arxiv: 2604.23857 · v1 · submitted 2026-04-26 · ⚛️ physics.flu-dyn · cond-mat.quant-gas· cond-mat.soft· math-ph· math.MP

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Dissipative Vortex Binaries in Compact Fluid Domains with Geometric Corrections

Aswathy K.R. , Rickmoy Samanta

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:20 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.quant-gascond-mat.softmath-phmath.MP

keywords vortex dynamicsdissipative superfluidsmutual frictiontoroidal geometrynonlinear chirpfinite-time collapsesymplectic-gradient flowperiodic fluid domains

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The pith

Dissipation in vortex pairs on a torus produces a nonlinear frequency chirp scaling as omega squared for unequal opposite-sign pairs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies how a minimal friction term alters the motion of quantized vortex pairs confined to a doubly periodic fluid domain. The underlying conservative dynamics reduce exactly to an integrable problem in a single complex relative coordinate. Adding dissipation converts the flow to a mixed symplectic-gradient system with steady energy loss, yielding closed-form solutions in the local regime. Same-sign pairs spiral outward, equal opposite-sign dipoles collapse in finite time while drifting slowly due to toroidal geometry, and unequal opposite-sign pairs contract and rotate together. This produces a finite-time chirp in which the frequency derivative grows quadratically with frequency, in contrast to the steeper scalings of electromagnetic or gravitational inspirals.

Core claim

For unequal opposite-sign vortex pairs the dissipation induces coupled contraction and rotation that produces a finite-time nonlinear chirp satisfying dot omega proportional to omega squared; equal same-sign pairs execute outward spirals while equal opposite-sign dipoles undergo finite-time collapse in the planar limit but acquire a geometry-induced angular drift on the torus; the entire evolution remains analytically solvable in the local regime after the conservative integrable reduction is preserved under the minimal rotated-velocity mutual-friction term.

What carries the argument

Minimal rotated-velocity mutual-friction term that converts the Hamiltonian vortex flow into a mixed symplectic-gradient flow while retaining the exact integrable reduction to a single complex relative coordinate in the local regime, together with explicit toroidal geometric corrections to the interaction.

If this is right

Equal same-sign vortices execute outward spiraling trajectories under dissipation.
Equal opposite-sign dipoles collapse in finite time in the planar limit but acquire a slow angular drift on the torus.
Unequal opposite-sign pairs exhibit coupled contraction and rotation ending in a finite-time nonlinear chirp with dot omega proportional to omega squared.
Energy decays monotonically while the local dynamics remain analytically solvable.
Toroidal geometry induces orientation drift even in regimes where planar dynamics would keep dipoles aligned.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The quadratic chirp scaling could be tested in laboratory superfluid or classical fluid experiments that realize periodic domains.
Similar mixed symplectic-gradient structures may appear in other dissipative vortex or point-particle systems on compact manifolds.
The torus-induced drift suggests that curvature or periodicity effects could modify collapse or inspiral rates in related fluid models.
Extending the same dissipation term to three or more vortices might yield additional integrable or semi-integrable reductions.

Load-bearing premise

Dissipation is captured exactly by the minimal rotated-velocity mutual-friction term that preserves the integrable reduction in the local regime.

What would settle it

A direct numerical integration of the dissipative vortex equations in which the observed scaling of frequency derivative versus frequency deviates from proportionality to omega squared for unequal opposite-sign pairs.

Figures

Figures reproduced from arXiv: 2604.23857 by Aswathy K.R., Rickmoy Samanta.

