Formation of Fine Structures in Incompressible Hall Magnetohydrodynamic Turbulence Simulations

Abstract

Hall magnetohydrodynamic simulations are often carried out to study the subjects of instabilities and turbulence of space and nuclear fusion plasmas in which sub-ion-scale effects are important. Hall effects on a structure formation at a small scale in homogeneous and isotropic turbulence are reviewed together with a simple comparison to a (non-Hall) MHD turbulence simulation. A comparison between MHD and Hall MHD simulations highlights a fine structure in Hall MHD turbulence. This enhancement of the fine structures by the Hall term can be understood in relation to the whistler waves at the sub-ion scale. The generation and enhancement of fine-scale sheet, filamentary, or tubular structures do not necessarily contradict one another.

1. Introduction

Hall magnetohydrodynamic (MHD) equations are variants of MHD equations and can be derived from either Braginskii’s two-fluid model or the Vlasov–Boltzmann equation, neglecting electron inertia, finite ion Larmor radius effects, and other fine-scale effects [1].

The Hall term appears in the generalized Ohmic law to represent the effects of finite ion-skin-depth or ion-inertia-scale. Hereafter, we refer to a scale shorter than the ion-skin-depth as the sub-ion scale. The introduction of the Hall term changes the frozen-in condition, conservation laws, inviscid invariants (energy, magnetic helicity, and hybrid helicity), the linear dispersion relation of waves, and consequent plasma dynamics, including dynamos and magnetic reconnection. This modification has encouraged many studies, from Alfvén waves, whistle waves, and double-Beltrami flows to more fundamental and mathematical subjects [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. These changes enrich the physics related to the sub-ion scale.

The Hall MHD model is widely used to study various subjects such as the Kelvin–Helmholtz instability, Rayleigh–Taylor, and some other instabilities, sometimes together with finite Larmor radius effects (gyro-viscosity) [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38].

The Hall term also changes various aspects of turbulence, including dynamo action [39,40,41,42,43] and magnetic reconnection [44,45,46].

The applicability of the Hall MHD model to plasma dynamics has been also investigated [47,48,49,50]. Schnack et al. [48] have shown that Hall MHD can be derived for studying plasma physics in the inner heliosphere, shock tubes, and other subjects. In Ref. [49], it is shown that the Hall MHD model can be a reliable one for studying the ion temperature gradient instability for a certain scale range. Papini et al. [50] carried out turbulence simulations of both Hall MHD and Hall MHD-PIC hybrid simulations. They concluded that the Hall MHD model, to a large extent, captures the scale from the MHD scale to the sub-ion scale.

There are also many studies of Hall MHD turbulence. See Refs. [47,51,52,53,54,55,56] and references therein for an overview of Hall MHD turbulence with a wide scope. Hereafter, we simply focus on a small category of numerical simulations of turbulence often called homogeneous and isotropic incompressible Hall MHD turbulence, mainly in the context of fine structure formation. Studies on other subjects are mentioned briefly.

This paper is organized as follows. In Section 2, numerical simulations of Hall MHD turbulence are introduced. In Section 3, topics of the energy spectra of Hall MHD turbulence are reviewed briefly. In Section 4, formation of coherent or fine structures in incompressible Hall MHD turbulence is presented, together with a comparison to (non-Hall) MHD turbulence. Section 5 presents a summary.

2. Numerical Simulations of Hall MHD Turbulence

Incompressible Hall MHD equations can be described as follows:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial u_i}{\partial t}}& =& -{\displaystyle \frac{\partial }{\partial x_j}}\left(u_i{u}_j\right)-{\displaystyle \frac{\partial p}{\partial x_i}}+{ϵ}_{ijk}{J}_j{B}_k+\nu {\displaystyle \frac{{\partial }^2{u}_j}{\partial x_j\partial x_j}}+f_i^u,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial B_i}{\partial t}}& =& -{ϵ}_{ijk}{\displaystyle \frac{\partial }{\partial x_j}}\left[-{ϵ}_{klm}\left(u_l-{ϵ}_H{J}_l\right)B_m+\eta J_k\right]+f_i^b,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial u_i}{\partial x_i}}& =& 0,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial B_i}{\partial x_i}}& =& 0.\hfill \end{array}$$

(4)

The symbols $B_i$, $J_i={ϵ}_{ijk}{\partial }_j{B}_k$, and $u_i$ represent the i-th components of the magnetic field, current density, and velocity field vectors, respectively. The symbol ${ϵ}_{ijk}$ is Levi-Civita’s anti-symmetric tensor. Equations (1)–(4) are already normalized by some representative variables. See Refs. [57,58] for the normalization. The Hall parameter ${ϵ}_H$ represents the ratio of the ion-skin-depth to the system size.

