this effect, however, is very small. From equation (11), the maximum distance is achieved at 45°. Thus, the difference in distance is

and

For a calf jumping at 40° (a 5° difference), the decrease in distance jumped is only 0.015 (1.5%), and even if the calf jumped at 30° the difference in distance crossed by the center of mass, while out of the water, is 0.134 (13.4%). For 50°, from equation (13), the difference in distance crossed is again only 0.015 (1.5%) as the decrease is symmetrical with respect to the angular difference from the maximal 45°. The distance the mother can cross, when moving at 2.4 m/sec is, from equation (11), above, about 59 cm, and at 4 m/sec it is 1.63 m. Thus, even at the higher speed, a 15° error by the calf will result in only a 22 cm longitudinal displacement in landing.

Next, steps 2 and 4 are examined. If re-entry is at same penetration angle (the angle between the animal's long axis and the horizon) for both mother and calf, the only differences that may occur are a result of the spray energy being proportional to animal mass (equation 2 from

Here, _{j }is proportional to animal mass so, all other parameters being equal, the energy lost by the smaller calf is less, but can be a higher proportion of the calf's energy store. For example, neonates can have less than 40% of the oxygen-storage capacity of adult dolphins

The stretched straight dolphin was modeled as a 6:1 ellipsoidal body. The splash produced by such a shape, when penetrating water, is highly dependent on the angle between the body longitudinal axis and the angle of penetration, with the lowest value being, naturally, when these angles are equal (see Figure

Optimal exit and entry require that the animal change orientation in mid-air from roughly 45° above the horizon when exiting, to roughly 45° below the horizon for re-entry. In practice, dolphins are observed to exit at 30-45°

Comparison with existing observations

Observations of drafting in the literature have been sparse and mainly anecdotal, and almost no data of the accuracy and detail required were found. Data collected from flights over spinner dolphin groups

Geometric parameters of drafting mother-calf pairs from Figure

Case

Calf length _{1 }(cm)

Length ratio _{2}/_{1}

Lateral displacement

Longitudinal displacement

_{1}

_{2}

H1

119

1.59

26.0

35.3

0.22

0.19

H2

114

1.67

58.5

27.7

0.51

0.15

DL

113

1.68

36.9

58.1

0.33

0.31

DR

162

1.17

30.3

63.3

0.19

0.33

A2

128

1.48

46.2

41.1

0.36

0.22

A3

168

1.13

54.3

5.4

0.32

0.03

A4

123

1.55

33.5

67.1

0.27

0.35

A5

132

1.44

52.8

60.7

0.40

0.32

A6

128

1.48

61.6

56.5

0.48

0.30

BR

168

1.13

41.9

53.1

0.25

0.28

The case name is from the picture marking (the letter) in Figure _{2 }is assumed as 190 cm. No error estimates are presented as these values are taken as indicative only.

First, I assumed the larger animal in each pair to be the mother, and took its size to be 1.90 m _{1 }in Table _{2}/_{1}. Using the same scaling, the lateral distance between center-lines of the mother and calf _{1 }were obtained. Next, the longitudinal displacement of centers of mass _{2}. This displacement _{2 }was used to find the value of thrust (_{2}. From Figure _{2 }= 0.35, so that, for example, for pair A2, _{2 }= 0.22, so that _{max }= 0.90.

Next, we find the maximal thrust (_{max}) for this case from Figure _{2}/_{1 }and _{1}. Taking again pair A2 as the example, we have _{2}/_{1 }= 1.48 and _{1 }= 0.36. From Figure

_{max}*0.9 =

Where, in this case,

So that

Recalling that the drag on a newborn calf coasting at 2.4 m/sec was estimated at 12 N, and that the drag coefficient does not change because of geometric similarity, the drag increases simply with surface area (length squared). We can thus estimate the drag on a calf of length _{1 }coasting at 2.4 m/sec by equation (16)

which for the calf of pair A2 is 21.8 N.

So, we finally determine that, in this case, the drafting thrust is 7.3/21.8 = 0.33 of the force required for the calf to coast. This value appears as _{req }in Table

The thrust force increment on the calf due to drafting

Case

_{max}

_{req}

H1

0.82

3.99

11.7

0.62

H2

0.71

3.66

3.1

0.18

DL

0.98

3.65

8.2

0.48

DR

0.99

7.39

20.5

0.58

A2

0.9

4.62

7.3

0.33

A3

0.02

7.94

0.2

0.005

A4

1

4.26

11.5

0.57

A5

0.98

4.9

7.7

0.33

A6

0.96

4.61

4.9

0.22

BR

0.94

7.94

15.3

0.41

_{max }is the fraction of the maximal thrust possible, obtained from inserting the value of _{2 }from Table _{req }represents the thrust transferred from the mother as a fraction of the total thrust required of the calf.

The lateral suction force (Bernoulli attraction) on a calf

Case

_{max}

H1

0.65

34.2

H2

0.8

10.2

DL

0.3

8.2

DR

0.22

21.9

A2

0.6

15.8

A3

1

33.3

A4

0.17

7.0

A5

0.27

6.5

A6

0.34

5.2

BR

0.4

27.3

The value _{max }is obtained from inserting the value of _{2 }from Table

Figure

Conclusions

Drafting has been shown to enable adult dolphins to help their young by reducing the forces required of the young for swimming. Several separate hydrodynamic effects join to produce this interaction. Under ideal conditions, the drafting force can counteract a large part of the drag experienced by a neonate calf. Examination of aerial photographs of eastern spinner dolphin mother-calf pairs shows that the predicted preferred positions for the calf to maximally benefit from these hydrodynamic effects are found in most cases. There is a need for more controlled experimental data to be able to improve the current model, especially where the effects of viscosity and free surface penetration are concerned, and to ascertain whether burst-and-coast motions are found when dolphins flee tuna fishermen. The clear implication for dolphin chases is that long chases at high speeds will result in an increased probability of separation of mother-calf pairs, as a result of a combination of fatigue on the calf's side, decreased help from the mother due to the larger body oscillations by both mother and calf, and the increased probability of erroneous leaping.

Materials and methods

Aerial photography

Images were taken in the eastern tropical Pacific Ocean from a Hughes 500D helicopter flying at approximately 60 knots (around 110 km/h) at about 250 m altitude (Figure

Additional data file

The following is provided as an additional file: a brief overview of slender body theory used for the calculation of flow around a pair of slender bodies (Additional data file

A brief overview of slender body theory used for the calculation of flow around a pair of slender bodies

A brief overview of slender body theory used for the calculation of flow around a pair of slender bodies

Click here for additional data file

Acknowledgements

I thank Elizabeth Edwards for asking the questions that resulted in this paper and her careful and helpful monitoring of the project, F. Archer, W.F. Perrin, and S. Reilly for useful discussions, W. Perryman and K. Cramer for permitting the use of their unpublished photographs (Figure