J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Contributions of small scales to statistics of Hall MHD
turbulence
Hideaki MIURA
Department of Simulation Science, National Institute for Fusion Science, Gifu 509-5292, Japan(Received: 18 November 2009 / Accepted: 5 February 2010)

A direct numerical simulation of decaying, homogeneous and isotropic turbulence of the incompressible Hall magnetohydrodynamic equations is carried out to clarify its statistical natures. Contributionsof small scales to the statistics are examined. It is shown that the probability density function (PDF)of the enstrophy density is well characterized by the Gaussian distribution when the short wave number
coeﬃcients are removed, even though the vortex ﬁeld shows intermittent structures. It is also shownthat the local vortex structures are aligned to the magnetic ﬁeld lines especially when the turbulent ﬁeld
is under developing and small scales are going to be excited. The alignment is lost in the relaxation
process, suggesting the small scale current density ﬁeld is less aﬀected by the dissipations than thevorticity ﬁeld.
Keywords: Hall MHD turbulence, statistics, vortex structures

1. Introduction

gate along the magnetic ﬁeld lines, and also by slower
dynamics in the direction perpendicular to the magnetic ﬁeld lines [5–7]. In the Hall MHD turbulence,
the interaction time scales and frequency of interactions are altered because high wave number WhistlerAlfvén waves propagate much faster than the Alfvénwaves. Then the power spectra and local structuresin the Hall MHD turbulence are modiﬁed from thoseof non-Hall MHD turbulence.
In the previous article [3], we have reported
some spectral properties of the Hall MHD turbulence through the direct comparison with the nonHall MHD turbulence of the same dissipative coeﬃcients and the same initial condition. In this article we report results of further numerical analyses on
the Hall MHD turbulence. Our interests in this article are vortex structures and their statistical properties with/without small scale Fourier coeﬃcients, because such analyses on scale separations can provide
basic information to model the equations in terms ofso called eddy viscosity and/or eddy currents, which
we often require for various numerical studies. This
paper is organized as follows. In Sec.2, a direct numerical simulation of the Hall MHD equations is carriedout. Basic information of the simulation and some
aspects of the numerical simulations are shown. In
Sec.3, statistical properties of the turbulent ﬁeld arereported. Summary is presented in Sec.4.

Plasma turbulence is commonly observed in various phenomena such as solar coronas, solar winds andfusion experiments. Since macroscopic behaviors ofplasma turbulence are well described by the one-ﬂuid
magnetohydrodynamics (MHD) equations, it is a suitable model to initiate studies of plasma turbulence.
MHD turbulence is often characterized by a powerlaw of energy spectra and/or intermittent structures,as neural ﬂuid turbulence is. While theories of MHD
turbulence by Iroshnikov and Kraichnan predict thek −3/2 power law of the energy spectrum, observations of solar winds appear to show rather the classical k −5/3 Kolmogorov spectrum. (See Ref. [1] andreferences there in.) The scaling property of the turbulent energy spectra has been extensively studied bymeans of direct numerical simulation (DNS), too. Although the numerical works provide us invaluable information on turbulence, due to a limited resolution
achievable in numerical simulations, the energy spectrum obtained by a DNS of MHD turbulence should be
essentially scaled at the near-dissipation range, where
one-ﬂuid MHD ordering may not necessarily be appropriate. Thus in order to study properties of turbulencein such small scales, a more precise model is required.
An alternative and the simplest framework of theplasma turbulence might be the Hall MHD equations,in which the Hall term is added to the induction equation. The Hall term is quadratic to the magnetic
ﬁeld and varies some aspects of (non-Hall) MHD turbulence. For example, we expect emergence of the
Whistler-Alfvén waves, in stead of the Alfvén waves
in non-Hall MHD turbulence [4]. Nonlinear interactions in the non-Hall MHD turbulence is often understood by the collisions of Alfvén waves which propa-

2. Direct numerical simulation
A DNS of the decaying, homogeneous andisotropic turbulence of the incompressible Hall MHD
equations is carried out in this section. The incompressible Hall MHD equations are described as∂Vi∂tauthor’s e-mail: miura.hideaki@nifs.ac.jp

535

= −

∂
(Vi Vj − Bi Bj )∂xj
©2010 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchH. Miura, Contributions of Small Scales to Statistics of Hall MHD Turbulence

