NIFS PROC 88
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NIFS PROC 88


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in \ufb01g.12.3.
The series expansion of the plasma dispersion function Z(\u3b6) is
Z(\u3b6) = i\u3c01/2
kz
|kz | exp(\u2212\u3b6
2)\u2212 2\u3b6
(
1\u2212 2
3
\u3b62 +
4
15
\u3b64 + · · ·
)
in the case of |\u3b6| \ufffd 1 (hot plasma). The asymptotic expansion of Z(\u3b6) is
Z(\u3b6) = i\u3c01/2\u3c3
kz
|kz| exp(\u2212\u3b6
2)\u2212 1
\u3b6
(
1 +
1
2
1
\u3b62
+
3
4
1
\u3b64
+ · · ·
)
\u3c3 = 0 for Im\u3c9 > 0, \u3c3 = 2 for Im\u3c9 < 0,
but \u3c3 = 1 for |Im \u3b6||Re \u3b6| <\u223c \u3c0/4, and |\u3b6| \ufffd 1
in the case of |\u3b6| \ufffd 1 (cold plasma) (ref.[1],[2]). Refer to appendix C for the derivation.
The part i\u3c01/2(kz/|kz |) exp(\u2212\u3b62) of Z(\u3b6) represents the terms of Landau damping and cyclotron
damping as is described later in this section.
When T \u2192 0, that is, \u3b6n \u2192 ±\u221e, b \u2192 0, the dielectric tensor of hot plasma is reduced to the
dielectric tensor (10.9)\u223c(10.13) of cold plasma.
In the case of b = (kx\u3c1\u3a9)2 \ufffd 1 (\u3c1\u3a9 = vT\u22a5/\u3a9 is Larmor radius), it is possible to expand e\u2212bXn
by b using
In(b) =
(
b
2
)n \u221e\u2211
l=0
1
l!(n + l)!
(
b
2
)2l
=
(
b
2
)n ( 1
n!
+
1
1!(n + 1)!
(
b
2
)2
+
1
2!(n + 2)!
(
b
2
)4
+ · · ·
)
.
The expansion in b and the inclusion of terms up to the second harmonics in K give
Kxx =1 +
\u2211
j
(
\u3a0j
\u3c9
)2
\u3b60
(
(Z1 + Z\u22121)
(
1
2
\u2212 b
2
+ · · ·
)
+ (Z2 + Z\u22122)
(
b
2
\u2212 b
2
2
+ · · ·
)
+ · · ·
)
j
,
150
12.3 Dielectric Tensor of Hot Plasma, Wave Absorption · · · 151
Kyy =1 +
\u2211
j
(
\u3a0j
\u3c9
)2
\u3b60
(
Z0(2b+ · · ·) + (Z1 + Z\u22121)
(
1
2
\u2212 3b
2
+ · · ·
)
+ (Z2 + Z\u22122)
(
b
2
\u2212 b2 + · · ·
)
+ · · ·
)
j
,
Kzz =1\u2212
\u2211
j
(
\u3a0j
\u3c9
)2
\u3b60
(
2\u3b60W0(1\u2212 b+ · · ·) + (\u3b61W1 + \u3b6\u22121W\u22121)(b+ · · ·)
+(\u3b62W2 + \u3b6\u22122W\u22122)
(
b2
4
+ · · ·
)
+ · · ·
)
j
,
Kxy = i
\u2211
j
(
\u3a0j
\u3c9
)2
\u3b60
(
(Z1 \u2212 Z\u22121)
(
1
2
\u2212 b+ · · ·
)
+ (Z2 \u2212 Z\u22122)
(
b
2
+ · · ·
)
+ · · ·
)
j
,
Kxz = 21/2
\u2211
j
(
\u3a0j
\u3c9
)2
b1/2\u3b60
(
(W1 \u2212W\u22121)
(
1
2
+ · · ·
)
+ (W2 \u2212W\u22122)
(
b
4
+ · · ·
)
+ · · ·
)
j
,
Kyz = \u221221/2i
\u2211
j
(
\u3a0j
\u3c9
)2
b1/2\u3b60
(
W0
(
\u22121 + 3
2
b+ · · ·
)
+ (W1 +W\u22121)
(
1
2
+ · · ·
)
+ (W2 \u2212W\u22122)
(
b
4
+ · · ·
)
+ · · ·
)
j
(12.22)
Kyx = \u2212Kxy, Kzx = Kxz, Kzy = \u2212Kzy
where
Z±n \u2261 Z(\u3b6±n), Wn \u2261 \u2212 (1 + \u3b6nZ(\u3b6n)) ,
\u3b6n = (\u3c9 + n\u3a9)/(21/2kz(\u3baTz/m)1/2).
When x\ufffd 1, ReW (x) is
ReW (x) = (1/2)x\u22122(1 + (3/2)x\u22122 + · · ·).