In Equations (1)–(4), the Ohmic law $E_k=-{ϵ}_{klm}{u}_l{B}_m+\eta J_k$ in resistive MHD equations has been replaced by the generalized Ohmic law $E_k=-{ϵ}_{klm}{u}_l^e{B}_m+\eta J_k$, where $E_k$ is the k-th component of the electric field, and $u_k^e=u_k-{ϵ}_H{J}_k$ is the electron velocity.

Among the various differences between the Hall MHD model and the MHD one, the emergence of whistler waves that propagate along magnetic field lines with a high frequency of

$$\begin{array}{c}\hfill {\omega }^2={\left({\displaystyle \frac{V_A^2}{\Omega }}\right)}^2{k}^2{k}_{\Vert }^2\end{array}$$

(5)

is a key feature of a Hall MHD simulation, where $k_{\Vert }$, $V_A$, $\Omega$, and $\omega$ are the wave vector parallel to the background magnetic field, Alfvén velocity, ion cyclotron frequency, and frequency, respectively [48].

The appearance of whistler waves brings about not only an excitation of high wave-number Fourier coefficients of the magnetic field but also a severe restriction in the time-step width to satisfy the stability condition of the dispersive wave. Furthermore, we need to sufficiently resolve the nonlinear transport of the magnetic field energy from low-wave-number to high-wave-number Fourier coefficients. One needs a high resolution (many grid points in real space) to resolve both the MHD scale and sub-ion scale and an extremely small time-step width in a Hall MHD turbulence simulation to satisfy the requirements.

In order to overcome the severe requirements and to elucidate the physics of turbulence by simulations, we often use hyper-diffusivity, which is added to the right-hand side of Equations (1) and (2). We also use this classic technique in this article. However, we can also consider carrying out a large eddy simulation (LES) instead of the hyper-diffusivity. Although we do not use numerical data from an LES in this article, it is worth considering carrying out an LES to overcome the severe numerical restrictions. See Appendix A for an outline of our LES.

3. Energy Spectra and Related Topics in Hall MHD Turbulence

There are many theoretical and numerical studies on MHD and Hall MHD turbulence. With respect to the theoretical aspects of MHD turbulence, especially in relation to hydrodynamic turbulence, see textbooks [59,60,61].

Among various subjects of turbulence studies, scaling laws in energy spectra are of central interest. Space satellite measurements report scaling laws of the frequency f by $f^{-1.7}$ in the MHD-scale (the scale larger than ion-skin-depth) and $f^{-2.6}$ in the sub-ion scale (the scale smaller than the ion-skin-depth) of solar winds and coronal plasma [62,63]. Recent advancements in Parker Solar Probe (PSP) measurements provide further information. Since a study using POP data exceeds the scope of this article, see reviews and articles such as Refs. [64,65,66,67,68].

The existence of scaling regimes are closely related to the classic phenomenology of MHD turbulence, such as Kolmogorov’s theory, which predicts $E_M\left(k\right)\sim k^{-5/3}$, and the Iroshnikov–Kraichnan theory, which predicts $E_M\left(k\right)\sim k^{-3/2}$ at the MHD scale, where $E_M\left(k\right)$ is the magnetic energy spectrum, while the kinetic energy spectrum is represented by $E_K\left(k\right)$ (see Refs. [60,61] for these topics). The phenomenology of turbulence under a guiding magnetic field is given by Goldreich and Sridhar as $E_M\left(k_{\perp }\right)\sim k_{\perp }^{\alpha }$ [69]. Succeeding theories and simulations support basic views given by Goldreich and Sridhar, which are typically seen in the decomposition of the wave-number vector to parallel ($k_{\Vert }$) and perpendicular ($k_{\perp }$) components [70,71,72,73,74,75,76]. Nevertheless, in the context of homogeneous and isotropic turbulence, we consider Kolmogorov’s or Iroshnikov-Kraichnan’s phenomenology rather than these more recent theories because of the absence of the guiding magnetic field.