∂Vj
∂xj
∂Bi
∂t
ji

∂
∂xi

�
�
1
∂ 2 Vip + B i Bi + ν, (1)2∂xj ∂xj

= 0,
∂
[
∂xj
�kmn (Vm − �jm ) Bn − ηjk ] ,∂Bk= �ijk
,
∂xj

(a)

6

EK
EM
EK+EM

5

(2)

4
EK,EM

−

= �ijk

3
2

(3)

1

(4)

0

where Vi , Bi , ji are the i-th component of the velocity
ﬁeld vector V , the magnetic ﬁeld vector B, and thecurrent density vector j, respectively. We also introduce the vorticity vector ﬁeld ω, of which i-th component is given as ωi = �ijk ∂Vk /∂xj . The symbols �ijk ,ν, η are the Eddington’s anti-symmetric tensor, theviscosity and the resistivity.
The equations (1)-(4) are solved numerically by
the pseudo-spectral method and the Runge-KuttaGill scheme under the triple periodic boundary con3
dition over (2π) box. The number of grid points is
N 3 = 5123 . The aliasing errors in the pseudo-spectral
computations are removed by the 2/3-truncation. Because of the 2/3-truncation, the maximum wave number in this simulation is limited to kmax = 170. The
Hall parameter, the viscosity and the resistivity are
set as � = 0.05, ν = η = 2 × 10−3 , respectively. The
initial conditions
� by the energy spectrum
� are given
Eα (k) ∝ k 4 exp −2k 2 /k02 (k0 = 2 is given here) andthe random phases for both the velocity vector ﬁeld
and the magnetic vector ﬁeld.
In this section we see some averaged quantities
and energy spectra. (A few of them have been shownin the previous work [3]. We see again for the sakeof understanding the nature of turbulence better.)
In Fig.1(a), time evolutions of the kinetic energy
EK = �Vi Vi � /2 (solid line) and the magnetic energy
EM = �Bi Bi � /2 (dashed line) are shown. The symbol �·� denotes the volume average. In Fig.1(a), thetotal energy ET = EK + EM (dotted line) decays
monotonically, while each of EK and EM does not
behave monotonically because EK and EM exchange
their energies each other. In Fig.1(b), time evolutionsof the enstrophy Q = �ωi ωi � /2 and the total current
J = �ji ji � /2 are shown. Both Q and J are peaked
at t = T0 � 0.5. Hereafter we make use of the timestamp T0 , and study the turbulent ﬁeld at t = T0 /2,
T0 , 2T0 , 3T0 and 4T0 in this article. In Fig.1(c), theTaylor’s micro-scale Reynolds number RλV for the velocity ﬁeld and its counter part to the magnetic ﬁeld
RλM� are shown. At t = T0 , RλV � 89 and RλM� � 93. A
diﬀerence between the two Reynolds numbers is clear
after t = T0 . The magnetic Reynolds number RλM� isalmost constant for t ≥ T0 , while RλV keeps decaying
slowly.
In Figs.2(a) and (b), the kinetic energy spectrum

0

0.5

1

1.5

2

2.5

3

t

(b) 800

Q
J

700
600

Q,J

500
400
300
T0/2

200
100

T0

0
0

2T0

0.5

1

3T0
1.5

4T0
2

2.5

3

t

(c)

500

V

ReMλ
Re λ

450
400

V

Re λ, Re

M

350
300
250
200
150
100
50
0

0

0.5

1

1.5

2

2.5

3

t

Fig. 1 Time evolutions of (a)EK , EM , EK = EK + EM ,(b)Q and J, and (c)RλV , RλM

EK (k) and the magnetic energy spectrum EM (k) are
shown at t = T0 /2, T0 , 2T0 , 3T0 and 4T0 , respectively. The former appears to be scaled by k −5/3 (thesolid line), while the latter appears to be scaled byk −7/3 rather than by k −5/3 . (Refer to Ref. [2, 3] on
the examination of the scaling indices by the compensated energy spectra.) The scaling properties of thetwo spectra look almost unchanged after t = T0 . In
Fig.2(c), the transfer functions of the kinetic energy
and the magnetic energy, TK and TM respectively, areplotted at t = T0 . The two transfer functions are deﬁned as
TK (k)=�[k]

�i (k)A536�

�i (k),
V�i (k)∗ A

= F −

∂
(Vi Vj − Bi Bj )
∂xj

(5)

