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Interplay of KelvinHelmholtz and superradiant instabilities of an array of quantized vortices in a twodimensional BoseEinstein condensate
by Luca Giacomelli, Iacopo Carusotto
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Authors (as registered SciPost users):  Iacopo Carusotto · Luca Giacomelli 
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Preprint Link:  https://arxiv.org/abs/2110.10588v2 (pdf) 
Date submitted:  20211116 09:28 
Submitted by:  Giacomelli, Luca 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We investigate the various physical mechanisms that underlie the dynamical instability of a quantized vortex array at the interface between two counterpropagating superflows in a twodimensional BoseEinstein condensate. Instabilities of markedly different nature are found to dominate in different flow velocity regimes. For moderate velocities where the two flows are subsonic, the vortex lattice displays a quantized version of the hydrodynamic KelvinHelmholtz instability (KHI), with the vortices rolling up and corotating. For supersonic flow velocities, the oscillation involved in the KHI can resonantly couple to acoustic excitations propagating away in the bulk fluid on both sides. This makes the KHI rate to be effectively suppressed and other mechanisms to dominate: For finite and relatively small systems along the transverse direction, the instability involves a repeated superradiant scattering of sound waves off the vortex lattice; for transversally unbound systems, a radiative instability dominates, leading to the simultaneous growth of a localized wave along the vortex lattice and of acoustic excitations propagating away in the bulk. Finally, for slow velocities, where the KHI rate is intrinsically slow, another instability associated to the rigid lateral displacement of the vortex lattice due to the vicinity of the system's boundary is found to dominate.
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Report #2 by Anonymous (Referee 2) on 2022217
(Invited Report) Cite as: Anonymous, Report on arXiv:2110.10588v2, delivered 20220217, doi: 10.21468/SciPost.Report.4447
Report
Authors study the instability of a vortex street in a twodimensional BoseEinstein condensate (BEC) by numerically solving the GrossPitaevskii (GP) equation and the Bogoliubov equations. They initially prepare a Gaussian potential between two counterpropagating superflows and observe the nonequilibrium dynamics after removing the potential. The counterpropagating state decays into a vortex street and makes distinct dynamics depending on the initial relative velocity \Delta v between the two superflows; the nonequilibrium dynamics are dominated by complicated motion of vortices for the subsonic superflows while the acoustic excitations are radiated to the bulk for the supersonic case.
Authors focus the aspects of KelvinHelmholtz instability (KHI) and the superradiance due to dynamical instability of this system. I think the connection of the hydrodynamic instability with superradiance can be an interesting topic. However, the argument and discussion by authors do not have enough evidence which can prove their theory, especially, with respect to the Bogoliubov analysis. I would like to describe the reason why I think so below.
(1) Point vortex model
Authors try to analyze the stationary solution of an array of quantized vortices in terms of the theory of KHI in an unconventional formalism. This system should be considered more fundamentally as a vortex street (not Karman’ vortex street). It is also known that the system is unstable against a transverse shift of vortices (see some textbook on hydrodynamics). Then, I would like to ask authors to compare the accuracy of their numerical analysis with the result of the point vortex model by removing the trapping potential. I think the situation considered is too complicated to understand such a fundamental aspect of the system.
(2) Width of the shear layer
The smallest length scale, which determine hydrodynamics in a superfluid is the healing length \xi. However, the width \delta_v defined by Eq. (3) is no such a restriction. The interpretation of the width is crucial to characterize the fundamental property of this system because it is used in Eq. (11) for justifying the numerical analysis in Fig. 2.
A vortex street looks like a velocity shear layer if it is viewed on the length scale much longer than the distance between neighboring vortices. Therefore, I can agree with authors expectation that theory of KHI is applicable to this system under some condition. Concretely speaking, it is natural to expect that KHI would be realized when the wave number k_H of the unstable mode is much smaller than n_{vort}. However, the value of k_H, considered by authors, seems to be similar to n_{vort}~Mv/hbar as is mentioned in the last paragraph of Sec. 4.1.
