Hall MHD waves: A fundamental departure from their MHD counterparts

It is thoroughly established that ideal (or resistive) magnetohydrodynamics (MHD) is a cornerstone of plasma physics,^1–7 with applications ranging from space science and astrophysics to nuclear fusion and aerospace engineering. It is important, however, to recall that MHD is not a universally valid theory, as it is derived under a particular set of assumptions,⁴ which apply to a subset of plasmas.

In particular, when the plasma is chiefly collisionless and we focus on phenomena occurring at length scales comparable to, or smaller than, the ion skin depth, it is necessary to go “beyond MHD” and avail ourselves of the models comprising extended MHD.⁷ One of the most crucial models in this regard is Hall MHD (HMHD), which entails the inclusion of a Hall current term within the MHD induction equation. HMHD has been utilized in a variety of contexts to uncover diverse phenomena, as briefly summarized later.

The basics of HMHD were delineated and elaborated decades ago,^8–12 and a bevy of equilibrium solutions have been subsequently derived.^13–19 Variational and Hamiltonian formulations of MHD have attracted extensive attention^20–29 partly because they have enabled formal analysis of the underlying mathematical structure of HMHD, such as its conserved quantities. In tandem, a multitude of papers on waves in HMHD have analyzed its mathematical characteristics;^30–39 of this category, a special class of waves, termed linear-nonlinear waves, was the subject of Refs. 40 and 41.

From an astrophysical standpoint (e.g., protoplanetary disks), HMHD has been invoked in conjunction with the famous magnetorotational instability (MRI) to analyze how the former influences the evolution of the latter.^42–45 Other applications of HMHD to real-world systems include generic energy conversion mechanisms,^46–48 large- and small-scale dynamo processes,^49–53 the unified dynamo-reverse dynamo mechanism,^54,55 small-scale turbulence,^56–65 fast reconnection,^66–73 and even newly emerging disciplines like astrobiology.^74,75 We mention, however, that fast magnetic reconnection may also be mediated by resistive MHD mechanisms such as the plasmoid instability.^2,76–78 Many of these studies were conducted under the incompressible assumption, which is physically reasonable in select circumstances and can help simplify the analytical or numerical modeling.

Hence, it is apparent that HMHD is a key model in plasma physics. Thus, elucidating its fundamental properties would be of much significance. This short note accordingly demonstrates, through a readily accessible calculation, that linear waves associated with incompressible HMHD are considerably more interesting and involved than what seems to be generally realized and appreciated. Given that HMHD remains a very active area of investigation both for linear and nonlinear studies (as indicated in the preceding paragraphs), a compact—yet essentially complete—delineation of the fundamental waves (normal modes) of the incompressible HMHD system ought to be valuable to myriad researchers.

In a nutshell, this endeavor of working out the salient physics of (incompressible) HMHD waves is the chief objective of our current work—even if some/all aspects of the derivation were to have been reported before, at the minimum, this pedagogical treatment serves to synthesize together disparate “branches” of incompressible HMHD waves into a coherent whole and may additionally offer novel insights into the nature of these waves and their deviations from their ideal MHD counterparts.

These two modes are completely along expected lines, thus serving as a consistency check.

One of the key distinguishing features of (incompressible) HMHD waves is evident from the relations Eqs. (8) and (9). Unlike in the case of linear MHD waves, the quantities $bz$ and $(k×b)·êz∝jz$ are no longer independent and must either be zero or nonzero together. In other words, the Hall current lashes together the temporal evolution of the z-components of the magnetic field and of the current density; in consequence, HMHD, which represents a singular perturbation of ideal MHD,^12,35,79 opens the door for novel physics. This striking facet of the coupling is attested from a careful consideration of either Eqs. (12) and (13) or Eqs. (8) and (9).

When both these quantities (⁠ $bz$ and $jz$⁠) are nonzero, we may duly proceed with the dispersion relation given by Eq. (14). However, a more radical feature induced by the Hall current is the existence of an entirely new class of modes for which $bz=0=(k×b)·êz$—it is this class of modes that we will chiefly dwell on in this paper. Note that this solution [ $bz=0=(k×b)·êz$] is fully consistent with Eqs. (8) and (9) because it simultaneously converts the RHS and LHS of these equations to zero. We can approach $bz=0$ through an alternative path: the incompressibility of the velocity field corresponds to $∇·v=0$⁠, which yields $k·v=0$ in the Fourier domain. On plugging this expression in Eq. (12), we see that $bz=0$⁠.

