Comment on “Supersonic regime of the Hall-magnetohydrodynamics resistive tearing instability” [Phys. Plasmas 19, 072519 (2012)]

In a recent paper,¹ Ahedo and Ramos extended their earlier analysis (Ref. 2) of the Hall-magnetohydrodynamics (MHD) resistive tearing instability to the case of a very low plasma thermal pressure, when the supersonic regime of instability is reached. Their major findings can be summarized as follows. First, the dispersion relation and, hence, the instability growth rate, derived in Ref. 2 for the subsonic regime, remain the same in the supersonic case for an arbitrary large value of the instability Mach number, i.e., all the way down to $β→0$⁠. Second, in the supersonic case, the electron MHD (EMHD) regime of this instability cannot be reached within the validity range of the multiple-scale asymptotic matching analysis, and, therefore, the strictly $β$ = 0 limit is always in the single-fluid dispersion relation regime. These conclusions are in apparent contradiction with the results of Refs. 3 and 4, where the transition to the β = 0 Hall-MHD regime and the respective dispersion relations have been reported.

Thus, in order to resolve this issue, recall the governing equations of the system under discussion, which in notations of Ref. 4 read

Only relevant terms are retained here, with $x$ being the non-dimensional length scaled by $L$—the global size of the system. In these equations, it is assumed that $x≪1$⁠, which is the case for the internal current sheet. Here, $ϕ$ and $χ$ are, respectively, the non-dimensional stream function and velocity potential, $ψ1$ is the perturbation of the poloidal magnetic flux function, and $b$ is the perturbation of the “out-of-plane” magnetic field component. Since the main interest is in the instability dependence on the parameters $β$ and $di≡c/ωpiL$⁠, in what follows it is assumed, for simplicity, that the perturbation wave vector $k$ and the guide field parameter $ε$ are both of the order of unity.

The tearing mode dispersion equation is determined by plasma dynamics inside the resistive internal current sheet. Therefore, this instability is supersonic, if its growth time, $γ−1$⁠, is short compared to the sound transit time $τs=L(Δx)/cs$⁠, where $(Δx)$ ≪ 1 is the scaled width of the resistive reconnection layer (the “innermost” diffusive layer of width $d1$⁠, in notations of Ref. 1). Since the speed of sound is equal to $cs∼β1/2VA$⁠, the supersonic regime occurs when $γ−1≪L(Δx)/β1/2VA$⁠, i.e., it requires

This criterion is evident from Eq. (3) for the plasma compression rate $χ″=(∇→⋅v→)$⁠. The latter is driven by the guide field perturbation $b$ (the term on the right-hand side of this equation), the effect of which is balanced by the plasma inertia and the perturbed thermal pressure (the first and the second terms, respectively, on the left-hand side of Eq. (3)). Under condition (5), the inertial effect dominates, which yields the supersonic compression rate $χs″=−b″/(γτA)2$⁠. Therefore, in this regime, the dynamics of instability ceases to depend on the thermal pressure parameter $β$⁠. In the opposite, quasistatic limit, the inertia plays no role, and the compression rate is equal to $χq″=b/β$⁠. As shown in Ref. 4, the supersonic condition (5) has a universal form, $β≤S−2$⁠, which is valid for both the single-fluid and Hall-MHD regimes.

In Ref. 1, another definition of the supersonic regime has been formally imposed, namely that the parameter $M≡γ/kcs∼γL/cs$ (called there the Mach number) is large, $M≫1$⁠, i.e., $β≪(γτA)2$⁠. Clearly, the proper condition (5) is more restrictive, because the width of the resistive layer is always small, $(Δx)≪1$⁠. Therefore, the regimes with $M≫1$⁠, which are considered in Ref. 1 as supersonic ones, are actually subsonic in their physical essence. Hence, it is not surprising that the authors arrived there to the subsonic dispersion relation (Eq. (40) of Ref. 1). Speaking formally, the very derivation of this equation assumes the quasistatic plasma compressibility relation $χq″=b/β$⁠, which, as discussed above, does not hold when $β≤S−2$⁠. In this case, the relation $χ″=χs″$ should be used instead, and the respective results of Ref. 4 completely agree with Ref. 3, where the strict condition $β=0$ has been adopted from the outset.

