J.
6 (2004)
J. Plasma
Plasma Fusion
FusionRes.
Res.SERIES,SERIES,Vol.
Vol.
6 (2004)130–133
000–000

Two-Fluid Nonlinear Simulation of Self-Organization ofPlasmas with Flows
NUMATA Ryusuke, YOSHIDA Zensho and HAYASHI Takaya1
Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa 277-8561, Japan
1
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 9 December 2003 / Accepted: 6 July 2004)

Abstract
A three-dimensional Hall-Magnetohydrodynamic (Hall-MHD) simulation code has been developed to study theself-organization process in a two-fluid plasma. An appreciable amount of flow with a component perpendicular to
the magnetic field is created in the two-fluid plasma. An adjustment of the ion helicity, which includes the Hall term,
characterizes the two-fluid relaxation process.
Keywords:
two-fluid MHD, self-organization, double Beltrami equilibrium, plasma flow, nonlinear 3D simulation

1. Introduction

2. Theoretical background

The Hall effect in a two-fluid plasma is represented by asingular perturbation term scaled by the ion skin depth,bringing about far richer structures created by flow-fieldcoupling. Recent theory [1] predicts creation of a double
Beltrami (DB) field that is in marked contrast with the Taylor
relaxed state of the single-fluid MHD; the former may havea variety of flows.
In this paper, we study the dynamical process of the selforganization in two-fluid plasmas. We have developed anonlinear three-dimensional (3D) simulation code. Startingfrom an unstable initial condition, through violent dynamics,
total fields are rearranged, and new structure is reorganized.
We can characterize such transient processes by some macroscopic quantities, that are “ideal constants of motion”, i.e.
energy and helicities. The ideal constants of motion conservein the ideal limit. Dissipation, however, allows “adjustment”
in these quantities. The adjustment of the ion helicity may
play a role to create flows perpendicular to the magnetic field[2].
In quasi-static turbulence, perpendicular flows aredescribed by a statistical average of fluctuations (Reynoldsstress) [3]. Similar for generation of magnetic field, fluctuations can yield dynamo effects. A lot of studies related tothose processes have been done in the single-fluid model.
However, how the Hall effect alters those processes is still
undeveloped (in recent studies [4], the Hall effects for
dynamo processes have been addressed). In the self-organization process where drastic change of global structure occurs,
we cannot define the statistical mechanical ensemble offluctuations.

2.1 Double Beltrami equilibrium
In an incompressible two-fluid model, the macroscopic
evolution equations of a plasma can be cast into coupled
vortex equations,w j - — ¥ (U j ¥ w j ) = 0 ( j = 1, 2),t(1)where j = 1, 2 indicate an electron and an ion. A pair of generalized vorticities and the corresponding flows are definedbyw1 =B,
w2
U1 = V - e— ¥ B, U2==B + e— ¥ V ,V,(2)

where B is the magnetic field, V is the ion flow velocity, ande  di /L (L: system size, di  c/wpi : ion collisionless skindepth, c: speed of light, wpi : ion plasma frequency). The simplest equilibrium solution to (1) is given under the “Beltramicondition”, that demands alignment of the vorticities along
the corresponding flows;

B = a(V - e— ¥ B), B + e— ¥ V = bV ,(3)where a, b are constants. The general solution to (3) is given
by a linear combination of two Beltrami fields,

B = C+ G+ + C- G- ,V = ( a -1 + el + )C+ G+ + ( a -1 + el - )C- G- ,(4)©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: numata@ppl.k.u-tokyo.ac.jp1130

Numata R. et al., Two-Fluid Nonlinear Simulation of Self-Organization of Plasmas with Flows

where G± are the Beltrami functions satisfying — ¥ G± =l±G±, and C± are constants. The parameters l±, which are theeigenvalues of the curl operators, characterize the spatial
scales of the vortices G±.
In the vortex dynamics equations (1), the general steadystates are given by

U j ¥ w j = —j j ( j = 1, 2),d ( E - m1¢ H1 - m 2¢ H2 ) = 0,

where m¢1, m¢2 are Lagrange multipliers, leads the DB field.
The relaxation process is realized by minimizing perturbations, which is scaled by the generalized enstrophy F, withappropriate adjustments of macroscopic variables E, H1, H2.