With the derivation of the linear wave dispersion relations of incompressible HMHD out of the way, we are in a position to provide a deeper perspective and underscore the inherent significance of these findings:

• Perhaps the most important of all is the emergence of a totally new branch of purely circularly polarized (Beltrami) waves/modes exhibiting the polarization of

  $b=bR±ibI, |bR|=|bI|, bR·bI=0,$

  (27)

  $v=−kzωb, bz=0,$

  (28)

  which are rendered possible only via the Hall current; this branch does not exist in ideal MHD. It is the expression of the fact that the Hall term is a singular perturbation that imparts novel physics to HMHD with respect to ideal MHD. The formulation of this singular Beltrami branch has been outlined earlier: note that Eq. (27) is an outcome of the text between (19) and (23), whereas Eq. (28) is obtained from Eq. (11) after setting $bz=0$ in this expression. These modes are Beltrami in nature, since they satisfy the Beltrami equation as highlighted in Eq. (23) and the subsequent discussion.

• The circularly polarized waves referenced earlier are endowed with unique mathematical characteristics—conspicuously different from those of the standard linear waves—because, for each individual wave (viz., for a given $k$⁠), the nonlinear terms in the HMHD wave system can be demonstrated to vanish exactly.⁴⁰ Thus, even though our analysis entailed a linear derivation, these waves are actually wave solutions of HMHD with arbitrary amplitude and have been discussed in several papers under the nomenclature of linear-nonlinear waves.^39–41,80 In this regard, these modes ought to be perceived as the HMHD counterparts of the Alfvén–Walen solutions $b=±v$⁠, which are arbitrary amplitude wave solutions documented in ideal MHD.^81,82

• The aforementioned linear-nonlinear waves exhibit independence from the waves represented by the dispersion relation (14) despite the seeming partial resemblance of Eqs. (14) and (24). Let us, for instance, examine (14) for what would be approximately a shear Alfvén wave (⁠ $ω2≈kz2$⁠) in MHD. For the sake of simplicity, we shall compare the two sets of waves in the regime of $k2≫kz2$ and $εk≪1$⁠. The first set of Eqs. (8), (9), and (14) collectively yields

  $ωkz=±11+ε2kz2,$

  (29)
  with the polarization (after some simple algebra)

  $bx=−kxkx2+ky2[kz+ik2εωkz]bz,$

  (30)

  $by=−kykx2+ky2[kz−ik2εωkz]bz,$

  (31)

  which is manifestly distinct from the nonlinear-linear waves, to wit, the set of Beltrami modes described by Eqs. (26)–(28).

These two branches of linear modes are, loosely speaking, mutually orthogonal to each other, implying that no linear transitions can take place among them; for instance, the Beltrami modes associated with the linear–nonlinear waves exhibit $bz=0$⁠, whereas the other class of modes evinces $bz≠0$⁠, as is evident from inspecting Eqs. (30) and (31).

The two sets of waves each have their own pros. The linear–nonlinear wave solutions have been used to explain the potential energy spectra in solar wind turbulence through Kolmogorov-type scaling arguments⁸⁰ as well as to explain an observed absence of equipartition (of magnetic and kinetic energy) in Hall turbulence and dynamos.^45,52 Likewise, the other set of waves have been invoked in understanding certain properties of Hall turbulence.^30–34,36

This phenomenon would be essentially ported over to the more general case of extended MHD when electron inertia, for example, is included; the reason is that the Hall current still remains and induces analogous effects, albeit with (smaller) corrections stemming from the finite electron mass and skin depth.^39,41,80 Hence, all studies “beyond MHD” (i.e., with two-fluid contributions) must properly account for the fundamentally altered structure of the linear modes, in fact, of the incompressible branch that constitutes a powerful new addition to the repertoire of Alfvénic waves.

This work was supported by U.S. DOE under Grant Nos. DE- FG02-04ER54742 and DE-AC02-09CH11466.