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# NIFS PROC 88

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```\u3bc\u2032 \u2261 mm
mM
, \u3bc \u2261 \u3a9M
\u3a9m
=
mmZM
mMZm
.
Figure 12.4 shows the ion-ion hybrid resonance layer; K\u22a5\u2212N2\u2016 = 0, the L cuto\ufb00 layer; L\u2212N2\u2016 = 0,
and R cuto\ufb00 layer; R\u2212N2\u2016 = 0, of a tokamak plasma with the two ion species D+ (M ion) and
H+ (m ion). Since the Kzz component of the dielectric tensor is much larger than the other
component, even in a hot plasma, the dispersion relation of a hot plasma is (ref.[4])
Kxx \u2212N2\u2016 Kxy
\u2212Kxy Kyy \u2212N2\u2016 \u2212N2\u22a5
= 0. (12.26)
When we use the relation Kyy \u2261 Kxx +\u394Kyy, |\u394Kyy| \ufffd |Kxx|,
N2\u22a5 =
(Kxx \u2212N2\u2016 )(Kxx +\u394Kyy \u2212N2\u2016 ) +K2xy
Kxx \u2212N2\u2016
\u2248
(Kxx + iKxy \u2212N2\u2016 )(Kxx \u2212 iKxy \u2212N2\u2016 )
Kxx \u2212N2\u2016
.
154
12.4 Wave Heating in Ion Cyclotron Range of Frequency 155
Fig.12.4 L cuto\ufb00 layer(L = N2\u2016 ), R cuto\ufb00 layer (R = N
2
\u2016 ), and the ion-ion hybrid resonance layer
(K\u22a5 = N2\u2016 ) of ICRF wave in a tokamak with two ion components D
+, H+. The shaded area is the region
of N2\u22a5 < 0.
When \u3c92 is near \u3c92IH, Kxx is given by
Kxx = \u2212\u3a0
2
i
\u3c92
(
mM
2mm
\u3b7m\u3b60Z(\u3b61) +
\u3b7M\u3c9
2
\u3c92 \u2212\u3a92M
)
.
The resonance condition is Kxx = N2\u2016 . The value of Z(\u3b61) that appears in the dispersion equation
is \ufb01nite and 0 > Z(\u3b61) > \u22121.08. The condition
\u3b7m \u2265 \u3b7cr \u2261 21.08
mm
mM
21/2N\u2016
vTi
c
(
\u3b7M\u3c9
2
\u3c92 \u2212\u3a92M
+N2\u2016
\u3c92
\u3a02i
)
is necessary to obtain the resonance condition. This point is di\ufb00erent from the cold plasma disper-
sion equation (note the di\ufb00erence between Kxx and K\u22a5).
It is deduced from the dispersion equation (12.26) that the mode conversion (ref.[4]) from the
fast wave to the ion Berstein wave occurs at the resonance layer when \u3b7m \u2265 \u3b7cr. When the L cuto\ufb00
layer and the ion-ion hybrid resonance layer are close to each other, as shown in \ufb01g.12.4, the fast
wave propagating from the outside torus penetrates the L cuto\ufb00 layer partly by the tunneling e\ufb00ect
and is converted to the ion Bernstein wave. The mode converted wave is absorbed by ion cyclotron
damping or electron Landau damping. The theory of mode conversion is described in chapter 10
of (ref.[1]). ICRF experiments related to this topic were carried out in TFR.
When \u3b7m < \u3b7cr, K\u22a5 = N2\u2016 cannot be satis\ufb01ed and the ion-ion hybrid resonance layer disappears.
