NIFS PROC 88
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NIFS PROC 88


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is 1/h = 1 + 2kz\u3c1i(1 + p)s and
\u3a02e (x)
\u3a92e
=
\u3a02M.C.
\u3a92e
\u2261 p
1 + 2kz\u3c1i(1 + p)s
.
Accordingly, the mode conversion occurs at the position satisfying
\u3c92
\u3a02i
=
(
1\u2212 \u3c9
2
|\u3a9i|\u3a9e
)
+
N\u2016vTe2
\u221a
3
c
\u239b\u239dTi
Te
+
1
4
(
\u3c92
\u3a9i\u3a9e
)2\u239e\u23a01/2
157
158 12 Wave Propagation and Wave Heating
Fig.12.6 Array of waveguides to excite a lower hybrid wave (slow wave).
and the value of k2\u22a5\u3c1
2
i at this position becomes
k2\u22a5\u3c1
2
i |M.C. =
mi
me
kz\u3c1i
s
.
If the electron temperature is high enough at the plasma center to satisfy vTe > (1/3)c/N\u2016, the
wave is absorbed by electrons due to electron Landau damping.
After the mode conversion, the value N\u22a5 becomes large so that c/N\u22a5 becomes comparable to the
ion thermal velocity (c/N\u22a5 \u223c vTi). Since \u3c9 \ufffd |\u3a9i|, the ion motion is not a\ufb00ected by the magnetic
\ufb01eld within the time scale of \u3c9\u22121. Therefore the wave with phase velocity c/N is absorbed by
ions due to ion Landau damping. When ions have velocity vi larger than c/N\u22a5 (vi > c/N\u22a5), the
ions are accelerated or decelerated at each time satisfying vi cos(\u3a9it) \u2248 c/N\u22a5 and are subjected to
stochastic heating.
The wave is excited by the array of waveguides, as shown in \ufb01g.12.6, with an appropriate phase
di\ufb00erence to provide the necessary parallel index N\u2016 = kzc/\u3c9 = 2\u3c0c/(\u3bbz\u3c9). In the low-density
region at the plasma boundary, the component of the electric \ufb01eld parallel to the magnetic \ufb01eld is
larger for the slow wave than for the fast wave. Therefore the direction of wave-guides is arranged
to excite the electric \ufb01eld parallel to the line of magnetic force. The coupling of waves to plasmas
is discussed in detail in (ref.[11]) and the experiments of LHH are reviewed in (ref.[12]).
For the current drive by lower hybrid wave, the accessibility condition (12.30) and c/N\u2016 \ufffd vTe
are necessary. If the electron temperature is high and \u3baTe \u223c 10 keV, then vTe/c is already \u223c 1/7.
Even if N\u2016 is chosen to be small under the accessibility condition, (12.30), the wave is subjected to
absorbtion by electron damping in the outer part of the plasma, and it can not be expected that
the wave can propagate into the central part of the plasma.
When the value of N\u2016 is chosen to be N\u2016 \u223c (1/3)(c/vTe), electron heating can be expected and
has been observed experimentally. Under the condition that the mode conversion can occur, ion
heating can be expected. However, the experimental results are less clear than those for electron
heating.
12.6 Electron Cyclotron Heating
The dispersion relation of waves in the electron cyclotron range of frequency in a cold plasma
is given by (10.79). The plus and minus signs in (10.79) correspond to ordinary and extraordinary
waves, respectively. The ordinary wave can propagate only when \u3c92 > \u3a02e as is clear from (10.86)
(in the case of \u3b8 = \u3c0/2). This wave can be excited by an array of waveguides, like that used
for lower hybrid waves (\ufb01g.12.6), which emits an electric \ufb01eld parallel to the magnetic \ufb01eld. The
phase of each waveguide is selected to provide the appropriate value of the parallel index N\u2016
= kzc/\u3c9= 2\u3c0c/(\u3c9\u3bbz).
The dispersion relation of the extraordinary wave is given by (10.87). When \u3b8 = \u3c0/2, it is given
by (10.52). It is necessary to satisfy \u3c92UH > \u3c9
2 > \u3c92L, \u3c9
2
LH. As is seen from the CMA diagram of
\ufb01g.10.5, the extraordinary wave can access the plasma center from the high magnetic \ufb01eld side (see
\ufb01g.12.7) but can not access from the low \ufb01eld side because of \u3c9 = \u3c9R cuto\ufb00. The extraordinary
wave can be excited by the waveguide, which emits an electric \ufb01eld perpendicular to the magnetic
\ufb01eld (see sec.10.2a).
