Sound-ring radiation of expanding vortex clusters

Open Access

Sound-ring radiation of expanding vortex clusters

August Geelmuyden, Sebastian Erne, Sam Patrick, Carlo F. Barenghi, and Silke Weinfurtner

Phys. Rev. Research 4, 023099 – Published 6 May 2022

Abstract

We investigate wave-vortex interaction emerging from an expanding compact vortex cluster in a two-dimensional Bose-Einstein condensate. We adapt techniques developed for compact gravitational objects to derive the characteristic modes of the wave-vortex interaction perturbatively around an effective vortex flow field. We demonstrate the existence of orbits or sound rings, in analogy to gravitational light rings, and compute the characteristic spectrum for the out-of-equilibrium vortex cluster. The spectrum obtained from numerical simulations of a stochastic Gross-Pitaevskii equation exhibiting an expanding vortex cluster is in excellent agreement with analytical predictions. Our findings are relevant for two-dimensional quantum turbulence, the semiclassical limit around fluid flows, and rotating compact objects exhibiting discrete circulation.

Received 9 June 2021
Accepted 3 December 2021

DOI:https://doi.org/10.1103/PhysRevResearch.4.023099

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Fluids & classical fields in curved spacetime Vortex flows Vortices in superfluids

Gravitation, Cosmology & AstrophysicsFluid DynamicsAtomic, Molecular & OpticalInterdisciplinary Physics

Authors & Affiliations

August Geelmuyden ^1,*, Sebastian Erne ^1,2,†, Sam Patrick ^1,3,‡, Carlo F. Barenghi^4,§, and Silke Weinfurtner^1,5,∥

¹School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom
²Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria
³Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada, V6T 1Z1
⁴Joint Quantum Centre Durham-Newcastle, School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom
⁵Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom

^*august.geelmuyden@nottingham.ac.uk
^†sebastian.erne@gmail.com
^‡sampatrick31@googlemail.com
^§carlo.barenghi@newcastle.ac.uk
^∥Silke.Weinfurtner@nottingham.ac.uk

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 4, Iss. 2 — May - July 2022

Subject Areas

Condensed Matter Physics

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Simulations of a decaying vortex cluster. (a) The $x$ coordinate of individual vortices over time in a single simulation. The minimal cluster size $r_{m}$ (red dotted line) and the largest sound-ring radius $r_{s r} (m = 0)$ (blue dashed line) is shown, together with $r_{s r} (m)$ at $m \in {- 5, - 15, - 25, ..., - 65}$ (short blue dashed lines). The colored regions illustrate the time windows of three qualitatively different stages: initial multiwinded vortex (stage I), multiwinded vortex that has decayed in many singly winded vortices forming a disordered cluster (stage II), and cluster that has expanded (stage III). (b)–(g) The density $ρ \equiv {| Ψ |}^{2}$ (top row) and phase $ϕ \equiv arg (Ψ)$ (bottom row) at times [see horizontal black dashed lines in (a)] $t = 1000$ [(b) and (e)], $t = 2650$ [(c) and (f)], and $t = 4300$ [(d) and (g)].
Reuse & Permissions
Figure 2
Measured background. (a) Measured velocities $v = Im \nabla ln Ψ$ at the initial (red dot-dashed curve) and final (blue dashed curve) time frame as a function of radius $r$ . The ensemble-averaged azimuthal mean is plotted with the standard deviation over the ensemble shaded. (b) Azimuthally averaged density profile $ρ^{(a)} (r)$ at the initial (red dot-dashed curve) and final (blue dashed curve) time frame. The density is compared with the uniform Thomas-Fermi density $ρ ≃ 1 - U (r)$ (yellow dotted line). In both panels, the model is fitted to find the core size $R$ (vertical lines) and plotted as black curves on top.
Reuse & Permissions
Figure 3
Analytical predictions. Three qualitatively different regions shaded for $ℓ = 29$ : waves bound to the core region (red), waves trapped outside (blue), and waves that communicate between the core and the boundary (yellow). (a) Radial phase-space trajectories for waves with azimuthal number $m = - 15$ . The black curves correspond to the trajectories of waves that oscillate at the sound-ring frequency $ω_{s r}$ . (b) The effective potential $ω^{\pm} (r)$ (solid black curves) for $m = - 15$ . The symmetry $- ω_{D}^{\pm} (m) = ω_{D}^{\mp} (- m)$ means that negative frequencies inform us of $m = + 15$ . Frequencies at three key radii are labeled: the Rankine radius $R$ (vertical dotted line), the sound-ring radius $r_{s r} (m, ℓ)$ , and the potential boundary $r_{B}$ . (c) Allowed frequencies $ω$ over azimuthal number $m$ for all radii $r$ in the system. For a single $m = \pm 15$ , this corresponds to (b) projected onto the frequency axis. Here, the solid black curve is the sound-ring frequency $ω_{sr}$ , and the dashed black line is the frequency $ω^{+} (R)$ of waves with turning point at the Rankine radius. The diamonds signal which modes stagnate at the sound ring (black), counter-rotate at the core (red), corotate at the core (yellow), and corotate at the boundary (blue) for $| m | = 15$ .
Reuse & Permissions
Figure 4
Simulation results. (a) Ensemble-averaged squared amplitudes for $m = \pm 15$ ; white curves show the turnover frequencies $ω^{\pm} (r)$ for the average Rankine radius $R$ (vertical white dashed line). (b) Radially averaged spectrum from $r_{a} = 40$ [see white vertical line segment in (a)] to the boundary $r_{B} = 120$ . (c) Relative amplitudes of (b) at selected values for $m$ . Here, $Δ m = 10$ corresponds to an amplitude of 0.1 in that $m$ channel. Black solid horizontal lines are depicting the sound-ring frequencies, and the shaded regions for $m > 0$ are the delimiting frequencies $ω^{+} (R)$ from the initial ( $R_{0} = 26.2 \pm 0.3$ at time $\sim 2400$ ) and final ( $R_{1} = 32.7 \pm 0.5$ at time $\sim 2900$ ) ensemble-averaged core size. The black dashed lines show the delimiting frequency $ω^{+} (R)$ at the average core size $R = (R_{0} + R_{1}) / 2$ . As expected, it peaks at the frequencies corresponding to core sizes at earlier times.
Reuse & Permissions
Figure 5
Vortex-tracking results. Left: density of the first time frame in in the analyzed window of a single realization. White lines are drawn along curves of constant phase. White circular dots are the detected phase singularities, and white crosses are the locations of the phase singularities with inverted winding with respect to the net rotation of the cluster. Middle: phase information of the data shown in the left panel. The dots and crosses are the same as in the left panel. Right: traced trajectories of phase singularities (vortices) in the corotating frame. Black dots are the initial phase singularities shown in the left and middle panels.
Reuse & Permissions

Physical Review Research