Response to “Comment on ‘Supersonic regime of the Hall-magnetohydrodynamics resistive tearing instability’” [Phys. Plasmas 20, 014703 (2013)]

As clearly stated in our Ref. 1, the results derived in that work are guaranteed to apply to the range of equilibrium parameters such that the ion inertial layer of the Hall-tearing instability is much narrower than the equilibrium length scale. Together with our adopted definition of “supersonic regime,” i.e., $γ≥O(kcs)$⁠, this corresponds to the parameter domain bounded by the conditions $β≪O(k−2di−2)$ and either $β≤O(S−6/5)$ or $β≤O(S−2k4di4)$⁠. The condition $β≪O(k−2di−2)$ corresponds to the ion inertial layer characteristic scale (which in our supersonic regime is the ion sound gyroradius) being much narrower than the equilibrium scale. The conditions $β≤O(S−6/5)$ and $β≤O(S−2k4di4)$ correspond, respectively, to $γ≥O(kcs)$ in the parametric regions labeled PR1 and PR5 in Fig. 3 of Ref. 1. Unlike Ref. 2 and the above Comment by Vekstein and Kusano⁶ which offer only rough arguments based on order of magnitude estimates, our work carried out a systematic and rigorous analysis and our conclusions were supported by the explicit solutions thus obtained. Our analysis shows unequivocally that the structure of the Hall-tearing equations and their normal mode eigenfunctions change qualitatively for the parameters that correspond to our definition of subsonic-supersonic transition, $γ=O(kcs)$⁠. However, due to some finely tuned relationships between the real and imaginary parts of the eigenfunctions, the growth rate dispersion relation ends up being insensitive to those changes, which underscores the importance of a rigorous treatment of this problem and the danger of relying on simple-minded estimates. In any case, the fact that the authors of the above Comment might prefer a different, more restrictive definition of “supersonic regime” turns out to be completely irrelevant because the domain of our definition extends all the way to $β=0$ and contains the domain of theirs.

In the domain of interest here [again defined by $β≪O(k−2di−2)$ and either $β≤O(S−6/5)$ or $β≤O(S−2k4di4)$], we obtained an explicit solution of the Hall-resistive system which yields the closed-form dispersion relation given by Eq. (40) of Ref. 1. Our rigorous solution is valid in that whole domain, in particular all the way down to $β=0$ with no behavior change for $β≤O(S−2)$⁠. The dispersion relation tends asymptotically to the Furth-Killeen-Rosenbluth form³ for $β≪O(S−4/5k−2di−2)$⁠, tends asymptotically to the Drake-Lee form⁴ for $O(S−4/5k−2di−2)≪β≪O(k−2di−2)$⁠, and interpolates between the two for $β=O(S−4/5k−2di−2)$⁠, a result that disagrees with the conclusions of Ref. 2 pertaining the lowest $β$ portions of such parameter domain of interest. It also disagrees with the $β=0$ result of Ref. 5, but the reason for this discrepancy is easy to pinpoint: the very simplified model adopted in Ref. 5 ignores the coupling between the plasma compression and the magnetic compression, which is retained by our model. The reason for the discrepancy between Refs. 1 and 2 has to be subtler since these two works appear to be based on essentially the same model. The above Comment by Vekstein and Kusano does not provide any clarification because it basically just repeats the reasoning of Ref. 2 and, like Ref. 2, does not produce any actual solution either analytical or numerical. Reference 2 and the above Comment only put forward arguments based on order of magnitude estimates. There is always an element of vagueness in this approach, which certainly can never be a substitute for an explicit, mathematically proven solution like the one obtained by us. As a matter of fact, our simpler than expected solution is the consequence of several finely tuned identities and cancellations that would not have been anticipated with back of the envelope estimates.

As for the regime where the ion inertial layer of the Hall-tearing instability becomes comparable to or wider than the equilibrium length scale, i.e., $β≥O(k−2di−2)$ for our supersonic case, we agree that the growth rate should be governed by the resistive sublayer. However, the macroscopic scale outer solution and the $Δ′$ definition should now be revisited to include the ion inertial terms. Not having worked out this problem in detail down to an explicit solution and being mindful of the potential pitfalls, we chose out of prudence and mathematical consistency to leave this regime outside our discussion. We just reaffirm the validity of our results for $β≪O(k−2di−2)$⁠.

Summarizing, we stand by our mathematically proven result for the range of parameters defined by $β≪O(k−2di−2)$ and either $β≤O(S−6/5)$ or $β≤O(S−2k4di4)$⁠, in disagreement with some of the conclusions of Ref. 2 which were only inferred from order of magnitude estimates and miss crucial fine details of the problem. The Comment by Vekstein and Kusano just repeats those crude estimates without showing any actual solution to substantiate them, therefore we do not deem that it provides a valid clarification or counterargument.