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# NIFS PROC 88

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```approximation, we can put
B \u221d 1/R, p = p(r), and \u2202/\u2202s = (\u2202\u3b8/\u2202s)\u2202/\u2202\u3b8 = (\u3b9/(2\u3c0R))\u2202/\u2202\u3b8, where \u3b9 is the rotational transform
angle. When s increases by 2\u3c0R, \u3b8 increases by \u3b9. Then (6.42) is reduced to
\u3b9
2\u3c0R
\u2202j\u2016
\u2202\u3b8
= \u2212\u2202p
\u2202r
2
RB
sin \u3b8
65
66 6 Equilibrium
Fig.6.9 P\ufb01rsch-Schlu¨ter current j\u2016 in a toroidal plasma.
i.e.,
\u2202j\u2016
\u2202\u3b8
= \u22124\u3c0
\u3b9B
\u2202p
\u2202r
sin \u3b8, j\u2016 =
4\u3c0
\u3b9B
\u2202p
\u2202r
cos \u3b8. (6.43)
This current is called the P\ufb01rsch-Schlu¨ter current (ref.[5]) (\ufb01g.6.9). These formulas are very
important, and will be used to estimate the di\ufb00usion coe\ufb03cient of a toroidal plasma. The P\ufb01rsch-
Schlu¨ter current is due to the short circuiting, along magnetic-\ufb01eld lines, of toroidal drift polariza-
tion charges. The resultant current is inversely proportional to \u3b9.
Let us take the radial variation in plasma pressure p(r) and \u3b9 to be
p(r) = p0
(
1\u2212
(
r
a
)m)
,
\u3b9(r) = \u3b9
respectively; then j\u2016 is
j\u2016 = \u2212
4\u3c0mp0
B\u3b9a
(
r
a
)m\u22121
cos \u3b8.
Let us estimate the magnetic \ufb01eld B\u3b2 produced by j\u2016. As a/R is small, B\u3b2 is estimated from the
corresponding linear con\ufb01guration of \ufb01g.6.9. We utilize the coordinates (r, \u3b8\u2032, \u3b6) and put \u3b8 = \u2212\u3b8\u2032
and j\u2016 \u2248 j\u3b6 (\u3b9 is assumed not to be large). Then the vector potential A\u3b2 = (0, 0, A\u3b2\u3b6 ) for B\u3b2 is
given by
1
r
\u2202
\u2202r
\u239b\u239dr\u2202A\u3b2\u3b6
\u2202r
\u239e\u23a0+ 1
r2
\u22022A\u3b2\u3b6
\u2202\u3b8\u2032 2
= \u2212\u3bc0j\u3b6 .
When A\u3b2\u3b6 (r, \u3b8
\u2032) = A\u3b2(r) cos \u3b8\u2032, and parameters s = m \u2212 1, \u3b1 = 4\u3c0mp0\u3bc0/(B\u3b9) = m\u3b20B/(\u3b9/2\u3c0)
(\u3b20 is beta ratio at the center) are used, we \ufb01nd
1
r
\u2202
\u2202r
(
r
\u2202A\u3b2
\u2202r
)
\u2212 A
\u3b2
r2
=
\u3b1
a
(
r
a
)s
.
