J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)473–476
000–000

Magnetic Structures inside the Separatrix on Field-Reversed
Configuration Plasmas
GOTA Hiroshi, TAKAHASHI Tsutomu and NOGI Yasuyuki
College of Science and Technology, Nihon University, Tokyo 101-8308, Japan(Received: 9 December 2003 / Accepted: 23 February 2004)

Abstract
Separatrix shapes and internal structures of a field-reversed configuration (FRC) plasma are determined bycomparing the measured magnetic fluxes with the Grad-Shafranov equation. This analysis also suggests a width of anedge-layer plasma wρi (w is the number of ion gyro-radius ρi ), a beta value at the separatrix βs , and the formation ofmagnetic islands near the field null. In order to confirm these findings by the magnetic method, a radiation power
density of the FRC is measured. It is found that the w and βs values agree well with those obtained by the radiation
measurement. However, the positions of the magnetic islands do not coincide with those appearing in an intensity
contour map of the radiation. Asymmetrical profiles of the radiation are also discussed with respect to thenonconcentricity of the pressure constant surface. The measured profiles can be explained when the center of the FRC
is shifted by 0.1–0.2rs (rs is a separatrix radius) without any shift of the separatrix surface.
Keywords:
field-reversed configuration, theta pinch, compact torus, separatrix shape, internal structure, magnetic structure,magnetic island
the separatrix βs [7]. It is found from the calculation that themagnetic islands are observed near the field null at theformation phase. Moreover, although they are coalesced onceby the axial contraction of the plasma to attain theequilibrium, they appear again at the quiescent phase.
In order to confirm the βs value and the formation of themagnetic islands optically, a radiation profile of the FRC ismeasured. The profile is analyzed by the Abel inversion
method to obtain a radial profile of the radiation power
density i(r). When the plasma temperature is assumed to beuniform, βs and the thickness of the edge-layer plasma canbe estimated from i(r). Then, we compare the two values
obtained from the optical and magnetic methods. Moreover,
the optical detector array is shifted along the vacuum vesselin order to observe the profile of line-integrated radiation
intensity at several axial positions. The obtained i(r) contours
on the r-z plane are compared to the internal structures fromthe magnetic method. Finally, the nonconcentricity of i(r)contours on the x-y plane is discussed.

1. Introduction
A field-reversed configuration (FRC) is one of thesimplest toroidal plasma configurations. The FRC plasma isconfined within a separatrix by a poloidal magnetic field thatis generated by plasma diamagnetic toroidal currents, and has
no toroidal field. It is well known that FRC plasma has many
advantages including its compact, simple geometry, and its
having a high plasma beta and being a natural divertor.
Therefore, the physics of FRC’s has been studied fromexperimental and theoretical points of view [1-4].
The separatrix shape of a FRC plasma is determined
approximately by an excluded flux method; it is derived fromthe combination of a magnetic flux φ inside the vacuum vessel
and an axial field Bz outside the plasma [5]. Although anexcluded flux radius r∆φ by this method can be measured
easily and is used as a standard diagnostic system in FRCexperiments, it is known that r∆φ does not coincide with thereal separatrix shape, particularly in end regions. Recently, inorder to measure the separatrix shape rs (z) and the internalstructure, the iterative method has been proposed [6], where
the separatrix shape of the FRC is determined by comparing
iteratively measured magnetic fluxes with a solution of theGrad-Shafranov equation. The magnetic structure inside theseparatrix can also be obtained by solving the equation using
the determined separatrix shape rs (z) and the beta value at

2. Experimental apparatus
FRC plasmas are produced by a negative-biased thetapinch device called NUCTE-III, as illustrated in Fig. 1(a) [8].
The theta-pinch coil consists of 28 coil elements; a centerregion has a 0.17 m radius and a mirror one has a 0.15 m

Corresponding author’s e-mail: gota@pyxis.phys.cst.nihon-u.ac.jp1473©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchGota H. et al., Magnetic Structures inside the Separatrix on Field-Reversed Configuration Plasmas(a)(b)

Fig. 1(a) Experimental device NUCTE-III and (b) arrangementof the optical detector array.

