This is a vortex-impulse model of the same jellyfish jet: instead of balancing a quasi-steady momentum flux against drag, it tracks the actual formation of the shed vortex ring — the doughnut-shaped rotating parcel of water each contraction leaves behind — and derives thrust from the rate the ring accumulates circulation and hydrodynamic impulse.
Vorticity flux (slug model): dΓ/dt = ½·u_jet²
Formation number: F = ∫u_jet dt / D_orifice
Ring impulse (thin ring): I = ρ_water·Γ·π·R²
Ring self-speed (Saffman): U_ring = Γ/(4πR)·[ln(8R/a_core) − ¼]
Swim reaction: m_eff·(dv/dt) = dI/dt − ½ρ_water·C_d·A_frontal·v·|v|
- Formation number — the Gharib–Rosenfeld–Dabiri result that a starting jet can only feed a coherent vortex ring up to a universal length-to-diameter ratio of about 4; push more fluid through the same stroke and the excess sheds as a trailing jet instead of rolling into the ring. Real jellyfish and squid strokes sit close to this threshold — it is the same saturation effect the companion 3D bell-contraction model's own theory notes call out as "simplified away."
- Ring circulation Γ — the rotational strength accumulated in the ring this stroke; it sets both the ring's self-propagation speed and the impulse it carries away, which by Newton's third law is exactly the forward push delivered to the animal.
- Duty cycle and frequency control how fast fluid is forced through the orifice each stroke, which drives both Γ and the formation number together — a faster, shorter power stroke reaches the pinch-off threshold sooner.
- Bell diameter sets the orifice size (and hence the vortex ring's radius) and the added mass being accelerated; drag sets how quickly the animal decelerates between pulses.
Watch the side-view: each pinch-off spawns a pair of counter-rotating vortex cores (the 2D cross-section of the ring) that drift apart and downward on their own self-induced velocity while the animal glides upward on the reaction.