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# NIFS PROC 88

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```equilibrium equation (6.9),
2
a2
\u222b a
0
(
\u3bc0p0 +
B2z
2
)
rdr =
(
\u3bc0p0 +
B2z +B2\u3b8
2
)
r=a
we can convert the necessary condition for stability to
a2B2\u3b8 (a) > 4\u3bc0
\u222b a
0
rp0dr
i.e.,
\u3b2\u3b8 \u2261
(
2\u3bc0
B2\u3b8
)
2\u3c0
\u3c0a2
\u222b
p0rdr < 1. (8.112)
Next let us study the stability of the mode m = 0 in the reversed \ufb01eld pinch. It is assumed that
Bz reverses at r = r1. The substitution
\u3ber(r) = \u3bbr 0 \u2264 r \u2264 r1 \u2212 \u3b5, \u3ber(r) = 0 r > r1 + \u3b5
105
106 8 Magnetohydrodynamic Instabilities
into (8.82) yields
Wp =
\u3c0
2\u3bc0
\u3bb2
\u222b r1
0
rdr
(
4B2z \u2212 2B\u3b8
d
dr
(rB\u3b8) + k2r2B2z
)
.
Using the equilibrium equation (6.8), we obtain the necessary condition for stability:
\u3bc\u221210 r
2
1B
2
\u3b8 (r1) > 8
\u222b r1
0
rp0dr \u2212 4r21p0(r1).
If p0(r1) \u223c 0, we have the condition
\u3b2\u3b8 <
1
2
. (8.113)
8.4 Hain-Lu¨st Magnetohydrodynamic Equation
When the displacement \u3be is denoted by
\u3be(r, \u3b8, z, t) = \u3be(r) exp i(m\u3b8 + kz \u2212 \u3c9t)
and the equilibrium magnetic \ufb01eld B0 is expressed by
B(r) = (0, B\u3b8(r), Bz(r))
the (r, \u3b8, z) components of magnetohydrodynamic equation of motion are given by
\u2212\u3bc0\u3c1m\u3c92\u3ber = ddr
(
\u3bc0\u3b3p(\u2207 · \u3be) +B2 1
r
d
dr
(r\u3ber) + iD(\u3be\u3b8Bz \u2212 \u3bezB\u3b8)
)
\u2212
(
F 2 + r
d
dr
(
B\u3b8
r
)2)
\u3ber \u2212 2ikB\u3b8
r
(\u3be\u3b8Bz \u2212 \u3bezB\u3b8), (8.114)
\u2212\u3bc0\u3c1m\u3c92\u3be\u3b8 = im
r
\u3b3\u3bc0p(\u2207 · \u3be) + iDBz 1
r
d
dr
(r\u3ber) + 2ik
B\u3b8Bz
r
\u3ber \u2212H2Bz(\u3be\u3b8Bz \u2212 \u3bezB\u3b8), (8.115)
\u2212\u3bc0\u3c1m\u3c92\u3bez = ik\u3b3\u3bc0p(\u2207 · \u3be)\u2212 iDB\u3b8 1
r
d
dr
(r\u3ber)\u2212 2ikB
2
\u3b8
r
\u3ber +H2B\u3b8(\u3be\u3b8Bz \u2212 \u3bezB\u3b8) (8.116)
where
F =
m
r
B\u3b8 + kBz = (k ·B), D = m
r
Bz \u2212 kB\u3b8, H2 =
(
m
r
)2
+ k2,
\u2207 · \u3be = 1
r
d
dr
(r\u3ber) +
im
r
\u3be\u3b8 + ik\u3bez.
When \u3be\u3b8, \u3bez are eliminated by (8.115),(8.116), we \ufb01nd
d
dr
(
(\u3bc0\u3c1m\u3c92 \u2212 F 2)
\u394
(\u3bc0\u3c1m\u3c92(\u3b3\u3bc0p+B2)\u2212 \u3b3\u3bc0pF 2)1
r
d
dr
(r\u3ber)
)
+
[
\u3bc0\u3c1m\u3c9
2 \u2212 F 2 \u2212 2B\u3b8 ddr
(
B\u3b8
r
)
\u2212 4k
2
\u394
B2\u3b8
r2
(\u3bc0\u3c1m\u3c92B2 \u2212 \u3b3\u3bc0pF 2)
+r
d
dr
(
2kB\u3b8
\u394r2
(
m
r
Bz \u2212 kB\u3b8
)
(\u3bc0\u3c1m\u3c92(\u3b3\u3bc0p+B2)\u2212 \u3b3\u3bc0pF 2)
)]
\u3ber
= 0 (8.117)
106
8.5 Ballooning Instability 107
Fig.8.11 Ballooning mode
where \u394 is
\u394 = \u3bc20\u3c1
2
m\u3c9
4 \u2212 \u3bc0\u3c1m\u3c92H2(\u3b3\u3bc0p+B2) + \u3b3\u3bc0pH2F 2.