Although I think this is the most fundamental aspect of this system to be examined first, authors start with unconventional formulation of KHI with no kind explanation. For example, the result of Eq. (10) looks quite different from the conventional result obtained by considering the stability of interface tension wave (I do not have the book of Ref [1]). However, there is no explanation about the difference in the manuscript. Furthermore, authors continues to extend the theory for the case with a finite width of the shear layer. The velocity field is not potential flow (vortex free) inside the shear layer in classical fluid, but it is fully potential flow in superfluids. How to connect the two systems? If authors try to apply approximately, what is the condition that can be safely applied?
(3) Interpretation of the results in Fig. 2
As authors suggest in the text, there could be the localized modes at the shear layer or vortices and the acoustic mode (phonons) propagating in the bulk fluid. Although a lot of branches are plotted for the real part in Fig. 2, there is no explanation about which branch corresponds to the localized or bulk modes. Thereby, the authors’ assertion that the resonance occurs between the bulk and localized modes lacks persuasiveness. This is contrast to the explanation on the resonance and the superradiance due to the dynamical instability of a multiply quantized vortex in superfluid in Ref. [14], or originally and more quantitatively by Takeuchi, et. al., J. Phys. Soc. Jpn. 87, 023601 (2018).
The style of the plot also makes it difficult for readers to understand. The difficulty might come from the introduction of Eq. (12) without enough explanation. Where does this relation come from? The relation of momentum and wave number can be complicated in the Bogoliubov analysis because of the complicated property of the norm in the presence of unstable mode with complex eigenvalue. This is related to the form of the Bovoliubov transformation and its normalization condition. The corresponding form of Eq. (6) is different from the conventional transformation, i.e., \delta\psi =uv^*, which make readers confused more.
I am not also satisfied with the fact that the “adjustment” is not so small for the fitting curve in the leftmost panel of Fig. 2. In such a difficult situation, I am sorry that I could not read the content of Sec. 4.2 seriously.
(4) Previous works on KHI
Baggaley and Parker revealed the nonequilibrium dynamics for the case with small and large number of vortices in a very similar situation of singlecomponent superfluids in Ref. [12]. Since authors investigate a very similar situation, they should explain distinction between the result of Ref. [12] and their result more clearly. Additionally, although authors give attention to the radiation of acoustic excitations propagating away in the bulk fluid for the supersonic case, a very similar phenomenon has been observed for KHI in immiscible binary BECs by Kokubo et. al. Phys. Rev. A 104, 023312 (2021). According to this paper, the acoustic excitations are shock waves forming Machconelike structure. A similar pattern is also observed in the rightmost panel in Fig. 1. This is just a nonlinear effect although authors try to describe such acoustic radiation as superradiance in terms of the linear analysis based on the Bogoliubov theory. How to explain this inconsistency?
Author: Luca Giacomelli on 20220218 [id 2220]
(in reply to Report 2 on 20220217)
 We thank the referee for their insight. In the following we respond to each of their remarks, that we think mostly stem from a misinterpretation of our aims when using results from hydrodynamics. We modified the text so to make the rationale of our comparisons more clear. We include here a pdf of the manuscript where the changes and additions are highlighted in red.
(1) Point vortex model Authors try to analyze the stationary solution of an array of quantized vortices in terms of the theory of KHI in an unconventional formalism. This system should be considered more fundamentally as a vortex street (not Karman’ vortex street). It is also known that the system is unstable against a transverse shift of vortices (see some textbook on hydrodynamics). Then, I would like to ask authors to compare the accuracy of their numerical analysis with the result of the point vortex model by removing the trapping potential. I think the situation considered is too complicated to understand such a fundamental aspect of the system.
