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# NIFS PROC 88

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```(r, \u3b8, \u3b6), coupling constant
becomes 2.5\ufffd instead of 2\ufffd (ref.[8]). The energy integral from (14.49) without coupling term of
m±1 modes is reduced to following equation by use of partial integral:
G(\u3c9,Em) \u2261 P
\u222b a
0
dr r
((
r2
(
dEm
dr
)2
+ (m2 \u2212 1)E2m
)(
\u3c92
v2A
\u2212 k2\u2016m
)
\u2212 \u3c92rE2m
d
dr
1
v2A
)
= Em(r\u2212s )Cm(r
\u2212
s )\u2212 Em(r+s )Cm(r+s ) (14.51)
where
Cm(r) =
(
\u3c92
v2A
\u2212 k2\u2016m
)
r2
dEm
dr
, Em(a) = 0.
The radius r = rs is singular at which (\u3c92/vA)2\u2212 k2\u2016m = 0 and P is principal value of the integral.
From this formulation, it is possible to estimate the damping rate of TAE and is given by (ref.[8])
\u3b4\u3c9
\u3c9
= \u2212i\u3c0 sign(\u3c90)Cm(rs)
2
r3s
\u2223\u2223\u2223\u2223 \u2202\u2202r (\u3c92v2
A
\u2212 k2\u2016m
)\u2223\u2223\u2223\u2223\u3c90 \u2202G\u2202\u3c90 . (14.52)
Since \u3c90\u2202G/\u2202\u3c90 > 0, Im(\u3b4\u3c9) < 0. This is called continuum damping.
14.2b Instability of TAE Driven by Energetic Particles
Dynamics of energetic particles must be treated by kinetic theory. Basic equations will be
described according to Betti and Freidberg (ref.[9]).
\u2202fj
\u2202t
+ v · fj + qj
mj
(E + v ×B) · \u2207vfj = 0, (14.53)
177
178 14 Instabilities Driven by Energetic Particles
\u2202nj
\u2202t
+\u2207 · (njuj) = 0, (14.54)
mj
\u2202
\u2202t
(njuj) +\u2207 · Pj = qjnj(E + uj ×B), (14.55)
Pj = mj
\u222b
vvfjdv, (14.56)
B1 = \u2207× (\u3be\u22a5 ×B), (14.57)
\u3bc0j1 = \u2207B1 = \u2207×\u2207× (\u3be\u22a5 ×B), (14.58)
j1 ×B + j ×B1 =
\u2211
j
(\u2207P1j \u2212 i\u3c9mj(n1juj + nju1j)) \u2248
\u2211
j
(
\u2207P1j \u2212 \u3c1\u3c92\u3be\u22a5j
)
. (14.59)
Fj is equilibrium distribution function of axisymmetric torus. Fj(\u3b5, p\u3d5) is assumed to be a function
of constants of motion \u3b5 and p\u3d5, where
\u3b5 =
mj
2
v2 + qj\u3c6, p\u3d5 = mjRv\u3d5 + qj\u3c8, \u3c8 = RA\u3d5 (14.60)
RBZ =
\u2202\u3c8
\u2202R
, RBR = \u2212\u2202\u3c8
\u2202Z
,
\u2202f1j
\u2202t
+ v · f1j + qj
mj
(v ×B) · \u2207vf1j = \u2212 qj
mj
(E + v ×B1) · \u2207vFj (14.61)
\u2207vFj = \u3d5\u2c6\u2202p\u3d5
\u2202v\u3d5
\u2202Fj
\u2202p\u3d5
+ (\u2207v\u3b5)\u2202Fj
\u2202\u3b5
= \u3d5\u2c6mjR
\u2202Fj
\u2202p\u3d5
+mjv
\u2202Fj
\u2202\u3b5
. (14.62)
The solution is obtained by integral along the particle orbit (refer appendix sec.C.2)
f1j = \u2212 qj
mj
\u222b t
\u2212\u221e
(E + v ×B1) · \u2207vFjdt\u2032. (14.63)
It is assumed that the perturbations are in the form of
Q1 = Q1(R,Z) exp i(n\u3d5\u2212 \u3c9t).
The second term mjv(\u2202Fj/\u2202\u3b5) of the right-hand side of (14.62) contributes to the integral
\u2212 qj
mj
\u222b t
\u2212\u221e
(E + v ×B1) ·mjv\u2202Fj
\u2202\u3b5
dt\u2032 = \u2212qj\u2202Fj
\u2202\u3b5
\u222b t
\u2212\u221e
E · vdt\u2032.