The absorbed power by Landau damping (including transit time damping) may be estimated by
the terms associated with the imaginary part G0 of \u3b60Z(\u3b60) in (12.22) of Kij :
G0 \u2261 Im\u3b60Z(\u3b60) = kz|kz |\u3c0
1/2\u3b60 exp(\u2212\u3b620 ).
Since
(ImKyy)0 =
(
\u3a0j
\u3c9
)2
2bG0,
(ImKzz)0 =
(
\u3a0j
\u3c9
)2
2\u3b620G0,
151
152 12 Wave Propagation and Wave Heating
(ReKyz)0 =
(
\u3a0j
\u3c9
)2
21/2b1/2\u3b60G0
the contribution of these terms to the absorption power (12.17) is
P ab0 = 2\u3c9
(
\u3a0j
\u3c9
)2
G0
(
\ufffd0
2
)(
|Ey|2b+ |Ez|2\u3b620 + Im(E\u2217yEz)(2b)1/2\u3b60
)
. (12.23)
The 1st term is of transit time damping and is equal to (11.16). The 2nd term is of Landau
damping and is equal to (11.10). The 3rd one is the term of the interference of both.
The absorption power due to cyclotron damping and the harmonic cyclotron damping is obtained
by the contribution from the terms
G±n \u2261 Im\u3b60Z±n = kz|kz|\u3c0
1/2\u3b60 exp(\u2212\u3b62±n)
and for the case b\ufffd 1,
(ImKxx)±n = (ImKyy)±n =
(
\u3a0j
\u3c9
)2
G±n\u3b1n,
(ImKzz)±n =
(
\u3a0j
\u3c9
)2
2\u3b62±nG±nb\u3b1nn
\u22122,
(ReKxy)±n = \u2212
(
\u3a0j
\u3c9
)2
G±n(±\u3b1n),
(ReKyz)±n = \u2212
(
\u3a0j
\u3c9
)2
(2b)1/2\u3b6±nG±n\u3b1nn\u22121,
(ImKxz)±n = \u2212
(
\u3a0j
\u3c9
)2
(2b)1/2\u3b6±nG±n(±\u3b1n)n\u22121,
\u3b1n = n2(2 · n!)\u22121
(
b
2
)n\u22121
.
The contribution of these terms to the absorbed power (12.17) is
P ab±n = \u3c9
(
\u3a0j
\u3c9
)2
Gn
(
\ufffd0
2
)
\u3b1n|Ex ± iEy|2. (12.24)
Since
\u3b6n = (\u3c9 + n\u3a9i)/(21/2kzvTi) = (\u3c9 \u2212 n|\u3a9i|)/(21/2kzvTi)
the term of +n is dominant for the ion cyclotron damping (\u3c9 > 0), and the term of \u2212n is dominant
for electron cyclotron damping (\u3c9 > 0), since
\u3b6\u2212n = (\u3c9 \u2212 n\u3a9e)/(21/2kzvTe).
The relative ratio of E components can be estimated from the following equations:
(Kxx \u2212N2\u2016 )Ex +KxyEy + (Kxz +N\u22a5N\u2016)Ez = 0,
\u2212KxyEx + (Kyy \u2212N2\u2016 \u2212N2\u22a5)Ey +KyzEz = 0, (12.25)
(Kxz +N\u22a5N\u2016)Ex \u2212KyzEy + (Kzz \u2212N2\u22a5)Ez = 0.
152
12.4 Wave Heating in Ion Cyclotron Range of Frequency 153
For cold plasmas, Kxx \u2192 K\u22a5, Kyy \u2192 K\u22a5, Kzz \u2192 K\u2016, Kxy\u2192 \u2212iK×, Kxz \u2192 0, Kyz \u2192 0 can
be substituted into (12.25), and the relative ratio is Ex : Ey : Ez = (K\u22a5 \u2212 N2) ×(K\u2016 \u2212 N2\u22a5) :
\u2212iKx(K\u2016 \u2212N2\u22a5) : \u2212N\u2016N\u22a5(K\u22a5 \u2212N2).
In order to obtain the magnitude of the electric \ufb01eld, it is necessary to solve the Maxwell equation
with the dielectric tensor of (12.19). In this case the density, the temperature, and the magnetic
\ufb01eld are functions of the coordinates. Therefore the simpli\ufb01ed model must be used for analytical
solutions; otherwise numerical calculations are necessary to derive the wave \ufb01eld.