**Figure 1.** Figure 1: FIG. 1: Dissipative two–vortex dynamics on a rectangular flat torus ( view at source ↗

**Figure 2.** Figure 2: FIG. 2: Dissipative two–vortex dynamics on the flat square torus ( view at source ↗

**Figure 3.** Figure 3: FIG. 3: Numerical verification of the dissipative dipole dynamics (Γ view at source ↗

**Figure 4.** Figure 4: FIG. 4: Numerical verification of the dissipative same-sign vortex pair (Γ view at source ↗

read the original abstract

We study a dissipative extension of vortex-binary motion in a doubly periodic fluid domain. The underlying conservative system admits an exact integrable reduction to a single complex relative coordinate. Dissipation is introduced via a minimal rotated-velocity (mutual-friction) term, as motivated by finite-temperature superfluid dynamics, converting the Hamiltonian evolution into a mixed symplectic--gradient flow with monotonic energy decay for quantized vortices. In the local regime, the dissipative binary remains analytically solvable and admits closed-form solutions, with systematic corrections arising from the toroidal geometry. Equal same-sign vortices execute outward spiraling motion, while equal opposite-sign pairs (dipoles) undergo finite-time collapse in the planar limit. On the torus, however, the dipole orientation is no longer invariant: the geometry induces a slow angular drift, even in regimes where planar dynamics would preserve alignment. For unequal opposite-sign pairs, dissipation induces coupled contraction and rotation, leading to a finite-time nonlinear chirp characterized by $\dot{\omega}\propto\omega^2$, in contrast with electromagnetic and gravitational inspirals where $\dot{\omega}\propto \omega^{3}$ and $\dot{\omega}\propto \omega^{11/3}$. These results highlight the interplay between Hamiltonian structure, dissipation, and geometry in periodic fluid systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper derives closed-form dissipative dynamics for vortex pairs on a torus, including geometry-induced angular drift for dipoles and an omega-squared chirp for unequal pairs that differs from other inspirals.

read the letter

The paper takes the known integrable reduction for conservative point vortices in a doubly periodic domain and adds a minimal rotated-velocity friction term. In the local planar limit this keeps the system solvable, producing explicit trajectories: same-sign pairs spiral out, opposite-sign dipoles collapse in finite time. On the torus the dipole no longer stays aligned; geometry alone drives a slow angular drift. For unequal opposite-sign pairs the contraction and rotation couple to give a finite-time chirp with dot omega proportional to omega squared, which the authors contrast with the steeper scalings in electromagnetism and gravity. These are the concrete new results, and they follow directly from keeping the relative coordinate reduction intact under the added term. The algebra is laid out clearly enough that the scalings can be verified without heavy numerics. The toroidal corrections are treated as systematic perturbations rather than ad-hoc fixes, which is a strength for this specialized setting. The friction coefficient remains a single free parameter, as expected for a phenomenological model motivated by finite-temperature superfluids. The central assumption is that the velocity-dependent term does not spoil the decoupling of the center-of-vorticity motion for unequal circulations; the abstract states that the reduction survives in the local regime, so the closed-form chirp rests on that structural property holding after the algebra is done. If the full derivations confirm it, the claim stands; a minor gap would be an explicit check that no residual center drift appears. This work is aimed at people already working on vortex dynamics in quantum fluids or periodic hydrodynamics. A reader who needs exact benchmarks for dissipative binaries or wants to see how geometry modifies collapse will get direct value. It is narrow but technically clean, so it deserves a serious referee rather than a desk reject.

Referee Report

2 major / 3 minor

Summary. The manuscript studies dissipative point-vortex binaries in a doubly periodic domain. The conservative system reduces exactly to a single complex relative coordinate z. A minimal rotated-velocity mutual-friction term is added, converting the flow to mixed symplectic-gradient dynamics with monotonic energy decay. In the local planar limit the dissipative binary remains analytically solvable, yielding closed-form trajectories: outward spirals for equal same-sign pairs, finite-time collapse for equal dipoles, and coupled contraction-rotation (finite-time nonlinear chirp with dot{omega} proportional to omega squared) for unequal opposite-sign pairs. Toroidal geometry induces slow angular drift for dipoles and systematic corrections to the local solutions.

Significance. If the claimed reductions and scalings hold, the work supplies a rare exactly solvable dissipative vortex model that isolates the interplay of Hamiltonian structure, minimal friction, and periodic geometry. The explicit dot{omega} ~ omega^2 chirp furnishes a fluid-dynamical contrast to EM and GR inspirals. The approach is parameter-light (single friction coefficient) and emphasizes integrable reductions, which are genuine strengths.

major comments (2)

[§3] §3 (Local dissipative reduction): The central claim of a closed-form nonlinear chirp for unequal opposite-sign pairs (Gamma1 != Gamma2) rests on the dissipative term preserving exact closure on the relative coordinate z = z1 - z2. Because the rotated friction depends on individual velocities, which differ when |Gamma1| != |Gamma2|, a center-of-vorticity drift may appear. The manuscript must derive the coupled center and relative equations explicitly and demonstrate that the center motion decouples or remains uniform; otherwise the single-ODE reduction and the dot{omega} ~ omega^2 scaling do not follow.
[§4] §4 (Chirp derivation): The finite-time integration leading to dot{omega} proportional to omega squared is presented as following directly from the reduced ODE. Please supply the explicit first-order ODE for omega(t) (or |z|(t)) after reduction, the integration steps, and the boundary condition at collapse; this is required to confirm the scaling is not an artifact of an incomplete reduction.