As in the MHD scale, a scaling law in the sub-ion scale is also studied extensively. An incompressible Hall MHD turbulence simulation is often expected to show $E_M\left(k\right)\propto k^{-7/3}$. Numerical simulations of Hall MHD turbulence support the $k^{-7/3}$ law at the sub-ion scale, whether the simulation is a shell model [77,78] or a three-dimensional numerical simulation [42,57,58,79,80,81,82].

Although the scaling law $E_M\left(k\right)\propto k^{-7/3}$ has been widely accepted, another scaling law in the sub-ion scale of the magnetic energy spectrum $E_M\left(k\right)\sim k^{-8/3}$ has been reported by Meyrand and Galtier for electron MHD turbulence [83]. Below we see that an incompressible Hall MHD simulation can show the scaling law of $E_M\left(k\right)\sim k^{-8/3}$ as well.

In Figure 1, $E_K\left(k\right)$ and $E_M\left(k\right)$ are obtained in a forced Hall MHD turbulence simulation, with the magnetic Prandtl number $P{r}_M=\nu /\eta =100$ plotted together with compensated energy spectra. The external force is given to the equation of motion with a fixed amplitude at $k\le 3$ with random Fourier phases. The amplitude of the external force is predetermined at a level at which the force does not cause an excessive raise of the tail of the energy spectrum. The abscissa is normalized by the ion-skin-depth so that the region ${ϵ}_{H}k>1$ represents the sub-ion scale. This computation is similar to those presented in Ref. [84] by the pseudo-spectral method and the Runge–Kutta–Gill scheme. We use either the FFTE [85], P3DFFT [86], or FT3D [87] three-dimensional libraries, depending on the supercomputer. A hyper-diffusivity is also used to suppress a very high-k region. (Another scaling of $E_M\left(k\right)\sim k^{-5/2}$ has been reported in Ref. [84], without using hyper-diffusivity.) At the sub-ion scale, $E_K\left(k\right)\propto k^{-17/3}$, while $E_M\left(k\right)\propto k^{-8/3}$ rather than the well-known $E_M\left(k\right)\propto k^{-7/3}$ scaling law. As we see later, coherent structures in MHD and Hall MHD turbulence are often considerably influenced by the magnetic Prandtl number $P{r}_M$. We need to clarify a detailed setup of a numerical simulation in order to compare our numerical results to an observational result and choose an appropriate $P{r}_M$.

While scaling laws in the energy spectrum have been studied extensively, those in the real-space structure functions (von Kármán–Howarth-like scaling laws, or a so-called exact relation of correlation functions), intermittency, and extended self-similarity are closely related to the scaling laws in the energy spectrum. With respect to the structure functions of MHD turbulence, a formulation given by Politano and Pouquet [88,89] for the third-order structure function (von Kármán–Howarth equation) and succeeding works provides an important foundation [90,91,92,93,94,95,96,97,98,99,100,101,102].

There are other important keywords for Hall MHD turbulence. Anisotropy, (inverse) cascade, dissipation, and energy transfer are the typical important keywords of both MHD turbulence and Hall MHD turbulence. We do not review these subjects and move on to the formation of fine structures. See earlier studies [41,42,43,82,103,104,105,106,107,108,109,110,111,112,113] for these subjects.

There are also interesting subjects such as a closure theory, as in Ref. [114], which should be studied further.

4. Formation of Coherent Structures

4.1. Discussions of Coherent Structures in MHD Turbulence

Formation of coherent structures has been one of important subjects of MHD turbulence. Recent studies discuss the emergence of coherent structures more closely, sometimes reporting not only current sheets but also filaments of currents and/or vortices [115,116,117,118].

From a point of view of numerical simulations, the emergence of coherent structures is closely related to energy dissipation, energy transfer, intermittency, and other phenomena. In the context of structure formation, a coherent structure can mean not only a large-scale structure but also a fine, small-scale one. In this article, we include both of these meanings in the expression coherent structures. See also textbook [59], reviews [119,120,121,122,123,124], and articles [46,104,117,125,126,127,128,129,130,131] for coherent structures in (non-Hall) MHD turbulence.

