J. Plasma Fusiorr Res. SERIES, Vol. 5 (2002) 413417

Modification of the Magnetic Field Structure in the Vicinityof the X-Point by the Strong Mirror Field for a Field-Reversed
Gonfiguration (FRC) with the Thick Edge-Layer Plasma
SUZUKI Yukihisa, OKADA Shigefumir and GOTO Seiichir
Department of Electronics and Information Engineering, Graduate School of Engineering,
Tolcyo Metropolitan University, 1-1 Minami Oosawa, Hachioji, Tolqto 192-0397, Japan1

Plasma Physics lnboratory, Graduate Schoot of Engineering, Osaka University,
Yamada-Oka 2-1, Suita, Osaka 565-0871, Japan(Received: 1l December 2O0l / Accepted: 6 September 2002)

Abstract
Mrrdification of the magnetic field structure in the vicinity of the x-points and changes of theseparat::ix shape are investigated under the pressure effects due to an edge-layer plasma together with amirror iield by the two-dimensional (2-D) MHD equilibrium solutions of field-reversed configuration(FRC) ,:btained from the Grad-Shafranov equation. To explore the coupling pressure effects caused byedge-layer plasma and mirror field, the equilibrium calculations are performed by the combinations ofseveral values of mirror ratio (R-) and of edge-layer width (E), respectively. A summary of results forpresent study is as follows. In the condition of weak mirror field (1.0<R.< 1.6), ry=g surface opens up in
the axi,rl direction, namely 2-D FRC equilibrium with closed separatrix is lost when 6 is above thecritical value. When the mirror field is strong enough as R-> 1.6, F=0 surface never opens up for any d.
These original results make it clear that large magnetic curvature produced by the strong mirror fieldenhanct:s the magnetic stress around the x-point, so that the ends ofFRC are effectively sustained by thisenhanct:d magnetic stress, which counteracts the edge-layer plasma pressure effect.

Keywords:FRC, I\,IHD equilibrium, Grad-Shafranov equation, mirror field, edgeJayer plasma, axial force balance

1. lntroduction
A field-re versed configuration [,2] (FRC) is anedge-layer plasma is shown in ref. 5. In this study, it isfound that V=0 surface opens and the closed 2-D FRC
equilibrium is lost when the thickness increases further.
Here, t4 (r,z) denotes poloidal magnetic flux function,and closed W=0 surface corresponds to separatrix inordinary FRC equilibria. In ref. 6, the dependence of

axisymmetric t:longated compact toroid with negligible
toroidal magnetic field. It has an extremely high average
B value which reaches 0.9 or higher [3]. Here the valueof B is the ratio of the plasma pressure to the external
magnetic pressure. Because of such a characteristic,
FRC plasma is thought to be influenced by the change

the separatrix shape on B, is shown. Here B, denotes
the local B value on the separatrix. This study indicatesthat axial pressure balance does not hold unless ry=0
surface opens up, as the value of p, is above the criticalof external ma:lnetic field and the edge-layer plasma.
Previousll', there were a few studies in which edgelayer effects were taken into account [4-6]. The relation
between the st:paratrix shape and the thickness of thevalue.
@2002 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorrespondinq author's e-mail: suzuki@ eei.metro-u.acjp4t3

Suzuki Y. et at., Modiftcation of the Magnetic Field Structure in the Viciniy of the X-Point for FRC with the Thick EL Plasma

appropriate as the pressure model [13]. The recent FIX
experiment [2] shows that an exponentially decaying
pressure profile is suitable to represent the equilibriumin the open field region. It is required that the current

