NIFS PROC 88
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NIFS PROC 88


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a special case; in most cases the plasma
current decreases gradually at the boundary. Let us consider the case of a di\ufb00use-boundary plasma
whose parameters in the equilibrium state are
p0(r), B0(r) = (0,B\u3b8(r), Bz(r)).
The perturbation \u3be is assumed to be
\u3be = \u3be(r) exp(im\u3b8 + ikz).
The perturbation of the magnetic \ufb01eld B1 = \u2207× (\u3be ×B0) is
B1r = i(k ·B0)\u3ber, (8.69)
B1\u3b8 = ikA\u2212 ddr (\u3berB\u3b8), (8.70)
B1z = \u2212
(
imA
r
+
1
r
d
dr
(r\u3berBz)
)
(8.71)
97
98 8 Magnetohydrodynamic Instabilities
where
(k ·B0) = kBz + m
r
B\u3b8, (8.72)
A = \u3be\u3b8Bz \u2212 \u3bezB\u3b8 = (\u3be ×B0)r. (8.73)
Since the pressure terms \u3b3p0(\u2207 · \u3be)2 + (\u2207 · \u3be)(\u3be · \u2207p0) = (\u3b3 \u2212 1)p0(\u2207 · \u3be)2 + (\u2207 · \u3be)(\u2207 · p0\u3be) in the
energy integral are nonnegative, we examine the incompressible displacement \u2207 · \u3be = 0 again, i.e.,
1
r
d
dr
(r\u3ber) +
im
r
\u3be\u3b8 + ik\u3bez = 0. (8.74)
From this and (8.73) for A, \u3be\u3b8 and \u3bez are expressed in terms of \u3ber and A as
i(k ·B)\u3be\u3b8 = ikA\u2212 B\u3b8
r
d
dr
(r\u3ber), (8.75)
\u2212i(k ·B)\u3bez = imA
r
+
Bz
r
d
dr
(r\u3ber). (8.76)
From \u3bc0j0 = \u2207×B0, it follows that
\u3bc0j0\u3b8 = \u2212dBzdr , (8.77)
\u3bc0j0z =
dB\u3b8
dr
+
B\u3b8
r
=
1
r
d
dr
(rB\u3b8). (8.78)
The terms of the energy integral are given by
Wp =
1
4
\u222b
Vin
(
\u3b3p0|\u2207 · \u3be|2 + (\u2207 · \u3be\u2217)(\u3be · \u2207p0) + 1
\u3bc0
|B1|2 \u2212 \u3be\u2217 · (j0 ×B1)
)
dr
=
1
4
\u222b (
\u2212p1(\u2207 · \u3be) + 1
\u3bc0
|B1|2 \u2212 j0(B1 × \u3be\u2217)
)
dr, (8.79)
WS =
1
4
\u222b
S
|\u3ben|2 \u2202
\u2202n
(
B20,ex
2\u3bc0
\u2212 B
2
0,in
2\u3bc0
\u2212 p0
)
dS, (8.80)
WV =
1
4\u3bc0
\u222b
Vex
|B1|2dr. (8.81)
\u3be\u3b8 and \u3bez can be eliminated by means of (8.75) and (8.76) and dBz/dr and dB\u3b8/dr can be
eliminated by means of (8.77) and (8.78) in (8.79). Then Wp becomes
Wp =
1
4
\u222b
Vin
(k ·B)2
\u3bc0
|\u3ber|2 +
(
k2 +
m2
r2
)
|A|2
\u3bc0
+
1
\u3bc0
\u2223\u2223\u2223\u2223B\u3b8 d\u3berdr + \u3ber
(
\u3bc0jz \u2212 B\u3b8
r
)\u2223\u2223\u2223\u22232 + 1\u3bc0
\u2223\u2223\u2223\u2223\u3berBzr +Bz d\u3berdr
\u2223\u2223\u2223\u22232
+
2
\u3bc0
Re
(
ikA\u2217
(
B\u3b8
d\u3ber
dr
+
(
\u3bc0jz \u2212 B\u3b8
r
)
\u3ber
)
\u2212 imA
\u2217
r2
(
\u3berBz + rBz
d\u3ber
dr
))
+ 2Re
(
\u3be\u2217rj0z
(
\u2212B\u3b8 d\u3berdr \u2212
\u3ber\u3bc0jz
2
+ ikA
))
dr.