3. Simulation model(5)

To test the two-fluid self-organization in a more general
dynamical framework, we consider a compressible, dissipative Hall-MHD plasma, governed bywhere jj is a certain scalar field corresponding to the energy
density. The Beltrami condition (3) gives a special class ofsolution such that both side of (5) equal to zero. The “generalized Bernoulli condition” —jj = 0 demands that the energy
density is uniform in space. From the original electron andion momentum equations, we find that the generalized
Bernoulli condition is given by the relation,

V2+ p = const.
2n= -—  (nV ),t(nV )= -—  (nVV ) - —p + (— ¥ B) ¥ B
t11+(— 2 V + —(—  V )),
3Re

(6)

2.2 Variational principle

Be
1 2= — ¥ ÈÍV ¥ B - (— ¥ B) ¥ B +— B,
tnÎ
 Rm

The self-organization process of a plasma into relaxedstates may be discussed by a variational principle. Taylor [5]assumed the minimization of the magnetic energy under theconstraint of the magnetic helicity, and obtained the forcefree state. This model is based on the “selective decay”
implying that one of the constants of motion of the ideal limit
decays faster than the others.
The variational principle for the two-fluid MHD requiresa more generalized and rigorous arguments [2]. There arethree global ideal invariants:

E=1( B2 + V 2 )dx ,
2

Ú

(7)

Ú A  Bdx,

(8)

Ú(9)

H1 =H2 = ( A + eV )  ( B + e— ¥ V ) dx ,

12

Ú

(13)

(14)

(15)

p= - ( V  — ) p - g p—  V
t(16)
141+| — ¥ V |2 +| —  V |2 +| — ¥ B |2 ,3 Re
ReRm
where n is the number density, p is the pressure, Re, Rm are theReynolds number and the Magnetic Reynolds number, and g is
the ratio of the specific heat. We have normalized variables by
the scale length L and an appropriate measure of the magnetic
field B0 and the density n0 (the ion mass mi is assumed to beunity) as x Æ Lx, B Æ B0 B, n Æ n0n, V Æ VAV, t Æ tAt, p Æ(B20/m0)p, where VA  B0/ m 0 mi n0 is the Alfvén velocity, tA L/VA. The quasi-neutrality n = ni = ne, and zero electron pressure
are assumed. The equations are solved by the finite difference
and the Runge-Kutta-Gill methods.
The simulation domain is a rectangular box with size 2a¥ 2a ¥ 2pR, surrounded by a rigid perfect conducting wall.
The system is periodic along the z axis. The boundary
conditions in x, y directions arerepresenting the total energy, the electron helicity, and the ionhelicity. The minimization of a generalized enstrophy (measure of the complexity),

F=(12)

2— ¥ ( A + eV ) dx ,(10)n  B = 0, n ¥ (— ¥ B) = 0, V = 0 at x , y = ± a.

with keeping E, H1, and H2 constant is carried out throughthe variationd ( F - m 0 E - m1 H1 - m 2 H2 ) = 0,(17)

To assure tangential components of the electric field vanish,
we set e = 0 at the wall.
In order to confirm the validity of the simulation code,
we have solved dispersion relations of purely transversal wave
in a periodic domain. In an ideal limit, the dispersion relation
for small perturbations around a uniform equilibrium, (B0 =(0, 0, B0), V0 = 0, n0, p0 are constants), is given by(11)

where m0, m1, m2 are Lagrange multipliers. The generalsolution to the Euler-Lagrange equation is given by a linear
combination of three Beltrami functions, which is not anequilibrium in general. The adjustment between three ideal
constants of motion,

w=2131VA È22 42Í ± ek ± e k + 4 k ,Í
2 Î(18)Numata R. et al., Two-Fluid Nonlinear Simulation of Self-Organization of Plasmas with Flows

which describes the Alfvén whistler/Alfvén ion cyclotron
wave. In the limit e Æ 0, we recover the dispersion relationof the shear Alfvén wave. Figure 1 shows the dispersion
relation (18) for e = 0, 0.1, 0.05. The Hall term yields adispersion effect to split the Alfvén wave in the electron andion branches. The numerical results show good agreementswith analytical curves.

the perpendicular and the parallel components of the kineticenergy. The perpendicular component remains more than 10%
for e = 0.1, while it does only few percent for e = 0. This
result agrees with the theoretical prediction, and highlights
the difference from the Taylor relaxation in the single-fluid
MHD. The simulation result demonstrates creation of quasistatic stable state ( t @ 0), where the energy density (p + –21 V 2)
is almost homogenized (fluctuation of the energy density issmall, |— (p + –21 V 2)| / | j ¥ B| @ 0.1). Hence, the homogeneityof the Beltrami parameters implies the creation of the DBfield. In Fig. 4, we show the distribution of the Beltrami
parameters a, b in the poloidal plane. The figures show thata, b are almost uniform in the magnetic columns.
We have investigated the variational principle in theHall-MHD plasma. Figure 5 shows the time evolution of themacroscopic quantities: E, H1, H¢2  H2 – H1, and

4. Nonlinear simulations
The initial condition for nonlinear simulation was a 2D
force free equilibrium that is also an equilibrium of the singlefluid model [6];Bx = -By =1( k1 B1 cos k2 x sin k1 y + k2 B2 cos k1 x sin k2 y), (19)k01
( k2 B1 sin k2 x cos k1 y + k1 B2 sin k1 x cos k2 y),k0

