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# NIFS PROC 88

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```surface passing X points is the separatix surface.
\u3bas =
2Zx
Rmax \u2212Rx =
AZx
R
\u3b2p0 \u2261 p(R, 0)\u2212 pb
B2z (Rx, 0)/2\u3bc0
=
a
a+ (R2/R2x)b
.
When A and \u3bas are speci\ufb01ed, \u3b2p0 is \ufb01xed. To avoid this inadequateness, Weening (ref.[2]) added
an additional particular solution r2 ln(r2/R2\u3b1)\u2212 r2 to Solovev solution; that is,
\u3c8 =
b+ d
2
(
1\u2212 r
2
R2x
)
z2 +
a+ (R2/R2x)(b+ d)
8R2
(
(r2 \u2212R2)2\u2212(R2 \u2212R2x)2
)
(6.17)
\u2212d
4
(
r2 ln
r2
R2x
\u2212 (r2 \u2212R2x)
)
.
When the plasma boundary is chosen to be the separatrix \u3c8(r, z) = 0, the aspect ratioA, elongation
ratio\u3bas, and central poloidal beta \u3b2p0 are
Z2x
R2x
= \u22121
2
(
a
b+ d
+
R2
R2x
)(
1\u2212 R
2
x
R2
)
R2max
R2
=
(
2\u2212 R
2
x
R2
)
+
2d[x ln x/(x\u2212 1)\u2212 1]
a+ (R2/R2x)(b+ d)
, x \u2261 R
2
max
R2x
1
A
=
Rmax/R \u2212Rx/R
2
, \u3bas =
AZx
R
\u3b2p0 =
a
a+ (R2/R2x)(b+ d)
[
1 +
2d(ln(R2/R2x)\u2212 (1\u2212R2/R2x))
(a+ (R2/R2x)(b+ d))(1\u2212R2x/R2)
]
.
The magnetic surface \u3c8, the magnetic \ufb01eld B and the pressure p in translationally symmetric
system (\u2202/\u2202z = 0) are given by
\u3c8 = Az(r, \u3b8),
Br =
1
r
\u2202\u3c8
\u2202\u3b8
, B\u3b8 = \u2212\u2202\u3c8
\u2202r
, Bz =
\u3bc0
2\u3c0
I(\u3c8),
p = p(\u3c8).
56
6.3 Tokamak Equilibrium 57
Fig.6.3 Toroidal coordinates.
The equilibrium equation is reduced to
1
r
\u2202
\u2202r
(
r
\u2202\u3c8
\u2202r
)
+
1
r2
\u22022\u3c8
\u2202\u3b82
+ \u3bc0
\u2202p(\u3c8)
\u2202\u3c8
+
\u3bc20
8\u3c02
\u2202I2(\u3c8)
\u2202\u3c8
= 0,
j =
1
2\u3c0
I \u2032B + p\u2032ez,
\u394\u3c8 + \u3bc0jz = 0.
It is possible to drive the similar equilibrium equation in the case of helically symmetric system.
6.3 Tokamak Equilibrium (ref.[3])
The equilibrium equation for an axially symmetric system is given by (6.14). The 2nd and 3rd
terms of the left-hand side of the equation are zero outside the plasma region. Let us use toroidal
coordinates (b, \u3c9, \u3d5) (\ufb01g.6.3). The relations between these to cylindrical coordinates (r, \u3d5, z) are
r =
R0 sinh b
cosh b\u2212 cos\u3c9 , z =
R0 sin\u3c9
cosh b\u2212 cos\u3c9 .
The curves b = b0 are circles of radius a = R0 sinh b0, centered at r = R0 coth b0, z = 0. The curves
\u3c9 =const. are also circles with the center at r = 0, a = R0 cos\u3c9/ sin\u3c9. When the magnetic-surface
function \u3c8 is replaced by F , according to
\u3c8 =
F (b, \u3c9)
21/2(cosh b\u2212 cos\u3c9)1/2
the function F satis\ufb01es
\u22022F
\u2202b2
\u2212 coth b\u2202F
\u2202b
+
\u22022F
\u2202\u3c92
+
1
4
F = 0
outside the plasma region. When F is expanded as
F = \u3a3gn(b) cosn\u3c9,
the coe\ufb03cient gn satis\ufb01es
d2gn
db2
\u2212 coth bdgn
db
\u2212
(
n2 \u2212 1
4
)
gn = 0.
57
58 6 Equilibrium
Fig.6.4 The coordinates r, z and \u3c1, \u3c9\u2032
There are two independent solutions:(
n2 \u2212 1
4
)
gn = sinh b
d
db
Qn\u22121/2(cosh b),
(
n2 \u2212 1
4
)
fn = sinh b
d
db
Pn\u22121/2(cosh b).