H. Miura, Contributions of Small Scales to Statistics of Hall MHD Turbulence(a) 100

function TK (k) is negative at k < 20 and positive atk > 20. The proﬁle of TK (k), negative at low wave
numbers and positive in large wave numbers, is quite
typical when the forward energy transfer is dominant.
On the other hand, the magnetic energy transfer function TM (k) changes its sign twice. It is positive at
the two regions, k < 3 and k > 20, and negativeat 3 ≤ k ≤ 20. There should be the inverse energy
transfer toward the ﬁrst region k < 3. The region
3 ≤ k ≤ 20 is considered to be the source of the energy to the other two regions k < 3 and k > 20.
In Fig.3(a), isosurfaces of the enstrophy density
Ω2 = ωi ωi /2 are shown. The threshold of the isosurfaces is four times of the deviation above the meanvalue of Ω2 . In Fig.3(b), isosurfaces of the current
density I 2 = ji ji /2 are drawn by the threshold of fourtimes of the deviation above the mean value. The
isosurfaces of both Ω2 and I 2 exhibit sheet-like structures, as they do in the non-Hall MHD turbulence.
However, large-scale structures of the two quantities(especially the enstrophy density) are rather tubularin the Hall MHD turbulence. In fact, the isosurfacesof the two quantities which are drawn only with their
Fourier coeﬃcients k ≤ 32, in Fig.3(c), are either elongated ellipsoids or tubes. (The magnetic ﬁeld lines arealso drawn there.) From these observations, we consider that some basic properties of the Hall MHD turbulence are diﬀerent between large and small scales.
For a convenience, we assume k = 32 as a wave number to distinguish the large and small scales in thisarticle. Although we do not have a theory to determine the threshold wave number as k = 32 uniquely,
it is one of a typical scale by which we can ﬁnd clear
change of spatial structures in the processes of visualizations. Hereafter we study turbulent statistics by
the use of the low-pass/high-pass ﬁlters with k = 32
cut-oﬀ. For a convenience, we refer to the data withthe all Fourier coeﬃcients as the full resolution andthe data only with k ≤ 32 Fourier coeﬃcients as thelimited resolution.

-1

10

10-2
-3

EK(k)

10

10-4
10-5

k-5/3
t=T0/2t= T0
t=2 T0t=3 T0t=4 T0

-6

10

-7

10

-8

10

1

10

100
k

(b) 100
-1

10

-2

10

-3

EM(k)

10

-4

10

k-5/3
-7/3
k
t=T0/2t= T0
t=2 T0t=3 T0t=4 T0

-5

10

-6

10

-7

10

-8

10

1

10

100
k

(c)

0.02
0.01

Ti(k) (Hall MHD)

0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07

TK(k) DNS
TM(k) DNS

-0.08
-0.09

1

10

100
k

Fig. 2 Time evolutions of (a)the kinetic energy spectrum EK (k) and (b)the magnetic energy spectrum
EM (k). (c)The energy transfer functions for the
kinetic energy TK (k) and for the magnetic energy
TM (k) at t = T0 .

−

∂
∂xi

�
��
1
p + B j Bj
,
2

(6)

3. Statistics w/wo high wave number
coeﬃcientsandTM (k) =�[k]

�i (k) =C�i (k)∗ C
�i (k),B

In this section, we study of some statistics withhigh wave number Fourier coeﬃcients (the full resolution statistics) and without them (the limited resolution statistics). In Fig.4, the probability density
function (PDF) of the I 2 at t = T0 is shown bothfor the full resolution (solid line) and the limited resolution(dashed line), in with the abscissa (a)I 2 and(b)I. While the PDFs of both the full resolution andthe limited resolution are concave up in Fig.4(a), they
have straight tails in Fig.4(b). It indicates(7)

F [�ijk�∂{�kmn (Vm − �jm ) Bn } , (8)∂xj�where the symbols [k] , F [] and ∗ denote the shell
average over the wave number vector k, the Fouriertransform, and the complex conjugate, respectively.
The variables with �· are the Fourier coeﬃcients. Here
we note characteristics of the two transfer functions
TK (k) and TM (k) brieﬂy. The kinetic energy transfer

P (I 2 ) ∝537

exp (−I)(9)H. Miura, Contributions of Small Scales to Statistics of Hall MHD Turbulence(a)10-1(a)j2/2
j2/2 for k<=32

10-2
-3

10

10-4
-5

10

0

500

1000

1500

2

2000

j /2

(b)10-1

2

j /2
2
j /2 for k<=32

-2

10

(b)

-3

10

-4

10

-5

10

0

5

10

15

20

2

25

30

35

40

sqrt(j /2)

Fig. 4 The PDF of the current density I 2 = ji ji /2. The
abscissa is I 2 in (a) and I in (b).

while the PDF of the limited resolution appears��(11)
P (Ω2 ) ∝ exp −Ω2 .

