Instability of a Vortex Ring due to Toroidal Normal Fluid Flow in Superfluid 4He

Bhimsen K. Shivamoggi

REVIEW 3 major objections 3 minor 37 references

Instability of a Vortex Ring due to Toroidal Normal Fluid Flow in Superfluid 4He

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper shows that a toroidal normal-fluid flow drives vortex rings in superfluid 4He unstable, at a growth rate independent of ring radius.

desk verdict Plausible ring instability undermined by an incorrect approximate solution: Eq. (8) doesn't solve Eq. (7), so the paper's quantitative claims don't stand as written. read the letter →

arxiv 1908.04359 v3 pith:6XYC7WL7 submitted 2019-08-12 cond-mat.other

Bhimsen K. Shivamoggi This is my paper · ORCID

classification cond-mat.other

keywords superfluid4HevortexringsmutualfrictionDonnelly-GlabersoninstabilitylocalinductionapproximationnormalfluidflowKelvinwavestoroidal

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a self-propelling vortex ring in superfluid 4He, which is stable when the surrounding normal fluid is at rest, becomes unstable when the normal fluid flows around the ring in the azimuthal (toroidal) direction. The author linearizes the mutual-friction-coupled vortex motion around a circular ring and derives a dispersion relation whose imaginary part changes sign once $U_\theta$ is present, so small sinusoidal perturbations grow exponentially. The instability is presented as the ring analogue of the Donnelly-Glaberson instability of Kelvin waves on a straight vortex filament. A further claim is that the growth rate is essentially independent of the ring radius, obtained by taking the mode number $m$ proportional to $r_0$.

What carries the argument

The load-bearing object is the HVBK (two-fluid mutual-friction) equation (1) in the local induction approximation, with the normal-fluid velocity taken as a purely toroidal flow $U=U_\theta \hat e_\theta$. Linearizing around a circular ring of radius $r_0$ and assuming perturbations of the form $e^{i(m\theta-\omega t)}$ gives the dispersion relation (7); the small-$\alpha$ approximation (8) is what brings out the instability. The step $m=kr_0$, treating the mode number as proportional to the ring radius, is the mechanism that makes the predicted growth rate independent of $r_0$.

What would settle it

Take the quadratic dispersion relation (7) with a fixed integer mode, say $m=2$, and compute the imaginary part of $\omega$ for two different ring radii $r_0$ without using $m=kr_0$; if the growth rate changes with $r_0$, the radius-independence claim fails. Alternatively, solve the linearized equations (5) numerically with $U_\theta\neq0$ and see whether small perturbations grow in time faster than the normal fluid decays.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the mutual friction between a vortex ring and a normal-fluid flow tangent to the ring's circulation (a toroidal flow) turns the ring's self-propelled motion into an instability. Starting from the HVBK (two-fluid mutual-friction) equation in the local induction approximation, the paper obtains the linearized equations (5) for perturbations $\hat r(\theta,t)$ and $\hat z(\theta,t)$, and from them the quadratic dispersion relation (7). For small mutual-friction coefficient $\alpha$, the solution (8) has an imaginary part proportional to $\alpha U_\theta$; when $U_\theta$ exceeds a threshold, that imaginary part produces exponential growth rather than decay. Taking $m=kr_0$, the paper concludes that the growth rate does not depend on $r_0$.

Load-bearing premise

The paper's claim that the growth rate is independent of the ring radius rests on writing the perturbation mode number as proportional to the radius; for a closed ring the mode number is just an integer, so without that imposed proportionality the growth rate generally changes with radius.

Editorial extensions

If this is right

If a toroidal normal-fluid flow is present, sinusoidal perturbations of a vortex ring grow exponentially rather than decay, so the ring becomes unstable.
The instability is qualitatively the ring counterpart of the Donnelly-Glaberson instability of Kelvin waves on a vortex filament.
If the growth rate is indeed independent of ring radius, large and small rings are destabilized equally quickly by the same toroidal flow.
The instability develops only when its growth time is shorter than the viscous decay time of the normal-fluid flow, a condition favoring large $U_\theta$ and large rings.
In the limits $U_\theta\to0$ and $\alpha\to0$, the dispersion relation reduces to the earlier mutual-friction decay result and to the classical hydrodynamics result, respectively.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: The radius-independent growth rate is not a robust consequence of the linearized equations, because for a closed ring $m$ is an integer independent of $r_0$; solving (7) exactly for fixed $m$ generally gives an imaginary part that varies with the ring radius.
Editorial inference: The instability mechanism suggests an experimental route: a beam of vortex rings exposed to a controlled azimuthal normal-fluid flow should show growing ring distortion; the paper does not propose such an experiment.
Editorial inference: If the analogy with the Donnelly-Glaberson instability holds, the same toroidal flow could also destabilize higher-order ring distortions and possibly drive a ring toward a turbulent tangle; that extension is not worked out here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: the paper has a plausible physical idea, but the central quantitative result is wrong. Equation (8) is not the small-α solution of the paper's own dispersion relation (7). The reader's stress-test is correct: expanding (7) with ω = ω0 + αω1 gives Im ω = α(2m^2−1)[Uθ/(2 r0 sqrt(m^2−1)) − γ/(2 r0^2)] for the branch that can go unstable, which has different prefactors from (8). The advertised growth rate, the radius-independence claim, and condition (9) are therefore unsupported.

