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# NIFS PROC 88

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Fig.6.7 Equilibrium of forces acting on a toroidal plasma.
where Lp is the self-inductance of the current ring:
Lp = \u3bc0R
(
ln
8R
a
+
li
2
\u2212 2
)
.
Accordingly, the hoop force is
Fh =
\u3bc0I
2
p
2
(
ln
8R
a
+
li
2
\u2212 1
)
.
The outward force Fp exerted by the plasma pressure is (\ufb01g.6.7)
Fp = \u3008p\u3009\u3c0a22\u3c0.
The inward (contractive) force FB1 due to the tension of the toroidal \ufb01eld inside the plasma is
FB1 = \u2212
\u3008B2\u3d5\u3009
2\u3bc0
2\u3c02a2
and the outward force FB2 by the pressure due to the external magnetic \ufb01eld is
FB2 =
B2\u3d5v
2\u3bc0
2\u3c02a2.
The force FI acting on the plasma due to the vertical \ufb01eld B\u22a5 is
FI = IpB\u22a52\u3c0R.
Balancing these forces gives
\u3bc0I
2
p
2
(
ln
8R
a
+
li
2
\u2212 1
)
+ 2\u3c02a2
(
\u3008p\u3009+ B
2
\u3d5v
2\u3bc0
\u2212 \u3008B
2
\u3d5\u3009
2\u3bc0
)
+ 2\u3c0RIpB\u22a5 = 0,
61
62 6 Equilibrium
Fig.6.8 Volume element of a toroidal plasma.
and the amount of B\u22a5 necessary is
B\u22a5 =
\u2212\u3bc0Ip
4\u3c0R
(
ln
8R
a
+
li
2
\u2212 1 + \u3b2p \u2212 12
)
,
where \u39b = \u3b2p + li/2\u2212 1. Eq.(6.9) is used for the derivation.
6.4 Poloidal Field for Tokamak Equilibrium
The plasma pressure and the magnetic stress tensor are given by (ref.[4])
T\u3b1\u3b2 =
(
p+
B2
2\u3bc0
)
\u3b4\u3b1\u3b2 \u2212 B\u3b1B\u3b2
\u3bc0
.
Let us consider a volume element bounded by (\u3c9, \u3c9 + d\u3c9), (\u3d5,\u3d5 + d\u3d5), and (0, a) as is shown in
\ufb01g.6.8. Denote the unit vectors in the directions r, z, \u3d5 and \u3c1, \u3c9 by er,ez,e\u3d5 and e\u3c1,e\u3c9, respectively.
The relations between these are
e\u3c1 = er cos\u3c9 + ez sin\u3c9, e\u3c9 = ez cos\u3c9 \u2212 er sin\u3c9, \u2202e\u3c9
\u2202\u3c9
= \u2212e\u3c1, \u2202e\u3c1
\u2202\u3c9
= e\u3c9.
(Here \u3c9 is the same as \u3c9\u2032 of sec.6.3). Let dS\u3c1, dS\u3c9, dS\u3d5 be surface-area elements with the normal
vectors e\u3c1,e\u3c9,e\u3d5. Then estimate the forces acting on the surfaces dS\u3d5(\u3d5), dS\u3d5(\u3d5+d\u3d5) ; dS\u3c9(\u3d5),
dS\u3c9(\u3c9 + d\u3c9); and dS\u3c1(a). The sum F \u3d5 of forces acting on dS\u3d5(\u3d5) and dS\u3d5(\u3d5+ d\u3d5) is given by
F \u3d5 = \u2212d\u3c9d\u3d5
\u222b a
0
(
T\u3d5\u3d5
\u2202e\u3d5
\u2202\u3d5
+ T\u3d5\u3c9
\u2202e\u3c9
\u2202\u3d5
+ T\u3d5\u3c1
\u2202e\u3c1
\u2202\u3d5
)
\u3c1d\u3c1
= \u2212d\u3c9d\u3d5
\u222b a
0
(
T 0\u3d5\u3d5 (e\u3c9 sin\u3c9 \u2212 e\u3c1 cos\u3c9)\u2212 T 0\u3d5\u3c9e\u3d5 sin\u3c9
)
\u3c1d\u3c1.