Greco et al. [127] have discussed a relation between the Probability Density Function (PDF) of the current density and the coherent structures of the current density. In Figure 2, Figure 3 of Ref. [127] was reprinted with permission from Greco et al., Astrophysical Journal 691:L111-L114 (2009). Copyright (2009) by the Institute of Physics. In the figure, PDF of the current density from the 2D simulation in Ref. [127], and magnetic field lines (contours of constant magnetic potential) corresponding to the regions I-III of the PDF. The PDF of the current density has a skewed profile with a tail much wider than the Gaussian, showing a clear tendency of spatial intermittency. It shows a relation between the parts of the PDF (the head, body, and tail) and magnetic field lines. This work reveals a contribution of coherent structures to the statistics of the current and thus to the intermittency of turbulence.

Sahoo et al. have carried out numerical simulations of both freely decaying and forced turbulence over a range of $P{r}_M$ [118]. In that work, the isosurfaces of the current density, vorticity, and energy dissipation are presented. A high-$P{r}_M$ simulation tends to give fine structures of the current density and vorticity, while the large-scale structures are found in the isosurfaces of the magnetic energy dissipation of a high-$P{r}_M$ simulation. In Figure 3, isosurfaces of the current density of MHD turbulence with various magnetic Prandtl numbers with or without external forcing, presented in Ref. [118] as Figure 27, are reproduced with permission by the authors and the Institute of Physics. These views are presented in the literature together with PDFs of the velocity and magnetic field increments, the plots of the velocity and magnetic field vector gradient invariants, and some other important statistical information.

Zhdankin et al. have highlighted statistics of the current sheets [129] as well as the typical role of coherent turbulence structures in the energy dissipation [130]. They present numerical procedures to identify the current sheets in Ref. [129] and study the orbit of magnetic field lines on the current sheets. In Ref. [130], they studied energy dissipation in MHD turbulence and related the study to coherent structures. They report that the main part of the energy dissipation occurs in large-scale coherent structures (current sheets), while the dissipation spreads over a wide range of the wave-number from the inertial range to the dissipation range and exhibits coherent structures (current sheets) and small reconnection events (nanoflares).

While the current sheets are recognized as the center of interest in various subjects related to MHD turbulence, rolls of vortices and currents due to a local Kelvin–Helmholtz instability in turbulence have been reported [126]. In Figure 4, the roll-up of a current sheet in an MHD turbulence simulation is presented. This figure, a reproduction of Figure 2 from [126] used with permission, clearly captures a fine-scale roll of current in a high-resolution MHD turbulence simulation. We also note that, in a different regime of physical parameters (numerical settings of simulations), vortex tubes can be more dominant. For example, Kitiashvili et al. [132], Kivotides [133], and Silva et al. [134] report vortex tubes in solar convection.

4.2. Fine Structures in Hall MHD Turbulence

Coherent or fine structures in Hall MHD turbulence are often studied as an extension of numerical studies of (non-Hall) MHD turbulence. As Marino and Sorriso-Valvo [56] have reviewed recently, a role of the Hall term in coherent structures is a focus of discussion. While many works support the formation of a large-scale magnetic structure (or a large current sheet), some works present formation of a fine structure such as current and vortex filaments. A large-scale magnetic structure or a large current sheet is considered to explain a large jump in magnetic field signals reported by spacecraft observations. As we have seen in the work of Zhdankin et al. [130], a current sheet involves a wide range of scale from the largest scale of the inertial range to the dissipation scale. However, simultaneously, we often observe fine structures as well. Formation of fine structures does not necessarily contradict the formation of a large-scale structure.

To the best knowledge of the authors, formation of coherent or fine structures is often different between a freely decaying simulation and a forced simulation, whether it is Hall MHD turbulence or (non-Hall) MHD turbulence. Various aspects of a freely decaying turbulence simulation depend on details of the initial condition. The ratio of the kinetic and magnetic energies and the three inviscid invariants, including influences of helicity [55,135,136] in particular, are considered crucial roles there. Although it is difficult to assert the nature of coherent or fine structures in a freely decaying turbulence because of this initial condition dependence, freely decaying turbulence simulations are carried out partially because a decay law in turbulence stimulates wide interest (whether it is MHD or Hall MHD) [60]. On the other hand, a forced turbulence simulation is influenced by the nature of an external force. The formation can be also different between turbulence with and without magnetization.

