Proceedings of ITC/ISHW2007

Spectrum properties of Hall MHD turbulence
Dan HORI1) , Hideaki MIURA1)
1)

National Institute for Fusion Science, Toki 509-5292, Japan(Received 15 October 2007;)

Aiming to understand roles of the two ﬂuid eﬀect on plasma dynamics, direct numerical simulations of Hall
MHD turbulence are carried out. A comparison between the numerical results of the Hall MHD turbulence tothose of one-ﬂuid MHD turbulence reveal that Hall term modiﬁes small scale properties of MHD turbulence.
Keywords: MHD turbulence, Two ﬂuid eﬀect, Hall term, Energy spectrum, Vortex structure
DOI: 10.1585/pfr.1.001

1Introduction

In recent years, roles of two ﬂuid eﬀects have attracted
attention in fusion plasmas [1, 2], as well as astrophysical plasmas [3, 4]. A two-ﬂuid model contains various
physical eﬀects [5], and therefore exhibits rich phenomena. It may be sometimes expected that the two-ﬂuid effects contribute to transfer the energy from larger scales
to lower scales and enhance the dissipation. However, sofar as the authors recognize, the two-ﬂuid eﬀects on theenergy transfer among scales has not been studied suﬃciently. One subject in which the energy transfer among
the scales are well formulated and studied should be theisotropic turbulence of a neutral, incompressible ﬂuid, as
are seen in enormous number of works after the famous
Kolmogorov’s theory (see Ref. [6] and references therein)
and MHD turbulence [7]. In studies of turbulence, the energy transfer among the scales are studied in the context of
the Fourier energy spectrum and the energy transfer functions. Even in the long history of studies of isotropic turbulence, there remain some aspects of the energy transfer,
such as the localness/non-localness, remains imperfect understandings. Compared to these preceding subjects, rolesof the two-ﬂuid eﬀect in turbulence (whether it is fully developed or not) are not studied very well.
For studies of two-ﬂuid eﬀects among scales, a simpler model is more preferable so that the eﬀects are distinguished easily from the other eﬀects. Hall magnetohydrodynamics (Hall MHD) provides a minimal model which
expresses two-ﬂuid eﬀects:∂u∂t∂B∂tthe 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 scale parameter ε = li /L0 is called Hallp
Mi /µ0 ni e2 (e is an elementaryparameter, where li =charge) is the ion skin depth.
A excellent tool to study the energy spectrum is theshell model proposed by Yamada and Ohkitani [8] for aneutral ﬂuid turbulence. The shell model approach mimics
the dynamics of the basic equations as sparse and artiﬁcial Fourier mode couplings. Although detailed dynamics
of the original equations are abandoned, it achieves very
high Reynolds number, which direct numerical simulation(DNS) can not achieve even by the most powerful supercomputer of the recent years. Recently, one of the authors(D.H.) have developed a new shell model for Hall MHD,and performed numerical simulations both the Hall MHD
case (ε , 0) and the single ﬂuid MHD case (ε = 0). [9, 10]
The shell model computations predict a modiﬁcation of theenergy spectrum by the Hall eﬀects, suggesting the energy
transfer from the large scales to small scales.
Based on these understandings, we conduct numerical studies of Hall MHD turbulence by means of the DNSof both MHD and Hall MHD equations in order to study
the eﬀects of Hall term on the energy transfer in isotropic
turbulence. We ﬁrst review the computational results ofthe shell model for MHD and Hall MHD turbulence very
brieﬂy. Then, some basic views in the two kinds of turbulence are studied. Summary appears in the ﬁnal section.