P\u3bd(x) and Q\u3bd(x) are Legendre functions. If the ratio of the plasma radius to the major radius
a/R0 is small, i.e., when eb0 \ufffd 1, then gn and fn are given by
g0 = eb/2, g1 = \u221212e
\u2212b/2, f0 =
2
\u3c0
eb/2(b+ ln 4\u2212 2), f1 = 23\u3c0e
3b/2.
If we take terms up to cos\u3c9, F and \u3c8 are
F = c0g0 + d0f0 + 2(c1g1 + d1f1) cos\u3c9,
\u3c8 =
F
21/2(cosh b\u2212 cos\u3c9)1/2 \u2248 e
\u2212b/2(1 + e\u2212b cos\u3c9)F.
Use the coordinates \u3c1, \u3c9\u2032 shown in \ufb01g.6.4. These are related to the cylindrical and toroidal
coordinates as follows:
r = R0 + \u3c1 cos\u3c9\u2032 =
R0 sinh b
cosh b\u2212 cos\u3c9 z = \u3c1 sin\u3c9
\u2032 =
R0 sin\u3c9
cosh b\u2212 cos\u3c9 .
When b is large, the relations are
\u3c9\u2032 = \u3c9,
\u3c1
2R0
\u2248 e\u2212b.
Accordingly the magnetic surface \u3c8 is expressed by
\u3c8 =c0 +
2
\u3c0
d0(b+ ln 4\u2212 2)
+
[(
c0 +
2
\u3c0
d0(b+ ln 4\u2212 2)
)
e\u2212b +
(
4
3\u3c0
d1e
b \u2212 c1e\u2212b
)]
cos\u3c9
=d\u20320
(
ln
8R
\u3c1
\u2212 2
)
+
(
d\u20320
2R
(
ln
8R
\u3c1
\u2212 1
)
\u3c1+
h1
\u3c1
+ h2\u3c1
)
cos\u3c9.
In terms of \u3c8, the magnetic-\ufb01eld components are given by
rBr = \u2212\u2202\u3c8
\u2202z
, rBz =
\u2202\u3c8
\u2202r
,
rB\u3c1 = \u2212 \u2202\u3c8
\u3c1\u2202\u3c9\u2032
, rB\u3c9\u2032 =
\u2202\u3c8
\u2202\u3c1
.
58
6.3 Tokamak Equilibrium 59
From the relation
\u2212d
\u2032
0
\u3c1
= rB\u3c9\u2032 \u2248 R\u2212\u3bc0Ip2\u3c0\u3c1 ,
the parameter d\u20320 can be taken as d\u20320 = \u3bc0IpR/2\u3c0. Here Ip is the total plasma current in the \u3d5
direction. The expression of the magnetic surface is reduced to
\u3c8 =
\u3bc0IpR
2\u3c0
(
ln
8R
\u3c1
\u2212 2
)
+
(
\u3bc0Ip
4\u3c0
(
ln
8R
\u3c1
\u2212 1
)
\u3c1+
h1
\u3c1
+ h2\u3c1
)
cos\u3c9\u2032 (6.18)
where R0 has been replaced by R. In the case of a/R \ufffd 1, the equation of pressure equilibrium
(6.9) is
\u3008p\u3009 \u2212 pa = 12\u3bc0 ((B
2
\u3d5v)a + (B
2
r +B
2
z )a \u2212 \u3008B2\u3d5\u3009).
Here \u3008 \u3009 indicates the volume average and pa is the plasma pressure at the plasma boundary. The
value of B2r +B
2
z is equal to B
2
\u3c9\u2032 . The ratio of \u3008p\u3009 to \u3008B2\u3c9\u2032\u3009/2\u3bc0 is called the poloidal beta ratio \u3b2p.
When pa = 0, \u3b2p is
\u3b2p = 1 +
B2\u3d5v \u2212 \u3008B2\u3d5\u3009
B2\u3c9\u2032
\u2248 1 + 2B\u3d5v
B2\u3c9\u2032
\u3008B\u3d5v \u2212B\u3d5\u3009.
B\u3d5 and B\u3d5v are the toroidal magnetic \ufb01elds in the plasma and the vacuum toroidal \ufb01elds respec-
tively. When B\u3d5 is smaller than B\u3d5v, the plasma is diamagnetic, \u3b2p > 1. When B\u3d5 is larger
than B\u3d5v, the plasma is paramagnetic, \u3b2p < 1. When the plasma current \ufb02ows along a line of
magnetic force, the current produces the poloidal magnetic \ufb01eld B\u3c9\u2032 and a poloidal component of
the plasma current appears and induces an additional toroidal magnetic \ufb01eld. This is the origin of
the paramagnetism.
