Fluid dynamics · Hamiltonian vortex motion

Point-Vortex Dynamics Studio

How do a handful of point vortices move one another? Each vortex simply drifts with the flow that all the others create. Two vortices of the same sign circle a common center; two of opposite sign travel together. Two such opposite-sign pairs can leapfrog through each other, and with four or more vortices, partners can swap chaotically.

Drag point vortices around a two-dimensional phase plane and compare a corotating pair, a translating dipole, leapfrogging dipoles and four interacting vortices. The velocity field, the tracers, the minimum distance between vortices and the Hamiltonian drift all update together.

Vortex count $2$

Total circulation

\sum_i\Gamma_i

$4.00\,\mathrm{m^2\,s^{-1}}$

Minimum separation

d_{\min}

$2.00\,\mathrm m$

Hamiltonian

H

$-0.44$

Relative drift

|\Delta H/H_0|

$0.00\,\%$

Orbit familyCorotating pair

Vortex workbench

Compose an orbit from signed circulation

Fixed-time position mapSample vortex A at equal physical-time intervals. This is a stroboscopic map, not a Poincaré crossing section.Velocity vectorsSample the instantaneous softened Biot–Savart field.Passive tracersAdvect massless markers without feeding back on the vortices.Invariant historyPlot minimum distance and numerical Hamiltonian drift.Periodic boundariesUse a minimum-image periodic square instead of the unbounded-plane view.Sampling interval

\Delta t_s

0.50\,\mathrm{s}

Circulation magnitude

|\Gamma|

2.00\,\mathrm{m^2\,s^{-1}}

Core regularization

\varepsilon

0.08\,\mathrm m

Background strain

S

0.00\,\mathrm{s^{-1}}

Evolution speed

s_t

1.00\times

The Periodic vortex row preset sums its infinite set of images analytically. The flow repeats horizontally and is unbounded vertically; it is not a doubly periodic box. The other periodic presets still use the minimum-image model, which is an approximation and is labeled as one. The Hamiltonian includes the potential of the background strain. Under an imposed strain or a periodic boundary, the usual impulse invariants need not be conserved.

Drag any vortex in the phase plane to set a new initial condition. When you let go, the conserved quantities are measured afresh from that state.

This two-dimensional inviscid model moves every vortex with the velocity $\dot{\mathbf x}_i$ induced by all the others. The core scale $\varepsilon$ removes the numerical singularity but changes close encounters.

Physics tutorial

Point vortices: complex flow from minimal degrees of freedom

BackgroundIn a two-dimensional inviscid flow, a small, concentrated patch of vorticity looks like a point from far away, described only by its position and circulation. A point vortex does not move itself. It is carried along by the velocity that all the other vortices induce through the Biot–Savart law.

Why it mattersThe model is a classic workbench for translating dipoles, corotating pairs, leapfrogging dipoles, and few-body chaos. Aref and collaborators described it in their 2007 review as a classical mathematics playground connecting fluids, Hamiltonian mechanics, and dynamical systems.

Start with the essentials

Focus question: How can nothing more than positions and signed circulations produce stable orbits and complicated exchanges?
One-sentence intuition: Vortex $i$ moves with the sum of the velocities induced by every other vortex. Two vortices of the same sign circle a common center; two of opposite sign travel together at right angles to the line joining them.

Core mathematical model

Soft-core point-vortex equations

\dot x_i=-\frac{1}{2\pi}\sum_{j\ne i}\Gamma_j\frac{y_i-y_j}{r_{ij}^2+\varepsilon^2},\qquad \dot y_i=\frac{1}{2\pi}\sum_{j\ne i}\Gamma_j\frac{x_i-x_j}{r_{ij}^2+\varepsilon^2}

The core scale removes the singularity at zero distance; outside the core, the classical point-vortex velocity returns.

Regularized Hamiltonian

H=-\frac{1}{4\pi}\sum_{i<j}\Gamma_i\Gamma_j\ln\!\left(r_{ij}^2+\varepsilon^2\right)

With no background strain, and away from jumps at the periodic boundary, it is conserved and gives a sensitive check on the numerical integration.

Exact speeds for two softened vortices

\omega_{++}=\frac{\Gamma}{\pi(d^2+\varepsilon^2)},\qquad U_{+-}=\frac{|\Gamma|d}{2\pi(d^2+\varepsilon^2)}

Two equal vortices of the same sign rotate at this rate; two equal vortices of opposite sign form a dipole that moves at this speed.

Common difficulties

A vortex does not push itself along

Typical misconceptionThe visible spin of a point vortex gives that vortex its own forward velocity.

Better mental modelWith symmetric softening, the velocity a vortex induces on itself is zero. Its motion comes entirely from the other vortices and the background flow.

Softening is not viscous diffusion

Typical misconceptionA larger softening core is the same as a more viscous fluid.

Better mental modelSoftening only smooths the near-field singularity; it does not spread the core in time or dissipate circulation.

Run the experiment

01
Check a corotating pair
Choose Corotating pair with zero background strain and watch pair distance and Hamiltonian drift.
What to observe: The pair rotates about its midpoint, the separation stays nearly constant and the numerical drift stays small.
02
Reverse one circulation
Switch to Translating dipole and compare the trails with passive tracers.
What to observe: The two vortices keep their separation and travel together, carrying the surrounding fluid along in a dipole pattern.
03
Break the integrable symmetry
Choose Leapfrogging pairs or Perturbed quadrupole, then drag one core.
What to observe: Which vortices swap places now depends sensitively on the starting layout. The conserved quantities still tell real sensitivity apart from numerical error.

Conceptual inspiration: FluidMotion. Original simulations and graphics; the model limits are described above.