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

Shear ﬂow eﬀects on double tearing mode global magnetic
reconnection
Thibaut VOSLION1,2,3) , Olivier AGULLO1,2) , Peter BEYER1,2) , Masatoshi YAGI3) ,Sadruddin BENKADDA1,2) , Xavier GARBET4) , Kimitaka ITOH5) , Sanae-I. ITOH3)
1)

France Japan Magnetic fusion Laboratory, LIA 336, france-JapanLPIIM, CNRS, Universit de Provence, Marseille 13397, France
3)
RIAM,Kyushu University, Kasuga 6-1, Kasuga 816-8580, Japan
4)
CEA, IRFM, f-13108 Saint Paul Lez Durance, France
5)
National Institute for Fusion Science , Oroshi - cho, Toki -shi , 509-52 Japan
2)(Received: 20 November 2009 / Accepted: 7 March 2010)

The dynamics of a global reconnection in the presence of a poloidal shear ﬂow which is located in
between magnetic islands is investigated. Diﬀerent linear regimes are identiﬁed according to the value ofthe resistivity and the distance between the low-order resonant surfaces. It is found that the presence ofa small shear ﬂow aﬀects and signiﬁcantly delays the global reconnection processes. It is shown that this
delay is linked to a breaking of symmetry imposed by the existence of the shear ﬂow and the generationof a mean poloidal ﬂow in the resistive layers.
Keywords: Nonlinear double tearing - shear ﬂow - Kelvin-Helmholtz instability - tokamak

1. Introduction

1
0.03

In tokamaks, the internal transport barrier isobserved in reversed magnetic shear conﬁgurations
where two low-order rational surfaces exist [1]. In suchconﬁgurations, toroidal and poloidal ﬂows, as well as
temperature and density gradients can coexist close to
the plasma core [2]. Note that such gradient can originate diﬀerent resistive instabilities which are currently
observed [3]. In fact, there is evidence that zonal ﬂows
exist and that the shear ﬂow is localized in between
resonant surfaces where the double tearing instability
can grow. The observed maxima of the ﬂow velocity
are weak compared with the Alfvén velocity vA . Nevertheless, such ﬂows can be important with respect tothe turbulence level because the associated radial electric ﬁeld can be linked to the formation of an ITB[4].
In fact, such conﬁgurations, allowing eventually thegeneration of a strong internal barrier, are part of the
ITER scenarios for advanced conﬁnement[5].
Previous works have investigated the inﬂuence ofa Bickley jet on a tearing instability[6, 7] and on adouble tearing instability [8]. More recently, Bierwage et al. [9] have studied the inﬂuence of the corerotation amplitude - for a given sheared ﬂow proﬁle- on the stability of magnetohydrodynamic (MHD)
and Kelvin-Helmholtz instabilities (KHI). These studies are mainly linear and a cylindrical geometry isused. They have shown that KHI can develop at large
poloidal mode numbers and are strongly enhanced
when the core rotation passes a critical value of the order of 10−2 vA in the case where the distance between
the resonant surfaces is small. They also found that
below this critical value the growth rates of both the

0

0

V0

B0

0.5

−0.5
−0.03
−1
−3

−2

−1

0
x

1

2

3

Fig. 1 Equilibrium proﬁles of the magnetic (dased lines)
and velocity ﬁelds (solid line).

MHD and KH mode decrease with the amplitude of
the core velocity. Recentently, Ishii et al. [10] havestudied the diﬀerent growth phases of weakly coupled
non linear DTM with KHI stable rotation proﬁle. Additionally, It as been shown in [11] that micro instabilities can play a crucial role on DTM dynamics.
In this paper, we focus on the case where the shear
ﬂow is in between the two tearing instabilities. Within
the framework of MHD, a 2D slab geometry is used.
It is found that the nature of the dynamics depends
drastically on the distance between the low-order resonant surfaces. The impact of the ﬂow on the nonlinear
process leading to a global magnetic reconnection of
the system is studied. The origin and the role of the
dynamically generated m = 0 poloidal ﬂow turbulence
are investigated as well as the dynamics at small scales
once the reconnection has taken place.