This equation was derived by Hain-Lu¨st (ref.[14]). The solution of (8.117) gives \u3ber(r) in the region
of 0 < r < a. The equations for the vacuum region a < r < aw (aw is the radius of wall) are
\u2207×B1 = 0, \u2207 ·B1 = 0
so that we \ufb01nd
B1 = \u2207\u3c8, \ufffd\u3c8 = 0
and
\u3c8 = (bIm(kr) + cKm(kr)) exp(im\u3b8 + ikz),
B1r =
\u2202\u3c8
\u2202r
=
(
bI \u2032m(kr) + cK
\u2032
m(kr)
)
exp(im\u3b8 + ikz). (8.118)
B1r in the plasma region is given by
B1r = i(k ·B)\u3ber = iF\u3ber
and the boundary conditions at r = a are
B1r(a) = iF\u3ber(a), (8.119)
B\u20321r(a) = i(F
\u2032\u3ber(a) + F\u3be\u2032r(a)), (8.120)
and the coe\ufb03cients b,c can be \ufb01xed.
To deal with this equation as an eigenvalue problem, boundary conditions must be imposed on
\u3ber; one is that \u3ber \u221d rm\u22121 at r = 0, and the other is that the radial component of the perturbed
magnetic \ufb01eld at the perfect conducting wall B1r(aw) = 0. After \ufb01nding suitable \u3c92 to satisfy these
conditions, the growth rate \u3b32 \u2261 \u2212\u3c92 is obtained (ref.[15]).
8.5 Ballooning Instability
In interchange instability, the parallel component k\u2016 = (k ·B)/B of the propagation vector is zero
and an average minimum-B condition may stabilize such an instability. Suydam\u2019s condition and
the local-mode stability condition of toroidal-system are involved in perturbations with k\u2016 = 0. In
this section we will study perturbations where k\u2016
= 0 but |k\u2016/k\u22a5| \ufffd 1. Although the interchange
instability is stabilized by an average minimum-B con\ufb01guration, it is possible that the perturbation
107
108 8 Magnetohydrodynamic Instabilities
with k\u2016
= 0 can grow locally in the bad region of the average minimum-B \ufb01eld. This type of
instability is called the ballooning mode (see \ufb01g.8.11). The energy integral \u3b4W is given by
\u3b4W =
1
2\u3bc0
\u222b
((\u2207× (\u3be ×B0))2 \u2212 (\u3be × (\u2207×B0)) · \u2207 × (\u3be ×B0)
+\u3b3\u3bc0p0(\u2207 · \u3be)2 + \u3bc0(\u2207 · \u3be)(\u3be · \u2207p0))dr.
Let us consider the case that \u3be can be expressed by
\u3be =
B0 ×\u2207\u3c6
B20
, (8.121)
where \u3c6 is considered to be the time integral of the scalar electrostatic potential of the perturbed
electric \ufb01eld. Because of
\u3be ×B0 = \u2207\u22a5\u3c6
the energy integral is reduced to
\u3b4W =
1
2\u3bc0
\u222b (
(\u2207×\u2207\u22a5\u3c6)2 \u2212
(
(B0 ×\u2207\u22a5\u3c6)× \u3bc0j0
B20
)
\u2207×\u2207\u22a5\u3c6
+\u3b3\u3bc0p0(\u2207 · \u3be)2 + \u3bc0(\u2207 · \u3be)(\u3be · \u2207p0)
)
dr.
\u2207 · \u3be is given by
\u2207 · \u3be = \u2207 ·
(
B0 ×\u2207\u3c6
B20
)
= \u2207\u3c6 · \u2207 ×
(
B0
B20
)
= \u2207\u3c6 ·
((
\u2207 1
B2
)
×B + 1
B2
\u2207×B
)
.
The 2nd term in ( ) is negligible compared with the 1st term in the low beta case. By means of
\u2207p0 = j0 ×B0, \u3b4W is expressed by
\u3b4W =
1
2\u3bc0
\u222b
(\u2207×\u2207\u22a5\u3c6)2 + \u3bc0\u2207p0 · (\u2207\u22a5\u3c6×B0)
B20
(
B0 · \u2207 ×\u2207\u22a5\u3c6
B20
)
\u2212\u3bc0(j0 ·B0)
B20
\u2207\u22a5\u3c6 · \u2207 ×\u2207\u22a5\u3c6+ \u3b3\u3bc0p0
(
\u2207
(
1
B20
)
· (B0 ×\u2207\u22a5\u3c6)
)2
+
\u3bc0\u2207p0 · (B0 ×\u2207\u22a5\u3c6)
B20
(
\u2207
(
1
B20
)
· (B0 ×\u2207\u22a5\u3c6)
)
dr.