(c)

It can be interesting because the PDF of the large scalevortices looks that of a simple Gaussian random ﬁeld
even though they apparently have intermittent vortex
structures in Fig.3(c). It is also interesting to ﬁnd
diﬀerences between I 2 and Ω2 , because the diﬀerence
may come from the quadratic nature of the Hall term
to the magnetic ﬁeld, which we miss in the non-Hall
MHD equations. In the non-Hall MHD equations, thecurrent density ﬁeld j and the vorticity ﬁeld w areclosely related with each other through the Elsässer
variables V ± B, while they are not in the Hall MHD
equations due to the quadratic nature.
Next, we study the alignment of the local vortex
structures and the magnetic ﬁeld lines which is reported ﬁrst in Ref. [3]. As in the reference, there is atendency that the magnetic ﬁeld lines are tangential tothe isosurfaces of the enstrophy density. The tendency
is investigated here more quantitatively by means of
the PDF of the angles between the surface orientationand the magnetic ﬁeld lines. We can characterize theorientation normal to the surfaces of Ω2 as the gradient of quantity. The PDF of the angle between themagnetic ﬁeld lines and the gradient vector,��
B · (∇Ω2 )−1(12)
θB,∇Ω2 = cos|B| |(∇Ω2 )|Fig. 3 Visualization of turbulent ﬁeld. The isosurfaces of(a)the enstorphy density, and (b)the current density for the entire computational box. (c)The isosurfaces of the enstorphy density and the current
density, and the magnetic ﬁeld lines of only k ≤ 32
Fourier coeﬃcients are drawn.

at large I 2 , whether the resolution is full or limited.
In Fig.5, the PDFs of Ω2 of the full resolution andthe limited resolution are shown with the abscissa of(a)Ω2 and (b)Ω. In contrast to the PDFs of I 2 , weﬁnd that the PDF of the full resolution is
P (Ω2 ) ∝exp (−Ω)is shown In Fig.6 for (a)the full resolution and (b)

(10)

538

H. Miura, Contributions of Small Scales to Statistics of Hall MHD Turbulence(a) 10

(a)

-1
2
ω /2
ω2/2 for k <= 32

0.5

t=0
t=T0/2
t= T0
t=2T0
t=3T0
t=4T0

0.45
0.4

10-2
PDF(θ)

0.35
0.3

10

-3

10

-4

0.15

10

-5

0.05

0.25
0.2
0.1

0

500

1000

1500

0

2000

ω2/2

0

π/4

π/2

π*3/4

θ

(b) 10

-1

10

-2

0.35

10

-3

0.25

10

-4

0.15

10

-5

0.05

(b)

2

ω /2
ω2/2 for k <= 32

0.4

t=0
t=T0/2
t= T0
t=2T0
t=3T0
t=4T0

PDF(θ)

0.3

0.2

0.1

0

5

10

15

20

25

30

35

40

0

2

sqrt(ω /2)

0

π/4

π/2

π*3/4

θ

Fig. 5 The PDF of the enstrophy density Ω2 = ωi ωi /2.
The abscissa is Ω2 in (a) and Ω in (b).

Fig. 6 The angles between the gradient of the enstrophy
density and the magnetic ﬁeld lines of (a) the full
resolution and (b) the limited resolution.

for the limited resolution. In Fig.6(a), the PDF atθ � π/2 becomes the largest at t = T0 /2, in the midtime of the turbulence evolution, and it becomes small
rapidly after that. That is, the alignment of the magnetic ﬁeld lines and the vortex structures are achieved
when the turbulence is developing. In Fig.6(b), thePDF for the limited resolution are shown. Though
the basic proﬁle of the PDF is similar between thefull resolution and the limited resolution, the peaks of
the PDFs at the angle π/2 become less sharp withoutk > 32 coeﬃcients. Our current understanding is thatthe alignment is contributed mainly by the small scale
structures, which are excited in the course of development of the turbulence, and that the alignment is lostwhen the small scales are dissipated away quickly inthe relaxation process.
When we turn our eyes to the relaxation processof t ≥ T0 , it is natural to consider about the helicity,
since it has been considered that the conservation of
the helicity can play a key role in the relaxation process in studies of non-Hall MHD turbulence. In theHall MHD equations, we have three quantities to beconserved in the ideal limit, the total energy
ET =

