Plasma and Fusion Research: Regular Articles

Volume 11, 2401052 (2016)

Experimental Study of Hall Eﬀect on a Formation Process of an
FRC by Counter-Helicity Spheromak Merging in TS-4∗)
Yasuhiro KAMINOU, Michiaki INOMOTO and Yasushi ONO
Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan(Received 29 November 2015 / Accepted 2 March 2016)
Hall eﬀect on counter-helicity spheromak merging is experimentally investigated in a low-S ∗ region, where
S is the ratio of plasma minor radius to ion skin depth. Direct measurement of magnetic field, ion flow, electrondensity revealed that Hall eﬀect aﬀect not only merging reconnection phase but also relaxation phase throughformation of ion flow and current density profiles. Hall eﬀect breaks the symmetry of the “case-O” and “case-I”
counter-helicity spheromak merging, which are defined by combinations of poloidal flux and toroidal flux ofinitial spheromaks. Formation of toroidal current profile and radial outflow pattern are diﬀerent in case-O andcase-I during merging process, and that results in great diﬀerence of profiles of electron density after merging.
∗c 2016 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: magnetic reconnection, field-reversed configuration, hall eﬀect, plasma merging, spheromak, highbeta
DOI: 10.1585/pfr.11.2401052

1. Introduction
A field-reversed configuration (FRC) [1] is an attractive configuration because of its high-beta. High-beta magnetic confinement is important for economical fusion reactor, and the simply-connected structure of an FRC enables to make a fusion reactor simple. The counter-helicity
spheromak merging (CHSM) [2–4] is one of the methodof creating an FRC, and it has an advantage that it caneasily generate a large-flux FRC, compared with FieldReversed Theta Pinch (FRTP) method. Direct formationof an FRC by FRTP need high-voltage fast capacitor bankand fast formation time, while CHSM need middle-voltage
capacitor bank because a spheromak can be made slowly
by a magnetized coaxial plasma gun or a flux-core, andslow formation enables a plasma to get large poloidal magnetic flux. And it also contains some complicated phenomenon in the merging process, such as magnetic reconnection [5] and relaxation. In both phenomenon, it is considered or proved that Hall eﬀect or the kinetic eﬀect plays
some important roles [6–9]. In magnetic reconnection,
it is known that Hall eﬀect enhances reconnection rate,that is revealed by Particle-in Cell (PIC) simulations [10]or experiments [5]. And in theoretical predictions, highbeta plasma is generated with strong ion flow by relaxation process [11]. Some experiments suggest that MHD
is not related to the stability of an FRC [3, 8], and ion flow
through magnetic reconnection enhances the stability of an
FRC [12]. However, in CHSM, the relationship between
merging reconnection process and global structures of theformed FRC have not been investigated, especially in a
author’s e-mail: kaminou@ts.t.u-tokyo.ac.jp∗) This article is based on the presentation at the 25th International Toki
Conference (ITC25).

condition that the Hall eﬀect is strong. The index of Halleﬀect is S ∗ , which is defined as the ratio between plasma
characteristic length and ion skin depth, and Hall eﬀect isstrong in low S ∗ plasma. Especially, it has not been investigated the formation process and diﬀerence between twocases of CHSM, where the diﬀerence can be strong in alow-S ∗ condition. The two cases are called as “case-O”
and “case-I”, which are defined by combinations of a polarity of poloidal flux and toroidal flux [6]. Figure 1 shows
schematics of CHSM method and two cases of CHSM. InCHSM, two spheromaks with opposite toroidal magnetic
flux merge together, and then an FRC with no toroidal
magnetic flux is generated. In this method, there exist twopatterns of CHSM with the combinations of a polarity ofFig. 1 A schematic of counter-helicity spheromak merging
method and an illustration of “case-O” and “case-I”
counter-helicity merging [14].