There have been some measurements anddiscussion of the edge-layer plasma [7-10]. In ref. 7
plasma density was measured and found to extend
outside the separatrix to form the edge-layer. Moredirectly, in a recent study on the FRC Injection
Experiment tlll (FIX) at Osaka University, thethickness of the edge-layer plasma is inferred from theplasma pressure to be approximately 30-4O 7o of theseparatrix radius [2]. A feature of the FRC confined in
the FIX machine is the existence of a large mirror ratioas R*=1,-$ at both ends. The value of R- denotes theratio of the magnetic field strength in the mirror region
to that in the midplane without the FRC plasma. In thisstudy, the terms of "strong mirror field" or "large mirror
ratio" are used to refer to R.>2.0, which indicates theoperation parameter of FIX. The terms of "weak mirror
field" or "small mirror ratio" indicate the condition of
1.0< R.<2.0, which is used most of other FRC device.
It is expected that a high edge-layer pressure can besustained by a strong mirror field. However, in theprevious numerical studies, a strong mirror field has notbeen considered as to its effects onthe edge-layer.
In this paper, the following new results are foundas for those coupling pressure effects by 2-D numerical
MHD equilibria. In the case of strong mirror conditionas Rm>1.6, V/=0 surface never opens up for any edgelayer plasma width. Since the magnetic stress around thex-point is enhanced by large magnetic curvature due tothe strong mirror field, the ends of FRC are effectively
sustained no matter how the edge-layer plasma pressure
effect is large.

(b)

10(c)

10Fig. 1 A series of FRC equilibria for 50=2.6, F.=1.0 and
e=40.0: (a) y=30.0; lbl y=20.0; (cl y=2.0. Contoursof the poloidal flux function y(r,zl are shown; contour intervals are 0.1 for r/>0 and 0.002 for ty<O,respectively. The thick contour line denotes thesurface tg=O .

1.

(a) lt'7, T::'-'8.

2. Numerical Method'\,

qntI=O

8:

oThe effect of the edge-layer plasma on theequilibria is investigated using a previously developed
2-D numerical MHD equilibrium code [3]. In thiscode, the Grad-Shafranov equation without toroidal field(b)

is solved to obtain the equilibrium solution. The$.

equation is written in the cylindrical coordinate systemo(r,0,d as follows:

L.V=-r'W
where(c)(1)- qr{).0',a ll i_l* d'a.- '0,\r
3rl dz2o4z

10Fig. 2 A series of FRC equilibria for 50=2.6. R.=2.8 and
e=40.0: (a) y=30.0; (bl y=20.0; lcl y=2.A. Contoursof the poloidal flux function wlr,zl are shown; contour intervals are 0.1 for rlr>0 and 0.004 for vr<O,

Here, tg is the poloidal flux function and p(yr) denotes
the scalar pressure profile function which depends onlyon y. To obtain thin and elongated equilibria as given in
the experimental data of FIX, p(V)= -c(ry+l/2ery2l isrespectively.

4t4

Suzuki Y. et al., Modification of the Magnetic Field Structure in the Viciniy of the X-Point for FRC with the Thick EL plasma

density is continuous across the separatrix. For reasonsmentioned above, we choose the followins model as thety(r,z) are shown in Fig. 1 for R.= 1.0 and a vaiety of yvalues. Figures (a), (b) and (c) correspond to /=30.0,

pressure profile [4].
dp (v)=dV[ -r(t + evl (v' s o),(v\ -ce-YV >oitl.

20.0 and 2.0, respectively. The separatrix surface,defined by t/=0, is denoted by a bold line. When
T=30.0, the qr=Q surface is closed, and a solution withelliptical separatrix shape is obtained. For the case of
T=20.0, however, the r4=g surface is open. When y is2.0, the solution becomes z-independent as shown inFig. I (c). In the case of a thick edge-layer,2<y<2O, anequilibrium solution with closed V=0 surface cannot beobtained. The equilibrium shape is sensitive to the edgelayer pressure distribution, when R-=1.0. In this case,the so-called x-point easily goes to infinity with increasing edge-layer thickness. These results are consistent
with the previous study presented in refs. 5 and 6.

Q)

Here e and Tarc constant parameters, and c is an eigenvalue to be determined by the global constraint that is
the area (56) inside of the separatrix. This global constraint is necessary to obtain non-trivial solutions, since
the solution of the eigenvalue equation derived from eq.

(l) generally gives a variety of FRC equilibria. The
separatrix radius can be controlled by e as described inref. 15. When e is larger, the separatrix radius becomes

smaller. The pressure distribution in the edge-layer isgoverned by 7; which specifies the width of the edge
plasma. An equilibrium with a thick edge-layer is pro-

Equilibria with a strong mirror field, R,=2.8, areshown in Fig. 2. In the same manner as above, Figs.
2(a), (b) and (c) correspond to /=30.0, 20.0 and2.0,
respectively. The t4=Q surface does not open up in
every case and the separatrix shape does not changemuch with this strong mirror case. The values of x. andl, change little in these three cases, which are.r. - 0.61

duced by cho,tsing small values for 7. Note that wedefine ty<0 inside the separatrix and y>0 outside theseparatrix.

