© 2002 by The Society for Integrative and Comparative Biology
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Dynamics of Dolphin Porpoising Revisited1
1 Faculty of Aerospace Engineering Technion, Haifa 32000, Israel
Porpoising is the popular name for the high-speed surface piercing motion of dolphins and other species, in which long, ballistic jumps are alternated with sections of swimming close to the surface. The first analysis of this behavior (Au and Weihs, 1980) showed that above a certain "crossover" speed this behavior is energetically advantageous, as the reduction in drag due to movement in the air becomes greater than the added cost of leaping.
Since that publication several studies documented porpoising behavior at high speeds. The observations indicated that the behavior was more complex than previously assumed. The leaps were interspersed with relatively long swimming bouts, of about twice the leap length. In the present paper, the possibility of dolphins using a combination of leaping and burst and coast swimming is examined. A three-phase model is proposed, in which the dolphin leaps out of the water at a speed Uf, which is the final speed obtained at the end of the burst phase of burst and coast swimming. The leap is at constant speed and so the animal returns to the water at Uf, goes to a shallow depth and starts horizontal coasting while losing speed, till it reaches Ui. At that point it starts active swimming, accelerating to Uf. It then starts the next leap. Ranges of speeds for which this three-stage swimming is advantageous are calculated as a function of animal and physical parameters.
Notation
C—Constant defined in equation (12)
CD—Coasting drag coefficient
D—Drag
g—Gravitational acceleration
H—Height of jump
J—Energy required for jump
k—Ratio of swim length to jump length
l—Distance
L—Total distance (eq. 28)
m—Added mass
M—Animal mass
M1—Total mass
r—Coefficient defined in eq. (22)
R—Ratio of energies, for three-phase swimming
R2—Ratio of energies, for burst and coast swimming
t—Time
T—Thrust
U—Speed
V—Body volume
W—Weight
—Emergence (=return) angle
ß—Swim / coast drag penalty ratio
—Surface effects drag ratio
—Density of seawater and cetacean.
Subscripts
a—air
av—Average
b—Burst phase
c—Coast phase
e—Reference (maximal) thrust
f—Final, at end of burst
i—Initial, at start of burst
j—Jump phase
n—Nominal reference thrust
o—Optimal
s—Surface swimming
w—Water
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