In this case a fast wave excited by the loop antenna outside the torus can pass through the R cuto\ufb00
region (because the width is small) and is re\ufb02ected by the L cuto\ufb00 layer and bounced back and
forth in the region surrounded by R = N2\u2016 and L = N
2
\u2016 . In this region, there is a layer satisfying
\u3c9 = |\u3a9m|, and the minority m ions are heated by the fundamental ion cyclotron damping. The
majority M ions are heated by the Coulomb collisions with m ions. If the mass of M ions is l
times as large as the mass of m ions, the M ions are also heated by the lth harmonic ion cyclotron
damping. This type of experiment was carried out in PLT with good heating e\ufb03ciency. This
is called minority heating. The absorption power Pe0 due to electron Landau damping per unit
volume is given by (12.23), and it is important only in the case \u3b60 \u2264 1. In this case we have
Ey/Ez \u2248 Kzz/Kyz \u2248 2\u3b620/(21/2b1/2\u3b60(\u2212i)) and Pe0 is (ref.[5])
Pe0 =
\u3c9\ufffd0
4
|Ey|2
(
\u3a0e
\u3c9
)2 (k\u22a5vTe
\u3a9e
)2
2\u3b60e\u3c01/2 exp(\u2212\u3b620e). (12.27)
The absorption power Pin by the n-th harmonic ion cyclotron damping is given by (12.24) as
follows;
Pin =
\u3c9\ufffd0
2
|Ex + iEy|2
(
\u3a0i
\u3c9
)2 ( n2
2× n!
)(
b
2
)n\u22121
× \u3c9
21/2kzvTi
\u3c01/2 exp
(
\u2212(\u3c9 \u2212 n|\u3a9i|)
2
2(kzvTi)2
)
. (12.28)
155
156 12 Wave Propagation and Wave Heating
The absorption power due to the second harmonic cyclotron damping is proportional to the beta
value of the plasma. In order to evaluate the absorption power by (12.27) and (12.28), we need
the spatial distributions of Ex and Ey. They can be calculated in the simple case of a sheet model
(ref.[6]).
In the range of the higher harmonic ion cyclotron frequencies (\u3c9 \u223c 2\u3a9i, 3\u3a9i), the direct excitation
of the ion Bernstein wave has been studied by an external antenna or waveguide, which generates
a high-frequency electric \ufb01eld parallel to the magnetic \ufb01eld (ref.[7]).
12.5 Lower Hybrid Wave Heating
Since |\u3a9i| \ufffd \u3a0i in a tokamak plasma (ne \u2265 1013, cm\u22123), the lower hybrid resonance frequency
becomes
\u3c92LH =
\u3a02i +\u3a9
2
i
1 +\u3a02e /\u3a92e + Zme/mi
\u2248 \u3a0
2
i
1 +\u3a02e /\u3a92e
.
There are relations \u3a9e \ufffd \u3c9LH \ufffd |\u3a9i|, \u3a02i /\u3a02e = |\u3a9i|/\u3a9e. For a given frequency \u3c9, lower hybrid
resonance \u3c9 = \u3c9LH occurs at the position where the electron density satis\ufb01es the following condition:
\u3a02e (x)
\u3a92e
=
\u3a02res
\u3a92e
\u2261 p, p = \u3c9
2
\u3a9e|\u3a9i| \u2212 \u3c92 .
When the dispersion equation (10.20) of cold plasma is solved about N2\u22a5 using N
2 = N2\u2016 + N
2
\u22a5,
we have
N2\u22a5 =
K\u22a5K\u2dc\u22a5 \u2212K2× +K\u2016K\u2dc\u22a5
2K\u22a5
±
\u23a1\u23a3(K\u22a5K\u2dc\u22a5 \u2212K2× +K\u2016K\u2dc\u22a5
2K\u22a5
)2
+
K\u2016
K\u22a5
(K2× \u2212 K\u2dc2\u22a5)
\u23a4\u23a61/2 ,
where K\u2dc\u22a5 = K\u22a5 \u2212 N2\u2016 . The relations h(x) \u2261 \u3a02e (x)/\u3a02res, K\u22a5 = 1 \u2212 h(x), K× = ph(x)\u3a9e/\u3c9,
K\u2016= 1\u2212\u3b2\u3a0h(x), \u3b2\u3a0 \u2261 \u3a02res/\u3c92\u223cO(mi/me), \u3b1 \u2261 \u3a02res/(\u3c9\u3a9e) \u223c O(mi/me)1/2 and \u3b2\u3a0h\ufffd 1 reduce
