J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Hall eﬀects on local structures in decaying MHD turbulence
Hideaki MIURA and Dan. HORI1)
Department of Simulation Science, National Institute for Fusion Science, Gifu 509-5292, Japan1)
Japan Aerospace Exploration Agency, Sagamihara, Kanagawa 229-8510, Japan(Received: 27 August 2008 / Accepted: 18 November 2008)

Influences of the Hall eﬀects on local structures in magnetohydrodynamic turbulence are studied throughthe direct numerical simulations of decaying isotropic turbulence. Direct comparisons of the numerical results
between the incompressible, single-fluid- and the Hall-magnetohydrodynamic equations reveal that the highwavenumber region of the magnetic energy spectrum is considerably modified by the Hall eﬀects. It is also
shown that the structures which are composed of low-wavenumber velocity components are modified by the Halleﬀect from the sheet to tubular structures. It suggests that the basic plasma motions in MHD turbulence can be
modified from shearing to swirling motions by the Hall eﬀect.
Keywords: isotropic turbulence, small scale structures, Hall eﬀects

1. Introduction

numerical simulations (DNSs) of the Hall MHD equations
to study the turbulent spectra, the numerical resolution arelimited and the scaling property remains as a conjecture.
The purposes of this article are twofolds. The firstone is to obtain further insights on the scaling propertiesof the Hall MHD turbulence through the DNSs. The otherone is to clarify eﬀects of the Hall term on local structures.
For these purposes we carry out DNSs of both the singlefluid MHD and Hall MHD decaying isotropic turbulence.
Since the isotropic turbulence is an idealized system inthe limit of small-scales, the DNSs are essentially dissipative. The equations are discretized by means of the pseudospectral method with 2/3-truncation de-aliasing techniqueunder the triple-periodic boundary condition and marched
into the time direction by the use of the Runge-Kutta-Gillscheme. The initial condition is given by the energy spectrum proportional to k2 exp(−(k/k0 )2 ) (here k0 = 2) andrandom phases both for the velocity and magnetic fields.
The number of grid points is N 3 = 5123 . The parameters
are set as � = 0.05, ν = μ = 2 × 10−3 , aiming to put the ion
skin depth in between the energy injection scale k0 and thedissipation scale.

The single-fluid magnetohydrodynamic (MHD) equations have been applied to study many phenomena, including instabilities and/or turbulence, since many plasma phenomena are well described by the MHD equations. However, some of the phenomena can be out of the scope of theequations, and roles of two-fluid eﬀects have attracted attention in some research areas such as fusion plasmas [1,2]and astrophysical plasmas [3, 4].
One of the simplest fluid models of plasma motions
with two-fluid eﬀect is the incompressible Hall MHDequations, which can be described as∂u∂t∂B∂t=−(u · ∇)u − ∇p + j × B + ν∇2 u,

(1)

=

�
�
∇ × (u − � j) × B + η∇2 B,(2)where B is the magnetic field (normalized by a representative value B0 ), j = ∇ × B is the current (normalized by
B0 /L0 ; L0 is the characteristic length), u is the velocity√(normalized by the Alfvén speed VA = B0 / μ0 ni Mi ; μ0 is
the permeability of vacuum, Mi is the ion mass and ni is
the ion number density, which is assumed to be constantfor simplicity), ν is the viscosity and η is the resistivity(normalized by VA L0 ), and p is the pressure (normalizedby B20 /μ0 ). The velocity field u satisfies the incompressible continuity equation ∇ · u = 0 and the pressure p is
given as a solution of the Poisson equation which comesfrom the divergence of eq.(1).
� The Hall parameter is givenas � = li /L0 where li = Mi /μ0 ni e2 (e is an elementarycharge) is the ion skin depth.
Properties of the Hall MHD equations and/or Hall
MHD turbulence has been studied numerically as well as
theoretically. [2, 4–7] In our previous work [7], the authors
have proposed a possible new scaling of the magnetic energy spectra in the Hall MHD turbulence, in which themagnetic energy spectrum is scaled by k−7/3 in the highwavenumber region. Although we have carried out direct

