NIFS PROC 88
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NIFS PROC 88


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r2dr
the maximum ballooning stable beta is
\u3b2 = 0.6
a
R
(
1
a3
\u222b a
0
1
q3
dq
dr
r3dr
)
.
Under an optimum q pro\ufb01le, the maximum beta is given by (ref.[19])
\u3b2max \u223c 0.28 a
Rqa
(qa > 2) (8.124)
where qa is the safety factor at the plasma boundary. In the derivation of (8.124) qa > 2, q0 = 1
are assumed.
It must be noti\ufb01ed that the ballooning mode is stable in the negative shear region of S, as is
shown in \ufb01g.8.12. When the shear parameter S is negative (q(r) decreases outwardly), the outer
lines of magnetic force rotate around the magnetic axis more quickly than inner ones. When the
pressure increases, the tokamak plasma tends to expand in a direction of major radius (Shafranov
shift). This must be counter balanced by strengthening the poloidal \ufb01eld on the outside of tokamak
plasma. In the region of strong pressure gradient, the necessary poloidal \ufb01eld increases outwardly,
so on outer magnetic surfaces the magnetic \ufb01eld lines rotate around the magnetic axis faster than
on inner ones and the shear parameter becomes more negative (ref.[20]).
In reality the shear parameter in a tokamak is positive in usual operations. However the fact
that the ballooning mode is stable in negative shear parameter region is very important to develope
tokamak con\ufb01guration stable against ballooning modes. Since
r
Rq
=
B\u3b8
B0
=
1
B0
\u3bc0
2\u3c0r
\u222b r
0
j(r)2\u3c0rdr
110
8.6 \u3b7i Mode due to Density and Temperature Gradient 111
the pro\ufb01le of safety factor q(r) is
1
q(r)
=
R
2B0
(
\u3bc0
\u3c0r2
\u222b r
0
j2\u3c0rdr
)
\u2261 \u3bc0R
2B0
\u3008j(r)\u3009r .
Therefore a negative shear con\ufb01guration can be realized by a hollow current pro\ufb01le. The MHD
stability of tokamak with hollow current pro\ufb01les is analyzed in details in (ref.[21]).
8.6 \u3b7i Mode due to Density and Temperature Gradient
Let us consider a plasma with the density gradient dn0/dr, and the temperature gradient
dTe0/dr, dTi0/dr in the magnetic \ufb01eld with the z direction. Assume that the ion\u2019s density be-
comes ni = ni0 + n\u2dci by disturbance. The equation of continuity
\u2202ni
\u2202t
+ vi · \u2207ni + ni\u2207 · vi = 0
is reduced, by the linearization, to
\u2212i\u3c9n\u2dci + v\u2dcr \u2202n0
\u2202r
+ n0ik\u2016v\u2dc\u2016 = 0. (8.125)
It is assumed that the perturbation terms changes as exp i(k\u3b8r\u3b8+k\u2016z\u2212\u3c9t) and k\u3b8, k\u2016 are the \u3b8 and
z components of the propagation vector. When the perturbed electrostatic potential is denoted by
\u3c6\u2dc, the E ×B drift velocity is v\u2dcr = E\u3b8/B = ik\u3b8\u3c6\u2dc/B. Since the electron density follows Boltzmann
distribution, we \ufb01nd
n\u2dce
n0
=
e\u3c6\u2dc
kTe
. (8.126)
The parallel component of the equation of motion to the magnetic \ufb01eld
nimi
dv\u2016
dt
= \u2212\u2207\u2016pi \u2212 en\u2207\u2016\u3c6
is reduced, by the linearization, to
\u2212i\u3c9nimiv\u2dc\u2016 = \u2212ik\u2016(p\u2dci + en0\u3c6\u2dc). (8.127)
Similarly the adiabatic equation
\u2202
\u2202t
(pin
\u22125/3
i ) + v · \u2207(pin\u22125/3i ) = 0
is reduced to
\u2212i\u3c9
(
p\u2dci
pi
\u2212 5
3
n\u2dci
ni
)
\u2212 ik\u3b8\u3c6\u2dc
B
\u239b\u239d dTi0dr
Ti0
\u2212 2
3
dn0
dr
n0
\u239e\u23a0 = 0. (8.128)
Let us de\ufb01ne the electron drift frequencies \u3c9\u2217ne, \u3c9\u2217Tee and the ion drift frequency \u3c9
\u2217
ni, \u3c9
\u2217
T i by
\u3c9\u2217ne \u2261 \u2212
k\u3b8(\u3baTe)
eBne
dne
dr
, \u3c9\u2217ni \u2261
k\u3b8(\u3baTi)
eBni
dni
dr
,
\u3c9\u2217T e \u2261 \u2212
k\u3b8
eB
d(\u3baTe)
dr
, \u3c9\u2217T i \u2261
k\u3b8
eB
d(\u3baTi)
dr
.
111
112 8 Magnetohydrodynamic Instabilities
The ratio of the temperature gradient to the density gradient of electrons and ions is given by
\u3b7e \u2261 dTe/dr
Te
ne
dne/dr
=
d lnTe
d lnne
, \u3b7i \u2261 dTi/dr
Ti
ni
dni/dr
=
d lnTi
d lnni
respectively. There are following relations among these values;
\u3c9\u2217ni = \u2212
Ti
Te
\u3c9\u2217ne, \u3c9
\u2217
T e = \u3b7e\u3c9
\u2217
ne, \u3c9
\u2217
T i = \u3b7i\u3c9
\u2217
ni.