2. Model equations
We use a two ﬁeld model corresponding to a reduced (MHD) description of the ﬂuid equations that

author’s e-mail: thibaut.voslion@univ-provence.fr

574©2010 by The Japan Society of Plasma
Science and Nuclear Fusion Research

T. Voslion et al., Shear Flow Effects on Double Tearing Mode Global Magnetic Reconnection

provides a minimal framework to study the impact of aradially sheared ﬂow φ0 (x) on a double tearing mode.
The model consists of a set of two coupled equations
for the ﬂuctuations of the electrostatic potential φ andmagnetic ﬂux ψ. The equilibrium magnetic ﬁeld in thez− direction is given by the constant B0z and in thepoloidal by B0 (x) = ψ0� (x)y. The equilibrium current
is therefore j0 = ψ0�� (x). The time evolution of the two
ﬁelds is described by

1"(A.U.)

0.8

0

(2)

where η is the resistivity, ν the viscosity, ω = ∇2⊥ φ
the vorticity, and j = ∇2⊥ ψ the current density ﬂuctuation. The equations are normalized by τA = L⊥ /vA
for the time, L⊥ B0z for ψ, and L⊥ vA for φ where τA
is the Alfvén time and L⊥ is a magnetic shear length.
The poloidal equilibrium magnetic ﬁeld B0 (x) is
chosen as
B0 = tanh�x − x �
1r

aB− tanh�x − x �
2r

aB

−3

10

−2

!

−1

10

10

0.8
0.75
0.7
0.65

$

0.6
0.55
0.5
0.45
0.4

,

0.35

(3)

0.3
0

where the parameter aB = 0.5 controls the width ofthe proﬁle, x ∈ [−Lx /2, Lx/2] and (x1r , x2r = −x1r )are the locations of the resonant surfaces where double
tearing instabilities develop. The poloidal equilibrium
shear ﬂow is given by
v0 = Av tanh(x/av ) ,−4

10

Fig. 2 Scheme for α calculation: Without shear ﬂow (blueline), γ(log(η)) is a line and α the associated slope.
With a shear ﬂow, two diﬀerent branches exist. For
the low resistivity branch, KH is unstable (greenline) and for the high resistivity one, KH is stable(red line).

(1)∂t ψ + [φ + φ0 , ψ + ψ0 ] = ηj ,

0.4
0.2

∂t ω + [φ + φ0 , ω + ω0 ] = [ψ + ψ0 , j + j0 ]+ν∇2⊥ ω ,

0.6

1

2

3

4

#x

5

6

7

8

9

Fig. 3 α as a function of the distance between rationalsurfaces δx.

3. Coexistence of linear instabilities
It is well known that a constant magnetic ﬁeld
stabilizes a shear ﬂow instability when the initial ﬂow
is a vortex sheet [14], as far as the Alfvén velocity exceeds the amplitude of the vortex sheet. In [8], it isshown numerically that when the ﬂow is forced with avelocity ﬁeld v(x) = A(1/ cosh(x/a) − 1)y, KHI coexist with the magnetic tearing whenever a < 1. In thelatter work, the double tearing mode in the presenceof a shear ﬂow in cases where no KHI develops has
also been studied, while in [9] this has been done in acase where KHIs are present. However, the conditions
at which KHI and DTM coexist are not yet clear.
It is instructive to focus on the role of both the
distance between the resonances δx and the resistivity
for a ﬁxed amplitude of the ﬂow Av . In the absence of
shear ﬂow, the growth rate of the resistive DTM scaleswith the resistivity like η α with α = 3/5 in the limit ofvery ideally stable MHD conditions and α = 1/3 when
the system enters an ideally marginally stable regime.
Ideally Marginally stable means that the free magnetic energy of the ideal mode λH is zero[14, 15]. In
the limit of closed resonant surfaces, the latter can belinked to the tearing instability parameter ∆� . λH depends on the magnetic equilibrium proﬁle and therefore on δx. Fig. 3 shows the dependance of growthrate on the resistivity in the case with/without shear(4)

where Av = 0.03 and av = 0.2. The resulting proﬁles are shown in ﬁg. (1). The dashed lines represent four typical equilibrium magnetic proﬁles wehave used, corresponding to δx = x2r − x1r ={π/2, 3π/4, π, 5π/4} (cyan, red, green and blue respectively). The purple curve indicates V0 . Equations (1–2) are solved numerically using a ﬁnite diﬀerence scheme in the x-direction, including an Arakawa
algorithm for an accurate conservation of the Poisson brackets [.,.] and a pseudo-spectral method in
the y−direction, including an appropriate de-aliasing
scheme[12, 13]. The number of grid points in the x
direction range from Nx = 256 to Nx = 2048, witha typical spatial resolution of the order dx = 5.10−3 .
Lx and Ly are the box size in the x and y directions
respectively. We ﬁnd that these diﬀerent proﬁles giverise to very diﬀerent dynamics. For instance, whenδx = 5π/4, no global reconnection occurs and complex
nonlinear dynamics linked to the poloidal rotation of
the islands are observed. When δx = π/2, islands do
not rotate and there is a global reconnection process
in between the resonant surfaces. In this paper, we
focus mainly on the latter situation.

