Fluid dynamics · Hamiltonian vortex motion

Point-Vortex Dynamics Studio

Drag point vortices in a two-dimensional phase plane and compare a corotating pair, translating dipole, leapfrogging dipoles, and four-vortex interaction. The velocity field, tracers, minimum distance, and Hamiltonian drift 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

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.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

Drag any vortex in the phase plane to create a new initial condition; the conservation baseline resets at release.

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 two-dimensional inviscid flow, a compact vorticity patch can be approximated from afar by a point carrying only position and circulation. A point vortex has no self-induced translation; all other vortices jointly advect it through the Biot–Savart velocity.

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 positions and signed circulations alone assemble stable orbits and complicated exchanges?
One-sentence intuition: Vortex $i$ moves with the sum of velocities induced by every other vortex. Like signs rotate about a common center; opposite signs translate perpendicular to their line of centers.

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 zero-distance singularity while recovering the classical point-vortex velocity outside the core.

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 periodic wrap discontinuities, it provides a sensitive numerical-conservation check.

Exact softened two-vortex speeds

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

Equal like-signed vortices rotate; equal opposite-signed vortices form a translating dipole.

Common difficulties

A core does not propel itself

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

Better mental modelIts symmetric self-induced velocity is zero. Translation comes entirely from other vortices and the background flow.

Softening is not viscous diffusion

Typical misconceptionIncreasing core regularization is equivalent to increasing physical viscosity.

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

Run the experiment

01
Verify 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 while separation remains nearly constant and numerical drift stays small.
02
Reverse one circulation
Switch to Translating dipole and compare the trails with passive tracers.
What to observe: Both vortices preserve their separation and translate together while carrying a dipolar flow pattern.
03
Break the integrable symmetry
Choose Leapfrogging pairs or Perturbed quadrupole, then drag one core.
What to observe: Exchange becomes sensitive to initial geometry while invariants still separate physical sensitivity from numerical error.