The boundary conditions are as follows: y-0 at
r=0, drltldz=0 at z=0 and z-zr, and yr=V/wau at thechamber wall (see Fig. 2(a)). In this paper tg isnormalized by ty*uu. The mirror field is generated by
reducing the wall radius r*2 at zzlzlzr, because the flux

and /, - 9.4. The separatrix length does not change muchwith the edge-layer thickness when the mirror ratio islarge. It is found that the mirror field plays a significant
role for the equilibrium, because with a large mirror
ratio the equilibrium shape is independent of the edge-

is conserved inside the chamber wall as shown in Fie. 2(a). Hence, the mirror ratio is given by (r*1h*2)2.

layer width compared with the case of R-= 1

The approximate existence region of the
3. Results

equilibrium from this calculation is shown in Fig. 3. In

Numerical calculations are performed for thefollowing cast:s: R-=1.0, 1.2, 1.6,2.O, and 2.8. The
range of variation of 7 examined in this study is
1.0<y<40.0 lbr each value of R,. The edge-layer

(ii)
l./21

thickness is cbanged from thick to thin by varying thisparameter 7 The coordinate (r,z) is normalized by the+ +of!

tsrzld).s{a0.-5-5

wall radius r*1 in this study. The cross-sectional area.S6
inside of the separatrix is calculated under that metric.

\(r)

lslzl
is finile value

The global constraint 56 determines the separatrix length
1,. When Se is larger, equilibrium with large /, isobtained. To investigate the effect of mirror field, a longand thin equililrium as ll2 l"-9 and.x.-0.4 is needed inthe condition of R.=1.Q, because mirror field isgenerated at z>4.3. Here, l, is the separatrix length{oi'l').9tIIlll

0. l0

normalized by x., and x, is the ratio of the separatrix
radius to the u'all radius r.1. For reasons stated above,
the global constraint Ss is fixed at 2.6 and the value of s
is chosen to be 40.0 in all of the calculations.
To investigate the dependence of the equilibrium
shape on 7, the examples of the poloidal flux function

Il.o 1.2 1.4 t.6 1.8 2.o 2.2 2.4 2.6n",

Fis 3 The region in which 2-D FRC equilibria exist withrespect to the parameters 6and F.. The value of/"/zr goes to +- in region (ii). FRC equilibria canexist in region (i).
415

Suzuki Y. er al., Modification of the Magnetic Field Structure in the Viciniy of the X-Point for FRC with the Thick EL Plasma

this figure, the vertical and the horizontal axes denoteedge-layer width and R-, respectively, and each line
shows a contour of constant-1. lz1. The edge-layer widthis defined by d at the mid-plane which satisfies
tB. Here r. satisfies r/(r,,0)=Q,B(r) denotes
B(r.+d)=s
local beta value, and ft denotes the BQ) on the ry-0
surface. When Tbecomes smaller, edge-layer width 6
increases. The value of 6'is normalized by the wall
radius in Fig. 3. The three square markers are criticalvalues of 6" calculated at each R*. The bold line

edgelayu

indicates the inferred boundary that divides theparameter region into one in which the value of l"/zr

IIFLZ-12t1\----rl\

goes to +- from that in which l,/er is finite. Equilibria,'Itpotnt5'5

with closed W=0 surface cannot exist above and to theleft of the bold line. In the weak mirror region, fromR-= 1.0 to 1.6, the critical value of 6" increases from 0.2
to 0.37 as R- increases. In the parameter region of thisfigure (6<0.37), the critical boundary disappears forR.>1.6. It is found that only a moderate mirror field is
required to sustain the 2-D FRC equilibrium with a thickedge-layer of 6-0.37.