2. Numerical Simulations� �In Fig.1(a), the mean kinetic energy E K = u2 /2 and� �the mean magnetic energy E M = B2 /2 are plotted as afunction of the time for the MHD and the Hall MHD simulations, respectively, where �·� denotes the volume average.
The thick lines represent the total energy ET = E K + E M .
While the the total energy ET decays monotonically both in
the MHD and the Hall MHD runs, the E K (E M ) increases(decreases) at the initial phase of the simulations, showing that there exists an energy exchange between E K andE M . The increase of E K and decrease of E M are clearer inthe Hall MHD simulations than in the MHD simulations,suggesting the enhancement of the energy exchange by� the�Hall eﬀect. The time evolutions of the enstrophy Q = ω2� �where ω = ∇×v is the vorticity and the current J = j2 are

author’s e-mail: miura.hideaki@nifs.ac.jp73©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

shown in Fig.1(b). The thick lines denote Q + J. Since weset ν = η in our DNS, the total dissipation rate �T = νQ+ηJis proportional to Q + J. Here we compare the two kinds ofsimulations at the time when �T is the same to each other,that is t = 1 for the MHD simulation and t = 1.04 for theHall MHD simulation.

(a)Spectra(magnitude is arbitrarily normalized)

H. Miura and D. Hori, Hall Effects on Local Structures in Decaying MHD Turbulence

010-1

10-2

1010-3
10-4

5/3

10-5

EK+EM, EK, EM

54-6

10

MHD, EK+EMHall MHD, EK+EMMHD, EKHall MHD, EKMHD, EM
Hall MHD, EM

kEM(k)

7/3EM(k)k1

10100k(b)0103-1

102-2

10-3

101-410000.5

11.522.531200
1000-610kEK(k)5/3kEM(k)7/3EM(k)k-7

10MHD, Q+JHall MHD, Q+JMHD, Q
Hall MHD, QMHD, JHall MHD, J

1400

5/3-510t(b) 1600

Q+JEK(k)5/3-7

10Spectra(magnitude is arbitrarily normalized)(a) 6k110100k(c)3MHD, T(k)Hall MHD, T(k)

2.5
2800

1.5Transfer

600
400
200

10.5
0-0.5

000.5

11.5t22.5

3-1-1.5-2Fig. 1 (a)Time evolutions of the kinetic energy ET , magnetic energy E M and the total energy ET . (b)Time evolutions ofthe enstrophy Q, current J and sum of them Q + J.

020406080100120140160kFig. 2 Kinetic and magnetic energy spectra of (a)MHD and(b)Hall MHD simulations. The spectra are compensatedas k5/3 EK (k), k5/3 E M (k) and k7/3 E M (k). Magnitudes of thespectra are normalized by appropriate values so that they
can be distinguished from each other clearly. (c)The energy transfer functions of the MHD and the Hall MHD
simulations.

In Fig.2(a), the kinetic and magnetic energy spectra,E K (k) and E M (k) respectively, of the MHD simulation att = 1 are shown. Here k is the wavenumber. Fig.2(b) isthe Hall MHD counterparts at at t = 1.04. Hereafter weomit the time stamps of the simulations, assuming t = 1
for the MHD simulation and 1.04 for the Hall MHD simulation. The spectra E K (k) and E M (k) are compensatedby some typical powers of the wavenumber as k5/3 E K (k)(white boxes), k5/3 E M (k) (the black boxes), and k7/3 E M (k)(the white circles). Although some scalings other than the
k5/3 and k7/5 have been proposed [8], here we concentrateon studying a possible scaling we have proposed [7]. The
kinetic energy spectrum E K (k) appears to have a scaling by
k−5/3 , as has been pointed out in many works (see Ref. [8]and references therein). On the other hand, E M (k) may appear two scalings, as has been proposed by the authors, [7]although the scaling regimes are too short and the scalings
can not be conclusive. By a comprison of the compensatedmagnetic energy spectra in Fig.2(a) and (b), the tail of theE M (k) in the Hall MHD appears less steep than that in theMHD turbulence.
The change of the energy spectra at high wavenumbers of the magnetic field by the Hall eﬀect may be char-acterized by the energy transfer function T (k) [8, 9] which
describes the conservative energy transfer among various
wavenumber k by the nonlinear terms. In Fig.2(c), the energy transfer function for the total energy E K (k) + E M (k) ofboth the MHD and the Hall MHD simulations are shown.
In the MHD case, T (k) is positive at small k and negative at large k, suggesting that the backward energy transfer from larger to lower wavenumbers are dominant at thistime of the simulation. On the other hand, in the Hall
MHD case T (k) is negative at 10 ≤ k ≤ 50 and positiveat both k < 10 and k ≥ 50. The transfer T (k) is large andpositive especially at the large wavenumber regions, making the resolution of this simulation being marginal. (Recall that the Fourier coeﬃcients at the high wavenumber
regime is damped by the viscous and the Ohmic dissipationterms, which are not plotted here.) The diﬀerence of T (k)74H. Miura and D. Hori, Hall Effects on Local Structures in Decaying MHD Turbulence