When the forces acting on dS\u3c9(\u3c9) and dS\u3c9(\u3c9+d\u3c9) are estimated, we must take into account the
di\ufb00erences in e\u3c9, T\u3c9\u3b1, dS\u3c9 = (R + \u3c1 cos\u3c9) d\u3c1 d\u3d5 at \u3c9 and \u3c9 + d\u3c9. The sum F \u3c9 of forces is
F \u3c9 = \u2212d\u3c9d\u3d5
\u222b a
0
(
T\u3c9\u3c9
\u2202
\u2202\u3c9
(e\u3c9 (R+ \u3c1 cos\u3c9))
+T\u3c9\u3d5
\u2202
\u2202\u3c9
(e\u3d5 (R+ \u3c1 cos\u3c9)) + T\u3c9\u3c1
\u2202
\u2202\u3c9
(e\u3c1 (R+ \u3c1 cos\u3c9))
+
\u2202T\u3c9\u3c9
\u2202\u3c9
Re\u3c9 +
\u2202T\u3c9\u3d5
\u2202\u3c9
Re\u3d5 +
\u2202T\u3c9\u3c1
\u2202\u3c9
Re\u3c1
)
d\u3c1
= d\u3c9d\u3d5
(
Re\u3c1
\u222b a
0
T 0\u3c9\u3c9d\u3c1
)
+ d\u3c9d\u3d5
[
e\u3c1
(
cos\u3c9
\u222b
T 0\u3c9\u3c9\u3c1d\u3c1+R
\u222b
T (1)\u3c9\u3c9 d\u3c1
)
+e\u3c9
(
sin\u3c9
\u222b
T 0\u3c9\u3c9\u3c1d\u3c1\u2212R
\u222b
\u2202T
(1)
\u3c9\u3c9
\u2202\u3c9
d\u3c1
)]
62
6.4 Poloidal Field for Tokamak Equilibrium 63
+d\u3c9d\u3d5e\u3d5
(
sin\u3c9
\u222b
T 0\u3c9\u3d5\u3c1d\u3c1\u2212R
\u222b
\u2202T
(1)
\u3c9\u3d5
\u2202\u3c9
d\u3c1
)
+d\u3c9d\u3d5
(
\u2212e\u3c9R
\u222b
T (1)\u3c9\u3c1 d\u3c1\u2212 e\u3c1R
\u222b
\u2202T
(1)
\u3c9\u3c1
\u2202\u3c9
d\u3c1
)
= d\u3c9d\u3d5
(
Re\u3c1
\u222b a
0
T 0\u3c9\u3c9d\u3c1
)
+d\u3c9d\u3d5e\u3c1
(
cos\u3c9
\u222b
T 0\u3c9\u3c9\u3c1d\u3c1+R
\u222b (
T (1)\u3c9\u3c9 \u2212
\u2202T
(1)
\u3c9\u3c1
\u2202\u3c9
)
d\u3c1
)
+d\u3c9d\u3d5e\u3c9
(
sin\u3c9
\u222b
T 0\u3c9\u3c9\u3c1d\u3c1\u2212R
\u222b (
T (1)\u3c9\u3c1 +
\u2202T
(1)
\u3c9\u3c9
\u2202\u3c9
)
d\u3c1
)
+d\u3c9d\u3d5e\u3d5
(
sin\u3c9
\u222b
T 0\u3c9\u3c9\u3c1d\u3c1\u2212R
\u222b
\u2202T
(1)
\u3c9\u3d5
\u2202\u3c9
d\u3c1
)
.
As B\u3c1(a) = 0, the force F \u3c1 acting on dS\u3c1(a) is
F \u3c1 = \u2212e\u3c1T\u3c1\u3c1(R+ a cos\u3c9)ad\u3d5d\u3c9 = e\u3c1(\u2212T 0\u3c1\u3c1Ra\u2212 (T (1)\u3c1\u3c1 Ra+ T 0\u3c1\u3c1a2 cos\u3c9)).