Furthermore, in Ref. 1 it is claimed that the asymptotic boundary matching method (internal/external solutions), which all the results rely upon, is not applicable for a large enough values of the parameter $di$⁠, when the width $d2$ of the so called “intermediate” ion inertial layer becomes comparable to the global length $L$⁠. Thus, their conclusion that the electron MHD regime cannot exist in the zero-β plasma is based on the fact that the required value of $di$ is too large. Indeed, as shown in Refs. 3 and 4, the $β=0$ EMHD regime is reached at $di≥S3$⁠, which, according to the respective scaling relation for $d2$ (see below): $d2∼di3/4S−1/4L$⁠, formally yields $d2$ exceeding the global length. Similarly, in the case of the intermediate regime (labelled as regime (2b) in Ref. 4, and the semi-collisional one in Ref. 1), the width $d2∼β1/2diL$⁠. Hence, the restriction $di≪β−1/2$ has been imposed for this regime in Ref. 1.

We argue that the asymptotic matching method is valid irrespective of the value of $d2$⁠, providing only that the Lundquist number $S$ is large enough, so that the width $(Δx)≡d1$ of the resistive layer is small, $d1≪L$⁠, which is the case for all instability regimes depicted in Fig. 1 of Ref. 4. In order to support this point, we first clarify the physical meaning of the intermediate ideal layer in the Hall-MHD regimes of resistive tearing instability. Consider, for example, the so-called strong-Hall high-beta regime (denoted as PR3 in Ref. 2), which in Fig. 1 of Ref. 4 corresponds to $di>S−1/5,β>S−2/5$⁠. In this case, the internal solution may acquire a double layer structure, when the resistive layer is surrounded by another, much wider one, $d2≫d1$⁠, where the plasma resistivity plays no role, but the poloidal magnetic flux is still advected towards the reconnection site by the Hall effect [the last term on the right-hand side of Eq. (1)], while contribution of the plasma flow (the second term there) remains small. If so, Eq. (1) yields the following estimate for the required out-of-plane field perturbation $b$⁠:

According to Eq. (2), this magnitude of $b$ should be provided by the field generation due to Hall effect (the last term on its right-hand side), which is balanced by the first term there (the effect of plasma flow along the guide field), hence

which, together with Eq. (6), yields the current density

On the other hand, the very same current also accelerates the poloidal plasma flow, hence, as follows from Eq. (4),

(recall, that in this regime $(γτA)∼di1/2S−1/2$⁠). Therefore, the streamfunction $ϕ$ increases with distance $x$ as $ϕ∼ψ1(0)Sx3/di3$⁠, and its advection effect on the poloidal magnetic flux becomes comparable to that of the Hall term at

Hence, if $d2<1,$ at $x>d2$⁠, the plasma flow takes over, and the Hall term becomes not important there. Note, that since in the regime under discussion, for which $d1∼di−1/2S−1/2$ with $di>S−1/5$⁠, the width $d2$ of Eq. (8) is always larger than $d1$⁠. The important point is that this “outermost” layer makes no significant contribution to the total electric current of the internal solution. Indeed, according to Eqs. (7) and (8),

Therefore, it means that the matching condition of the internal and external solutions,

is determined entirely by the “innermost” resistive layer, whatever the magnitude of $d2$⁠. Clearly, this is also true for $di>S1/3,$ when Eq. (8) formally yields $d2>1.$ It means that in this case, the double-layer structure of the internal solution is absent, and the effect of the plasma flow on the magnetic field evolution is insignificant in the entire plasma domain. Therefore, in this limit, the resistive tearing instability can be completely described in the framework of the EMHD equations.⁵ Note that it does not require any revision of the macroscopic scale outer solution and, hence, the stability parameter $Δ′$⁠: in both cases (Hall-MHD and EMHD), the external solution is the same force-free magnetic equilibrium.

To summarize, our conclusion is that the results of Ref. 1 disagree with the earlier publications (Refs. 3 and 4) because of the inadequate definition of the supersonic regime of tearing mode, imposed in Ref. 1 as well as due to the improper interpretation of the double-layer structure in the Hall-MHD internal solution.