158
12.6 Electron Cyclotron Heating 159
Fig.12.7 The locations of electron cyclotron resonance (\u3c9 = \u3a9e), upperhybrid resonance (\u3c9 = \u3c9LH) and R
cut o\ufb00 (\u3c9 = \u3c9R) in case of \u3a9e0 > \u3a0e0, where \u3a9e0 and \u3a0e0 are electron cyclotron resonance frequency and
plasma frequency at the plasma center respectively (left-hand side \ufb01gure). The right-hand side \ufb01gure is the
CMA diagram near electron cyclotron frequency region.
The ion\u2019s contribution to the dielectric tensor is negligible. When relations b \ufffd 1, \u3b60 \ufffd 1 are
satis\ufb01ed for electron, the dielectric tensor of a hot plasma is
Kxx = Kyy = 1 +X\u3b60Z\u22121/2, Kzz = 1\u2212X +N2\u22a5\u3c7zz,
Kxy = \u2212iX\u3b60Z\u22121/2, Kxz = N\u22a5\u3c7xz, Kyz = iN\u22a5\u3c7yz,
\u3c7xz \u2248 \u3c7yz \u2248 2\u22121/2XY \u22121 vT
c
\u3b60(1 + \u3b6\u22121Z\u22121),
\u3c7zz \u2248 XY \u22122
(
vT
c
)2
\u3b60\u3b6\u22121(1 + \u3b6\u22121Z\u22121),
X \u2261 \u3a0
2
e
\u3c92
, Y \u2261 \u3a9e
\u3c9
, \u3b6\u22121 =
\u3c9 \u2212\u3a9e
21/2kzvT
, N\u22a5 =
k\u22a5c
\u3c9
.
The Maxwell equation is
(Kxx \u2212N2\u2016 )Ex +KxyEy +N\u22a5(N\u2016 + \u3c7xz)Ez = 0,
\u2212KxyEx + (Kyy \u2212N2\u2016 \u2212N2\u22a5)Ey + iN\u22a5\u3c7yzEz = 0,
N\u22a5(N\u2016 + \u3c7xz)Ex \u2212 iN\u22a5\u3c7yzEy + (1\u2212X \u2212N2\u22a5(1\u2212 \u3c7zz))Ez = 0.
The solution is
Ex
Ez
= \u2212 iN
2
\u22a5\u3c7xz(N\u2016 + \u3c7xz) +Kxy(1\u2212X \u2212N2\u22a5(1\u2212 \u3c7zz))
N\u22a5(i\u3c7xz(Kxx \u2212N2\u2016 ) +Kxy(N\u2016 + \u3c7xz))
,
Ey
Ez
= \u2212
N2\u22a5(N\u2016 + \u3c7xz)
2 \u2212 (Kxx \u2212N2\u2016 )(1\u2212X \u2212N2\u22a5(1\u2212 \u3c7zz))
N\u22a5(i\u3c7xz(Kxx \u2212N2\u2016 ) +Kxy(N\u2016 + \u3c7xz))
.
The absorption power P\u22121 per unit volume is given by (12.24) as follows:
P\u22121 = \u3c9X\u3b60
\u3c01/2
2
exp
(
\u2212(\u3c9 \u2212\u3a9e)
2
2k2zv2Te
)
\ufffd0
2
|Ex \u2212 iEy|2.
159
160 12 Wave Propagation and Wave Heating
When \u3c9 = \u3a9e, then \u3b6\u22121 = 0, Z\u22121 = i\u3c01/2, Kxx = 1+ ih, Kxy = h, \u3c7yz = \u3c7xz = 21/2X(vTe/c)\u3b60 =
X/(2N\u2016), \u3c7zz = 0, h \u2261 \u3c01/2\u3b60X/2. Therefore the dielectric tensor K becomes
K =
\u23a1\u23a3 1 + ih h N\u22a5\u3c7xz\u2212h 1 + ih iN\u22a5\u3c7xz
N\u22a5\u3c7xz \u2212iN\u22a5\u3c7xz 1\u2212X
\u23a4\u23a6 .
For the ordinary wave (O wave), we have
Ex \u2212 iEy
Ez
=
iN2\u22a5(O)N\u2016(N\u2016 + \u3c7xz)\u2212 i(1\u2212N2\u2016 )(1\u2212X \u2212N2\u22a5(O))
N\u22a5(O)(N\u2016h+ i\u3c7xz(1\u2212N2\u2016 ))
.