We are sorry but we did not manage to find stability analyses of the vortex street you mention in the hydrodynamics textbooks we consulted. If you could give us some direct references we would be interested in looking at them. For the point of the present work however we feel that additional comparisons with hydrodynamic models are not needed. Our aim was to further characterize what was identified as KHI in [12] ([18] in the new manuscript), before dealing with the other instabilities we identified. We feel that the comparison with finitewidth shear layers already shows a good parallelism with what is commonly called KHI. The main point we want to highlight is that the hydrodynamics results we report are not meant as being a good description of our system, it is just a reference model for a \textit{different system} that we take as a comparison for our numerical results. We tried to better highlight this in the text. We think it is interesting that many features of KHI are present also in our very different physical system.
(2) Width of the shear layer The smallest length scale, which determine hydrodynamics in a superfluid is the healing length $\xi$. However, the width $\delta_v$ defined by Eq. (3) is no such a restriction. The interpretation of the width is crucial to characterize the fundamental property of this system because it is used in Eq. (11) for justifying the numerical analysis in Fig. 2. A vortex street looks like a velocity shear layer if it is viewed on the length scale much longer than the distance between neighboring vortices. Therefore, I can agree with authors expectation that theory of KHI is applicable to this system under some condition. Concretely speaking, it is natural to expect that KHI would be realized when the wave number $k_H$ of the unstable mode is much smaller than $n_{vort}$. However, the value of $k_H$, considered by authors, seems to be similar to $n_{vort}\sim Mv/hbar$ as is mentioned in the last paragraph of Sec. 4.1.
Again we wish to stress that we are not using Eq. (11) (now eq. (12)) to \textit{justify} our numerical results, we are just taking the hydrodynamic model as a comparison to see which feature survive the very different setting. We agree that we consider values of the momenta of the perturbations comparable to the healing length, and that also the shear layer width is comparable to that. This in fact makes the hydrodynamic model not a good model for our system, that however we investigate directly by numerical means, and we take the hydrodynamics results as a comparison of what is usually understood as KHI.
Although I think this is the most fundamental aspect of this system to be examined first, authors start with unconventional formulation of KHI with no kind explanation. For example, the result of Eq. (10) looks quite different from the conventional result obtained by considering the stability of interface tension wave (I do not have the book of Ref [1]). However, there is no explanation about the difference in the manuscript. Furthermore, authors continues to extend the theory for the case with a finite width of the shear layer. The velocity field is not potential flow (vortex free) inside the shear layer in classical fluid, but it is fully potential flow in superfluids. How to connect the two systems? If authors try to apply approximately, what is the condition that can be safely applied?
We do not believe that the dispersion relation of the KHI we report is unconventional, since we also found it in other books. As an example we added a second book as a reference for the classical hydrodynamics results we mention (the one by Drazin and Reid [2]) where this result is also obtained in the introductory chapter. We added some specifications to give some missing information about the system for which that result is derived, that is an incompressible inviscid fluid of constant density without gravity. For what concerns the final questions we agree that the flow is not a potential one in the hydrodynamic model, while it surely is in the superfluid we are considering. But again, let us stress that we are not trying to apply that model to our system.
(3) Interpretation of the results in Fig. 2. As authors suggest in the text, there could be the localized modes at the shear layer or vortices and the acoustic mode (phonons) propagating in the bulk fluid. Although a lot of branches are plotted for the real part in Fig. 2, there is no explanation about which branch corresponds to the localized or bulk modes. Thereby, the authors’ assertion that the resonance occurs between the bulk and localized modes lacks persuasiveness. This is contrast to the explanation on the resonance and the superradiance due to the dynamical instability of a multiply quantized vortex in superfluid in Ref. [14], or originally and more quantitatively by Takeuchi, et. al., J. Phys. Soc. Jpn. 87, 023601 (2018).}
In Figure 2 the superradiant physics of the localized modes does not come significantly into play, since it matters for supersonic velocities and large systems, that we investigate in Section 4.2.2. In that context the presence of a localized mode is visible in Figure 6, and also in the newly added Appendix A2, where it can be also seen that negativeenergy localized modes exist, very similarly to the vortex case (for which we added also the work you mentioned as a reference [15]). About the plots of Figure 2, we think it is not really possible to point at modes and say which is localized and which is not, since the output of the diagonalization are modes of the whole system, in which resonances between localized and phononic modes appear as a single mode.