In the plasma region (r < a), the vector potential is
A\u3b2in =
(
\u3b1rs+2
((s+ 2)2 \u2212 1)as+1 + \u3b4r
)
cos \u3b8\u2032
66
6.7 Virial Theorem 67
and A\u3b2out outside the plasma region (r > a) is
A\u3b2out =
\u3b3
r
cos \u3b8\u2032,
where \u3b4 and \u3b3 are constants. Since B\u3b2r , B
\u3b2
\u3b8\u2032 must be continuous at the boundary r = a, the
solution for B\u3b2 inside the plasma is
B\u3b2r = \u2212
\u3b1
(s+ 2)2 \u2212 1
((
r
a
)s+1
\u2212 s+ 3
2
)
sin \u3b8\u2032,
B\u3b2\u3b8\u2032 = \u2212
\u3b1
(s+ 2)2 \u2212 1
(
(s+ 2)
(
r
a
)s+1
\u2212 s+ 3
2
)
cos \u3b8\u2032
and the solution outside is
B\u3b2r =
\u3b1
(s+ 2)2 \u2212 1
s+ 1
2
(
a
r
)2
sin \u3b8\u2032,
B\u3b2\u3b8 =
\u2212\u3b1
(s+ 2)2 \u2212 1
s+ 1
2
(
a
R
)2
cos \u3b8\u2032
(Br = r\u22121\u2202A\u3b6/\u2202\u3b8\u2032, B\u3b8\u2032 = \u2212\u2202A\u3b6/\u2202r). As is clear from (6.44), there is a homogeneous vertical-\ufb01eld
component
Bz =
\u2212(s+ 3)\u3b1
2((s+ 2)2 \u2212 1) =
\u2212(m+ 2)m
2((m+ 1)2 \u2212 1)
\u3b2
(\u3b9/2\u3c0)
B \u2261 \u2212fm \u3b20(\u3b9/2\u3c0)B = \u2212
m+ 2
m
fm
\u3008\u3b2\u3009
\u3b9/2\u3c0
,
where \u3008\u3b2\u3009 is the average beta value. This \ufb01eld causes the magnetic surface to be displaced by the
amount \u394. From (3.42), \u394 is given by
\u394
R
=
\u22122\u3c0Bz
\u3b9B
=
m
m+ 2
fm
(
2\u3c0
\u3b9
)2
\u3008\u3b2\u3009.
The condition \u394 < a/2 gives the upper limit of the beta ratio:
\u3008\u3b2\u3009c < 12
a
R
(
\u3b9
2\u3c0
)2
,
in the case of m = 2. This critical value is the same as (6.40).
6.7 Virial Theorem
The equation of equilibrium j ×B = (\u2207×B)×B = \u2207p can be reduced to
\u2211
i
\u2202
\u2202xi
Tik \u2212 \u2202p
\u2202xk
= 0 (6.45)
where
Tik =
1
\u3bc0
(BiBk \u2212 12B
2\u3b4ik). (6.46)
This is called the magnetic stress tensor. From the relation (6.45), we have\u222b
S
((
p+
B2
2\u3bc0
)
n\u2212 B(B · n)
\u3bc0
)
dS = 0 (6.47)
where n is the outward unit normal to the closed surface surrounding a volume V.
67
68 6 Equilibrium
Since the other relation\u2211
i
\u2202
\u2202xi
(xk (Tik \u2212 p\u3b4ik)) = (Tkk \u2212 p) + xk
\u2211
i
\u2202
\u2202xi
(Tik \u2212 p\u3b4ik) = (Tkk \u2212 p)
holds,it follows that\u222b
V
(
3p+
B2
2\u3bc0
)
dV =
\u222b
S
((
p+
B2
2\u3bc0
)
(r · n)\u2212 (B · r)(B · n)
\u3bc0
)
dS. (6.48)
This is called the virial theorem. When a plasma \ufb01lls a \ufb01nite region with p = 0 outside the region,
and no solid conductor carries the current anywhere inside or outside the plasma, the magnitude
of the magnetic \ufb01eld is the order of 1/r3, so the surface integral approaches zero as the plasma
surface approaches in\ufb01nity (r \u2192 0). This contradicts that the volume integral of (6.48) is positive
de\ufb01nite. In other words, a plasma of the \ufb01nite extent cannot be in equilibrium unless there exist
solid conductors to carry the current.
Let us apply the virial theorem (6.48) and (6.47) to a volume element of an axisymmetric plasma
bounded by a closed toroidal surface St formed by the rotation of an arbitrary shaped contour lt.
We denote the unit normal and tangent of the contour lt by n and l respectively and a surface
element of the transverse cross section by dS\u3d5. The volume and the surface element are related by
dV = 2\u3c0rdS\u3d5.
The magnetic \ufb01eld B is expressed by
B = B\u3d5e\u3d5 +Bp
where Bp is the poloidal \ufb01eld and B\u3d5 is the magnitude of the toroidal \ufb01eld and e\u3d5 is the unit
vector in the \u3d5 direction.