575

T. Voslion et al., Shear Flow Effects on Double Tearing Mode Global Magnetic Reconnection

ﬂow. Changing δx and η [16], we plot the power law
index α(δx) in ﬁg. (3) . As expected, without shear
ﬂow, the power law parameter α ranges in between the
two limit cases 1/3 and 3/5 (blue curve). When the
distance between two rational surfaces is large enough,we converge to the standard tearing law. Conversely
when δx becomes of the order of a typical magnetic
shear length L⊥ , we enter into a full DTM regime.
A global magnetic reconnection occurs nonlinearly insuch cases.
When a weak shear ﬂow is added, we have to discriminate between two regimes. A regime in which
the KHI is unstable appears, depending on the numerical value of the resistivity. This instability is radially localized in the vicinity of the layer where thevelocity shear is maximal x ∼ 0. Typically, it has
a high poloidal mode number but we observe thatit modiﬁes the growth of the tearing m = 1 mode
(km = m 2π/Ly ). In fact, this is not surprising because the DTM regime is linked to a strong radial
coupling between the two magnetic surfaces and thepresence of vortices in between them modiﬁes the nature of the interaction, and a priori should weakenit. In ﬁg. (3), the high and low resistivity cases, theKH stable (red curve) and KH unstable regimes (darkgreen curve) are shown. We observe that, for δx ≤ 2,
the presence of a KH instability in between the resonances ampliﬁes the power law index strongly suchthat it even exceeds the asymptotic regime α = 3/5.
On the other hand, we observe of strong decrease of
the power law index for 2 ≤ δx ≤ 4 in the KH stableregime, wich roughly follows the case without shear
ﬂow, but is even lower.
The reason is that the imposed shear ﬂow shownin ﬁg. (1) ideally leads to a poloidal rotation of theisland in both directions. The global poloidal rotation of the plasma modiﬁes the energy balance of thesystem and in fact reduces the growth rate of the magnetic instability. The viscosity, the resistivity, andalso the nature of the mode can prohibit such an ideal
scenario. In fact, the plasma starts to rotate aroundδx ∼ 2, but when the resonances are closer together,
given the weakness of the amplitude of ﬂow Av andthe strong interaction between the growing islands,
the plasma poloidal rotation is locked.

Kinetic Energy

0

10

−10

10

−20

Ln(E)

10

1000

mode 0mode 1mode 2mode 3mode 4mode 12
1250
1500

1000

mode 0mode 1mode 2mode 3mode 4mode 12
1250
1500

−30

10

−40

10

−50

10

0

250

500

750
t/%a
Magnetic Energy

0

10

−10

10

−20

Ln(E)

10

−30

10

−40

10

−50

10

0

250

500

t/%750aFig. 4 Energy of the poloidal modes versus time. δx =π/2.
Electrostatic Potential t = 220
6.28319

Perturbed Magnetic Flux = 220

−6

x 10
1.5

6.28319

2

1

1

0.5

0

y

y

0

3.14159

3.14159

−0.5

−1

−1

−2

−1.5
0
−3.141590x

3.14159

−5

x 10

0
−3.141590x

3.14159

Fig. 5 Snapshots of φ (Left) and ψ (Right) at t = 220τA .
δx = π/2.

phase (t < 160) and are destabilized in the quasilinear phase, where the exponential growth of the modes
satisﬁes the relation γm = mγ1 (t < 400) . Contrary
to the v0 = 0 case where the m = 0 mode is not generated in the linear and quasilinear phases, the relation
γ0 = 2γ1 is held in those phases. We will see that thegeneration of the m = 0 mode modiﬁes the dynamics of the system leading to the global reconnection in
between the resonances. In order to understand theorigin of this mode, it is instructive to observe thesnapshots of the electrostatic potential and the magnetic ﬂux in the quasilinear phase. They are shownin ﬁg. (5). Note that the structures are not symmetric with respect to a reﬂection at x = 0 and/or
y = constant as discussed in the previous section. Despite an imposed poloidal shear ﬂow, the islands do
not rotate poloidally. However, a symmetry is clearlybroken when compared with the v0 = 0 case. Indeed,in the latter case, the m = 1 mode satisﬁes a symmetry φ1 (x, −y, t) = −φ1 (x, +y, t) where