radius, and both regions are connected smoothly by taper
coils. Then, a mirror ratio of the field is RM = (0.17/0.15)2 ~–1.28 on the z-axis. The total length of the theta-pinch coil isapproximately 1.5 m; the center region is 1.0 m and the mirror
regions are 0.25 m. The strength of the negative bias field is–0.032 T, and that of the confinement field reaches 0.6 T on
the midplane (z = 0) at t = 4 µs and decays at t = 120 µs invacuum. The vacuum vessel consists of a transparent quartz
tube of 0.13 m radius and 2.0 m length. FRC plasmas areproduced by filling 15 mTorr deuterium into the vacuum
vessel.
Magnetic probes and flux loops are installed along thevacuum vessel to measure axial field Bz and magnetic flux φat multiple axial positions. These arrays are used fordetermining r∆φ (z) by the excluded flux method and rs (z) by
the iterative method. An array of optical detectors is shownin Fig. 1(b). A detector at x = x1 can measure the radiation
ranging in a wavelength 550±5 nm along an optical path
parallel to the y-axis, I(x1) = ∫ i( y)dy [9]. The line-integrated
electron density is measured by a 3.39 µm helium-neon laserinterferometer at z = 0.
Typical plasma parameters at z = 0 are the separatrix
radius rs (0) ~– 0.06, the full length ls ~– 0.8 m, the average beta
value 〈β 〉 = 1 – xs2/2 ~– 0.94 (xs : normalized separatrix radius
by coil radius rw ), the average electron density 〈ne〉 ~– 3 ×1021 m-3, the sum of electron and ion temperatures Te + Ti ~–200 eV, the configuration life time τφ ~– 70 µs, and the particle
confinement time τN ~– 60 µs during t = 20–30 µs. When Ti ~–Te ~– 100 eV is assumed, a gyro-radius of deuterium ion is ρi~– 4.1 mm in the vacuum field B0 = 0.5 T.

Fig. 2where µ0 and p(ψ ) are, respectively, the permeability of freespace and the plasma pressure. It is assumed that p(ψ ) has adifferent form corresponding to the magnetic regions ψ < 0(inside the separatrix), 0 ≤ ψ ≤ ψedge (edge-layer region), andψedge < ψ (vacuum region), as follows;⎧ ps exp {(− log β s ψ n )ψ } at ψ < 0 ,⎪⎪p(ψ ) = ⎨ ps exp {−2ψ ψ edge }at 0 ≤ ψ ≤ ψ edge , (2)⎪at ψ edge < ψ ,⎪⎩0where ps is the plasma pressure on the separatrix surface (ψ= 0), ψn and ψedge are flux functions at the magnetic fieldnull (r = R) and at the boundary of the edge-layer [r = rs (0) +wρi : w is the number of ion gyro-radius ρi ], respectively.
Since the true value of ψn is not known in advance, thecalculation is started using an appropriate value of ψn0 . The
solution of this calculation gives the flux at the field null ψ ′n0 .
When ψn0 ~– ψ ′n0 , ψn0 can be considered to be close to thetrue value of ψn . The pressure profile inside the separatrix isequivalent to the rigid rotor profile model under theassumption of uniform plasma temperature. Moreover, theprofile at the edge-layer satisfies the boundary condition p =ps at ψ = 0, and it is assumed that p = ps /e2 at ψ = ψedge .
In order to investigate the behavior of the FRC plasma,
the time evolution of the separatrix and the internal structureis determined by the iterative method using Eqs. (1) and (2).
On the basis of this calculation at the formation phase (t = 2–
10 µs), the flux contours are given in Fig. 2. It is seen thatthe closed separatrix surface starts to form at t = 2 µs in theregion under the taper coil (z/lc ~– 0.7: normalized length by

3. Internal structures
The separatrix shape can be determined with the edgelayer plasma in the open field region by the iterative method,
and the magnetic structure inside the separatrix is calculated
at the same time. For this calculation, the Grad-Shafranov
equation for the magnetic flux function ψ is used as follows;∂ 2ψ 1 ∂ψ ∂ 2ψ
dp (ψ )−+ 2 + µ0r 2=0,2dψr ∂r ∂z∂r

Time evolution of flux contour maps at the formation
phase. Dashed lines indicate the boundary of the edgelayer plasma (ψ = ψedge).

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Gota H. et al., Magnetic Structures inside the Separatrix on Field-Reversed Configuration Plasmas

uniform temperature. Therefore, i(r) ∝ ne (r) ∝ β (r). It isfound that the beta value at the separatrix (r = rs) is βs ~– 0.75
and the thickness of the edge-layer plasma is about 0.024 m
at t = 20 µs on the z = 0 plane. The βs value and the thicknessof the edge-layer plasma do not change significantly during t= 10–25 µs, and remain consistent with the values analyzed
by the magnetic method.
An intensity contour map on the r-z plane can be formed
from these i(r) profiles. Figure 4(a) shows the magnetic flux
contours at t = 20 µs on the discharge shown in Fig. 2, andFig. 4(b) depicts the contours of the radiation power densityi(r) normalized by i(0) at z = 0. Since the optical detectors
are not installed beyond z/lc > 0.43, the contours disappear
near the end region of the FRC. It is seen that the magnetic
islands in (a) and the high-density region in (b) exist atdifferent positions, and the separatrix shape in (b) is more
slender than that in (a). Next, the I(x) profiles at four axial
positions are compared with line-integrated square pressure
along the y-axis P(x) = ∫ p2(x, y)dy, where p(x, y) is thesolution given in Sec. 3. It is found that P(x) comparatively
agrees with I(x) at z = 0 and 0.11 m, but not I(x) at z = 0.22
and 0.33 m.
We discuss the asymmetry of the I(x) profiles in Fig. 3(c)and (d). For this purpose, the rigid rotor profile model for thedensity and the magnetic field is modified including
nonconcentricity as follows [10].