Let us notice two relations\u222b
r\u3b1(r · n)dSt = (\u3b1+ 3)
\u222b
r\u3b1dV (6.49)
\u222b
r\u3b1(er · n)dSt =
\u222b
\u2207 · (r\u3b1er)dV =
\u222b 1
r
\u2202
\u2202r
r\u3b1+1dV
= (\u3b1+ 1)
\u222b
r(\u3b1\u22121)dV = 2\u3c0(\u3b1+ 1)
\u222b
r\u3b1dS\u3d5 (6.50)
where er is the unit vector in the r direction. Applying (6.48) to the full torus surrounded by St,
we get\u222b (
3p+
B2\u3d5 +B2p
2\u3bc0
)
dV =
\u222b ((
p+
B2\u3d5 +B2p
2\u3bc0
)
(n · r)\u2212 Bn(B · r)
\u3bc0
)
dSt
=
\u222b ((
p+
B2l \u2212B2n
2\u3bc0
)
(n · r)\u2212 BnBl
\u3bc0
(l · r)
)
dSt +
\u222b
B\u3d5
2
2\u3bc0
(n · r) dSt, (6.51)
because of Bp = Bll + Bnn (see \ufb01g.6.10a). When the vacuum toroidal \ufb01eld (without plasma) is
denoted by B\u3d50, this is given by B\u3d50 = \u3bc0I/(2\u3c0r), where I is the total coil current generating the
toroidal \ufb01eld. By use of (6.50), (6.51) reduces to (ref.[6])\u222b (
3p+
B2p +B2\u3d5 \u2212B2\u3d50
2\u3bc0
)
2\u3c0rdS\u3d5
=
\u222b ((
p+
B2l \u2212B2n
2\u3bc0
)
(n · r)\u2212 BnBl
\u3bc0
(l · r)
)
dSt. (6.52)
68
6 References 69
Fig.6.10 Integral region of Virial theorem (a) (6.48) and (b) (6.47).
Applying (6.47) to the sector region surrounded by \u3d5 = 0, \u3d5 = \u394\u3d5 and St (see \ufb01g.6.10b) and
taking its r component gives (ref.[6])
\u2212\u394\u3d5
\u222b (
p+
B2
2\u3bc0
\u2212 B
2
\u3d5
\u3bc0
)
dS\u3d5 +
\u394\u3d5
2\u3c0
\u222b ((
p+
B2
2\u3bc0
)
(n · er)\u2212 (B · er)(B · n)
\u3bc0
)
dSt = 0
2\u3c0
\u222b (
p+
B2p \u2212B2\u3d5 +B2\u3d50
2\u3bc0
)
dS\u3d5 =
\u222b ((
p+
B2l \u2212B2n
2\u3bc0
)
(n · er)\u2212 BlBn
\u3bc0
(l · er)
)
dSt = 0.
(6.53)
Eqs. (6.52) and (6.53) are used to measure the poloidal beta ratio (6.18) and the internal plasma
inductance per unit length (6.23) of arbitrary shaped axisymmetric toroidal plasma by use of
magnetic probes surrounding the plasma.
References
[1] L. S. Solovev: Sov. Phys, JETP 26, 400 (1968).
N. M. Zueva and L. S. Solovev: AtomnayaEnrggia 24, 453 (1968).
[2] R. H. Weening: Phys. Plasmas 7, 3654 (2000).
[3] V. S. Mukhovatov and V. D. Shafranov: Nucl. Fusion 11, 605 (1971).
[4] V. D. Shafranov: Plasma Physics, J. of Nucl. Energy pt.C5, 251 (1963).
[5] D. P\ufb01rsch and A. Schlu¨ter: MPI/PA/7/62 Max-Planck Institut fu¨r Physik und Astrophysik, Mu¨nchen
(1962).
[6] V. D. Shafranov: Plasma Physics 13 757 (1971).
69
70
Ch.7 Di\ufb00usion of Plasma, Con\ufb01nement Time
Di\ufb00usion and con\ufb01nement of plasmas are among the most important subjects in fusion research,
with theoretical and experimental investigations being carried out concurrently. Although a general
discussion of di\ufb00usion and con\ufb01nement requires the consideration of the various instabilities (which
will be studied in subsequent chapters), it is also important to consider simple but fundamental
di\ufb00usion for the ideal stable cases. A typical example (sec.7.1) is classical di\ufb00usion, in which col-
lisions between electrons and ions are dominant e\ufb00ect. The section 7.2 describe the neoclassical
di\ufb00usion of toroidal plasmas con\ufb01ned in tokamak, for both the rare-collisional and collisional re-
gions. Sometimes the di\ufb00usion of an unstable plasma may be studied in a phenomenological way,
without recourse to a detailed knowledge of instabilities. In this manner, di\ufb00usions caused by
\ufb02uctuations in a plasma are explained in secs.7.3 and 7.4.```