When the function (6.18) is used, the magnetic \ufb01eld is given by
B\u3c9\u2032 =
1
r
\u2202\u3c8
\u2202\u3c1
=
\u2212\u3bc0Ip
2\u3c0\u3c1
+
(
\u3bc0Ip
4\u3c0R
ln
8R
\u3c1
+
1
R
(
h2 \u2212 h1
\u3c12
))
cos\u3c9\u2032,
B\u3c1 = \u2212 1
r\u3c1
\u2202\u3c8
\u2202\u3c9\u2032
=
(
\u3bc0Ip
4\u3c0R
(
ln
8R
\u3c1
\u2212 1
)
+
1
R
(
h2 +
h1
\u3c12
))
sin\u3c9\u2032.
(6.19)
The cross section of the magnetic surface is the form of
\u3c8(\u3c1, \u3c9\u2032) = \u3c80(\u3c1) + \u3c81 cos\u3c9\u2032.
When \u394 = \u2212\u3c81/\u3c8\u20320 is much smaller than \u3c1, the cross section is a circle whose center is displaced
by an amount (see \ufb01g.6.5)
\u394(\u3c1) =
\u3c12
2R
(
ln
8R
\u3c1
\u2212 1
)
+
2\u3c0
\u3bc0IpR
(h1 + h2\u3c12).
Let us consider the parameters h1 and h2. As will be shown in sec.6.4, the poloidal component B\u3c9\u2032
of the magnetic \ufb01eld at the plasma surface (r = a) must be
B\u3c9\u2032(a, \u3c9\u2032) = Ba
(
1 +
a
R
\u39bcos\u3c9\u2032
)
(6.20)
at equilibrium. a is the plasma radius and
\u39b = \u3b2p +
li
2
\u2212 1 (6.21)
59
60 6 Equilibrium
Fig.6.5 Displacement of the plasma column.
\u3c80(\u3c1\u2032) = \u3c80(\u3c1)\u2212 \u3c8\u20320(\u3c1)\u394 cos\u3c9, \u3c1\u2032 = \u3c1\u2212\u394cos\u3c9.
and \u3b2p is the poloidal beta ratio
\u3b2p =
p
(B2a/2\u3bc0)
(6.22)
and li is
li =
\u222b
B2\u3c9\u2032\u3c1d\u3c1d\u3c9
\u2032
\u3c0a2B2a
. (6.23)
The parameters h1 and h2 must be chosen to satisfy B\u3c1 = 0 and B\u3c9\u2032 = Ba(1+ (a/R)\u39b cos\u3c9\u2032) at
the plasma boundary, i.e.,
h1 =
\u3bc0Ip
4\u3c0
a2
(
\u39b+
1
2
)
, h2 = \u2212\u3bc0Ip4\u3c0
(
ln
8R
a
+ \u39b\u2212 1
2
)
. (6.24)
Accordingly \u3c8 is given by
\u3c8 =
\u3bc0IpR
2\u3c0
(
ln
8R
\u3c1
\u2212 2
)
\u2212 \u3bc0Ip
4\u3c0
(
ln
\u3c1
a
+
(
\u39b+
1
2
)(
1\u2212 a
2
\u3c12
))
\u3c1 cos\u3c9\u2032. (6.25)
The term h2\u3c1 cos\u3c9\u2032 in \u3c8 brings in the homogeneous vertical \ufb01eld
Bz =
h2
R
,
which is to say that we must impose a vertical external \ufb01eld. When we write \u3c8e = h2\u3c1 cos\u3c9\u2032 so
that \u3c8 is the sum of two terms, \u3c8 = \u3c8p + \u3c8e, \u3c8e and \u3c8p are expressed by
\u3c8e = \u2212\u3bc0Ip4\u3c0
(
ln
8R
a
+ \u39b\u2212 1
2
)
\u3c1 cos\u3c9\u2032 (6.26)
\u3c8p =
\u3bc0IpR
2\u3c0
(
ln
8R
\u3c1
\u2212 2
)
+
\u3bc0Ip
4\u3c0
((
ln
8R
\u3c1
\u2212 1
)
\u3c1+
a2
\u3c1
(
\u39b +
1
2
))
cos\u3c9\u2032. (6.27)
These formulas show that a uniform magnetic \ufb01eld in the z direction,
B\u22a5 = \u2212\u3bc0Ip4\u3c0R
(
ln
8R
a
+\u39b\u2212 1
2
)
, (6.28)
must be applied in order to maintain a toroidal plasma in equilibrium (\ufb01g.6.6). This vertical \ufb01eld
weakens the inside poloidal \ufb01eld and strengthens the outside poloidal \ufb01eld.
The amount of B\u22a5 (6.28) for the equilibrium can be derived more intuitively. The hoop force by
which the current ring of a plasma tends to expand is given by
Fh = \u2212 \u2202
\u2202R
LpI
2
p
2
\u2223\u2223\u2223\u2223
LpIp=const.
=
1
2
I2p
\u2202Lp
\u2202R
,
60
6.3 Tokamak Equilibrium 61
Fig.6.6 Poloidal magnetic \ufb01eld due to the combined plasma current and vertical magnetic \ufb01eld.```