[Paper Review] Strings and vortex rings

Steven S. Gubser, Revant Nayar+1 more|arXiv (Cornell University)|Aug 10, 2014

Black Holes and Theoretical Physics18 references3 citations

TL;DR

This paper provides the first analytical treatment of instabilities in circular vortex rings and head-on collisions of vortex rings in superfluids or low-viscosity fluids, using an effective string action with a Neveu-Schwarz three-form field. It derives exact expressions for instability bands and maximum amplification rates using elliptic integrals in the small-core limit, showing that the most unstable wavelength is parametrically larger than the dynamical cutoff scale.

ABSTRACT

We treat string propagation and interaction in the presence of a background Neveu-Schwarz three-form field strength, suitable for describing vortex rings in a superfluid or low-viscosity normal fluid. A circular vortex ring exhibits instabilities which have been recognized for many years, but whose precise boundaries we determine for the first time analytically in the small core limit. Two circular vortices colliding head-on exhibit stronger instabilities which cause splitting into many small vortices at late times. We provide an approximate analytic treatment of these instabilities and show that the most unstable wavelength is parametrically larger than a dynamically generated length scale which in many hydrodynamic systems is close to the cutoff. We also summarize how the string construction we discuss can be derived from the Gross-Pitaevskii lagrangian, and also how it compares to the action for giant gravitons.

Motivation & Objective

• To analytically determine the instability boundaries of a single circular vortex ring in the small-core limit.
• To investigate the enhanced instabilities arising in head-on collisions of two vortex rings, leading to fragmentation into smaller vortices.
• To derive the effective string action from the Gross-Pitaevskii Lagrangian and relate it to known string and vortex dynamics.
• To compare the results with giant graviton dynamics and previous hydrodynamic treatments, resolving discrepancies in the literature.

Proposed method

• Uses an effective string action with Neveu-Schwarz three-form field strength $H_3$ and a regulated interaction term to model vortex ring dynamics.
• Applies a perturbative ansatz with mode number $m$ to describe small deformations of a circular vortex ring: $\vec{X}(t,\theta) = \text{circular base} + \epsilon \cdot \cos(m\theta) \cdot \text{deformation modes}$.
• Expands the action to $O(\epsilon^2)$ to derive linearized equations of motion for radial and axial perturbations $r_m(t)$ and $z_m(t)$.
• Solves the resulting equations using exact expressions involving complete elliptic integrals to compute the potential terms $V_{r_m}$ and $V_{z_m}$.
• Derives the instability band and maximum amplification rate $\alpha_x$ by analyzing the effective potential and its dependence on $m$ at large $m$.
• Compares analytical results with numerical data from prior work, validating the large-$m$ approximation even at moderate $m \gtrsim 8$.

Experimental results

Research questions

• RQ1 What are the precise analytical boundaries of instability for a single circular vortex ring in the small-core limit?
• RQ2 How do head-on collisions of vortex rings lead to fragmentation into multiple smaller vortices, and what determines the most unstable mode?
• RQ3 What is the relationship between the effective string action used here and the Gross-Pitaevskii Lagrangian in superfluids?
• RQ4 How does the most unstable wavelength compare to the dynamically generated cutoff scale in hydrodynamic systems?
• RQ5 Why does the large-$m$ approximation for the amplification rate work well even at moderate $m$, and how does it resolve discrepancies with prior numerical studies?

Key findings

• The instability band for a single vortex ring is analytically determined by solving $V_{z_m} = 0$ and $V_{r_m} = 0$, yielding precise bounds on the radial scale $\ell_0/r$.
• The maximum amplification rate $\alpha_x$ is found to scale as $\alpha_x \simeq \frac{3}{4} \frac{\log m + 0.96351}{\tilde{V}_{\text{max}}}$, with $\tilde{V}_{\text{max}} \approx \log m + \gamma - \frac{1}{2} + 2\log 2 - \frac{17}{24m^2}$, matching numerical results closely even at $m=8$.
• The most unstable wavelength is parametrically larger than the dynamical cutoff scale $a$, indicating that the instability is not resolved by the microscopic core structure.
• The analytical treatment using elliptic integrals provides exact expressions for $V_{r_m}$ and $V_{z_m}$, enabling full control over perturbations in the small-core limit.
• The derived large-$m$ approximation for $\alpha_x$ matches the form $\alpha_x \simeq \frac{3}{4} \frac{\log m + 1}{\tilde{V}}$ observed numerically in prior work, resolving discrepancies by including $O(1)$ corrections.
• The effective string action is derived from the Gross-Pitaevskii Lagrangian under quasi-static and small-core approximations, establishing a bridge between superfluid vortex dynamics and string-theoretic models.