The style of the plot also makes it difficult for readers to understand. The difficulty might come from the introduction of Eq. (12) without enough explanation. Where does this relation come from? The relation of momentum and wave number can be complicated in the Bogoliubov analysis because of the complicated property of the norm in the presence of unstable mode with complex eigenvalue. This is related to the form of the Bovoliubov transformation and its normalization condition. The corresponding form of Eq. (6) is different from the conventional transformation, i.e., $\delta\psi =uv^*$, which make readers confused more.
The relation between hydrodynamic and Bloch momentum is a technical point depending on how one chooses to take the fluctuation field. This is what we explain in the discussion on page 9, that we tried to improve. This is not altered by the Bogoliubov dispersion, and in fact would be present also if we considered the socalled hydrodynamic approximation for fluctuations, that removes dispersion. It is a matter of convention and all the properties of the system, including dispersion and the presence of dynamical instabilities, come after and independently of this choice. Also, we do no think that our convention for the Bogoliubov transformation is unusual, this is the one used for example in the book by Pitaevskij and Stringari (ref. [24] section 5.6), that differs for a minus with respect to the one you wrote. This is also not related to out choice of the Bloch momentum. We added footnote 2 in page 9 to highlight these facts and try to avoid this confusion.
I am not also satisfied with the fact that the “adjustment” is not so small for the fitting curve in the leftmost panel of Fig. 2. In such a difficult situation, I am sorry that I could not read the content of Sec. 4.2 seriously.}
As we already pointed out, that curve is not a theoretical justification of our results, it just serves as a comparison with a model for a different system in which the KHI is usually studied. Moreover, the content of Section 4.2 does not mention those hydrodynamic models anymore, the analysis is purely based on our solutions of the Bogoliubov problem.
(4) Previous works on KHI Baggaley and Parker revealed the nonequilibrium dynamics for the case with small and large number of vortices in a very similar situation of singlecomponent superfluids in Ref. [12]. Since authors investigate a very similar situation, they should explain distinction between the result of Ref. [12] and their result more clearly.}
Our work starts from the results of Baggaley and Parker. We first characterized in a different way the KHI they observed and we then proceeded to consider higher velocity differences, in which novel physical mechanisms emerge that, to our knowledge, were not reported before. In ref. [12] (now [18]) only velocity differences below $2 c_s$ were considered. Also, the drift of vortices at smaller velocities was not reported. We tried to make more clear the comparison with that work in the introduction.
Additionally, although authors give attention to the radiation of acoustic excitations propagating away in the bulk fluid for the supersonic case, a very similar phenomenon has been observed for KHI in immiscible binary BECs by Kokubo et. al. Phys. Rev. A 104, 023312 (2021). According to this paper, the acoustic excitations are shock waves forming Machconelike structure. A similar pattern is also observed in the rightmost panel in Fig. 1. This is just a nonlinear effect although authors try to describe such acoustic radiation as superradiance in terms of the linear analysis based on the Bogoliubov theory. How to explain this inconsistency?
We think there is no inconsistency here. The Machconelike structure of that work may well be something very similar to what we observe (we were not aware of that work, that we added to the references of previous works on the KHI). At least in our case however that is not a shock wave, since it emerges as small amplitude fluctuations. This is also confirmed by the fact that those features can be captured by the linear Bogoliubov theory. It is hence firstly a linear phenomenon, since it involves acoustic radiation, that only becomes nonlinear when the amplitude of the unstable modes become large enough.
Attachment:
Report #1 by Anonymous (Referee 1) on 2022117 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.10588v2, delivered 20220117, doi: 10.21468/SciPost.Report.4191
Report
Report on “Interplay of KelvinHelmholtz and superadiant instabilities of an array of quantized vortices in a twodimensional BoseEinstein condensate”
The authors present a numerical and theoretical study of the instabilities occurring at the interface between two superfluids, examining a range of relative velocities from subsonic to supersonic. They find a range of novel behaviours that they investigate both through GPE simulations and diagonalisation in the Bogoliubov approximation.