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c 2016 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles

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poloidal flux and toroidal flux. One is called “case-O”,
in which the plasma current is positive (poloidal magnetic
flux is positive) and toroidal flux is positive at the +Z side,and toroidal flux is negative at the −Z side. The other iscalled “case-I”, in which the plasma current is positive andtoroidal flux is negative at the +Z side, and toroidal flux is
positive at the −Z side. The reconnection plane is tilted totoroidal direction because both poloidal field and toroidal
field reconnect, and the reconnection electric field and reconnection current have both toroidal and radial components. In the reconnection current sheet, Et and jt are negative (Et < 0, jt < 0) in both case-O and case-I, while Erand jr are negative (Er < 0, jr < 0) in case-O and positive(Er > 0, jt > 0) in case-I [13]. In plasma merging, the difference of the tilting direction of the reconnection plane between case-O and case-I breaks symmetry between the twocases because the merging plasma are tori [6]. The toroidal
eﬀect couples with Hall eﬀect during merging process andthat can aﬀect global structure of a merging-formed FRC.
However, the Hall eﬀect on a formed FRC has not been investigated, while only reconnection process of CHSM has
been studied in previous works [6, 13, 14]. In this paper,
we investigated influences of Hall eﬀect during merging
on the formed FRC after merging, especially a relationship between merging process and formation of current andelectron density profiles.

device [4] and flux cores. Two flux cores are installed ateach side of the vacuum vessel. They contain both poloidalcoils (PF coils) and toroidal coils (TF coils), and they create spheromaks around them by swinging both PF coils andTF coils. After the PF coil currents are reversed, the twospheromaks are cut oﬀ by the flux cores, and pushed toward the center of the vacuum vessel (Z = 0 m). Then, thetwo plasma merge together by magnetic reconnection, anda single plasma is formed. The typical parameters are as
follows: poloidal magnetic field Bpol ∼ 20 mT, and electron
density ne = 1020 ∼ 1021 m−3 , plasma current Ip ∼ 30 kA.
The electron temperature is about 5 ∼ 10 eV, and the ion
ramor radius (rL ) is rL ∼ 10 cm at the separatrix. In thiscondition, ions are weakly magnetized (s = a/rL ∼ 2, ais the plasma minor radius.) and the Hall eﬀect is strong(S ∗ = a/li ∼ 2, li is the ion skin depth).
For measurement of two-dimensional profiles of magnetic field and current profile, 2D magnetic probe arrays
were used. Magnetic field to z direction (Bz ) were firstly
measured in various R and Z positions, and then poloidal
magnetic flux was estimated by the integration of Bz (r, z)
with the assumption of axisymmetry. Finally, radial magnetic field (Br ) was got from equation (2).
 r
Ψ (r, z) =Bz (r, z) · 2πrdr,(1)rmin1 ∂Ψ.
(2)
2πr ∂zFor measurement of ion flow and electron density, AMach probe and a Langmuir probe are installed at the midplane (Z = 0 m). A Mach
 probe can measure ion mach
Br = −

2. Experimental Setup
Figure 2 shows a schematic of TS-4 plasma merginge)) by comparing ion satnumber (Mi = vi /cs , cs = γ(Tmi +Tiuration current at upstream electrode ( jup ) and downstream
electrode ( jdown ).

jup= exp(K Mi ),
jdown

(3)

K is a constant parameter, and we adopt K = 2.0 forweakly magnetized condition, which have been got by
SSX experiment [15]. In this experiment, T i = T e is assumed because little electron or ion heating occurs in thecondition that radiation loss is large and ions do not havegyro-motion with weak magnetic field. T e is measured
by a Langmuir probe [14]. The Langmuir probe and theMach probe can be swept shot-by-shot, and radial profilesof these quantities are got by assuming reproducibility.

3. Results & Discussions
3.1 Global structure of magnetic fields

Fig. 2 Schematics of a flux core and formation process of aspheromak by the flux core.

We report relation between current profile formationand the Hall eﬀect during merging process in this section.
Figures 3, 4 show poloidal flux contour (poloidal magneticfield lines) and toroidal magnetic field in case-O/I CHSM
measured by 2D magnetic probe arrays. In Fig. 3 (case-O),
the toroidal flux at +Z side is positive, and the toroidal flux
at −Z side is negative. In Fig. 4 (case-I), the toroidal flux

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Fig. 5 Time evolutions of the X-point/O-point radius (RX ) andthe separatrix radius (Rsep ) in case-O (blue) and case-I(red). The vertical lines show merging completion timeof case-O (blue) and case-I (red).