4. Structure of the m=1 DTM and generation of a mean poloidal ﬂow
In this section we focus on the nonlinear evolution of the m=1 DTM. Fig. 4 shows the time evolution of the kinetic and magnetic energy of the m ={0, 1, 2, 3, 4, 12} modes for the case with δx = π/2,
η = 10−3 , µ = 10−4 , Lx = 2π, and Ly = 2π, where
KH is linearly stable and only the m=1 DTM is unstable. The m > 1 modes are stable in the linear

φ1 (x, y, t)= φ̂1 (x, t) cos(α1 (x) + k1 y)(5)

is the m = 1 mode, and φ̂1 and α1 are the amplitude
and phase of the mode respectively. In other words,
when v0 = 0, α1 is constant because of the symme-

576

T. Voslion et al., Shear Flow Effects on Double Tearing Mode Global Magnetic Reconnection

0−3.14159

0x

Total Magnetic Flux = 470
6.28319

Electrostatic Potential t = 360

1.5

6.28319

Electrostatic Potential t = 440
0.03

6.28319

0.02

0.04

0.5

0.5

0.01

0.03

0

0

0

3.14159

−0.5

−0.5

−1

−1

−1.5
3.141590−3.141590x

3.14159y1

3.14159

0.02

−0.01

0.01

−0.02

−1.5
3.141590−3.14159

Fig. 6 Total magnetic ﬂux just before and at the end of
the global reconnection process. δx = π/2. (Left)t = 440τA . (Right) t = 470τA .

0

−0.03
0
x

0
−3.14159

3.141590x3.14159

Fig. 7 δx = π/2. Perturbed electrostatic potential justbefore the global reconnection. (Left) No shear
ﬂow. (Right) With the shear ﬂow
Total Mean Velovity at t = 380

Total Mean Velovity at t = 390

0.03

0.03

0.02

0.02

0.01

0.01

0

y

y

try. This is clearly false when v0 �= 0 as is seen inﬁg. 5. In fact, a straightforward calculation showsthat whenever α1 is constant, the projection of the
Poisson brackets [φ1 , ω1 ] on m = 0 mode is zero. forthe same reason the projection of [ψ1 , j1 ] will also be
zero when v0 = 0. From the breaking of poloidal parity as soon as v0 �= 0, it follows that the generation of
the m = 0 mode occurs at, of course, a growth rate
2γ1 through both the Reynolds and Maxwell stresses.

−0.01

−0.02

−0.02
−0.03

−0.03
−3.14
0.04

0
x

3.14

−3.14

Total Mean Velovity at t = 430
0.03
0.02

0.01
y

3.14

Total Mean Velovity at t = 440

0.02

0.01

0

0

−0.01

−0.01

−0.02

−0.02
−0.03

−0.03
−0.04
−3.14

0
x

0.04

0.03

Impact of the shear ﬂow on the
global reconnection process

When the islands are suﬃciently closed together,
the global reconnection occurs regardless of a shear
ﬂow, as is seen in ﬁg. 6 where the nonlinear interaction of the two magnetic islands deforms the DTM
structure (t = 440τA ) and forces them to vanish progressively (t = 470τA ). It is interesting to check if
this process is linked to the generation of the mean
velocity ﬂow ﬂuctuation ṽ0 = v0tot (x, t) − v0 (x) in thequasilinear phase of the m = 0 mode. A close examination of ﬁg. (4) studied together with snapshots ofthe magnetic ﬂux shows a correlation. Indeed, the ψ̂1component of the m = 1 mode starts to move radiallywhen the kinetic energy of the m = 0 mode, which
is linked to ṽ0 , becomes more important than that of
the m = 1 mode (step 1), in the δx = π/2 case att ∼ 395. The global reconnection process can thendevelop and occur in the time interval t ∈ [440, 480].
In fact, when the magnetic energy of the m = 0 modecrosses that of the m = 1 mode at t ∼ 448 (step2), the remaining islands are no longer topologicallylinked to their initial resonant surface (see right snapshot of ﬁg. (6)). At t = 478, the global reconnection
process is completed and the islands disappear (step3). Inspection of the maps of the electrostatic potential just before the global reconnection (see ﬁg. (7))
shows that the presence of the m = 0 mode, even if
it appears to be much more complex, does not substantially modify the plasma ﬂow compared with thecase when v0 = 0, where typical butterﬂy-like struc-

0

−0.01

y

5.