The contribution from the \ufb01rst term mjR(\u2202Fj/\u2202p\u3d5) is
\u2212 qj
mj
(\u222b t
\u2212\u221e
(E\u3d5mjR
\u2202Fj
\u2202p\u3d5
dt\u2032 +
\u222b t
\u2212\u221e
mjR(v ×B1)\u3d5 \u2202Fj
\u2202p\u3d5
dt\u2032
)
= \u2212qj \u2202Fj
\u2202p\u3d5
(\u222b t
\u2212\u221e
E\u3d5Rdt\u2032 +
\u222b t
\u2212\u221e
Rv ×
(
(\u2207×E)\u3d5
\u2212i\u3c9
)
dt\u2032
)
= \u2212qj \u2202Fj
\u2202p\u3d5
(\u222b t
\u2212\u221e
1
\u2212i\u3c9
\u2202(E\u3d5R)
\u2202t
dt\u2032 +
\u222b t
\u2212\u221e
(
n
\u3c9
(v ·E)\u2212 1
i\u3c9
(v · \u2207)(E\u3d5R)
)
dt\u2032
)
= \u2212qj \u2202Fj
\u2202p\u3d5
(\u222b t
\u2212\u221e
1
\u2212i\u3c9
d(E\u3d5R)
dt
dt\u2032 +
\u222b t
\u2212\u221e
n
\u3c9
(v ·E)dt\u2032
)
.
178
14.2 Toroidal Alfve´n Eigenmode 179
The solution is
f1j = \u2212qj
\u3c9
(
i
\u2202Fj
\u2202p\u3d5
RE\u3d5 +
(
\u3c9
\u2202Fj
\u2202\u3b5
+ n
\u2202Fj
\u2202p\u3d5
)\u222b t
\u2212\u221e
(E · v)dt\u2032
)
. (14.64)
Since
E1\u2016 = 0, \u2212 i\u3c9\u3be\u22a5 =
E\u22a5 ×B
B2
, E\u22a5 = i\u3c9(\u3be\u22a5 ×B),
RE\u3d5 = i\u3c9(\u3be\u22a5 ×B)\u3d5R = i\u3c9(\u3be\u22a5RBZ \u2212 \u3be\u22a5ZBR)R = \u2212i\u3c9(\u3be · \u2207\u3c8),
E · v = i\u3c9(\u3be\u22a5 ×B) · v = \u2212i\u3c9\u3be\u22a5 · (v ×B) = \u2212i\u3c9\u3be\u22a5 ·
mj
qj
dv
dt
= \u2212i\u3c9mj
qj
\u3be\u22a5 ·
dv
dt
= \u2212i\u3c9mj
qj
(
d(\u3be\u22a5 · v)
dt
\u2212 v · d\u3be\u22a5
dt
)
,
f1j becomes
f1j = \u2212qj \u2202Fj
\u2202p\u3d5
(\u3be · \u2207\u3c8) + imj
(
\u3c9
\u2202Fj
\u2202\u3b5
+ n
\u2202Fj
\u2202p\u3d5
)(
\u3be\u22a5 · v \u2212
\u222b t
\u2212\u221e
v · d\u3be\u22a5
dt
dt\u2032
)
= \u2212qj\u2202Fj
\u2202\u3c8
+ imj(\u3c9 \u2212 \u3c9\u2217j)\u2202Fj
\u2202\u3b5
(\u3be\u22a5 · v \u2212 sj) (14.65)
where
sj \u2261
\u222b t
\u2212\u221e
v · d\u3be\u22a5
dt
dt\u2032, \u3c9\u2217j \u2261 \u2212n\u2202Fj/\u2202p\u3d5
\u2202Fj/\u2202\u3b5
.