Fig. 3 Poloidal flux contour (line) with toroidal magnetic field(colored) in Case-O.

Fig. 6 Radial profiles of toroidal current ( jt ) in case-O (top) andcase-I (bottom).

Fig. 4 Poloidal flux contour (line) with toroidal magnetic field(colored) in Case-I.

at +Z side is negative, and the toroidal flux at −Z side ispositive. In both figures, toroidal plasma currents are positive. Figure 5 shows time evolutions of X-point (duringmerging) and O-point of the formed FRCs (after merging),
and the separatrix radius. In case-O, the X-point moves
outward in the radial direction, and in case-I, it moves inward during merging. The direction of the movement ofthe X-point is same as the radial component of the reconnection current. In case-O, radial current in the current
sheet is negative because the spheromak at the +Z side has
positive toroidal magnetic field and the spheromak at the−Z side has negative toroidal magnetic field. And the di-rection of the X-point movement is outward. On the otherhand, in case-I, radial current in the current sheet is positive because the spheromak at the +Z side has negative
toroidal magnetic field and the spheromak at the −Z sidehas positive toroidal magnetic field. And the direction ofthe X-point movement is inward. Coupled with previous
studies [16, 17] that Hall reconnection moves the X-pointin the same direction as the reconnection current, it is assumed that the X-point moves both toroidal (negative) andradial (negative in case-O and positive in case-I) directionin this experiment. These results are consistent with previous experiment in MRX [6]. By the motion of the Xpoint, it is found that the toroidal current profiles during
and after merging are diﬀerent between case-O and caseI. Figure 6 shows the profiles of toroidal current density( jt ) at Z = 0 m in case-O and case-I, which are measured
by 2D magnetic probe arrays. The direction of plasma

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current of the initial spheromaks are toroidally positive( jt > 0). These are same characteristics and diﬀerence
between case-O and case-I. The same characteristic is current sheet position during reconnection (t = 370 µs). Current sheet peak is located on R ∼ 0.45 m and the X-point is
R ∼ 0.5 m at t = 370 µs in case-O, while the current sheet
peak is located on R ∼ 0.45 m and the X-point is R ∼ 0.4 mat t = 370 µs in case-I. Thus, the same characteristic is thatthe current sheet peak shifts to the opposite direction to
the motion of the X-point. However, current profile aftermerging has diﬀerence between case-O and case-I. Hollow current profile is formed just after merging completion(t = 390 µs) in case-O. After that, the current profile relaxes to broad profile (t = 410 µs). On the other hand, incase-I, the current profile after merging is sharply peaked(t ≥ 390 µs). During merging, positive current arises at
the inboard side of the X-point, and this is caused by thecompression of the poloidal magnetic field (Bz ) by the inward displacement of the X-point (t ≤ 380 µs). The inward
displacement of the X-point enhances the magnetic pressure of the poloidal magnetic field in case-I, and then thestrong gradient of poloidal magnetic pressure makes positive toroidal current at the inboard side of the X-point(during merging) or O-point (after merging). At the outboard side, magnetic pressure becomes weak, and positive
toroidal current is suppressed. In case-O, compression ofpoloidal magnetic pressure is not strong because the direction of the X-point motion is outward and the outboard
side. Instead, the X-point motion makes the separatrix
larger. Thus, it is concluded that the hollow current profile in case-O and the peaked current profile in case-I aftermerging is derived from the Hall eﬀect in the merging process.

3.2

Fig. 7 Radial flow profiles at Z = 0 in case-O (top) and case-I(bottom).

