A preprint presents a moving-coordinate framework for slender vortex filaments and tubes that can accommodate strong core distortions while the flow remains slender. In a worked application, it derives the tilt, vorticity-surface shape and propagation speed of a steadily propagating ring with axial flow. The central shift is to make the coordinate system follow the vortex, moving much of the geometric evolution into the coordinate map while the equations describe the remaining flow.
A coordinate system that moves with the vortex
The formulation treats a moving coordinate grid explicitly. It separates the true fluid velocity u into a background velocity U and a relative velocity v, expressed as u = U + v. It is designed to work in any coordinate system: the metric g and volume form µ encode the geometry, while U represents time dependence. That separation lets the model keep coordinate motion and relative flow in the same set of equations.
The final Lagrangian coordinates are adapted so that vortex lines lie on tubular surfaces of constant r, with no vorticity flux through those surfaces. Under the imposed vorticity and relative-flow structure, the component vorticity equation collapses to one equation for the azimuthal component. The fixed axial component is wound by differential rotation and then axially transported.
The approximation depends on scale
The construction depends on a scale separation. The slender-vortex expansion sets the core scale a = O(1), the larger length scale ℓ much greater than 1, and ε = a/ℓ much less than 1. For the ring, the circumference is 2πℓ, the curvature is κ = ℓ^-1, and the minor radius is a, with ε = a/ℓ ≪ 1. In ordinary terms, the tube is being treated as narrow compared with the scale of the curve it follows.
The same expansion orders the ring's tilt: q = O(ε), qᵣ = O(ε), qθ = O(ε), while qz and qt are O(ε²). These are orders imposed by the slender expansion, identifying which components enter at leading and smaller scales.
The ring calculation turns geometry into equations
For the ring, a fluctuating-momentum relation fixes the tilt from curvature, radius and the relevant axial and azimuthal velocity scales. The calculation then eliminates an auxiliary shape variable and produces a differential equation for the ring-shape function f. This makes the vorticity-surface shape an output of the calculation rather than a feature inserted in advance.
That is the broader purpose of the framework. It is aimed at vortex-filament or tube models with very general dynamics and strong core distortions, so long as slenderness is retained. For a new geometry or dynamical process, however, the general equations still need problem-specific modeling and approximation.
Matching inside and outside the tube
The propagation speed comes from matching an inner stream function to an outer stream function. The reported result is Vc = Γκ/(4π)[log(8/(aκ)) − 1/4 − 16Q²/(3a²Γ²)]. When Q = 0, the calculation recovers Kelvin's result. The velocity is therefore tied to the ring's slender geometry and the assumptions used to match its inner and outer regions.
A framework still awaiting validation
This is a model calculation: vorticity is localized around an evolving curve C(t), and the ring is the worked configuration. The supplied analysis reports no statistical uncertainty, confidence interval or empirical error bound for the matched velocity, and no numerical error estimate for the shape equation.
The slender requirement marks a real limit: the coordinate construction can break down when aκ is approximately 1, a case outside the stated ε ≪ 1 regime. The worked ring is also a specific steady-propagation, constant-curvature configuration, not a general account of every transient or interacting vortex. Those restrictions define what the current derivation can and cannot claim.
The framework is presented as a basis for future simplified equations for vortex motion and interaction. The supplied analysis identifies quantitative comparison of the predicted ring velocity and surface shapes with independent simulations or experiments as a question for future work.
Paper data and sources
Original title: Vortex filament dynamics and vortex ring motion revisited
Authors: Andrew D Gilbert
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text