4.2.1. Decaying Homogeneous and Isotropic Turbulence

An advantage of studying freely decaying turbulence is that a comparison between MHD and Hall MHD turbulence is clear. By giving the same initial condition, $\nu$, $\eta$, and the time-step width, we can attribute any single difference to the Hall effect, although we need to take care with the difference in numerical resolution required by the two kinds of simulations.

In Figure 5, isosurfaces of the enstrophy density (green) $E_{\omega }=〈{\omega }_i{\omega }_i〉/2$ and the current density (gray) $J=〈J_i{J}_i〉/2$ in decaying isotropic simulations of (a) MHD (${ϵ}_H=0$) and (b) Hall MHD (${ϵ}_H=0.05$) turbulence are shown, where ${\omega }_i={ϵ}_{ijk}{\partial }_j{u}_k$ and $J_i={ϵ}_{ijk}{\partial }_j{B}_k$ are the i-th component of the vorticity and the current density, respectively. These two numerical simulations start from the same initial condition with the same kinetic and magnetic energies as those in Refs. [57,58] and are carried out with $\nu =\eta =5\times {10}^{-5}$ and $N^3={2048}^3$ grid points. The thresholds of $E_{\omega }$ and J are given by four times the deviation above the mean value of each quantity, respectively.

In Figure 5a, both green and gray isosurfaces are thin and fine sheets or threads. In comparison to simulations with a lower resolution (and thus larger $\nu$ and $\eta$) reported in earlier works [57,58], the current sheets are smaller and narrower. This suggests that the large-scale current sheets observed in MHD turbulence in Refs. [57,58] can be either unstable or covered by fine-scale structures (signals of high wave-number components), in a higher Lundquist number simulation.

In Figure 5b, the gray isosurfaces of J are either fine threads or fine filaments, while the green isosurfaces of $E_{\omega }$ are tubular or filamentary. These structures are much finer than those in Figure 5a, taking the shape of a tube (green) or a very fine filament (gray). As has been noted already, since both the MHD and Hall MHD simulations start from the same initial condition and the same diffusive coefficients, all differences between the two simulations are finally attributed to the Hall effect.

Based on these simulations, the Hall term is considered to enhance the filamentary or tubular structures of J and $E_{\omega }$, which represent the excitation of the small scale of the magnetic and velocity fields. A similar result regarding structure formation in comparison between freely decaying MHD and Hall MHD turbulence has been given for another initial condition, in which the kinetic and magnetic energy are the same but have different decaying properties [81]. Our current understanding is that the Hall term allows for the roll-up of a vortex sheet to a vortex tube through the change of the frozen-in condition, and then a current filament is formed through either an entrainment of a current sheet by a vortex tube or an excitation of whistler waves.

We note that Stawarz and Pouquet have also carried out freely decaying Hall MHD turbulence simulations [138]. They report that the Hall term makes the current sheets thinner, while the width of the vorticity makes them wider, although they are non-committal on the detail of vortex strictures in the work. They do not report formation of fine structures. We need further study to clarify what is a key difference between the simulations in Figure 5 and the results in Ref. [138].

An influence of the enhancement of a filamentary or tubular structure can reach a scale larger than the ion-skin-depth $1/{ϵ}_H$. In Ref. [58], it is shown that changes to the structures are observed even at the MHD scale. This indicates that the Hall effect, which changes only the physics in the sub-ion scale, can reach the MHD scale. Banerjee and Halder have studied the range of nonlinear interactions in Hall MHD turbulence recently [139]. Studying how far the Hall term can reach at a scale larger than the ion-skin-depth and how the Hall term can influence the large scale may be an interesting subject.

4.2.2. Forced Homogeneous and Isotropic Turbulence

Next, we see an example of structure formation in forced isotropic simulations.