Flow and density profiles

To investigate influences of Hall eﬀect on flow formation during merging and density profiles after merging, wemeasured radial profiles of radial ion flow, electron density,and electron temperature by a Mach probe and a Langmuirprobe (a triple probe) at Z = 0 m. We found that ion flowcaused by reconnection is strongly biased to one side, andthe direction depends on the polarity of the reconnecting
toroidal magnetic flux. Figure 7 shows radial profiles ofradial ion flow at Z = 0. In case-O, radial flow is strong
in negative direction. The flow is strongly negative at theinboard side of the X-point (R < 0.5 m), and weakly positive at the outboard side of the X-point. On the other hand,in case-I, radial flow is strong in positive direction. The
flow is weakly negative at the inboard side of the X-point(R < 0.4 m), and strongly positive at the outboard sideof the X-point. Thus, ion acceleration was observed during the merging process, and this acceleration seems to bestrongly related with Lorentz force ( j × B), which is modified by the Hall eﬀect. Figure 8 shows the time evolutionof Lorentz force calculated by radial profiles of toroidal
current and poloidal magnetic field from equation (4).

Fig. 8 Radial component of j × B force ( jt Bz ) profiles in case-O(top) and case-I (bottom).

( j × B)r = jt Bz .

(4)

During merging (t ≤ 380 µs), the sign of jt Bz is negative in almost all region in case-O. This is because themagnetic field (Bz ) is compressed in the outboard side of
the X-point by the displacement of the X-point which iscaused by the Hall eﬀect, and then the compressed poloidal
magnetic field makes positive toroidal current. At the inboard side of the X-point, the displacement of the X-point
enhances magnetic tension and negative toroidal current.
Therefore, the jt Bz force is negative and reconnection outflow is mainly accelerated to inward. The negatively biased
radial ion outflow pattern in case-O seemed to be causedin this way. On the other hand, in case-I, the jt Bz force
is positive in almost all region during reconnection. This2401052-4

Plasma and Fusion Research: Regular Articles

Volume 11, 2401052 (2016)tion process but also profiles of density profile and global
magnetic field profile of a merging-formed FRC.

4. SummaryFig. 9 Time evolution and radial profile of electron density incase-O/I. The dashed line shows X-point (during merging)/O-point (after merging) radius, and the solid line
shows separatrix radius.

Influences on formation of flow, current profile, andelectron density profile by the Hall eﬀect in the formation
process of a merging-formed FRC in a low-S ∗ condition
was investigated through comparing two cases of counterhelicity spheromak merging. In the TS-4 experiment, it
was observed that the X-point in case-O merging moves
outward, and the X-point in case-I merging moves inward.
The X-point movement generates compression of poloidal
magnetic pressure, and then the Lorentz force by this eﬀect
produces strong bias of the radial outflow, which aﬀects
electron density profile after merging completion. It wasalso found that hollow current profile is formed in case-O,
and peaked current profile is formed in case-I at merging
completion period. It is concluded that the current profiles
and flow patterns formed by Hall reconnection in merging
phase has large influence on global electron density profileof formed FRCs after merging completion.

Acknowledgement
is because the magnetic field (Bz ) is compressed in the inboard side of the X-point by the inward displacement of
the X-point which is caused by the Hall eﬀect, and thenthe compressed poloidal magnetic field makes strong positive toroidal current. At the outboard side of the X-point,
the displacement of the X-point enhances magnetic tension
and negative toroidal current. Therefore, the jt Bz force ispositive and reconnection outflow is mainly accelerated tooutward. The positively biased radial ion outflow pattern
in case-I seemed to be caused in this way. Electron density
profile after merging completion also has strong relationship with the Hall eﬀect. Figure 9 shows time evolution
and radial profiles of electron density at Z = 0 m in caseO and case-I. The solid red line shows a separatrix radiusof the plasma at Z = 0 m, and the dashed red line shows
the X-point of the magnetic reconnection (during merging)
and O-point of the formed FRC (after merging). In case-O,
the electron density peak moves inward, and it is trapped
inside the separatrix. On the other hand, the electron density peak moves outward and it is not trapped inside theseparatrix. It is assumed that the bulk ions moves withthe radial ion flow generated during the merging process,and large inertia of the ions enabled to leave the flow pattern after merging. Thus, these results demonstrated thatthe Hall eﬀect during merging aﬀects not only reconnec-This work was supported by JSPS Fellows (278028), JSPS A3 Foresight Program “Innovative Tokamak Plasma Startup and Current Drive in SphericalTorus”, and the NIFS Collaboration Research program(NIFS14KNWP004).
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