sj is reduced to
sj =
\u222b t
\u2212\u221e
(
v2\u22a5
2
\u2207 · \u3be\u22a5 +
(
v2\u22a5
2
\u2212 v2\u2016
)
\u3be · \u3ba
)
dt\u2032 (14.66)
as will be shown in the end of this subsection. The perturbed pressure tensor is
P1j =
\u222b
mjvvf1jdv = P1\u22a5jI + (P1\u2016j \u2212 P1\u22a5j)bb (14.67)
and \u2207P1j is given by (14.21b) and (14.22b). Then the equation of motion is
\u2212\u3c1\u3c92\u3be\u22a5 = F\u22a5(\u3be\u22a5) + iD\u22a5(\u3be\u22a5), (14.68)
F\u22a5(\u3be\u22a5) = j1 ×B + j ×B1 +\u2207(\u3be\u22a5 · \u2207P1), (14.69)
D\u22a5(\u3be\u22a5) = mj
\u222b (
v2\u22a5
2
\u2207\u22a5 +
(
v2\u2016 \u2212
v2\u22a5
2
)
\u3ba
)
mj(\u3c9 \u2212 \u3c9\u2217j)\u2202Fj
\u2202\u3b5
sjdv. (14.70)
F\u22a5(\u3be\u22a5) is the ideal MHD force operator for incompressible displacement. D\u22a5(\u3be\u22a5) contains the
contribution of energetic particles. Eqs.(14.68-14.70) describe the low frequency, \ufb01nite wave number
stability of energetic particle-Alfve´n waves in axisymmetric torus.
The energy integral of (14.68) consists of plasma kinetic energy normalization KM, ideal MHD
perpendicular potential energy \u3b4WMHD and the kinetic contribution to the energy integral \u3b4WK:
\u3c92KM = \u3b4WMHD + \u3b4WK. (14.71)
179
180 14 Instabilities Driven by Energetic Particles
where
KM =
1
2
\u222b
\u3c1|\u3be\u22a5|2dr,
\u3b4WMHD = \u221212
\u222b
\u3be\u2217\u22a5F\u22a5(\u3be\u22a5)dr,
\u3b4WK = \u2212 i2
\u222b
\u3be\u2217\u22a5D\u22a5(\u3be\u22a5)dr.
After a simple integration by parts, \u3b4WK can be written as
\u3b4WK =
i
2
\u2211
j
\u222b
(\u3c9 \u2212 \u3c9\u2217j)\u2202Fj
\u2202\u3b5
sj
ds\u2217j
dt
dvdr, (14.72)
since
ds\u2217j
dt
=
(
v2\u22a5
2
\u2207\u22a5 · \u3be\u2217 +
(
v2\u22a5
2
\u2212 v\u2016
)
\u3be\u22a5 · \u3ba
)
.
On the other hand ds\u2217j /dt is given by
ds\u2217j
dt
= i\u3c9\u2217s\u2217j +Ds
\u2217
j , D \u2261 (v · \u2207) +
qj
mj
(v ×B) · \u2207v.
With use of the notation sj \u2261 aj + icj (aj and cj are real), we have
sj
ds\u2217j
dt
= i\u3c9\u2217|sj|2 + i(cjDaj \u2212 ajDcj) + 12D(a
2
j + c
2
j ).
Contribution of the last term to the integral (14.72) by drdv is zero, since Fj and \u3c9\u2217j are functions
of the constants of motion \u3b5 and p\u3d5 and
\u3b4WK =
1
2
\u2211
j
\u222b
(\u3c9 \u2212 \u3c9\u2217j)\u2202Fj
\u2202\u3b5
(i\u3c9i|sj|2 +Rj)dvdr,
Rj = cjDaj \u2212 ajDcj \u2212 \u3c9r|sj|2.
The desired expression for the growth rate is obtained by setting the real and imaginary parts of
(14.71) to to be equal individually:
\u3c92r =
\u3b4WMHD
KM
+O(\u3b2). (14.73)
O(\u3b2) is the contribution of Rj term. In the limit of \u3c9i \ufffd \u3c9r, the imaginary part yields
\u3c9i \u2248 WK
KM
, WK \u2261 lim
\u3c9i\u21920
\u239b\u239d 1
4\u3c9r
\u2211
j
\u222b
(\u3c9 \u2212 \u3c9\u2217j)\u2202Fj
\u2202\u3b5
\u3c9i|sj|2dvdr
\u239e\u23a0 . (14.74)
Let us estimate (14.74). Since \u2207 · \u3be\u22a5 + 2\u3be\u22a5 · \u3ba \u2248 0 (refer (B.7) of App. B), sj is
sj = \u2212mj
\u222b t
\u2212\u221e
(
v2\u2016 +
v2\u22a5
2
)
(\u3ba · \u3be\u22a5)dt\u2032 = mj
\u222b t
\u2212\u221e
(
v2\u2016 +
v2\u22a5
2
)
\u3beR
R
dt\u2032
where
\u3beR = \u3ber cos \u3b8 \u2212 \u3be\u3b8 sin \u3b8 = \u3ber e
i\u3b8 + e\u2212i\u3b8
2
\u2212 \u3be\u3b8 e
i\u3b8 \u2212 e\u2212i\u3b8
2i
.