In a numerical simulation of MHD and Hall MHD turbulence, formation of the current sheets or large-scale magnetic field structures with a large jump are often reported. However, the emergence of the coherent structure can depend on the magnetic Prandtl number $P{r}_M=\nu /\eta$, as Sahoo et al. [118] have reported for MHD turbulence. In Figure 6, Figure 4 in Ref. [84] is presented (reused in accordance with the American Physical Society’s copyright policy). Isosurfaces of $E_{\omega }$ (green) and J (gray) in (a) forced Hall MHD turbulence of $P{r}_M=10$ (${ϵ}_H=0.20,\nu =1\times {10}^{-3},\eta =1\times {10}^{-4}$), (b) forced Hall MHD turbulence of $P{r}_M=100$ (${ϵ}_H=0.20,\nu =1\times {10}^{-2},\eta =1\times {10}^{-4}$), and (c) forced MHD turbulence of $P{r}_M=100$ (${ϵ}_H=0,\nu =1\times {10}^{-2},\eta =1\times {10}^{-4}$). One finds that J-isosurfaces in Figure 6a,b are filamentary, while those in Figure 6c are flattened. In this sense, the Hall term appears to enhance filamentary structures. This indicates that we need to pay attention to the value of $P{r}_M$ (or a corresponding concept in a kinetic simulation) if we study coherent structure formation of a phenomena in which $P{r}_M$ is supposed to be large, as we have seen in freely decaying simulations.

4.3. Energy Transfer and Coherent Structures

Here we consider Hall effects on the coherent or fine structures from the point of view of the energy spectrum. In Figure 7, (a) the kinetic energy spectrum $E_M\left(k\right)$ and (b) magnetic energy spectrum in a Hall MHD run (${ϵ}_H=0.05$, $0.025$) and MHD run (${ϵ}_H=0$) analyzed in Ref. [58] are presented. In order to see the MHD scale ($k<1/{ϵ}_H$) closely, a range of $k<50$ is presented in the linear–linear plot. (The log–log plot of the full range of the energy spectra can be seen in Ref. [58]). Although Figure 7a does not show a clear difference among the three runs, Figure 7b shows that the large-scale magnetic energy decreases as the Hall parameter ${ϵ}_H$ becomes larger. Although the current sheet covers a relatively wide range of the wave-number from the largest scale of the inertial range to the dissipation range as pointed out for MHD turbulence [130], the suppression of the magnetic energy at the MHD scale suggests the suppression of the size of the current sheets.

One important bit of information related to the filamentary structure formation, Figure 5 and Figure 6, is found in Ref. [140]. In the literature, the authors have extended an earlier study by Galtier [90] and expressed Hall MHD equations by the generalized Elsässer variable. Then, the nonlinear energy transfer of Equations (1)–(4) is expressed by the couplings of ion–cyclotron (IC) and whistler (Wh) modes. Time series of the energy transfer spectra for the IC and Wh modes in [140] are presented in Figure 8. (The figures are reused from [140]). While an active scale-to-scale energy transfer is observed in both the IC modes and the Wh modes, we can see in the figure that the energy exchange between the two modes is mainly from the IC mode to the Wh mode, and the transfer in the opposite direction is very small. This suggests that the magnetic field excited in the decay of Hall MHD turbulence is going to be occupied by the whistler waves. Since a propagation of a whistler wave forms a spiral magnetic field orbit elongated along a large-scale magnetic field line, it is natural that the excitation of the whistler wave brings about filamentary structures in a visualization of J.

This understanding is supported by observing magnetic field vector topology. The presence of elongated spiral magnetic field lines is consistent with the excitation of the whistler mode. We recall that spiral field lines in the velocity field are often visualized by the use of three invariants of the velocity gradient tensor P, Q, and R [141] (in the case of an incompressible fluid, $P\equiv 0$). The so-called Q-value is used quite frequently to visualize tubular structures in turbulence. In the same manner, we visualize spiral magnetic field vector topology by the use of the three invariants of the magnetic field gradient tensor.

In Figure 9, J (green) and the Q-value of the magnetic field (gray) are drawn in the same simulation as Figure 5b. We find that the two kinds of isosurfaces overlap with each other in the encircled regions. While J (green) tends to be sheet-like, fine structures of the Q-values (gray), which are supposed to arise from the whistler waves, add filamentary variations to the current sheets.

We recall that the $Q-R$ plots of the magnetic and the velocity field vectors in MHD turbulence have been already studied by Sahoo et al. [118] In Figure 10, contours of the joint PDFs of the $Q-R$ values of MHD turbulence with various magnetic Prandtl numbers with or without external forcing, presented in Ref. [118] as Figure 31, are reproduced with permission by the authors and the Institute of Physics. Recently, Yadav et al. have carried out the $Q-R$ analysis of both MHD an Hall MHD turbulence [142]. See Ref. [142] for a comparison of the $Q-R$ plots between MHD and Hall MHD simulations.