between the two simulations clearly shows that the Halleﬀects change the energy transfer. In the Hall MHD turbulence, the scales comparable to 1/� becomes the sourceof the energy which is distributed to the two wavenumber
regimes k < 10 and k > 50. The positive T (k) at k < 10in the Hall MHD turbulence is not reported in our earlywork [7], probably because the Hall parameter � = 0.1 inthe reference is too large to allow the positive T (k) at thelargest scales.

(a)3. Local structures
In order to see the influences of the modification of
the energy transfer by the Hall eﬀect in Fig.2(c), we observe spatial structures in the real (Cartesian-coordinated)
space. Though the authors have seen the structures simply
by drawing the isosurfaces of the enstrophy density andcurrent density [7], it is diﬃcult to clarify the Hall eﬀect
among many pieces of isosurfaces. In this article, aiming
at clarifying the large and scale structures, we introduce
the low-pass (< kcut ) and high-pass (> kcut ) filters. By
the use of filters a field quantity can be separate into thelower-wavenumber and higher-wavenumber components.
The cut-oﬀ wavenumber is set kcut = 50, where T (k) of theHall MHD turbulence changes its sign.
In Fig.3, isosurfaces of the lower-wavenumber vorticity components (ω<kcut )2 of (a)the MHD and (b)the Hall
MHD are drawn. The threshold of isosurface in the figures
is twice of the deviation above the mean value of ω2 . Acomparison between Fig.3(a) and (b) reveals that the vorticity field with the wavenumber k < kcut is more tubularin the Hall MHD turbulence than in the MHD turbulence.
The change of the structures are not easily found without
introducing the filters, since the fine structures of the vorticity are dominant when we plot ω2 . The change may berelated to the T (k) being positive for k < 10 and negative
for k > 10. The dominance of tubular structures in Fig.3(b)suggests that swirling motions are superior to the shearing
motions in the large scales due to the Hall eﬀects, like thevortical structures in the Navier-Stokes turbulence [10,11],although much more detailed analysis is require for furtherstudies.
Fig.4(a) and (b) are isosurfaces of the higher components (ω>kcut )2 of the MHD and the Hall MHD, respectively. The threshold is the same as the value for Fig.3.
We find that the the sheet-like structures of the vorticity
are torn into pieces in the Hall MHD turbulence.
In Figs.5, isosurfaces of the higher components of thecurrent of (a)the MHD and (b)the Hall MHD simulations
are drawn. (The lower components are omitted because we
do not observe a significant diﬀerence between the MHD
and the Hall MHD turbulence.) In comparison to Fig.5(a),the isosurfaces in Fig.5(b) are apparently torn into pieces,forming very fine structures. In the case of the current
density field, the Fourier amplitudes of the Hall MHD turbulence are apparently larger than those of the MHD tur-(b)Fig. 3 Isosurfaces of (ω<kcut )2 of (a)the MHD simulation at t = 1
and (b)the Hall MHD simulation at t = 1.04. The isosurfaces are drawn over a region with 1703 grid points out of
the entire 5123 region.

bulence, more than one degree of magnitude. (The amplitudes of the vorticity Fourier coeﬃcients of the MHD
simulation are slightly larger than those in the Hall MHD
turbulence at k > 80, although the diﬀerence of the amplitudes are very small.) The fine structures in Fig.4(b)may be attributed to the j × B-force in eq.(1), that is, thefine structures in the vorticity (and therefore in the velocity) fields are excited in association with the emergence ofthe fine structures in the magnetic field. It appears that theHall term influences the high wavenumber region of themagnetic field and the excited small scales of the magnetic
field then influences the small scales of the velocity field.