180
14.2 Toroidal Alfve´n Eigenmode 181
\u3ber and \u3be\u3b8 are (\u2207 · \u3be = (1/r)(\u2202(r\u3ber)/\u2202r)\u2212 i(m/r)\u3be\u3b8 \u2248 0)
\u3ber =
\u2211
m
\u3bem(r)e\u2212im\u3b8, \u3be\u3b8 = \u2212i
\u2211
m
(r\u3bem(r))\u2032
m
e\u2212im\u3b8.
Since the leading-order guiding center of orbits of energetic particles are given by
r(t\u2032) = r(t), \u3b8(t\u2032) =
v\u2016B\u3b8
rB\u3d5
(t\u2032 \u2212 t) + \u3b8(t), \u3d5(t\u2032) = v\u2016
r
(t\u2032 \u2212 t) + \u3d5(t)
perturbations along the orbit become
exp i(\u2212m\u3b8(t\u2032) + n\u3d5(t\u2032)\u2212 \u3c9t\u2032) = exp
(
i(\u2212mB\u3b8
rB\u3d5
v\u2016 +
nv\u2016
R
\u2212 \u3c9)(t\u2032 \u2212 t)
)
exp i(\u2212m\u3b8(t) + n\u3d5(t)\u2212 \u3c9t)
= exp
(\u2212i(\u3c9 \u2212 \u3c9m)(t\u2032 \u2212 t)) exp i(\u2212m\u3b8(t) + n\u3d5(t)\u2212 \u3c9t)
where
\u3c9m =
v\u2016
R
(
n\u2212 m
q
)
.
and
sj =
mj(v2\u2016 + v
2
\u22a5/2)
R
1
2
\u2211
m
(\u3bem\u22121 + \u3bem+1 \u2212 i\u3be\u3b8(m\u22121) + i\u3be\u3b8(m+1)) exp i(\u2212m\u3b8 + n\u3d5\u2212 \u3c9t)
×
\u222b 0
\u221e
exp(\u2212i(\u3c9 \u2212 \u3c9m)t\u2032\u2032dt\u2032\u2032
= i
mj
2R
(
v2\u2016 +
v2\u22a5
2
)\u2211
m
(
\u3bem\u22121 + \u3bem+1 \u2212 i(r\u3bem\u22121)
\u2032
(m\u2212 1) + i
(r\u3bem+1)\u2032
(m + 1)
)
exp i(\u2212im\u3b8 + n\u3d5\u2212 \u3c9t)
(\u3c9 \u2212 \u3c9m) . (14.75)
It is assumed that perturbation consists primarily of two toroidally coupled harmonics \u3bem and
\u3bem+1 and all other harmonics are essentially zero. Strong coupling occurs in a narrow region of
thickness \u223c \ufffda localized about the surface r = r0 corresponding to q(r0) = (2m + 1)/2n = q0. The
mode localization implies that \u3be\u2032m±1 terms dominate in (14.75). Substituting these results into the
expression for sj and maintaining only these terms which do not average to zero in \u3b8 leads to the
following expression for |sj|2:
|sj|2 =
m2j r
2
0
4R
(
v2\u2016 +
v2\u22a5
2
)2( |\u3be\u2032(m+1)|2
(m + 1)2
+
|\u3be\u2032m|2
m2
)(
1
|\u3c9 \u2212 \u3c9m|2 +
1
|\u3c9 \u2212 \u3c9m\u22121|2
)
since \u3c9m+1 = \u2212\u3c9m and \u3c9m+2 = \u2212\u3c9m\u22121. KM is given by
KM =
r20\u3c10
2
\u222b ( |\u3be\u2032m|2
m2
+
|\u3be\u2032m+1|2
(m + 1)2
)
dr. (14.76)
Using the relations \u3c9r \u2248 k\u2016vA, k\u2016 = 1/(2q0R), q0 = (2m+1/2n), we obtain the following expression
for growth rate:
\u3c9i
k\u2016vA
= lim
\u3c9i\u21920
\u2211```