Dynamics of dolphin porpoising revisited

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 U(f), 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 U(f), goes to a shallow depth and starts horizontal coasting while losing speed, till it reaches U(i). At that point it starts active swimming, accelerating to U(f). 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.NotationC-Constant defined in equation (12)C(D)-Coasting drag coefficientD-Dragg-Gravitational accelerationH-Height of jumpJ-Energy required for jumpk-Ratio of swim length to jump lengthl-DistanceL-Total distance (eq. 28)m-Added massM-Animal massM(1)-Total massr-Coefficient defined in eq. (22)R-Ratio of energies, for three-phase swimmingR(2)-Ratio of energies, for burst and coast swimmingt-TimeT-ThrustU-SpeedV-Body volumeW-Weightα-Emergence (=return) angleβ-Swim / coast drag penalty ratioγ-Surface effects drag ratioρ-Density of seawater and cetacean.Subscriptsa-airav-Averageb-Burst phasec-Coast phasee-Reference (maximal) thrustf-Final, at end of bursti-Initial, at start of burstj-Jump phasen-Nominal reference thrusto-Optimals-Surface swimmingw-Water.