4. Concluding remarks
We have studied both the scaling properties and thelocal structures of the Hall MHD turbulence through directcomparisons to the single-fluid MHD turbulence with thesame initial condition and the same dissipative parameters.
The DNS results appear being consistent with the scaling
proposed by Hori and Miura [7]. For the observations ofthe local structures, the introduction of the low-pass andhigh-pass filters work satisfactory. The observations of the75

H. Miura and D. Hori, Hall Effects on Local Structures in Decaying MHD Turbulence

(a)

(a)

(b)

(b)

Fig. 4 Isosurfaces of (ω>kcut )2 of (a)the MHD simulation at t = 1and (b)the Hall MHD simulation at t = 1.04.

Fig. 5 Isosurfaces of ( j>kcut )2 of (a)the MHD simulation at t = 1and (b)the Hall MHD simulation at t = 1.04.

vorticity field suggest that the vortex structures are significantly modified by the Hall eﬀect from sheets to tubes.
A tubular vortex is often formed through rolling-up of avortex sheet due to the Kelvin-Helmholtz (KH) instability.
However, it does not directly explain the diﬀerence of thevortex structures in between the MHD and the Hall MHD
turbulence unless the Hall eﬀects on the KH instability isshown. Concerning an eﬀect which exists in the Hall MHD
turbulence but does not in the MHD turbulence, we would
like to comment on the Whistler waves. In MHD turbulence, propagations and collisions of the Alfvén waves
along the magnetic field lines bring about the modificationsof the nonlinear interactions from the Navier-Stokes turbulence and thefore of the spatial structurs, too. (See Ref. [8],for example.) In the Hall MHD turbulence, high-wavenumber Whistler waves propagate along the magnetic fieldlines faster than the Alfvén waves and causes many more
collisions. We speculate that the wave collisions may berelated to the structure modifications, since many of thesestructures are observed being tangential to the magnetic
field lines (figures are omitted due to the page limitation).
It remains as a simple speculation here and will be studied more extensively with a detailed analysis on each oftubular vortices as are carried out in Ref. [10, 11].

This research was partially supported by ”Application of Fusion Theory to Space Plasmas” under theproject of Formation of International Network for Scientific Collaborations at National Institute for Fusion Science(NIFS)/National Institute for Natural Sciences, Japan. The
numerical work was performed on the NEC SX-7 “PlasmaSimulator” of NIFS.
[1] K. Itoh, K. Halides, and et. al., Phys. Plasmas 12, 062303(2005).
[2] S. M. Mahajan and Z. Yoshida, Phys. Plasmas 7, 635 (2000).
[3] S. Balbus and C. Terquem, Astrophys. J. 552, 235 (2001).
[4] J. Shiraishi, S. Ohsaki, and et. al., 12, 092901 (2005).
[5] P. D. Mininni, D. O. Gómez, and S. M. Mahajan, Astrophys.
J. 619, 1019 (2005).
[6] P. Mininni, A. Alexakis, and A. Pouquet, J. Plasma Phys. 73,377 (2007).
[7] D. Hori and H. Miura, to appear in Plasma Fusion Res.
(2008).
[8] D. Biskamp, Magnetohydrodynamic Turbulence, Cambridge
University Press, 2003.
[9] W.D. McComb, The Physics of Fluid Turbulence, Oxford
University Press, 1990.
[10] H. Miura and S. Kida, J. Phys. Soc. Japan, 1331 66 (1997)[11] H. Miura, J. Turbulence 5 (2004) N10.

76