1
1
�Vi Vi � + �Bi Bi � ,22ﬁeld by Bi = �ijk ∂Ak /∂xj , and the hybrid helicity
1
�(Bi + �ωi ) (Ai + �Vi )� .
(15)
2
In Fig.7, the time evolutions of the three quantities
ET , Hm and K are shown. The total energy ET , being multiplied by 1/5 in the ﬁgure so that the it can beeasily compared to the other quantities, decays quite
rapidly while the other two quantities vary relatively
slowly. It is reasonable to attribute the diﬀerence ofthe decay rates to their orders of the derivatives, inaccordance with a typical selective decay in non-Hall
MHD turbulence. Though all the three quantities are
not conserved for ﬁnite dissipations, their behaviors
might give us some insights to understand the statistics in Figs.4, 5 and 6. The conservation of Hm in
the ideal limit constrains the behaviors of the magnetic ﬁeld. The slower decay of Hm than ET implicitlymeans that the current density ﬁeld j is less sensitive
than the vorticity ﬁeld ω to the dissipations thanks tothe constraint. It is consistent with the behaviors of
RλV and RλM� in Fig.1(c), in which the former decays
after t ≥ T0 while the latter stays almost constant,showing that the small scales in the velocity ﬁeld are
dissipated while those of the magnetic ﬁeld are kept.
It might be a consequence of the insensitive naturethat the current density ﬁeld shows more intermittent
structures as in Fig.4(b) than the vorticity ﬁeld in
Fig.5(b), although this remains as a tentative conjecK=(13)

the magnetic helicity
Hm =1
�Ai Bi �
2(14)

where Ai is the vector potential to give the magnetic539

H. Miura, Contributions of Small Scales to Statistics of Hall MHD Turbulence

0.3(2004).
[5] P. Goldreich and H. Sridhar, ApJ 438, 763 (1995).
[6] P. Goldreich and H. Sridhar, ApJ 485, 680 (1997).
[7] J. Cho, A. Lazarian and E.T. Vishniac, ApJ 564, 291(2002).

(EK+EM)/20
Hm
K

0.25

ET, Hm, K

0.2
0.15
0.1
0.05
0
-0.05
-0.1

0

0.5

1

1.5
t

2

2.5

3

Fig. 7 The time variation of the three quantities (EK +EM )/20, Hm and K.

ture to be studied. The roles of the helicity and the
hybrid helicity in formation/relaxation of local structures should be studied further in next works.

4. Summary
Statistics in Hall MHD turbulence are investigated by making use of the low-pass ﬁlter with k = 32
cut-oﬀ wave number. It is found that vortex structuresin the large scales are well characterized by the Gaussian distribution in spite of their intermittent structures. It may help us to construct an eddy viscosity model by making use of some theories for randomﬁeld. On the other hand, the current density ﬁeld
represents more intermittent statistics than the vorticity ﬁeld, so that we need further analyses about
it. We have also found that the local vortex structures, especially of the small scales, are aligned to themagnetic ﬁeld lines when the turbulence level is underdeveloping. The alignment is lost in the decaying (relaxation) process, showing dominance of the turbulent
ﬁeld by the helicity and the hybrid helicity. In order toconstruct numerical models for the magnetic ﬁeld, weneed to take the inﬂuences of the helicity conservationeﬀectively.
This research was partially supported by “Application of Fusion Theory to Space Plasmas” under the project 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 performedon the NEC SX-8 “Plasma Simulator”, supported by
the NIFS collaborative work NIFS09KNXN154.

References[1] D. Biskamp, Magnetohydrodynamic Turbulence (Cambridge University Press, Cambridge, 2003).
[2] D. Hori and H. Miura, Plasma Fusion Res. 3 S1053(2008).
[3] H. Miura and D. Hori, J. Plasma Fusion Res. SERIES
8 73 (2009)[4] S. Ohsaki and S. Mahajan, Phys. Plasmas 11, 898

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