The paper extends the numerical results of Ref. [12], which used GPE simulations to investigate a similar geometry. Through the more complete analysis presented here, considerable additional insight is gained into the different regimes, while matching the qualitative results of Ref. [12] in the subsonic regime. In this regard, the paper is an important contribution to the study of shearlayer instabilities in 2D superfluids and will certainly serve as an important reference for future experimental realisations.
Despite some issues with the presentation of the qualitative conclusions in the article, I recommend that the paper be accepted after response to the below comments/improvements.
Major comments:
1. One of the regimes discussed is the “superradiant instability (SRI)”. It is unclear in the current presentation whether this process is indeed superradiant, or whether this is intended as descriptive terminology. There are more details in Sec. 4.2 discussing this point, but again the description appears mostly by analogy to the other studies in Refs. [5,13]. The authors have indeed identified the “possibility of having superradiant scattering”, but it unclear whether they can make the conclusion that the process is indeed superradiant.
2. In Figure 5, the rightmost plot extrapolates the behaviour of the large system to an infinite one. The description here is challenging to follow and it is not clear how the inferences to the infinite system are made. As a starting point, it would be good to produce an equivalent plot for the L_y=120 data. In particular, the origin of the blue line and its relation to the imaginary spectrum is unclear. Including the same for the L_y=120 data could make the connection more apparent. The discussion of the authors jumps between the large system and infinite system behaviour, and it cannot be sufficiently followed to understand the origins of the rightmost plot of Fig. 5, or how rigorously it follows from increasing the system size.
3. In Fig. 6, the authors investigate a system with absorbing boundaries, to simulate the effect of the infinite system where there are no edge reflections. On page 12, the authors mention that the density perturbations grow exponentially in time. This is not clear from any of the data presented and some additional plots or analysis is needed to support this statement. Again, in the discussion the “superradiant” aspects of this system are noted, but it is again unclear whether this extends beyond an analogy.
4. Page 13, “There is good agreement with the largest instability rates obtained for a finite but large system”. The agreement between the two data sets shown in the lower middle panel of Fig. 5 doesn’t appear particularly impressive – there is significant mismatch outside of error bars for the dominant instability.
Additional comments
5. Wording in the first sentence of the introduction “one of the most fundamental ones being” to “with one of the most fundamental being…”
6. The second paragraph of the introduction is difficult to understand as worded, perhaps defining a vortex sheet or better defining what is meant by “sharp” would help – clearly this is just that the shear layer size varies between the two regimes but would be helpful to state this.
7. In the third paragraph “see for example [3] for complete references”. It would be better to instead list relevant references here, rather than this nested approach which is not standard.
8. Second page “(RI) is analogous to the ergoregion instabilities…” without more being said this statement adds very little in terms of understanding. If the authors want to keep Ref. [14] here it requires a little more explanatory detail.
9. Page 4: There is a discussion about the shear layer width and its relation to the velocity, but there is no analysis of the shear layer in the GPE simulations. It would be useful to add a crosssection plots to Fig. 1, for the two velocities shown.
10. In the caption of Fig. 1 “clusterize” – not very standard terminology – just “cluster”?
11. End of page 4, there is a discussion on the shear configuration being a stationary state – yet the instability emerges in the GPE simulations shown. Some small discussion is needed on the seeding of the instability in the simulations – presume it is due to numerical noise?
12. Near Eqn. (8), page 6, \sigma_3 is not defined.
13. In the discussion following Eqns. (69), there is switching between \nu and \Delta\nu. It would seem that this is a typo, presumably the relative velocity \Delta\nu is the significant variable in all cases?