this to
N2\u22a5(x) =
\u3b2\u3a0h
2(1\u2212 h)
(
N2\u2016 \u2212 (1\u2212 h+ ph)±
[
(N2\u2016 \u2212 (1\u2212 h+ ph))2 \u2212 4(1\u2212 h)ph
]1/2)
. (12.29)
The slow wave corresponds to the case of the plus sign in (12.29). In order for the slow wave to
propagate from the plasma edge with low density (h \ufffd 1) to the plasma center with high density
(\u3a02e = \u3a0
2
res, h = 1), N\u22a5(x) must real. Therefore following condition
N\u2016 > (1\u2212 h)1/2 + (ph)1/2
is necessary. The right-hand side of the inequality has the maximum value (1 + p)1/2 in the range
0 < h < 1, so that the accessibility condition of the resonant region to the lower hybrid wave
becomes
N2\u2016 > N
2
\u2016,cr = 1 + p = 1 +
\u3a02res
\u3a92e
. (12.30)
If this condition is not satis\ufb01ed, the externally excited slow wave propagates into the position
where the square root term in (12.29) becomes zero and transforms to the fast wave there. Then
the fast wave returns to the low-density region (refer to \ufb01g.12.5). The slow wave that satis\ufb01es
the accessibility condition can approach the resonance region and N\u22a5 can become large, so that
the dispersion relation of hot plasma must be used to examine the behavior of this wave. Near
the lower hybrid resonance region, the approximation of the electrostatic wave, (13.1) or (C.36), is
applicable. Since |\u3a9i| \ufffd \u3c9 \ufffd \u3a9e, the terms of ion contribution and electron contribution are given
by (13.3) and (13.4), respectively, that is,
156
12.5 Lower Hybrid Wave Heating 157
Fig.12.5 Trace of lower hybrid wave in N2\u22a5 \u2212 h(x) (= \u3a02e (x)/\u3a02res) diagram for the case of
p = 0.353, N2\u2016cr = 1 + p = 1.353. This corresponds to the case of H
+ plasma in B = 3T, and
f = \u3c9/2\u3c0= 109 Hz. The electron density for the parameter \u3b2\u3a0 = 7.06× 103 (= \u3a02res/\u3c92) is
nres = 0.31× 1020 m\u22123.
1 +
\u3a02e
k2
me
Te
(1 + I0e\u2212b\u3b60Z(\u3b60)) +
\u3a02i
k2
mi
Ti
(1 + \u3b6Z(\u3b6)) = 0,
where \u3b60 = \u3c9/(21/2kzvTe), and \u3b6 = \u3c9/(21/2kvTi) \u2248 \u3c9/(21/2k\u22a5vTi). Since I0e\u2212b \u2248 1 \u2212 b + (3/4)b2,
\u3b60 \ufffd 1, \u3b6 \ufffd 1, 1 + \u3b6Z(\u3b6) \u2248 \u2212(1/2)\u3b6\u22122 \u2212 (3/4)\u3b6\u22124, we have(
3\u3a02i
\u3c94
\u3baTi
mi
+
3
4
\u3a02e
\u3a94e
\u3baTe
me
)
k4\u22a5 \u2212
(
1 +
\u3a02e
\u3a92e
\u2212 \u3a0
2
i
\u3c92
)
k2\u22a5 \u2212
(
1\u2212 \u3a0
2
e
\u3c92
)
k2z = 0. (12.31)
Using the notations \u3c1i = vTi/|\u3a9i| and
s2 \u2261 3
(
|\u3a9e\u3a9i|
\u3c92
+
1
4
Te
Ti
\u3c92
|\u3a9e\u3a9i|
)
= 3
(
1 + p
p
+
1
4
Te
Ti
p
1 + p
)
,
we have (
3\u3a02i
\u3c94
\u3baTi
mi
+
3
4
\u3a02e
\u3a94e
\u3baTe
me
)
=
\u3a02i
\u3c92
me
mi
v2Tis
2
\u3a9i
(
1 +
\u3a02e
\u3a92e
\u2212 \u3a0
2
i
\u3c92
)
=
1
1 + p
1\u2212 h
h
\u3a02i
\u3c92
Then the dimensionless form of (12.31) is
(k\u22a5\u3c1i)4 \u2212 1\u2212 h
h
mi
me
1
(1 + p)s2
(k\u22a5\u3c1i)2 +
(
mi
me
)2 1
s2
(kz\u3c1i)2 = 0. (12.32)
This dispersion equation has two solutions. One corresponds to the slow wave in a cold plasma and
the other to the plasma wave in a hot plasma. The slow wave transforms to the plasma wave at the
location where (12.31) or (12.32) has equal roots (ref.[8]-[10]). The condition of zero discriminant```