=−(u · ∇)u − ∇p + j × B + µ∇2 u,(1)2=∇ × (u −  j) × B + η∇2 B,(2)Here we review some numerical results reported inRefs. [9, 10] so that the background of our computational
work is well understood. Refer to the references for thedetails of the model and the scaling of the energy spectra
obtained by the computations.
The energy spectra for (a) the single ﬂuid MHD (hereafter we simply refer to MHD) and (b) Hall MHD (forε = 10−2 ) are shown in Fig. 1. As is observed in Fig. 1,where B is the magnetic ﬁeld (normalized by a representative value B0 , j (= ∇ × B) is the current (normalized byB0 /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 isauthor’s e-mail: hori.dan@nifs.ac.jp

Shell Model Computation Review

Proceedings of ITC/ISHW2007
10-2
10-6
Energy spectra

lent ﬁeld is often characterized by the vorticity rather thanthe velocity ﬁeld. It is partially because the vorticity ﬁeld
ω is invariant to the Galilean transform and partially because the vorticity ﬁeld represents small scale structures
much more than the velocity ﬁeld . For the same reason,
the current density j is considered to be more suitable forstudying the small scale properties than the magnetic ﬁeld.

Velocity field
Magnetic field
Total

10-4
10-8
10-10
10-12
10-14
10-16
10-18
10-20
10-22
10-1

(a)
100

101

100

103

104

105

106

105

106

Velocity field
Magnetic field
Total

10-2
10102

-4

Energy spectra

10-6
10-8
10-10
10-12
10-14-161010-18
10-20
10-22
10-1(b)100

101102103104
Wave number kFig. 1 Energy spectra obtained from the simulation of shellmodel. The MHD case is shown in the upper panel (a)and The Hall MHD case is done in the lower panel (b).

the scaling exponents of the magnetic ﬁeld is signiﬁcantly
modiﬁed in very high wavenumbers. In the case of theMHD, the magnetic ﬁeld spectrum is damped in the dissipation region in the same form as the velocity energy spectrum. However, in the Hall MHD spectrum, the magnetic
ﬁeld indicates −7/3 power-law in the small scale region.
These contrast may reﬂect the contribution of the Hall term
to the energy transfer in the wave number space.

3Direct numerical simulation of MHD
and Hall MHD turbulence

DNSs of the decaying MHD and Hall MHD turbulence
are carried out for the (2π)3 triple-periodic geometry. Spatial derivatives are approximated by the pseudo-spectrum
method and the variables are marched into the time direction by the Runge-Kutta-Gill scheme. The aliasing error isremoved by the 2/3-truncation method. The number of thegrid points are N 3 = 2563 , so that the maximum wavenumber available in this simulation work is kmax = 84. In thecase of Hall MHD case, Hall parameter is ε = 1 × 10−1 .
The dissipative coeﬃcients are ν = η = 1 × 10−2 . Boththe MHD and Hall MHD turbulence simulations start fromthe same initial conditions in which the velocity and magnetic ﬁelds have the energy spectrum roughly proportional
to k2 exp(−(k/k0 )2 ) (here k0 = 2) and random phases. No
external force is imposed so that the total energy of thevelocity and the magnetic ﬁelds decays monotonically totime.
In DNS studies of fully developed turbulence, turbu-Fig. 2 Time series of the enstrophy density and the current density averaged over the computational volume is shown in
the upper panel (a). Taylor scale Reynolds number Reλ isshown in the middle panel (b), and Taylor scale magnetic
Reynolds number is done in the lower panel (c).

In EFig. 2(a), time series ofD the Eenstrophy density|ω|2 /2 and the current density | j|2 /2 are shown for thetwo kinds of turbulence, where h·i denotes the volume average. A comparison on the enstrophy evolution shows thatthe enstrophy density is larger in the Hall MHD turbulence
than in the MHD turbulence with the same ν and η. We
also observe that the total current is larger in the MHD turbulence than in the Hall MHD turbulence. Another goodindex of fully developed turbulence is the Reynolds number Reλ based on the Taylor micro-scale [6]. In Figs. 2(b)and (c), we see the time evolution of Reλ and its counterpart for the magnetic ﬁeld, Rebλ for the two kinds of turbulence. We ﬁnd in both Fig. 2(b) and (c) that the Hall MHD
turbulence have larger micro-scale Reynolds numbers ReλDProceedings of ITC/ISHW2007
105