Bz = B1 cos k2 x cos k1 y + B2 cos k1 x cos k2 y,

(20)

(21)

where B1, B2 are the amplitudes of two kinds of Fouriermodes, k1 = n1p /(2a), k2 = n2p /(2a), k0 = k12 + k22 , and n1,n2 are arbitrary integers. A uniform density n0 = 1, and a
uniform pressure are assumed. The amplitude of the uniform
pressure is given by a parameter b = pdx/ B2dx. The initialcondition has a flow such that V0 = (0, 0, MABz/÷ n0), where
MA is the Alfén Mach number.
We carried out two simulation runs. The simulation
domain is implemented on 129 ¥ 129 ¥ 256 point grids. The
parameters are b = 3, MA = 0.5, and (a) e = 0.1, (b) e = 0.
Figure 2 shows isosurfaces of the toroidal magnetic field at
an initial time and t = 50tA, 75tA, 100tA. Two twisted columns
in different color represent the columns of Bz = ±0.3. Because
the initial condition is unstable against the ideal kink mode,
initially assigned perturbations grow exponentially and plasma
becomes turbulent. The magnetic reconnection process leads
rearrangement of plasma configuration in Alfvénic time scale.
The plasma settled into a twisted state after the turbulent
relaxation process, and it flows both in the toroidal andpoloidal direction as time goes. Figure 3 shows the ratio ofFig. 1

Dispersion relations of the Alfvén whistler/Alfvén ion
cyclotron wave.

Fig. 2 Isosurfaces of the toroidal magnetic field.

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Numata R. et al., Two-Fluid Nonlinear Simulation of Self-Organization of Plasmas with Flows

Fig. 3 Ratio of components of the kinetic energy.
Fig. 5

Fig. 4highlights the distinct character of the two-fluid relaxed state(the DB state). The relaxation process have also been investigated by means of the time evolution of the macroscopic
quantities. Since the ion helicity decays faster than the energy,
the selective decay concepts of the energy cannot work in thetwo-fluid model. The enstrophy (a measure of fluctuations)minimization leads the relaxation. During the relaxation
process, the ion helicity (the most fragile macroscopicquantity) is severely adjusted to obtain the DB state.
In the two-fluid relaxation, the ion helicity plays anessential role for the relaxation process. The ion helicity
includes the Hall term, and its counterpart in the single-fluid
model is the cross helicity HC = V · Bdx. The relaxation
model due to the variational principle by use of the crosshelicity only leads the single Beltrami magnetic field (theTaylor state) and the parallel flow. Thus, we conclude thatthe Hall term generates the perpendicular flow in the twofluid plasmas. Physically, the Hall current is induced bydeviation of ion motions from electron motions. The Beltrami
condition demands this Hall current to support the perpendicular flows.

Distribution of the Beltrami parameters a, b in the poloidal plane. The figures show that a, b are almost
uniform in the magnetic columns.

F¢ =1| — ¥ V | 2 dx ,2ÚTime evolution of the macroscopic quantities. The enstrophy minimization with the selective adjustment of
the ion helicity characterizes the relaxation in the twofluid plasma.

(22)

normalized by the initial values for the case (a). The minimization of (10) or (22) will give the same result. The energy
and two helicities monotonically decrease due to the dissipations. However, the dissipation rate of the ion helicity isfaster than that of others. Since the ion helicity includes thehighest order derivative term (V · — ¥ V), it is selectively
adjusted during the relaxation process. The generalized
enstrophy increases in the relaxation phase because it is not
an ideal constants of motion. After the relaxation phase t @20 ~ 40 tA, it decreases faster than other three quantities. The
relaxed state is characterized not by minimizing the energy,
but by minimizing the generalized enstrophy.

Acknowledge
This work was supported by JSPS Research Fellowships
for Young Scientists.

References[1] S.M. Mahajan and Z. Yoshida, Phys. Rev. Lett. 81, 4863
(1998).
[2] Z. Yoshida and S.M. Mahajan, Phys. Rev. Lett. 84,095001 (2002).
[3] P.W. Terry et al., Rev. Mod. Phys. 72, 109 (2003).
[4] P.D. Mininni et al., Astrophys. J. 587, 472 (2003).
[5] J.B. Taylor, Phys. Rev. Lett. 33, 1139 (1974).
[6] R. Horiuchi and T. Sato, Phys. Rev. Lett. 55, 211 (1985).

5. Summary
We have developed a Hall-MHD simulation code in a
3D rectangular domain, and have studied the self-organization
process of a flow-field coupled state. Comparing the relaxedstate in the two-fluid model with that in the single-fluid
model, an appreciable flow with a component perpendicular
to the magnetic field is created. The perpendicular flow

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