Counterintuitive Particle Confinement in a Helical Force-Free Plasma
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe present paper discusses structure of flux surfaces in helical force-free plasma and the corresponding dynamics of charged particles. As far as I can tell, the paper is free of major errors and the presented results are correct. The paper deals with, in my opinion, interesting and important topic in the theory of nonlinear dynamics and magnetic plasma confinement.
I would think that the paper can be published in Plasma, if the Authors address a couple of issues.
1. By itself, the non-conservation of the Adiabatic invariants in the systems with magnetic field curvature is nothing new. That should probably be better stated in the text.
2. For the systems with the non-conservation of Adiabatic invariant(s), it is important to identify where in the physical/phase space the Adiabatic invariant(s) changes: do these changes occur everywhere, or if they are localized somewhere. A plot of the time evolution of the Adiabatic invariant would help.
3. The reverse of the radial motion is not exactly clear. “the increase in gyroradius as the particle approaches the boundary allows particles to cross from the edge of one piece of flux rope to the edge of the neighboring piece.” So is it a probabilistic process (as the word “allows” indicates, or is always happens?
To conclude, for the paper to be accepted, provided the authors address the comments listed above.
Author Response
Please see the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsForce-free magnetic fields are often encountered in astrophysical and laboratory plasmas, and a lot of effort has been devoted in the literature to investigating them. The dynamics of the particle component of the plasma has received, instead, less attention. This manuscript reports a numerical study of charged particle motion in force-free magnetic equilibria produced in the SSX spheromak experiment.
The full dynamics of the particle is tracked, not its guiding-center approximation as often done in similar studies.
The results are analyzed in standard statistical form: statistics of trapped or escaping particles parameterized in terms of their energy and pitch angle, statistics of loss times; but also individual trajectories are scrutinized.
The overall conclusion is that a part of the trajectories can be understood in terms of standard guiding center dynamics, but also that for another fraction, the motion is not adiabatic, and differs from the guiding center approximation. The reason is attributed to the particle interacting with large magnetic field gradients, where the ratio between the Larmor radius and the field length scale is not negligible.
I think that the main theme of the paper, the investigation of ion dynamics in the spheromak geometry, is well motivated; the numerical study is carried out correctly, and the results--even though preliminary--are interesting since present evidence of non-adiabatic effects in the particle dynamics: an argument which is not often explored in magnetized plasma theory, where it is generally postulated the validity of the guiding center approximation, but no sufficient attention is paid to the conditions that make it valid (see to this regard, e.g., the recent study Chen-Zonca et al "Validity of gyrokinetic theory in magnetized plasmas", Communications Physics 2024).
In my opinion, the paper deserves to be published. There are, however, some points that need to be clarified in order to make the paper fully appreciable.
Here my list of questions and comments
1. "Plectonemic Taylor states". These are magnetic equilibria characterized by two intertwined helical structures of magnetic flux, like the DNA double helix. The authors report that (i) this is the magnetic equilibrium measured in SSX, and (ii) that this is one expected solution of force-free magnetic fields computed from the Taylor relaxation theory.
The double-helix is an admissible solution of Eq. (1) but, as far as I know, when referencing to non-axisymmetric solutions of the Taylor theory, one usually refers to single-helical ones (see, e.g., Ortolani-Schnack "Magnetohydrodynamics of plasma relaxation"). As a matter of fact, this is the first time I encounter such a double-helix state. Besides, by looking at the fig. 1 (a), the measured magnetic field displays evidence of a single helical structure, not a double one. it does not seem consistent with the numerically calculated fig. 1(b). I add that, even though not Taylor states, single helicity plasmas are routinely encountered in another laboratory device: Reverse Field Pinches, and there they appear as single-helix states, not double-helix ones [see, e.g., the review Marrelli et al, Nuclear Fusion 61, 023001 (2021)].
In summary, I think that these points should be addressed by the authors. (i) On the basis of Taylor theory, how do they arrive to select the double-helix solution rather than the single-helix one? (ii) They should provide better and more compelling experimental evidence of the existence of such magnetic structurein SSX. In my opinion, fig. 1a does not provide such evidence.
2. Figure 2 is totally incomprehensible to me. It does not even resemble a cylindrical geometry, rather a spherical one. I suggest removing it since it does not provide useful information.
3. The authors have employed a numerical code (PSI-Tet) in order to compute the field structure. I do not know the code, but the equation (1) is linear in the magnetic field; in cylindrical geometry its solutions are easy to compute in terms of Bessel functions and trigonometric functions. So, what is the point of employing numerical tools when analytical solutions are available? Perhaps is it to model finite-length effects?
4. In the Introduction (lines 52,53) the authors claim that MHD equilibria are not warranted in principle to be accurate solutions since ion Larmor orbits are not negligible with respect to the device size. However, this is contradicted at line 102, where this ratio is set to 1/100. This is not a large number; in tokamaks such a scale difference would justify the adoption of the adiabatic guiding center approximation for particle motion, rather than full orbits. Actually, as the authors themselves state, non-adiabatic effects appear in the strong-gradient regions. That is, adiabaticity of the motion breaks down--as expected--when the typical ratio rho/L between the ion Larmor radius rho and some typical length L becomes non negligible, but L has nothing to do with macroscopic scales. It is a local effect, whereas magnetic equilibria are determined on the basis of the large scale, global geometry.
In summary, in my opinion, there is a priori nothing surprising in the accuracy of the MHD for determining the equilibria.
5. Particle energy is provided in units normalized to the plasma temperature, but I was not able to find anywhere the numerical value of the temperature itself. Although not critical for the work, I think that providing it would be appreciated.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsI am not satisfied with the Authors' response to my comment #2. To begin with, Figures 15 and 16 hardly let a reader to assess the rate of change of the magnetic moment. I would definitely want to see a simple magnetic moment as a function of time plot, with another subplots showing the location of a particle to relate the change of the magnetic moment to the location.
That being said, the non-conservation of the magnetic moment does not mean that there is no adiabatic invariant in the system. Similar situation was discussed, for example in Adiabatic and chaotic charged particle motion in two-dimensional magnetic field reversals: 1. Basic theory of trapped motion, J Buchner, LM Zelenyi J. Geophys. Res 94 (11), 821
The presence of a different adiabatic invariant, that may "merge" with the magnetic moment in some of the part of the phase space, while becoming quite different in the other parts, could very well describe and explain the confinement.
Author Response
Please see attachment.
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors have satisfactorily replied to all my comments.
In my opinion, the paper now may be published.
Author Response
Comments 1: The authors have satisfactorily replied to all my comments.
In my opinion, the paper now may be published.
Response 1: Thank you very much for helping to improve the manuscript!