14. Fig. 3, \Gamma_{max} does not appear to be defined anywhere, similarly K_{max} is not explicitly defined.
15. Page 9, just after Eqn. (12), “red line in the rightmost lower plot” – the red line is in the leftmost plot.
Author: Luca Giacomelli on 20220218 [id 2219]
(in reply to Report 1 on 20220117)
We thank the referee for the careful reading of our manuscript and for the positive assessment of our results. In the following we reply point by point to their remarks, indicating the changes we made to the manuscript to address the issues they raised. We attach a pdf of the new manuscript with the main changes highlighted in red.
Major comments

The term superradiant was not intended as a descriptive terminology, the SRI stems from superradiant processes that occur in the system. We concluded this from the spectra, since the dynamical instabilities emerge from resonance of oppositenormed modes that are phononic propagating waves. However we probably have relied too much on our intuition in showing this. To back up our conclusions we added an Appendix (A1), with appropriate references in the main text, in which we report the result of numerical computations that show the occurrence of amplified scattering processes, for frequencies and Bloch momenta at which we found SRI. Our simulations involve the scattering of wavepackets in a system large enough that they do not interact with the boundaries. These simulations show that these sizable amplifications are stable processes in an infinite system, while they inevitably lead to SRI in a finite system.

We improved the discussion of the inference of the spectrum in the infinite system. In particular we included in the right plot of Figure 5 a plot of all the dynamically unstable modes of the largebutfinite system, that clearly distinguishes the instabilities that disappear for an unbound system from those that remain.

We changed Figure 6 to include a panel showing the exponential growth of (some of) the momentum components of the unstable growing perturbation. For what concerns the superradiant nature of the RI, we updated the discussion and, as for the SRI, we added in Appendix A2 a scattering numerical experiment in which we show how the RI is associated to superradiant scattering given by the existence of a negativeenergy mode localized around the array of vortices. Differently from the SRI case, here the amplified scattering cannot occur without starting the associated instability. The mechanism is completely analogous to the one we illustrated for multiply quantized vortices in [20].

What we meant with good agreement is that the measured instability rates for the setup mimicking an infinite system are in correspondence and comparable to the ones obtained by diagonalizing a large but finite system. An exact match is actually not to be expected and the mismatch in the dominant instability bubble can be ascribed to a finite size effect. Comparing with the panel for $L_y=60\xi$ one can see that the dominant bubble is higher in that case. The dominant instability is hence still a decreasing function of the system size, and a further decrease in the $L_y\to\infty$ limit is to be expected. We improved the discussion about this point.
Additional comments 56. We improved the first two paragraphs 7. We listed directly the relevant references 8. We added more details about that comparison. The parallelism is further explained in Section 4.2, when dealing with the RI. 9. We added two panels to Figure 1, showing the velocity profile obtained numerically and a superimposed plot of the hyperbolictangent profile. 10. Me modified the term. 11. We added a footnote commenting on this. The way in which the shear layer is created via a timedependent potential implies that fluctuations are present in the system. Also, as you pointed out, numerical noise is a further seed for instabilities. 1215. We defined the undefined symbols, uniformed the notation to $\Delta v$ and corrected the wrong indication for the red line.
Anonymous on 20211223 [id 2047]
Authors study the instability of a vortex street in a twodimensional BoseEinstein condensate (BEC) by numerically solving the GrossPitaevskii (GP) equation and the Bogoliubov equations. They initially prepare a Gaussian potential between two counterpropagating superflows and observe the nonequilibrium dynamics after removing the potential. The counterpropagating state decays into a vortex street and makes distinct dynamics depending on the initial relative velocity \Delta v between the two superflows; the nonequilibrium dynamics are dominated by complicated motion of vortices for the subsonic superflows while the acoustic excitations are radiated to the bulk for the supersonic case.
Authors focus the aspects of KelvinHelmholtz instability (KHI) and the superradiance due to dynamical instability of this system. I think the connection of the hydrodynamic instability with superradiance can be an interesting topic. However, the argument and discussion by authors do not have enough evidence which can prove their theory, especially, with respect to the Bogoliubov analysis. I would like to describe the reason why I think so below.