4.4. Additional Note on Coherent Structures

We add some notes on magnetic reconnection and scientific visualization and on the relationship between intermittency and phenomenology briefly here.

Magnetic reconnection is a representative phenomenon of the dynamic plasma phenomenon. This phenomenon influences turbulence through changing the topology of magnetic field lines. It is considered that the Hall term enhances the magnetic reconnection by thinning a current sheet, although the Hall term itself is not diffusive. In fact, Stawarz and Pouquet have reported that current sheets are finer and thinner in Hall MHD turbulence than in MHD turbulence [138]. There have been an enormous number of studies on this subject, and reviewing magnetic reconnection is outside the scope of this article. See Refs. [44,45,143,144,145,146,147,148,149,150,151,152,153,154] and references therein for information on turbulent reconnection, including Hall effects.

There have been efforts to identify reconnecting current sheets [155,156] and study magnetic field line topology in the current sheets [129], for example. Identifying reconnection events in a large-scale numerical simulation contains multiple steps: identifying the magnetic reconnection point, quantifying the change of the magnetic field topology, quantifying physical impacts of the magnetic reconnection on turbulence, viewing topological change of the magnetic field, and other physical quantities or conditions that are caused by the magnetic reconnection. Since each of the steps is possible, carrying out such processes in a massively parallel computation is not an easy task, and we need to build a new methodological approach to enable such a complex analysis.

For the purpose of visualizing such a complex phenomenon, an in situ visualization approach has been proposed for scientific visualizations [157,158]. One in situ visualization technique, the Four-Dimensional Street View (4DSV) has been implemented to visualize Hall MHD simulation results [159]. In the 4DSV visualization, many omnidirectional cameras are set to find an extreme event in a simulation. Such an approach may help identify reconnection events and watch formation of coherent structures in Hall MHD simulations.

Formation of coherent or fine structures is also closely related to classic Kolmogorov’s and Iroshnikov–Kraichnan’s phenomenology as well as to intermittency study by the use of the structure functions. In a strong Kolmogorov turbulence, filamentary current structures and tubular structures are likely to be formed. On the contrary, sheet structures are more usually formed in a simulation that obeys Iroshnikov–Kraichnan phenomenology. These two kinds of phenomenology can be distinguished by studying the scaling exponent ${\zeta }_p$ of the structure function of the Elsässer variable. We expect ${\zeta }_p\sim p/3$ for the p-th order structure function of Kolmogorov’s phenomenology, while we expect ${\zeta }_p\sim p/4$ for the Iroshnikov–Kraichnan phenomenology [59,88]. Although such a study can be found in earlier works, for example in Ref. [125], a study on Hall MHD turbulence including the sub-ion scale may not yet be sufficient, especially in the context of a fine structure formation. In order to elucidate the physics of formation of coherent structures, we need further studies in relation to phenomenology.

5. Summary

We have seen formation of fine structures in homogeneous and isotropic Hall MHD turbulence simulations. Although many works report the formation of large-scale magnetic field structures (current sheets), the results reviewed in this article show that there are occasions when the Hall term enhances filamentary structures. The enhancement of the fine and filamentary structures in freely decaying Hall MHD simulations reviewed in this article is considered to be due to the almost one-way energy transfer from the ion–cyclotron mode to the whistler mode. The transfer consequently enhances the whistler mode, which appears together with an elongated spiral magnetic field orbit (and thus a filamentary current structure). This may be true for many cases in which the Hall term appears to enhance filamentary structures.

We note that the enhancement of fine or filamentary structures and enhancement of large-scale structures do not necessarily contradict each other. In forced turbulence, there can be an enhancement of the Alfvén mode by an external force and the consequent formation of current sheets, while the energy transfer keeps enhancing the whistler modes at the sub-ion scale. Excitation of small-scale components in the magnetic field can easily mask the presence of the large-scale structures by fine current filaments or fine threads. We need further study to clarify how far the Hall effect can reach the large-scale side of the wave-number space through nonlinear energy transfer and how the Hall term can modify the large-scale structure by nonlinear interaction.