The integrand of Wp is reduced to
1
\u3bc0
(
k2 +
m2
r2
)
×
\u2223\u2223\u2223\u2223A+ ikB\u3b8((d\u3ber/dr)\u2212 \u3ber/r)\u2212 im(Bz/r)((d\u3ber/dr) + (\u3ber/r))k2 + (m2/r2)
\u2223\u2223\u2223\u22232
98
8.3 Instabilities of a Cylindrical Plasma 99
+
(
(k ·B)2
\u3bc0
\u2212 2jzB\u3b8
r
)
|\u3ber|2 + B
2
z
\u3bc0
\u2223\u2223\u2223\u2223d\u3berdr + \u3berr
\u2223\u2223\u2223\u22232 + B2\u3b8\u3bc0
\u2223\u2223\u2223\u2223d\u3berdr \u2212 \u3berr
\u2223\u2223\u2223\u22232
\u2212|ikB\u3b8((d\u3ber/dr)\u2212 (\u3ber/r))\u2212 im(Bz/r)((d\u3ber/dr) + (\u3ber/r))|
2
\u3bc0(k2 + (m2/r2))
.
Accordingly, the integrand is a minimum when
A \u2261 \u3be\u3b8Bz \u2212 \u3bezB\u3b8
= \u2212 i
k2 + (m2/r2)
((
kB\u3b8 \u2212 m
r
Bz
)
d\u3ber
dr
\u2212
(
kB\u3b8 +
m
r
Bz
)
\u3ber
r
)
.
Then Wp is reduced to
Wp =
\u3c0
2\u3bc0
\u222b a
0
( |(k ·B0)(d\u3ber/dr) + h(\u3ber/r)|2
k2 + (m/r)2
+
(
(k ·B0)2 \u2212 2\u3bc0jzB\u3b8
r
)
|\u3ber|2
)
rdr (8.82)
where
h \u2261 kBz \u2212 m
r
B\u3b8.
Let us next determine WS. From (6.8), it follows that (d/dr)(p0+(B2z +B
2
\u3b8 )/2\u3bc0) = \u2212B2\u3b8/(r\u3bc0).
B2\u3b8 is continuous across the boundary r = a, so that
d
dr
(
p0 +
B2z +B
2
\u3b8
2\u3bc0
)
=
d
dr
(
B2ez +B
2
e\u3b8
2\u3bc0
)
.
Accordingly we \ufb01nd
WS = 0 (8.83)
as is clear from (8.80).
The expression for WV can be obtained when the quantities in (8.82) for Wp are replaced as
follows: j \u2192 0, Bz \u2192 Bez = Bs(= const.), B\u3b8 \u2192 Be\u3b8 = Baa/r, B1r = i(k · B0)\u3ber \u2192 Be1r =
i(k ·Be0)\u3b7r. This replacement yields
WV =
\u3c0
2\u3bc0
\u222b b
a
((
kBs +
m
r
Baa
r
)2
|\u3b7r|2
+
|[kBs+(m/r)(Baa/r)](d\u3b7r/dr)+[kBs\u2212(m/r)(Baa/r)]\u3b7r/r|2
k2 + (m/r)2
)
rdr. (8.84)
By partial integration, Wp is seen to be
Wp =
\u3c0
2\u3bc0
\u222b a
0
(
r(k ·B0)2
k2 + (m/r)2
\u2223\u2223\u2223\u2223d\u3berdr
\u2223\u2223\u2223\u22232 + g|\u3ber|2
)
dr +
\u3c0
2\u3bc0
k2B2s \u2212 (m/a)2B2a
k2 + (m/a)2
|\u3ber(a)|2 (8.85)
g =
1
r
(kBz \u2212 (m/r)B\u3b8)2
k2 + (m/r)2
+ r(k ·B0)2 \u2212 2B\u3b8
r
d(rB\u3b8)
dr
\u2212 d
dr
(
k2B2z \u2212 (m/r)2B2\u3b8
k2 + (m/r)2
)
. (8.86)
Using the notation \u3b6 \u2261 rBe1r = ir(k ·Be0)\u3b7r, we \ufb01nd that
WV =
\u3c0
2\u3bc0
\u222b b
a
(
1
r(k2 + (m/r)2)
\u2223\u2223\u2223\u2223d\u3b6dr