What is genuinely new: the setup. Kiknadze and Mamaladze studied a superfluid ring with mutual friction but no normal flow; Arms and Hama did the hydrodynamic ring; the author's earlier work on straight filaments with axial normal flow doesn't cover a curved ring. Adding a toroidal Uθ to the ring is a legitimate gap to fill. The linearization to (5a,b) checks out, and (7) is correct to O(α). The qualitative instability—that a strong enough toroidal flow can destabilize the ring—survives the corrected calculation, so the core intuition isn't misplaced.

The soft spot is exactly where the reader puts it. Eq. (8) is not a typo; it's load-bearing. The abstract's headline claims about radius independence rest on (8) and on the m=kr0 substitution, which is itself a modeling choice for a closed ring. Even with that substitution, the paper's (8) gives a growth rate that depends on r0; the corrected expansion gives a radius-independent rate (~αk(Uθ−γk)) in the large-r0 limit, but that's not what is written. So the result as submitted is not established.

Other notes: the paper is short, the references are reasonable, and the appendix is a reprise of the straight-line case. No numerical or experimental content; this is a purely analytical claim.

Recommendation: I would not accept as is. But I would not desk-reject either. This is the kind of paper a competent referee can fix in one round: identify the wrong expansion, recompute, and check whether the corrected result still supports the abstract. It deserves serious referee time, but the likely outcome is major revision or rejection with a clear path to a corrected follow-up.

Referee Report

3 major / 3 minor

Summary. The manuscript treats the linear stability of a circular vortex ring in superfluid 4He when the normal fluid has a prescribed toroidal (azimuthal) velocity Uθ. Working in the local induction approximation supplemented by a mutual-friction force (HVBK model), the author derives two coupled linear equations for small displacements and obtains the dispersion relation Eq. (7). An approximate solution, Eq. (8), is then used to conclude that the ring is destabilized by the toroidal flow, in analogy with the Donnelly-Glaberson instability, and that the growth rate is essentially independent of the ring radius after setting m = kr0. The paper also gives a condition, Eq. (9), for the instability to develop before viscous decay of the normal-fluid flow, and an appendix re-derives the straight-filament analogue.

Significance. The physical question is well motivated: vortex rings are important in superfluid turbulence, and a normal-fluid-driven instability of a closed ring would be a useful extension of the Donnelly-Glaberson mechanism. The linearization leading to Eq. (7) is explicit and, on inspection, algebraically consistent; the use of the HVBK model and the decision to keep only the α mutual-friction term are clearly stated. The central problem is that Eq. (8) does not follow from Eq. (7), so the paper's quantitative claims are not established as written. The corrected small-α expansion still yields an instability for sufficiently strong Uθ and, in the large-radius limit with m = kr0, a radius-independent growth rate, so the qualitative idea appears plausible and worth pursuing in a revision.

major comments (3)

[Section 2, Eq. (8)] Equation (8) is not the small-α solution of Eq. (7). Writing ω = ω0 + αω1 with ω0 = ±σm√(m²−1) and expanding Eq. (7) to first order in α gives 2ω0ω1 + iσω0(2m²−1) − iσ(mUθ/r0)(2m²−1) = 0, so that Im ω = −ασ(2m²−1)/2 ± αUθ(2m²−1)/(2r0√(m²−1)). The printed Eq. (8) instead has −ασ(2m²−1) and ±αUθ(2m²−1)/(2r0m(m²−1)); both the mutual-friction prefactor and the Uθ term are incorrect. Substitution of Eq. (8) into Eq. (7) leaves a first-order-in-α residual, so the growth rate, the instability threshold, and condition (9) are not consequences of the dispersion relation derived in the paper.
[Section 2, after Eq. (8)] With the paper's substitution m = kr0, the Uθ contribution in Eq. (8) is O(αUθ/(kr0²)) and vanishes for large r0, while the printed damping is O(αγk²); Eq. (8) therefore does not yield a growing mode for large rings. The corrected expansion gives Im ω = α[−γ(2k²r0²−1)/(2r0²) ± Uθ(2k²r0²−1)/(2r0√(k²r0²−1))] ≈ αk(±Uθ − γk) in the large-r0 limit, which is radius independent and can be positive. The advertised instability and radius independence are thus supported only by the corrected calculation, not by the text as written.
[Section 2, m = kr0] For a closed vortex ring the azimuthal mode number m is an integer, so the substitution m = kr0 is an additional modeling assumption, not a direct consequence of the linearized equations. The paper should state that this amounts to holding the physical perturbation wavelength 2πr0/m fixed as r0 varies, and should identify the radius-independence claim as a leading-order large-r0 statement. As written, the claim is stronger than the analysis establishes.