12.4 Wave Heating in Ion Cyclotron Range of Frequency
The dispersion relation of waves in the ion cyclotron range of frequency (ICRF) is given by
(10.64);
N2\u2016 =
N2\u22a5
2[1\u2212 (\u3c9/\u3a9i)2]
(
\u2212
(
1\u2212
(
\u3c9
\u3a9i
)2)
+
2\u3c92
k2\u22a5v
2
A
±
\u23a1\u23a3(1\u2212 ( \u3c9
\u3a9i
)2)2
+ 4
(
\u3c9
\u3a9i
)2 ( \u3c9
k\u22a5vA
)4\u23a4\u23a61/2).
The plus sign corresponds to the slow wave (L wave, ion cyclotron wave), and the minus sign
corresponds to the fast wave (R wave, extraordinary wave). When 1\u2212 \u3c92/\u3a92i \ufffd 2(\u3c9/k\u22a5vA)2, the
dispersion relation becomes
k2z = 2
(
\u3c92
v2A
)(
1\u2212 \u3c9
2
\u3a92i
)\u22121
, (for slow wave)
k2z = \u2212
k2\u22a5
2
+
\u3c92
2v2A
. (for fast wave)
Since the externally excited waves have propagation vectors with 0 < k2z < (1/a)
2, k2\u22a5 > (\u3c0/a)
2
usually, there are constraints
\u3c92
v2A
2
(1\u2212 \u3c92/\u3a92i )
<
(
\u3c0
a
)2
,
n20a
2 < 2.6× 10\u22124B
2(1\u2212 \u3c92/\u3a92i )
A
for slow wave and
\u3c92
2v2A
>
(
\u3c0
a
)2
,
n20a
2 > 0.5× 10\u22122 (\u3a9i/\u3c9)
2
A/Z2
for the fast wave (ref.[3]), where n20 is the ion density in 1020 m\u22123, a is the plasma radius in meters,
and A is the atomic number.
An ion cyclotron wave (slow wave) can be excited by a Stix coil (ref.[1]) and can propagate and
heat ions in a low-density plasma. But it cannot propagate in a high-density plasma like that of a
tokamak.
The fast wave is an extraordinary wave in this frequency range and can be excited by a loop
antenna, which generates a high-frequency electric \ufb01eld perpendicular to the magnetic \ufb01eld (see
sec.10.2). The fast wave can propagate in a high-density plasma. The fast wave in a plasma with
a single ion species has Ex + iEy = 0 at \u3c9 = |\u3a9i| in cold plasma approximation, so that it is not
absorbed by the ion cyclotron damping. However, the electric \ufb01eld of the fast wave in a plasma
153
154 12 Wave Propagation and Wave Heating
with two ion species is Ex + iEy 
= 0, so that the fast wave can be absorbed; that is, the fast wave
can heat the ions in this case.
Let us consider the heating of a plasma with two ion species, M and m, by a fast wave. The
masses, charge numbers, and densities of the M ion and m ion are denoted by mM, ZM, nM and
mm, Zm, nm, respectively. When we use
\u3b7M \u2261 Z
2
MnM
ne
, \u3b7m \u2261 Z
2
mnm
ne
we have \u3b7M/ZM + \u3b7m/Zm = 1 since ne = ZMnM + Zmnm. Since (\u3a0e/\u3c9)2 \ufffd 1 in ICRF wave, the
dispersion relation in the cold plasma approximation is given by (10.64) as follows:
N2\u22a5 =
(R\u2212N2\u2016 )(L\u2212N2\u2016 )
K\u22a5 \u2212N2\u2016
,
R = \u2212\u3a0
2
i
\u3c92
(
(mM/mm)\u3b7m\u3c9
\u3c9 + |\u3a9m| +
\u3b7M\u3c9
\u3c9 + |\u3a9M| \u2212
\u3c9
|\u3a9M|/ZM
)
,
L = \u2212\u3a0
2
i
\u3c92
(
(mM/mm)\u3b7m\u3c9
\u3c9 \u2212 |\u3a9m| +
\u3b7M\u3c9
\u3c9 \u2212 |\u3a9M| +
\u3c9
|\u3a9M|/ZM
)
,
K\u22a5 = \u2212\u3a0
2
i
\u3c92
(
(mM/mm)\u3b7m\u3c92
\u3c92 \u2212\u3a92m
+
\u3b7M\u3c9
2
\u3c92 \u2212\u3a92M
)
,
\u3a02i \u2261
nee
2
\ufffd0mM
.
Therefore ion-ion hybrid resonance occurs at K\u22a5 \u2212N2\u2016 = 0, that is,
\u3b7m(mM/mm)\u3c92
\u3c92 \u2212\u3a92m
+
\u3b7M\u3c9
2
\u3c92 \u2212\u3a92M
\u2248 \u2212 \u3c9
2
\u3a02i
N2\u2016 \u2248 0,
\u3c92 \u2248 \u3c9IH \u2261 \u3b7M + \u3b7m(\u3bc
2/\u3bc\u2032)
\u3b7M + \u3b7m/\u3bc\u2032
\u3a92m,