\u2223\u2223\u2223\u22232 + 1r |\u3b6|2
)
dr. (8.87)
99
100 8 Magnetohydrodynamic Instabilities
The functions \u3ber or \u3b6 that will minimize Wp or WV are the solutions of Euler\u2019s equation:
d
dr
(
r(k ·B0)2
k2 + (m/r)2
d\u3ber
dr
)
\u2212 g\u3ber = 0, r \u2264 a, (8.88)
d
dr
(
1
r(k2 + (m/r)2)
d\u3b6
dr
)
\u2212 1
r
\u3b6 = 0, r > a . (8.89)
There are two independent solutions, which tend to \u3ber \u221d rm\u22121, r\u2212m\u22121 as r \u2192 0. As \u3ber is \ufb01nite at
r = 0, the solution must satisfy the conditions
r\u2192 0, \u3ber \u221d rm\u22121,
r = a, \u3b6(a) = ia
(
kBs +
m
a
Ba
)
\u3ber(a),
r = b, \u3b6(b) = 0.
Using the solution of (8.89), we obtain
WV =
\u3c0
2\u3bc0
1
r(k2 + (m/r)2)
\u2223\u2223\u2223\u2223d\u3b6dr \u3b6\u2217
\u2223\u2223\u2223\u2223b
a
. (8.90)
The solution of (8.89) is
\u3b6 = i
I \u2032m(kr)K \u2032m(kb)\u2212K \u2032m(kr)I \u2032m(kb)
I \u2032m(ka)K\u2032m(kb)\u2212K \u2032m(ka)I \u2032m(kb)
r
(
kBs +
m
a
Ba
)
\u3ber(a). (8.91)
The stability problem is now reduced to one of examining the sign of Wp +WV. For this we use
Wp =
\u3c0
2\u3bc0
\u222b a
0
(
f
\u2223\u2223\u2223\u2223d\u3berdr
\u2223\u2223\u2223\u22232 + g|\u3ber|2)dr +Wa,
Wa =
\u3c0
2\u3bc0
k2B2s \u2212 (m/a)2B2a
k2 + (m/a)2
|\u3ber(a)|2,
WV =
\u3c0
2\u3bc0
\u22121
r(k2 + (m/a)2)
\u2223\u2223\u2223\u2223d\u3b6dr \u3b6\u2217
\u2223\u2223\u2223\u2223
r=a
\u23ab\u23aa\u23aa\u23aa\u23aa\u23aa\u23aa\u23aa\u23aa\u23ac\u23aa\u23aa\u23aa\u23aa\u23aa\u23aa\u23aa\u23aa\u23ad
(8.92)
where
f =
r(kBz + (m/r)B\u3b8)2
k2 + (m/r)2
, (8.93)
g =
1
r
(kBz \u2212 (m/r)B\u3b8)2
k2 + (m/r)2
+ r
(
kBz +
m
r
B\u3b8
)2
\u2212 2B\u3b8
r
d(rB\u3b8)
dr
\u2212 d
dr
(
k2B2z \u2212 (m/r)2B2\u3b8
k2 + (m/r)2
)
. (8.94)
When the equation of equilibrium ddr (\u3bc0p+B
2/2) = \u2212B2\u3b8/r is used, (8.94) of g is reduced to
g =
2k2
k2 + (m/r)2
\u3bc0
dp0
dr
+ r(kBz +
m
r
B\u3b8)2
k2 + (m/r)2 \u2212 (1/r)2
k2 + (m/r)2
+
(2k2/r)(k2B2z \u2212 (m/r)2B2\u3b8 )
(k2 + (m/r)2)2
. (8.95)
8.3c Suydam\u2019s Criterion
100
8.3 Instabilities of a Cylindrical Plasma 101
The function f in the integrand of Wp in the previous section is always f \u2265 0, so that the term
in f is a stabilizing term. The 1st and 2nd terms in (8.94) for g are stabilizing terms, but the 3rd