minor comments (3)

[Section 2, first paragraph] There is a typo, 'assiciated', in the discussion of the mutual-friction term.
[Section 2, Eq. (9)] The derivation of condition (9) is not shown; because it is based on the incorrect Eq. (8), it should be re-derived and presented step by step.
[Section 2, m = kr0] The citation to Kiknadze and Mamaladze [30] for the substitution m = kr0 would benefit from a specific equation reference and a brief physical justification.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the instability calculation is self-contained given the HVBK/LIA model; self-citations are motivational and the stated m=kr0 ansatz is an explicit scaling assumption, not a restatement of the target result.

full rationale

The chain is: HVBK equation (1) + LIA + prescribed toroidal normal flow (4) -> linearized ring equations (5) -> dispersion relation (7) -> growth-rate expression (8) -> radius-independence via m=kr0. Each step is a stated model calculation; no parameter is fitted to the predicted growth rate, and no output quantity is an input renamed. The self-citations to Shivamoggi [3,4] motivate the choice U=Utheta and the neglect of alpha', but the Appendix re-derives the straight-filament destabilization from the same HVBK equation, and [33] independently supports the exclusion of axial flow; therefore these citations are not load-bearing. The radius-independence step uses the explicit substitution m=kr0 from Kiknadze and Mamaladze [30]; whether this is the right quantization for a closed ring, and whether Eq. (8) is the correct small-alpha solution of Eq. (7), are mathematical-correctness questions, not cases where the conclusion is identical to an assumption by construction. Footnote 6's HVBK validity caveat is a model limitation, not a circular step. No circular reduction was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper adds no data-fitting parameters. Its central calculation rests on standard LIA/HVBK modeling assumptions and on the author's earlier identification of the destabilizing normal-flow component (refs. [3,4]). The radius-independence claim additionally rests on the m = k r0 substitution, which couples the azimuthal mode number to the ring radius. There are no invented entities.

assumptions (5)

domain assumption The HVBK model applies and the normal-fluid flow U is prescribed as spatially uniform and time-independent, specifically U = U_theta i_theta.
Section 2: 'U is the normal fluid velocity (taken to be constant in space and time and prescribed)'. The vortex back-reaction on the normal fluid is neglected.
domain assumption Vortex rings are thin enough that the local induction approximation describes self-advection.
Section 1 and Eq. (1): the LIA resolves the Biot-Savart singularity; standard in the cited filament literature.
domain assumption The mutual-friction term proportional to alpha' is negligible because alpha > alpha' and it produces no significant effects.
Section 2, bullet list before Eq. (2); relies on Vinen and Niemela and the author's previous work.
domain assumption Perturbations are sinusoidal modes e^{i(m theta - omega t)} with integer m, and the identification m = k r0 is used to relate mode number to wavenumber.
Section 2, Eqs. (6) and the sentence 'If one takes m = k r0 (Kiknadze and Mamaladze)'. This substitution is a modeling choice that underlies the radius-independence claim.
domain assumption The destabilizing effect of the normal fluid on the vortex ring is produced by the toroidal flow component along the vorticity, following the author's earlier filament analysis.
Section 2, paragraph before Eq. (4), citing Shivamoggi [3,4] and the Appendix; the paper does not re-derive this from a microscale model.

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Cite this review

Pith. "Pith review of Instability of a Vortex Ring due to Toroidal Normal Fluid Flow in Superfluid 4He." pith.science (2026). https://pith.science/paper/6XYC7WL7

@misc{pith2026190804359,
  author       = {Pith},
  title        = {Pith review of: Instability of a Vortex Ring due to Toroidal Normal Fluid Flow in Superfluid 4He},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6XYC7WL7}},
  note         = {Machine review of arXiv:1908.04359}
}

read the original abstract

Vortex rings self-propelling in superfluid 4He are shown to be driven unstable by a toroidal normal fluid flow. This instability has qualitative similarities with the Donnelly-Glaberson instability of Kelvin waves on a vortex filament driven by the normal fluid flow along the vortex filament. The growth rate of the present instability is found to be independent of the radius of the vortex ring.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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