The equilibrium condition F \u3d5 + F \u3c9 + F \u3c1 = 0 is reduced to\u222b
T 0\u3c9\u3c9d\u3c1 = T
0
\u3c1\u3c1(a)a,
\u2202
\u2202\u3c9
\u222b
T (1)\u3c9\u3d5d\u3c1 =
2 sin\u3c9
R
\u222b
T 0\u3c9\u3d5\u3c1d\u3c1, (6.29)
\u222b (
T (1)\u3c9\u3c1 +
\u2202T
(1)
\u3c9\u3c9
\u2202\u3c9
)
d\u3c1 =
sin\u3c9
R
\u222b (
T 0\u3c9\u3c9 \u2212 T 0\u3d5\u3d5
)
\u3c1d\u3c1, (6.30)
cos\u3c9
\u222b
(T 0\u3d5\u3d5 + T
0
\u3c9\u3c9)\u3c1d\u3c1+R
\u222b (
T (1)\u3c9\u3c9 \u2212
\u2202T
(1)
\u3c9\u3c1
\u2202\u3c9
)
d\u3c1\u2212 T 0\u3c1\u3c1a2 cos\u3c9 \u2212 T (1)\u3c1\u3c1 Ra = 0. (6.31)
From T (1) \u221d sin\u3c9, cos\u3c9, it follows that \u22022T (1)/\u2202\u3c92 = \u2212T (1). So (6.30) is
\u222b (
\u2202T
(1)
\u3c9\u3c1
\u2202\u3c9
\u2212 T (1)\u3c9\u3c9
)
d\u3c1 =
cos\u3c9
R
\u222b
(T 0\u3c9\u3c9 \u2212 T 0\u3d5\u3d5)\u3c1d\u3c1.
Using this relation, we can rewrite (6.31) as
T (1)\u3c1\u3c1 (a) =
a
R
cos\u3c9
(
\u2212T 0\u3c1\u3c1(a) +
2
a2
\u222b a
0
T 0\u3d5\u3d5.\u3c1d\u3c1
)
. (6.32)
T\u3c1\u3c1 and T\u3d5\u3d5 are given by
T\u3c1\u3c1 = p+
B2\u3c9
2\u3bc0
+
B2\u3d5
2\u3bc0
, T\u3d5\u3d5 = p+
B2\u3c9
2\u3bc0
\u2212 B
2
\u3d5
2\u3bc0
. (6.33)
From (6.14), B\u3d5 is
B\u3d5 =
\u3bc0I(\u3c8)
2\u3c0r
=
\u3bc0I(\u3c8)
2\u3c0R
(
1\u2212 \u3c1
R
cos\u3c9 + · · ·
)
= B\u3d5(\u3c1)
(
1\u2212 \u3c1
R
cos\u3c9 + · · ·
)
. (6.34)
When B\u3c9(a) is written as B\u3c9(a) = Ba +B
(1)
\u3c9 , (6.33) and (6.34) yield the expression
63
64 6 Equilibrium
T (1)\u3c1\u3c1 (a) =
BaB
(1)
\u3c9
\u3bc0
\u2212 B
2
\u3d5v(a)
2\u3bc0
2
a
R
cos\u3c9.
On the other hand, (6.9) and (6.32) give T (1)\u3c1\u3c1 (a) as
T (1)\u3c1\u3c1 (a) =
a
R
cos\u3c9
(
\u2212pa \u2212 B
2
a
2\u3bc0
\u2212 B
2
\u3d5v(a)
2\u3bc0
+ \u3008p\u3009+ \u3008B
2
\u3c9\u3009
2\u3bc0
\u2212 \u3008B
2
\u3d5\u3009
2\u3bc0
)
=
a
R
cos\u3c9
(
B2a
2\u3bc0
li + 2(\u3008p\u3009 \u2212 pa)\u2212 B
2
a
\u3bc0
\u2212 B
2
\u3d5v(a)
\u3bc0
)
where li is the normalized internal inductance of the plasma per unit length (the internal inductance
Li of the plasma per unit length is given by \u3bc0li/4\u3c0). Accordingly, B
(1)
\u3c9 must be
B(1)\u3c9 =
a
R
Ba cos\u3c9
(
li
2
+
2\u3bc0(\u3008p\u3009 \u2212 pa)
B2a
\u2212 1
)
.
Ba is \u3c9 component of the magnetic \ufb01eld due to the plasma current Ip, i.e.,
Ba = \u2212\u3bc0Ip2\u3c0a .