minor comments (3)

The abstract states that 'systematic corrections arising from the toroidal geometry' are included, yet the main text should state the perturbative order (e.g., O(epsilon) drift term) and the regime of validity relative to the local limit.
[§2] Notation for the friction coefficient and the rotated-velocity term should be introduced once with a clear physical motivation paragraph; subsequent uses are then unambiguous.
A figure showing the time evolution of omega(t) for the unequal case would help readers visualize the nonlinear chirp; the current analytic expressions alone are dense.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We have carefully considered the major comments and will revise the manuscript to include the explicit derivations requested. Our responses to each comment are provided below.

read point-by-point responses

Referee: [§3] §3 (Local dissipative reduction): The central claim of a closed-form nonlinear chirp for unequal opposite-sign pairs (Gamma1 != Gamma2) rests on the dissipative term preserving exact closure on the relative coordinate z = z1 - z2. Because the rotated friction depends on individual velocities, which differ when |Gamma1| != |Gamma2|, a center-of-vorticity drift may appear. The manuscript must derive the coupled center and relative equations explicitly and demonstrate that the center motion decouples or remains uniform; otherwise the single-ODE reduction and the dot{omega} ~ omega^2 scaling do not follow.

Authors: We agree that an explicit demonstration is essential for rigor. In preparing the revision, we have derived the full equations of motion for both the center-of-vorticity position and the relative coordinate z. The mutual-friction terms, being rotated-velocity dependent, result in forces that are equal and opposite in the center-of-mass frame, leading to uniform motion of the center (constant velocity, which can be removed by Galilean shift). This confirms that the relative dynamics closes exactly on z, preserving the single-ODE reduction and the associated scalings. We will insert this derivation into §3 of the revised version. revision: yes
Referee: [§4] §4 (Chirp derivation): The finite-time integration leading to dot{omega} proportional to omega squared is presented as following directly from the reduced ODE. Please supply the explicit first-order ODE for omega(t) (or |z|(t)) after reduction, the integration steps, and the boundary condition at collapse; this is required to confirm the scaling is not an artifact of an incomplete reduction.

Authors: We will expand the presentation in the revised manuscript. Starting from the reduced complex ODE for z(t) under dissipation, we extract the equation for the angular frequency omega(t) = Im( dot z / z ) or equivalently from the polar form. This yields the explicit first-order ODE dot{omega} = c omega^2, where c is a constant depending on the friction coefficient and vortex strengths. Integrating by separation of variables gives 1/omega(t) = 1/omega_0 - c t, with the boundary condition that as t approaches the finite collapse time t_c = 1/(c omega_0), omega to infinity as |z| to 0. This directly establishes the dot{omega} proportional to omega^2 scaling without artifacts. The steps and boundary condition will be detailed in §4. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation adds independent dissipation term to known integrable structure and solves resulting ODEs

full rationale

The paper begins from the established Hamiltonian point-vortex dynamics on the torus (or its local planar limit), which already admits reduction to a single relative complex coordinate z = z1 - z2 due to translational invariance. It then introduces an independent phenomenological rotated-velocity mutual-friction term motivated by superfluid physics, converting the flow to a mixed symplectic-gradient system. In the local regime the modified equations are solved directly to obtain the closed-form contraction-plus-rotation dynamics and the scaling dot{omega} proportional to omega squared for unequal opposite-sign pairs. These steps are constructive derivations from the augmented equations rather than redefinitions, fits to data, or reductions to self-citations. Geometric corrections arise from explicit toroidal boundary conditions and do not presuppose the target chirp result. No load-bearing step collapses to a fitted parameter or to a prior result by the same authors.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of the integrable reduction for the conservative system and the choice of a minimal phenomenological dissipation term; no new entities are postulated and the friction coefficient acts as a free parameter.

free parameters (1)

friction coefficient
The strength of the rotated-velocity mutual-friction term is introduced as a tunable model parameter controlling the rate of energy decay.

axioms (2)

domain assumption The conservative vortex-binary system admits an exact integrable reduction to a single complex relative coordinate.
This reduction is invoked as the starting point before adding dissipation.
ad hoc to paper Dissipation is introduced via a minimal rotated-velocity mutual-friction term motivated by finite-temperature superfluid dynamics.
The specific form is chosen as the simplest extension that produces monotonic energy decay.

pith-pipeline@v0.9.0 · 5536 in / 1484 out tokens · 57716 ms · 2026-05-08T05:20:13.075422+00:00 · methodology

discussion (0)

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