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Superfluid vortex dynamics with an elliptical boundary
by Matteo Caldara, Andrea Richaud, Pietro Massignan, Alexander L. Fetter
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Matteo Caldara · Pietro Massignan |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202311_00039v2 (pdf) |
| Date submitted: | 2024-06-24 09:38 |
| Submitted by: | Caldara, Matteo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Recent advances in cold-atom platforms have made real-time dynamics accessible, renewing interest in the motion of superfluid vortices in two-dimensional domains. Here we show that the energy and the trajectories of arbitrary vortex configurations may be computed on a complicated (curved or bounded) surface, provided that one knows a conformal map that links the latter to a simpler domain (like the full plane, or a circular boundary). We also prove that Hamilton's equations based on the vortex energy agree with the complex dynamical equations for the vortex dynamics, demonstrating that the vortex trajectories are constant-energy curves. We use these ideas to study the dynamics of vortices in a two-dimensional incompressible superfluid with an elliptical boundary, and we derive an analytical expression for the complex potential describing the hydrodynamic flow throughout the fluid. For a vortex inside an elliptical boundary, the orbits are nearly self-similar ellipses.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Thank you for sending us the reports of the two Referees.
At the moment, we would decline the transfer to the journal SciPost Physics Core. Instead, we would like our resubmission to be considered for publication in the journal SciPost Physics.
In light of Referees' comments and requests, we have substantially modified our manuscript in order to highlight the original contributions that our work can bring to the field of superfluid vortex dynamics, as well as how it now fulfills the journal acceptance criteria. We summarised the general structure of the new version of our paper at the beginning of the reply to both Referees.
We have replaced the arXiv manuscript with the revised version that seeks to comply with the requests of both Referees. Our detailed point-by-point response to the Referees has been uploaded, too. In our response, we also explicitly reference the acceptance criteria.
In view of the considerable improvements that we have implemented in response to the points raised by the Referees, we are confident that this revised version of our paper can be accepted for publication in SciPost Physics.
Best regards,
The authors
List of changes
A detailed list of changes can be found at the beginning of the reply to both Referees that has been attached as a pdf file.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2024-7-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202311_00039v2, delivered 2024-07-05, doi: 10.21468/SciPost.Report.9342
Strengths
1. Novel approach for studying vortex dynamics in nontrivial geometries.
2. Interesting demonstration of method both inside and outside an ellipse.
Report
The updated manuscript has been significantly expanded, and now presents a much more general methodology for determining vortex dynamics and energetics in complex geometries. As such, I think the revised manuscript does open pathways for new resarch, and therefore satisfies the criteria for publication in SciPost Physics.
I have a few minor comments regarding the revisions, but once these have been addressed, I recommend the manuscript be accepted for publication.
Requested changes
The authors have addressed almost all my previous comments satsifactorily, although I have responses to some minor aspects of the first two points:
- The Jacobi theta function may be straightforward to calculate numerically, but mathematically it requires an infinite series to be defined, whereas sin(x) can be defined in terms of elementary operations (eg. as a ratio of two sides of a triangle, or via complex exponentiation). Since the theta function involves an infinite number of operations, I do not think it can strictly be referred to as "closed-form". Regardless, I leave the terminology up to the authors.
- A number of improvements have been made to the wording around the fluid compressibility in the introduction, but the second sentence still refers to superfluids having "negligible compressibility", which is not true for ultracold atomic gases (as I pointed out in my previous report). I suggest removing this.
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I also have a few minor comments regarding the new material:
- Could the title be made more general? The emphasis of the revised manuscript seems to be more on the general method, rather than on the specific case of an elliptical geometry, but this is not reflected in the title.
- Paragraph 2 of the introduction reads: "Vortices in two-dimensional films have simpler dynamics...". This seems to be implying that the dynamics are simpler than in three-dimensional systems, but the preceding text does not mention three-dimensional systems specifically, so the word "simpler" appears out of place. I suggest altering this wording slightly.
- The sentence directly after this reads: "Hence a point-vortex model applies". This does not immediately follow from the dynamics being restricted to 2D. A point-vortex approximation also requires core sizes to be small compared to other relevant lengthscales (eg. the system size, distances to other vortices). I think this is worth mentioning here.
- Paragraph 4 begins: "Most cold atom experimental platforms are able to produce essentially uniform systems". I am not sure if "most" is true, so I suggest weakening this claim to eg. "many".
- The acronym "BEC" is defined twice in the introduction.
- In paragraph 8 of the introduction, the authors refer to "the vortex interpretation of the boundary-value problem". I am not sure what this means. Can the authors clarify this sentence?
- Regarding Eq (10), it would be helpful to specify that * denotes complex conjugation.
- I was confused by the definitions of $n_0$ and $n_i$, introduced in Sec 3.1:
i) Following Eq (20), $n_0$ is described as the "quantized circulation around the boundary", although it is not clear exactly which boundary is being referred to. Shortly afterwards, following Eq (23), $n_i = n_0 - 1$ is described as "the circulation quantum around the circular boundary at R". So are $n_i$ and $n_0$ defined around different boundaries? My best guess is that $n_0$ is the winding around the w-plane (elliptical) boundary, while $n_i$ is the winding around the z-plane (circular) boundary, but this appears to be reversed in the caption of Fig 1. I suggest the authors clarify this.
ii) Is the difference of -1 between the two winding numbers $n_i$ and $n_0$ related to the pole at the origin of the z-plane? It would be useful to specify where this difference comes from.
- Following Eq (29), the authors point out that for small distances $d \ll a$ from the ellipse, the energy diverges as $\log(2d)$. Is this easy to see from Eq (29)? Which terms contribute or drop out in this limit?
- It would be useful to include color bars in the figures with color scale data, or at least a description of the color scale in the caption (eg. for Fig 2, blue $\rightarrow$ low energy, orange $\rightarrow$ high energy).
Recommendation
Ask for minor revision
Report #1 by Anonymous (Referee 2) on 2024-6-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202311_00039v2, delivered 2024-06-25, doi: 10.21468/SciPost.Report.9302
Strengths
1. Well-written, clear and concise manuscript
2. Of interest to a wide audience working in fluid flow in classical and quantum fluids.
3. Novel use of conformal transformations to derive analytical results for modelling complex vortex setups in bounded domains.
Report
The authors have submitted a revised manuscript that includes changes and responses to all of the reviewers’ comments. In doing so, they have adequately addressed the main criticism from both referees regarding the significance and novelty of the work and made significant improvements to the manuscript through revisions and additional research.
The revised introduction clarifies how the present work compares to previously published research. The revised manuscript significantly strides in using conformal transformations to study point vortex dynamics. It shows how the work can be applied to more general cases of point vortex dynamics in bounded domains.
Consequently, I am happy to recommend the publication of this work to SciPost Physics.
Requested changes
The Authors have addressed all changes previously outlined. No more changes are requested.
Recommendation
Publish (meets expectations and criteria for this Journal)
Author: Matteo Caldara on 2024-07-18 [id 4629]
(in reply to Report 1 on 2024-06-25)We are grateful to the Referee for their positive report and we are pleased with their final recommendation.
Author: Matteo Caldara on 2024-07-18 [id 4630]
(in reply to Report 2 on 2024-07-05)We would like to thank the Referee for their positive and constructive report. We are glad our revised manuscript is considered worth publication in SciPost Physics. We also thank the Referee for their new suggestions which we implemented in the new submitted version of our manuscript.
In the attached pdf document we provide a a complete point-by-point reply, followed by a "diff" file, highlighting the latest changes to our manuscript.
Attachment:
Reply_and_Diff.pdf