Let us use z coordinate as a length along a \ufb01eld line, r as radial coordinate of magnetic surfaces
and \u3b8 as poloidal angle in the perpendicular direction to \ufb01eld lines. The r, \u3b8, z components of \u2207p0,
B, and \u2207\u3c6 are approximately given by
\u2207p0 = (p\u20320, 0, 0), B = (0, B\u3b8(r), B0(1\u2212 rR\u22121c (z))),
\u2207\u3c6 = (\u2202\u3c6/\u2202r, \u2202\u3c6/r\u2202\u3b8, \u2202\u3c6/\u2202z), \u3c6(r, \u3b8, z) = \u3c6(r, z)Re(exp im\u3b8).
Rc(z) is the radius of curvature of the line of magnetic force:
1
Rc(z)
=
1
R0
(
\u2212w + cos 2\u3c0 z
L
)
.
108
8.5 Ballooning Instability 109
When Rc(z) < 0, the curvature is said to be good. If the con\ufb01guration is average minimum-B, w
and R0 must be 1 > w > 0 and R0 > 0. Since B\u3b8/B0, r/R0, r/L are all small quantities, we \ufb01nd
\u2207\u22a5\u3c6 = \u2207\u3c6\u2212\u2207\u2016\u3c6 \u2248 Re
(
\u2202\u3c6
\u2202r
,
im
r
\u3c6, 0
)
,
\u2207× (\u2207\u22a5\u3c6) \u2248 Re
(
\u2212im
r
\u2202\u3c6
\u2202z
,
\u22022\u3c6
\u2202z\u2202r
, 0
)
,
B0 ×\u2207\u22a5\u3c6 \u2248 Re
(\u2212im
r
B0\u3c6,B0
\u2202\u3c6
\u2202r
, 0
)
and \u3b4W is reduced to
\u3b4W =
1
2\u3bc0
\u222b
m2
r2
((
\u2202\u3c6(r, z)
\u2202z
)2
\u2212 \u3b2
rpRc(z)
(\u3c6(r, z))2
)
2\u3c0rdrdz
where \u2212p0/p\u20320 = rp and \u3b2 = p0/(B20/2\u3bc0). The 2nd term contributes to stability in the region
Rc(z) < 0 and contributes to instability in the region of Rc(z) > 0. Euler\u2019s equation is given by
d2\u3c6
dz2
+
\u3b2
rpRc(z)
\u3c6 = 0. (8.122)
Rc is nearly equal to B/|\u2207B|. Equation (8.122) is a Mathieu di\ufb00erential equation, whose eigenvalue
is
w = F (\u3b2L2/2\u3c02rpR0).
Since
F (x) = x/4, x\ufffd 1, F (x) = 1\u2212 x\u22121/2, x\ufffd 1,
we \ufb01nd the approximate relation
\u3b2c \u223c 4w(1 + 3w)(1\u2212 w)2
2\u3c02rpR0
L2
.
Since w is of the order of rp/2R0 and the connection length is
L \u2248 2\u3c0R0(2\u3c0/\u3b9)
(\u3b9 being the rotational transform angle), the critical beta ratio \u3b2c is
\u3b2c \u223c
(
\u3b9
2\u3c0
)2 (rp
R
)
. (8.123)
If \u3b2 is smaller than the critical beta ratio \u3b2c, then \u3b4W > 0, and the plasma is stable. The stability
condition for the ballooning mode in the shearless case is given by (ref.[16]).
\u3b2 < \u3b2c.
In the con\ufb01guration with magnetic shear, more rigorous treatment is necessary. According to the
analysis (ref.[17]-[19]) for ballooning modes with large toroidal mode number n\ufffd 1 and m\u2212nq \u223c 0
(see appendix B), the stable region in the shear parameter S and the measure of pressure gradient
\u3b1 of ballooning mode is shown in \ufb01g.8.12. The shear parameter S is de\ufb01ned by
S =
r
q
dq
dr
109
110 8 Magnetohydrodynamic Instabilities
Fig.8.12 The maximum stable pressure gradient \u3b1 as a function of the shear parameter S of ballooning
mode. The dotted line is the stability boundary obtained by imposing a more restricted boundary condition
on the perturbation (ref.[17]).
where q is the safety factor (q \u2261 2\u3c0/\u3b9: \u3b9 rotational transform angle) and the measure of pressure
\u3b1 = \u2212 q
2R
B2/2\u3bc0
dp
dr
.
The straight-line approximation of the maximum pressure gradient in the range of large positive
shear (S > 0.8) is \u3b1 \u223c 0.6S as is shown in \ufb01g.8.12. Since
\u3b2 =
1
B20/2\u3bc0
1
\u3c0a2
\u222b a
0
p2\u3c0rdr = \u2212 1
B20/2\u3bc0
1
a2
\u222b a
0
dp
dr```