Dynamics of dolphin porpoising revisited

Porpoising is the popular name for the high-speed surface piercingmotion of dolphins and other species, in which long, ballisticjumps are alternated with sections of swimming close to thesurface. The first analysis of this behavior (Au and Weihs,1980) showed that above a certain "crossover" speed this behavioris energetically advantageous, as the reduction in drag dueto movement in the air becomes greater than the added cost ofleaping. Since that publication several studies documented porpoisingbehavior at high speeds. The observations indicated that thebehavior was more complex than previously assumed. The leapswere interspersed with relatively long swimming bouts, of abouttwice the leap length. In the present paper, the possibilityof dolphins using a combination of leaping and burst and coastswimming is examined. A three-phase model is proposed, in whichthe dolphin leaps out of the water at a speed U_f, which is thefinal speed obtained at the end of the burst phase of burstand coast swimming. The leap is at constant speed and so theanimal returns to the water at U_f, goes to a shallow depth andstarts horizontal coasting while losing speed, till it reachesU_i. At that point it starts active swimming, accelerating toU_f. It then starts the next leap. Ranges of speeds for whichthis three-stage swimming is advantageous are calculated asa function of animal and physical parameters. Notation C—Constant defined in equation (12) C_D—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