0.05

1

y

y

3.14159

1.5

yTotal Magnetic Flux = 440
6.28319

−0.04
0
x

3.14

−3.14

0
x

3.14

Fig. 8 δx = π/2. Structure of v0tot (x, t). (Top-Left) t =380τA . (Top-Right) t = 390τA . (Bottom-Left) t =430τA . (Bottom-Right) t = 440τAtures are easily identiﬁed. To clarify the role of thegenerated poloidal mean ﬂow v0tot (x, t), it is interesting to focus on the time evolution of its structure.
Fig. (8) shows v0tot (x, t) at diﬀerent times. The redlines represent v0 = v0tot (x, 0) and the horizontal lines
show the position of the resonances. Initially, we observe that ṽ0 grows close to the resonances. Then,(step 1) occurs roughly when v0tot (xir , t) has crossed
zero (about t = 385) or some Alfvén time after thishas happened (about t = 395). Finally, the graphst = 430τA and t = 440τA show that during the global
reconnection, which starts at (step 2), the mean velocity oscillates around zero in between the resonances,approaching, in some sense, the case without imposed
shear ﬂow. These observations show, ﬁrst, that thenonlinear destabilization of the DTM occurs once thegenerated ṽ0 (x, t) compensates v0 in the vicinity of
the resonant surface, and second, that the global re-577

T. Voslion et al., Shear Flow Effects on Double Tearing Mode Global Magnetic Reconnection

Total mean Magnetic field at t = 430

Total mean Magnetic field at t = 440

1

1

0.5

0.5

0

0

Island position

15

−1

−3.14

0
x

−3.14

0.5

0.5

0

0

−0.5
−1

0
x

−10
0

3.14

Total mean Magnetic field at t = 476
1yy

Total mean Magnetic field at t = 450
1

3.14

−1

−3.14

0
x

200

400

600

800
t/%
a

1000

1200

1400

1600

Fig. 10δx = 3π/4. Time evolution of the poloidal positionsof the center of the two islands

over, in this latter case, the global reconnection corresponds to a structure-driven nonlinear instability ofDTM characterized by an abrupt growth after a longtime-scale evolution[17], such that, at t = 1610, theislands reach the wall. In both cases, however, thepresence of the shear ﬂow delays the time at which
global reconnection occurs. The delay is of about 15%
for δx = π/2 and 30% for δx = 3π/4. Note that forδx = π/2 the delay strongly increases with the shear
ﬂow amplitude reaching 116% for AV = 0.09 (just below KHI threshold). From these preliminary results
we estimate that below AV = 0.01 the delay becomes
negligible. A much more systematic study should be
performed, but this preliminary work shows that control of the global reconnection process through controlof the shear ﬂow in between the islands might be possible.

−0.5

0
x

Left island
Right island

−5

−1

3.14

5
0

−0.5

−0.5

−3.14

Amp

y

y

10

3.14

Fig. 9 δx = π/2. Structure of B0tot (x, t). (Top-Left) t =430τA . (Top-Right) t = 440τA . (Bottom-Left) t =450τA . (Bottom-Right) t = 476τA

connection process starts and occurs once the mean
velocity proﬁle in between the islands oscillates aroundzero.
It is instructive to follow the time evolution of
the mean magnetic ﬁeld B0tot (x, t). Fig. (9) shows
B0tot (x, t) at diﬀerent times during the global reconnection process, i.e. in between (step 2) and (step3). First, we can see that until (step 2), the proﬁle and also the positions of the resonant surfaces(B0tot (x, t) = 0) are roughly unchanged. This is followed by a ﬂattening of the proﬁle in between theresonant surfaces where the mean current decreases,
while in the vicinity of the resistive layers, the proﬁle becomes narrow signifying the generation of astrong mean current (see snapshot at t = 440τA andt = 450τA , respectively). This eﬀect is ampliﬁed until (step 3) occurs. It corresponds to the time when
B0tot (x, t) does not have a zero point at any location(see t = 476τA snapshot). In other words, there is noresonant surface anymore.
Let us consider also the case δx = 3π/4, keeping
all the other parameters identical to those used in thecase δx = π/2. The KH stable branch still exists andshows the similar behaviour as the case with δx = π/2.
We ﬁnd that the same behavior is observed. However,the situation is more complex because island rotation
is initially involved in the dynamics. In fact the (step1) event occurs at t = 660, when also the magnetic islands stop to rotate poloidally and the (step 2) event
occurs at t = 1600, after a long period where the islands do not move poloidally (see ﬁg.( 10)). More-