Velocity field
Magnetic field
Total

100
10-5(a)Energy spectra

10-10
10-15
10-20
10-25
10-30-35

10-40

10(a)

10-45
10-50
110100Velocity field
Magnetic field
Total

-1

Energy spectra

1010-210-3(b)10-4-510(b)

10-6
1

10100

Velocity field
Magnetic field
Total-1

Energy spectra

1010-210-3
10-4-510(c)

10-6
110
Wave number kFig. 3 Energy spectra in DNS of MHD and Hall MHD turbulence. The initial state is shown in the upper panel (a), theenergy spectra of the MHD is done in the middle panel(b), and the energy spectra of the Hall MHD is done in
the lower panel (c).

and Rebλ than the MHD turbulence, although the diﬀerence
between the two kinds of turbulence is smaller in (c) thanin (b). These observations suggest that the small scales inthe velocity (or the vorticity) ﬁeld is more excited in theHall MHD turbulence than in the MHD turbulence.
In Fig. 3(a) the energy spectra of the velocity and themagnetic ﬁeld vectors at the initial time are shown. Since
the energy is conserved as the total form u2 + B2 rather
than the kinetic and magnetic energies separately in theideal limit, we need to study not only the individual spectra u2k and b2k but also the total spectrum u2k + b2k as well.
Figs. 3(b) and (c) are the plots of the energy spectra of theMHD and Hall MHD turbulence, respectively. By comparing Figs. 3(b) and (c), clear diﬀerences between the twokinds of turbulence are seen. The magnetic energy spectrum, denoted by the ﬁlled boxes, at the middle (near-peak)wavenumber region is larger in Fig. 3(b) than in Fig. 3(c),while it is opposite in the high wavenumber region. It explains the observations in Fig. 2 that the total current islarger in the MHD turbulence because of the dominance inFig. 4 Isosurfaces of the enstrophy density in (a) MHD and(b) Hall MHD turbulence.

the middle wavenumber region. In the middle wavenumber region (say, 10 < k < 30), b2k > u2k in the MHD turbulence while b2k < u2k in the Hall MHD turbulence. In thehighest wavenumber region k > 40, the diﬀerence between
the MHD and Hall MHD turbulence becomes the clearest.
The decay of b2k obeys to rather a power-law than the exponential decay. It is similar to the earlier numerical results
shown in Fig. 1 and considered to be the direct inﬂuenceof the Hall term. The high wavenumber region of the velocity energy spectrum in the Hall MHD turbulence decays
more rapidly than that in the MHD turbulence, as if compensating the slow decay of the magnetic spectrum b2k . The
total spectrum u2k + b2k of the Hall MHD turbulence has also
weaker amplitudes in the 10 ≤ k ≤ 50 wavenumber region
than that of the MHD turbulence, but becomes larger in thehigher region k > 50.
In Figs. 4, isosurfaces of the enstrophy density of theMHD (a) and the Hall MHD turbulence (b) are shown. InFigs. 5, isosurfaces of the current density of the MHD (a)and the Hall MHD turbulence (b) are shown. The thresholds of the isosurfaces of the enstrophy density (currentdensity) are the same between the MHD and the Hall MHD
turbulence in Figs. 4 (Figs. 5), the volumes covered by theisosurfaces are clearly diﬀerent between the two kinds ofProceedings of ITC/ISHW2007

(a)

(b)

Fig. 5 Isosurfaces of the current density in (a) MHD and (b) Hall
MHD turbulence.

turbulence.

4Summary

We performed the direct numerical simulation of theboth Hall MHD and MHD turbulence to study the role ofthe two ﬂuid eﬀect. Various data show clear diﬀerence between the MHD and the Hall MHD turbulence. Time seriesof the enstrophy, the current, and the Taylor scale Reynolds
numbers show that small scale motions of the Hall MHD
turbulence is more excited than those of the MHD turbulence. In the small scale, the energy spectra of the magnetic ﬁeld in the Hall MHD turbulence is observed to be
power-law decay, while that of the MHD turbulence is tobe exponential decay.
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