(1) Point vortex model
Authors try to analyze the stationary solution of an array of quantized vortices in terms of the theory of KHI in an unconventional formalism. This system should be considered more fundamentally as a vortex street (not Karman’ vortex street). It is also known that the system is unstable against a transverse shift of vortices (see some textbook on hydrodynamics). Then, I would like to ask authors to compare the accuracy of their numerical analysis with the result of the point vortex model by removing the trapping potential. I think the situation considered is too complicated to understand such a fundamental aspect of the system.
(2) Width of the shear layer
The smallest length scale, which determine hydrodynamics in a superfluid is the healing length \xi. However, the width \delta_v defined by Eq. (3) is no such a restriction. The interpretation of the width is crucial to characterize the fundamental property of this system because it is used in Eq. (11) for justifying the numerical analysis in Fig. 2.
A vortex street looks like a velocity shear layer if it is viewed on the length scale much longer than the distance between neighboring vortices. Therefore, I can agree with authors expectation that theory of KHI is applicable to this system under some condition. Concretely speaking, it is natural to expect that KHI would be realized when the wave number k_H of the unstable mode is much smaller than n_{vort}. However, the value of k_H, considered by authors, seems to be similar to n_{vort}~Mv/hbar as is mentioned in the last paragraph of Sec. 4.1.
Although I think this is the most fundamental aspect of this system to be examined first, authors start with unconventional formulation of KHI with no kind explanation. For example, the result of Eq. (10) looks quite different from the conventional result obtained by considering the stability of interface tension wave (I do not have the book of Ref [1]). However, there is no explanation about the difference in the manuscript. Furthermore, authors continues to extend the theory for the case with a finite width of the shear layer. The velocity field is not potential flow (vortex free) inside the shear layer in classical fluid, but it is fully potential flow in superfluids. How to connect the two systems? If authors try to apply approximately, what is the condition that can be safely applied?
(3) Interpretation of the results in Fig. 2
As authors suggest in the text, there could be the localized modes at the shear layer or vortices and the acoustic mode (phonons) propagating in the bulk fluid. Although a lot of branches are plotted for the real part in Fig. 2, there is no explanation about which branch corresponds to the localized or bulk modes. Thereby, the authors’ assertion that the resonance occurs between the bulk and localized modes lacks persuasiveness. This is contrast to the explanation on the resonance and the superradiance due to the dynamical instability of a multiply quantized vortex in superfluid in Ref. [14], or originally and more quantitatively by Takeuchi, et. al., J. Phys. Soc. Jpn. 87, 023601 (2018).
The style of the plot also makes it difficult for readers to understand. The difficulty might come from the introduction of Eq. (12) without enough explanation. Where does this relation come from? The relation of momentum and wave number can be complicated in the Bogoliubov analysis because of the complicated property of the norm in the presence of unstable mode with complex eigenvalue. This is related to the form of the Bovoliubov transformation and its normalization condition. The corresponding form of Eq. (6) is different from the conventional transformation, i.e., \delta\psi =uv^*, which make readers confused more.
I am not also satisfied with the fact that the “adjustment” is not so small for the fitting curve in the leftmost panel of Fig. 2. In such a difficult situation, I am sorry that I could not read the content of Sec. 4.2 seriously.
(4) Previous works on KHI
Baggaley and Parker revealed the nonequilibrium dynamics for the case with small and large number of vortices in a very similar situation of singlecomponent superfluids in Ref. [12]. Since authors investigate a very similar situation, they should explain distinction between the result of Ref. [12] and their result more clearly. Additionally, although authors give attention to the radiation of acoustic excitations propagating away in the bulk fluid for the supersonic case, a very similar phenomenon has been observed for KHI in immiscible binary BECs by Kokubo et. al. Phys. Rev. A 104, 023312 (2021). According to this paper, the acoustic excitations are shock waves forming Machconelike structure. A similar pattern is also observed in the rightmost panel in Fig. 1. This is just a nonlinear effect although authors try to describe such acoustic radiation as superradiance in terms of the linear analysis based on the Bogoliubov theory. How to explain this inconsistency?