6. Conclusion
In this paper, we have presented a study of therole of a shear ﬂow in between magnetic islands onthe global magnetic reconnection processes. We ﬁrst
have identiﬁed the diﬀerent linear regimes which exist
when a shear ﬂow is present in between two surfaces
where DTM develop. We have seen that according to
the distance between the resonant surfaces and the resistivity of the plasma diﬀerent regimes exist. Whenthe distance is larger than the typical magnetic shear
length, no global reconnection occurs and the islands
rotate[18]. In this paper, we did not focus on thoseregimes. When it is of the order or smaller than the
typical shear length of the system, an island can belocked or rotate linearly according to the value of δx.
Moreover, the system can be KH stable or unstableaccording to the resistivity and, of course, to the amplitude of the shear ﬂow. We have shown that, independent of the regime (rotating island regime or not),a breaking of symmetry linked to the presence of theshear ﬂow leads to the generation of a mean poloidal
ﬂow at early times. We have shown that ﬂow develops ﬁrst in the vicinity of the resonant surfaces andis at the origin of the radial displacement and desta-

578

T. Voslion et al., Shear Flow Effects on Double Tearing Mode Global Magnetic Reconnection

bilization of the DTM structure. We have found thatthe presence of the shear ﬂow delays the global reconnection process, suggesting that its control might
be usefull for controlling the DTM. Further investigations are, however, necessary to precisely evaluate theimpact of the shear ﬂow and its amplitude, and also
the impact of the KH instability in the low resistivity
regimes, on the global reconnection processes of the
double tearing mode.
Acknowledgments: The authors wish to thank M.
Muraglia and G. Fuhr for fruitful discussions. T. V.
acknowledges the College Doctoral Franco-Japonais
for its fellowship. This work is partly supported by
LIA 336CNRS and by NIFS/NINS under the projectof formation of International Network for Scientiﬁc
Collaborations, and by Grant-in-Aid for Scientiﬁc Research (21224014).
[1] R.V. Budny et al., Plasma Phys. Control fusion 44,(2002)[2] R.C. Wolf, Plasma Phys. Control fusion 45, (2003)[3] S. Takeji et al., Nucl. fusion 42, 5 (2002)[4] R. E. Bell, PRL 81,1429 (1998)[5] A.C.C. Sips et al., ICPP, (2004)[6] L. Ofman, X.L. Chen and P.J Morrison, Phys. ﬂuids
B 3, (1991)[7] D. Biskamp, E. Schwarz, and A. Zeiler, Physics Of
Plasmas 5, 7 (1998)[8] L. Ofman, P.J. Morrison and R.S. Steinolfson, Phys.
ﬂuids B 5, (1992)[9] A. Bierwage et al., Phys. of Plasma 14 010704 (2007)[10] Y. Ishii, A.I. Smolyakov and M. Takechi , Nucl. fusion
49, 085006 (2009)[11] A. Ishizawa and N. Nakajima, Phys. of Plasma 14040702 (2007)[12] A. Arakawa, J. Comput. Phys 135, 103 114 (1997)[13] M. Muragliaet al., Nucl. fusion 49 055016 (2009)[14] Biskamp, Magnetic Reconnection in plasmas (Cambridge University Press, 2000)[15] G. Ara, B. Basu, B. Coppi, M.N. Rosembluth and
B.V. Waddel 112, 443 (1978)[16] M. Janvier, Y. Kishimoto, J. Li, Z.X Wang, private
communication, Kyoto University (may 2009)[17] Y, Ishii, M. Azumi, Y. Kishimoto, Phys Rev Lett. 89
205002 (2002)[18] T. Voslion, O. Agullo, P. Beyer, S. Benkadda and
X. Garbet, proceedings of the EPS Plasma conference
held in Creta (2008)579