r.\\f

seParatrix

4. Modification of the Magnetic Field Near
the X-Point
The details of the magnetic field structure and theaxial force balance are examined to interpret thenumerical equilibrium results. To investigate themagnetic structure of the FRC, we make a vector plot of
the magnetic field B around the end region. The

magnitudes of the vectors are normalized by theabsolute value of B (r*1,0). Figures 4(a) and (b)correspond to the conditions R.=1.!, 7=15.0 and
Rr=2.8, T=15.0, respectively. These are the cases witha thick edge-layer. The regions surrounded with thin-Fig. 4 The vector plot of the B field near the end regionof the FRC equilibrium with a thick edge-layerlor/=15.0. (al R-=1.2 and (b) F-=2.8 case.

lined circles are compared in both figures; the magnetic
curvature is larger for R-=!,.$ than for R^=1.2. Largemagnetic curvature produced by the strong mirror fieldenhances the magnetic stress at the end of the FRC.
Another point to notice is that the R-=2.8 case has alarger magnetic field strength than the R^=1.2 case, inthe region near the right side of the x-point marked bybold-lined circles. In this region the strength of themirror field is weakened by the edge pressure as for theweak mirror case of R^=1.2. When R^=1.$, there is astrong B field in this region caused by the strong mirror
field, in order to counteract the edge-layer plasma

According to the above discussion, the reasons how
the edge-layer pressure is supported by the mirror field
and how the 2-D equilibrium is formed can be explainedby considering the axial force balance. To calculate theaxial force balance precisely with respect to numericalequilibria of this study, the calculation region for this isshown in Fig. 5. In this figure, S1-S5 denote distinct
regions for the surface integral. The pressure balance inits general form is given byV.[T.-pI]=0.

pressure. The magnetic stress becomes stronger due tothe large magnetic curvature around the x-point, andconsequently the ends of the FRC are sustained by thestrons mirror field.

r.=* (""-;rr)
416

(3)

Suzuki Y. et al., Modification of the Magnetic Field Structure in the Viciniy of the X-Point for FRC with the Thick EL Plasma

where p is the scalar pressure, I is the unit tensor, and5. Conclusion

T- is the maglletic stress tensor. Neglecting the 0 com-The equilibrium calculation is performed for wideponent, the integration is transformed into a surface inte-

range of mirror ratio as 1.0<R,<2.8. The effects of thegral using Gauss's theorem. The axial force balance isedge-layer pressure for varying thickness of edge-layer

expressed by:

are examined for each mirror ratio by scanning theparameter 7. The region in which closed yr=Q surface

f l^2\I l-++,,lds
LPoJr, \equilibria exist is obtained for the parameter range
0.1<6<0.37 and 1<R.<2.8 due to the calculation. It isfound that the critical value of d" increases from 0.2 to0.37 as the mirror ratio increases from I to 1.6 in theweak mirror case. A mirror ratio in the range from 1.6
to 2.0 is required to sustain the equilibrium for a thickI.

( u, u."o, o *(ni - 4),i" o)asfr
J,.1, (- fr-o)o'=o'tr I['edge-layer of d-0.37. These numerical results can beexplained with an axial force balance calculation. A
simple model of axial force balance, which takes boththe edge and the mirror effects into consideration, showsthat the curvature of the magnetic field near the x-pointis enhanced by the strong mirror field. A sufficiently(4)\t- t) /V (/*, - l.)'+ R. (zr-2,\' .
sind=(r/n.
-Itencoso(=/R- (2,-z,t l r/ lf---t)'* R^lzr-z112

strong mirror field extends the region in which.

equilibrium solutions exist to the case of a thick edge-(5)layer.

The effect of the magnetic curvature is contained in
the second term on the left side of eq. (4). The mirror
effect appears in the third term, and the edge effect isincluded in p of this term. The second term depends onsin4 and cos4 shown in eq. (5). If R-- I and p =Q, si11sand B, on the surface,S3 is 0, so that the second term
becomes 0, and eq. (4) reduces to Barnes relationship[2]. When R^=l and p+O, the edge-layer pressure in thethird term directly influences the force balance. When
R- is increased beyond 1, the second and the third terms

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Fig. 5 A simpler model for the axial force balance. Boththe edge pressure and the mirror field are consid-ered in this model. Symbols from S, to Su denotedistinct regions for the surface integral.

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