When N\u2016 \ufffd 1 and the incident angle is nearly perpendicular, (10.82) gives 1 \u2212 X \u2212 N2\u22a5(O) =
(1\u2212X)N2\u2016 . Since \u3c7xz = X/2N\u2016, \u3c7xz \ufffd N\u2016. Therefore the foregoing equation reduces to
Ex \u2212 iEy
Ez
=
iN\u22a5(O)N\u2016\u3c7xz
N\u2016h+ i\u3c7xz
.
For extraordinary wave (X wave), we have
Ex \u2212 iEy
Ey
= \u2212
iN2\u22a5(X)N\u2016(N\u2016 + \u3c7xz)\u2212 i(1\u2212N2\u2016 )(1\u2212X \u2212N2\u22a5(X))
N2\u22a5(X)(N\u2016 + \u3c7xz)2 \u2212 (Kxx \u2212N2\u2016 )(1\u2212X \u2212N2\u22a5(X))
.
WhenN\u2016 \ufffd 1 and \u3c9 = \u3a9e, (10.83) gives 1\u2212X\u2212N2\u22a5(X) \u2248 \u22121+N2\u2016 . Since \u3c72xz = (2\u3c0)\u22121/2(vTe/cN\u2016)Xh\ufffd
h, the foregoing equation reduces to
Ex \u2212 iEy
Ey
=
\u2212(1 +N2\u22a5(X)N\u2016(N\u2016 + \u3c7xz))
h\u2212 i(1 +N2\u22a5(X)(N\u2016 + \u3c7xz)2)
\u223c \u22121
h
.
The absorption power per unit volume at \u3c9 = \u3a9e is
P\u22121(O) \u2248 \u3c9\ufffd02 |Ez|
2
hN2\u22a5(O)N
2
\u2016\u3c7
2
xz
(N\u2016h)2 + \u3c72xz
exp(\u2212\u3b62\u22121)
\u2248 \u3c9\ufffd0
2
|Ez |2 1(2\u3c0)1/2
(
\u3a0e
\u3c9
)2( vTe
cN\u2016
)
N2\u22a5(O)N
2
\u2016
N2\u2016 + (vTe/c)
2(2/\u3c0)
for ordinary wave and
P\u22121(X) \u223c \u3c9\ufffd02 |Ey|
2 1
h
=
\u3c9\ufffd0
2
|Ey|22
(
2
\u3c0
)1/2 (\u3a0e
\u3c9
)\u22122 (N\u2016vTe
c
)
.
for extraordinary wave (ref.[13]).
Since P (O) \u221d neT 1/2e /N\u2016, P (X) \u221d N\u2016T 1/2e /ne, the ordinary wave is absorbed more in the case of
higher density and perpendicular incidence, but the extraordinary wave has opposite tendency.
The experiments of electron cyclotron heating have been carried out by T-10, ISX-B, JFT-2,
D-IIID, and so on, and the good heating e\ufb03ciency of ECH has been demonstrated. Heating and
current drive by electron cyclotron waves are reviewed in (ref.[14]).
160
References 161
References
[1] T. H. Stix: The Theory of Plasma Waves, McGraw-Hill, New York 1962.
T. H. Stix: Waves in Plasmas, American Institute of Physics, New York, 1992.
[2] B. D. Fried and S. D. Conte: The Plasma Dispersion Function, Academic Press, New York 1961.
[3] B.D.Fried and S. D. Conte: The Plasma Dispersion Function Academic Press, New York 1961
[4] M. Porkolab: Fusion (ed. by E. Teller) Vol.1, Part B, p.151, Academic Press, New York (1981).
[5] J. E. Scharer, B. D. McVey and T. K. Mau: Nucl. Fusion 17, 297 (1977).
[6] T. H. Stix: Nucl. Fusion 15, 737 (1975)
[7] A. Fukuyama, S.Nishiyama, K. Itoh and S.I. Itoh: Nucl. Fusion 23 1005 (1983).
[8] M. Ono, T.Watari, R. Ando, J.Fujita et al.: Phys. Rev. Lett. 54, 2339 (1985).
[9] T. H. Stix: Phys. Rev. Lett. 15, 878 (1965).
[10] V. M. Glagolev: Plasma Phys. 14, 301,315 (1972).
[11] M. Brambilla: Plasma Phys. 18, 669 (1976).
[12] S. Bernabei, M. A. Heald, W. M. Hooke, R. W. Motley, F. J. Paoloni,