half coil length lc ) and the X-point of the main plasma moves
inward by axial contraction to attain equilibrium, while thetorn plasma remaining in the mirror region moves toward thecoil end and fades away.
Here, we notice the magnetic structure inside theseparatrix. It is seen that the closed field lines near the fieldnull are torn into three parts at t = 2 µs. These magnetic
islands coalesce into a large one by axial contraction at t = 6µs, and after that, the large island moves toward the midplanewith time to attain the equilibrium form. It is confirmed thatthese behaviors do not change significantly when the otherpressure profiles are assumed in solving Eq. (1) [7].

4. Radiation profile of FRC
In order to estimate the amount of the edge-layer plasma
and to determine the internal structures by means of an opticalmethod, the set of optical detectors shown in Fig. 1(b) isshifted along the vacuum vessel at four axial positions (z = 0,0.11, 0.22, and 0.33 m). The profiles I(x) at t = 20 µs are
depicted in Fig. 3, where (a) and (b) retain relatively
symmetrical forms against x = 0, but (c) and (d) are slightly
asymmetrical.
The radiation profiles can be analyzed by the Abelinversion method in order to estimate the radial profile of theradiation power density i(r). Since it is confirmed that theradiation consists mainly of bremsstrahlung [9], i(r) isapproximately proportional to n2e under the assumption of⎪ ⎡ ( x − δ x) 2 + y 2 ⎤ ⎫⎪2⎧
− 1⎥ ⎬ ,n(x, y) = nmsech ⎨ K ⎢R2⎦ ⎭⎪⎩⎪ ⎣(3)⎪⎧ ⎡ ( x − δ x) 2 + y 2 ⎤ ⎪⎫B(x, y) = B0tanh ⎨ K ⎢− 1⎥ ⎬ ,R2⎪⎩ ⎣⎦ ⎪⎭(4)

where n m is n at r = R, B 0 is B at the wall, and K =tanh -1 [B(r s )/B 0 ]. The value of δx which denotes theFig. 3

Profiles of the line-integrated radiation intensity at four
axial positions; (a) z = 0, (b) 0.11 m, (c) 0.22 m, and (d)0.33 m.

Fig. 4

Contour maps of (a) the magnetic flux and (b) theradiation power density at t = 20 µs.

Fig. 53475(a) Contour map of the plasma density at the x-y planeand (b) the line-integrated square density profile of anonconcentric FRC.

Gota H. et al., Magnetic Structures inside the Separatrix on Field-Reversed Configuration Plasmas

nonconcentricity of the n(x, y) and B(x, y) profiles is definedasδx =d(rs − x ) ,rs

is 0.75 and the thickness of the edge-layer plasma is 6 times
the ion gyro-radius at the quiescent phase. These findings
agree with those from the magnetic analysis.
From the radiation profiles at several axial positions, theintensity contour map on the r-z plane is produced, although
the profiles are asymmetrical except for that near the centerregion. It is found that the magnetic islands and the highintensity region exist at different positions. An explanation of
the asymmetry of the radiation profiles is attempted on thebasis of the nonconcentricity of the pressure constant surface.
It is found that the radiation profiles coincide with thenonconcentric profiles, the center of which is shifted fromthe equilibrium position by 0.1–0.2rs without any shift of theseparatrix surface.

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where d is the shift distance of the central axis of the FRC
from the z-axis. The n(x, y) contours and an I(x)r. r. = ∫ n2(x,y)dy profile in d = 0.2rs are depicted in Fig. 5(a) and (b),respectively. The profile (c) in Fig. 3 coincides well with theI(x)r. r. profile; moreover, the profile (d) can also agree with
the I(x)r. r. profile when d = 0.1rs . It is suggested from theseanalyses that the density contours inside the separatrix might
be the nonconcentric profile at end regions.

5. Summary
References

The separatrix shapes and the internal structures of FRC
plasma are determined at the same time by solving the GradShafranov equation, where the pressure term not only inside
the separatrix but also in the edge plasma region is assumed
by an exponential function of ψ. The closed separatrix surface
starts to form during t = 2 µs after the initiation of theconfinement field in the region under the taper coil. The Xpoint of the FRC moves inward by axial contraction, while
the torn plasma moves toward the coil end. It is noted thatthe closed field lines near the field null are torn into threeparts at the formation phase, and they coalesce into a large
island by axial contraction. The large island moves toward
the midplane with time.
In order to confirm the edge plasma parameters and theformation of the magnetic islands, the radiation profile of theFRC is observed. The profile is analyzed by the Abelinversion method in order to determine the radial profile of
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