When the poloidal ratio \u3b2p (recall that this is the ratio of the plasma pressure p to the magnetic
pressure due to Ba) is used, B
(1)
\u3c9 is given by
B(1)\u3c9 =
a
R
Ba cos\u3c9
(
li
2
+ \u3b2p \u2212 1
)
. (6.35)
6.5 Upper Limit of Beta Ratio
In the previous subsection, the value of B\u3c9 necessary for equilibrium was derived. In this
derivation, (a/R)\u39b < 1 was assumed, i.e.,(
\u3b2p +
li
2
)
<
R
a
.
The vertical \ufb01eld B\u22a5 for plasma equilibrium is given by
B\u22a5 = Ba
a
2R
(
ln
8R
a
+ \u39b\u2212 1
2
)
.
The direction of B\u22a5 is opposite to that of B\u3c9 produced by the plasma current inside the torus,
so that the resultant poloidal \ufb01eld becomes zero at some points in inside region of the torus
and a separatrix is formed. When the plasma pressure is increased and \u3b2p becomes large, the
necessary amount of B\u22a5 is increased and the separatrix shifts toward the plasma. For simplicity,
let us consider a sharp-boundary model in which the plasma pressure is constant inside the plasma
boundary, and in which the boundary encloses a plasma current Ip. Then the pressure-balance
equation is
B2\u3c9
2\u3bc0
+
B2\u3d5v
2\u3bc0
\u2248 p+ B
2
\u3d5i
2\u3bc0
. (6.36)
The \u3d5 components B\u3d5v, B\u3d5i of the \ufb01eld outside and inside the plasma boundary are proportional
to 1/r, according to (6.14). If the values of B\u3d5v, B\u3d5i at r = R are denoted by B0\u3d5v, B0\u3d5i respectively,
(6.36) may be written as
B2\u3c9 = 2\u3bc0p\u2212 ((B0\u3d5v)2 \u2212 (B0\u3d5i)2)
(
R
r
)2
.
64
6.6 P\ufb01rsch-Schlu¨ter Current 65
The upper limit of the plasma pressure is determined by the condition that the resultant poloidal
\ufb01eld at r = rmin inside the torus is zero,
2\u3bc0pmax
r2min
R2
= (B0\u3d5v)
2 \u2212 (B0\u3d5i)2. (6.37)
As r is expressed by r = R+ a cos\u3c9, (6.37) is reduced (with (rmin = R\u2212 a)) to
B2\u3c9 = 2\u3bc0pmax
(
1\u2212 r
2
min
r2
)
= 8\u3bc0pmax
a
R
cos2
\u3c9
2
.
Here a/R \ufffd 1 is assumed. From the relation \u222e B\u3c9ad\u3c9 = \u3bc0Ip, the upper limit \u3b2cp of the poloidal
beta ratio is
\u3b2cp =
\u3c02
16
R
a
\u2248 0.5R
a
. (6.38)
Thus the upper limit of \u3b2cp is half of the aspect ratio R/a in this simple model. When the rotational
transform angle \u3b9 and the safety factor qs = 2\u3c0/\u3b9 are introduced, we \ufb01nd that
B\u3c9
B\u3d5
=
a
R
(
\u3b9
2\u3c0
)
=
a
Rqs
,
so that
\u3b2 =
p
B2/2\u3bc0
\u2248 p
B2\u3c9/2\u3bc0
(
B\u3c9
B\u3d5
)2
=
(
a
Rqs
)2
\u3b2p. (6.39)
Accordingly, the upper limit of the beta ratio is
\u3b2c =
0.5
q2s
a
R
. (6.40)
6.6 P\ufb01rsch-Schlu¨ter Current
When the plasma pressure is isotropic, the current j in the plasma is given by (6.1) and (6.4) as
j\u22a5 =
b
B
×\u2207p
\u2207 · j\u2016 = \u2212\u2207 · j\u22a5 = \u2212\u2207 ·
(
B
B2
×\u2207p
)
= \u2212\u2207p · \u2207 ×
(
B
B2
)
.
Then j\u2016 is
\u2207 · j\u2016 = \u2212\u2207p ·
((
\u2207 1
B2
×B
)
+
\u3bc0j
B2
)
= 2\u2207p · \u2207B ×B
B3
(6.41)
\u2202j\u2016
\u2202s
= 2\u2207p · (\u2207B × b)
B2
, (6.42)
where s is length along a line of magnetic force. In the zeroth-order