and 4th terms may contribute to the instabilities. When a singular point
f \u221d (k ·B0)2 = 0
of Euler\u2019s equation (8.88) is located at some point r = r0 within the plasma region, the contribution
of the stabilizing term becomes small near r = r0, so that a local mode near the singular point is
dangerous. In terms of the notation
r \u2212 r0 = x, f = \u3b1x2, g = \u3b2, \u3b2 = 2B
2
\u3b8
B20
\u3bc0
dp0
dr
\u2223\u2223\u2223\u2223
r=r0
,
\u3b1 =
r0
k2r20 +m2
(
kr
dBz
dr
+ kBz +m
dB\u3b8
dr
)2
r=r0
=
rB2\u3b8B
2
z
B2
(
\u3bc\u2dc\u2032
\u3bc\u2dc
)2
r=r0
, \u3bc\u2dc \u2261 B\u3b8
rBz
.
Euler\u2019s equation (8.88) with (8.95) of g is reduced to
\u3b1
d
dr
(x2
d\u3ber
dx
)\u2212 \u3b2\u3ber = 0.
The solution is
\u3ber = c1x\u2212n1 + c2x\u2212n2 (8.96)
where n1 and n2 are given by
n2 \u2212 n\u2212 \u3b2
\u3b1
= 0, ni =
1± (1 + 4\u3b2/\u3b1)1/2
2
.
When \u3b1 + 4\u3b2 > 0, n1 and n2 are real. The relation n1 + n2 = 1 holds always. For n1 < n2, we
have the solution x\u2212n1, called a small solution. When n is complex (n = \u3b3 ± i\u3b4), \u3ber is in the form
exp((\u2212\u3b3 \u2213 i\u3b4) ln x) and \u3ber is oscillatory.
Let us consider a local mode \u3ber, which is nonzero only in the neighborhood \u3b5 around r = r0 and
set
r \u2212 r0 = \u3b5t, \u3ber(r) = \u3be(t), \u3be(1) = \u3be(\u22121) = 0.
Then Wp becomes
Wp =
\u3c0
2\u3bc0
\u3b5
\u222b 1
\u22121
(
\u3b1t2
\u2223\u2223\u2223\u2223d\u3bedt
\u2223\u2223\u2223\u22232 + \u3b2|\u3be|2)dt+O(\u3b52).
Since Schwartz\u2019s inequality yields (
\u222b
(uf(t) + g(t))2dt = Au2 + 2Bu+ C > 0, AC > B2)
\u222b 1
\u22121
t2|\u3be\u2032|2dt
\u222b 1
\u22121
|\u3be|2dt \u2265
\u2223\u2223\u2223\u2223\u222b 1\u22121 t\u3be\u2032\u3be\u2217dt
\u2223\u2223\u2223\u22232 = (12
\u222b 1
\u22121
|\u3be|2dt
)2
Wp is
Wp >
\u3c0
2\u3bc0
1
4
(\u3b1+ 4\u3b2)
\u222b 1
\u22121
|\u3be|2dt.
The stability condition is \u3b1+ 4\u3b2 > 0, i.e.,
r
4
(
\u3bc\u2dc\u2032
\u3bc\u2dc
)2
+
2\u3bc0
B2z
dp0
dr
> 0. (8.97)
r(\u3bc\u2dc\u2032/\u3bc\u2dc) is called shear parameter. Usually the 2nd term is negative, since, most often, dp0/dr < 0.
The 1st term (\u3bc\u2dc\u2032/\u3bc\u2dc)2 represents the stabilizing e\ufb00ect of shear. This condition is called Suydam\u2019s
101
102 8 Magnetohydrodynamic Instabilities
criterion (ref.[10]).