Then equations (8.125),(8.126),(8.127),(8.128) are reduced to
n\u2dci
n0
=
v\u2dc\u2016
\u3c9/k\u2016
+
\u3c9\u2217ne
\u3c9
e\u3c6\u2dc
\u3baTe
,
n\u2dce
n0
=
e\u3c6\u2dc
\u3baTe
,
v\u2dc\u2016
\u3c9/k\u2016
=
1
mi(\u3c9/k\u2016)2
(
e\u3c6\u2dc+
p\u2dci
n0
)
,
(
p\u2dci
pi0
\u2212 5
3
n\u2dc
n0
)
=
\u3c9\u2217ne
\u3c9
(
\u3b7i \u2212 23
)
e\u3c6\u2dc
\u3baTe
.
Charge neutrality condition n\u2dci/n0 = n\u2dce/n0 yields the dispersion equation (ref.[22]).
1\u2212 \u3c9
\u2217
ne
\u3c9
\u2212
(
vTi
\u3c9/k\u2016
)2 (
Te
Ti
+
5
3
+
\u3c9\u2217ne
\u3c9
(
\u3b7i \u2212 23
))
= 0.
(v2Ti = \u3baTi/mi). The solution in the case of \u3c9 \ufffd \u3c9\u2217ne is
\u3c92 = \u2212k2\u2016v2Ti
(
\u3b7i \u2212 23
)
.
The dispersion equation shows that this type of perturbation is unstable when \u3b7i > 2/3. This
mode is called \u3b7i mode.
When the propagation velocity |\u3c9/k\u2016| becomes the order of the ion thermal velocity vTi , the
interaction (Landau damping) between ions and wave (perturbation) becomes important as will be
described in ch.11 and MHD treatment must be modi\ufb01ed. When the value of \u3b7i is not large, the
kinetic treatment is necessary and the threshold of \u3b7i becomes \u3b7i,cr \u223c 1.5.
References
[1] G. Bateman: MHD instabilities, The MIT Press, Cambridge Mass. 1978.
[2] M. Kruskal and M. Schwarzschield: Proc. Roy. Soc. A223, 348 (1954).
[3] M. N. Rosenbluth, N. A. Krall and N. Rostoker: Nucl. Fusion Suppl. Pt.1 p.143 (1962).
[4] M. N. Rosenbluth and C. L. Longmire: Annal. Physics 1, 120 (1957).
[5] I. B. Berstein, E. A. Frieman, M. D. Kruskal and R. M. Kulsrud: Proc. Roy. Soc. A244, 17 (1958).
[6] B. B. Kadmotsev: Reviews of Plasma Physics 2, 153(ed. by M. A. Loentovich) Consultant Bureau, New
York 1966.
[7] K. Miyamoto: Plasma Physics for Nuclear Fusion (revised edition) Chap.9, The MIT Press, Cambridge,
Mass. 1988.
[8] M. D. Kruskal, J. L. Johnson, M. B. Gottlieb and L. M. Goldman: Phys. Fluids 1, 421 (1958).
[9] V. D. Shafranov: Sov. Phys. JETP 6, 545 (1958).
112
8 References 113
[10] B. R. Suydam: Proc. 2nd U. N. International Conf. on Peaceful Uses of Atomic Energy, Geneva, 31,
157 (1958).
[11] W. A. Newcomb: Annal. Physics 10, 232 (1960).
[12] V. D. Shafranov: Sov. Phys. Tech. Phys. 15, 175 (1970).
[13] D. C. Robinson: Plasma Phys. 13, 439 (1971).
[14] K. Hain and R. Lu¨st: Z. Naturforsh. 13a, 936 (1958).
[15] K. Matsuoka and K. Miyamoto: Jpn. J. Appl. Phys. 18, 817 (1979).
[16] R. M. Kulsrud: Plasma Phys. and Controlled Nucl. Fusion Research,1, 127, 1966 (Conf. Proceedings,
Culham in 1965 IAEA Vienna).
[17] J. W. Connor, R. J. Hastie and J. B. Taylor: Phys. Rev. Lett. 40, 393 (1978).
[18] J. W. Connor, R. J. Hastie and J. B. Taylor: Pro. Roy. Soc. A365, 1 (1979).
[19] J. A. Wesson, A. Sykes: Nucl. Fusion 25 85 (1985).
[20] J. M. Greene, M. S. Chance: Nucl. Fusion 21, 453 (1981).
[21] T. Ozeki, M. Azumi, S. Tokuda, S. Ishida: Nucl. Fusion 33, 1025 (1993).
[22] B. B. Kadomtsev and O. P. Pogutse: Reviews of Plasma Physics 5,304 (ed. by M. A. Leontovich)
Consultant Bureau, New York 1970.
113
114
Ch.9 Resistive Instability
In the preceding chapter we have discussed instabilities of plasmas with zero resistivity. In such
a case the conducting plasma is frozen to the line of magnetic force. However, the resistivity of
a plasma is not generally zero and the plasma may hence deviate from the magnetic line of force.
Modes which are stable in the ideal case may in some instances become unstable if a \ufb01nite resistivity
is introduced.
Ohm\u2019s law is
\u3b7j = E + V ×B. (9.1)
For simplicity we here assume that E is zero. The current density is j = V ×B/\u3b7 and the j ×B
force is
F s = j ×B = B(V ·B)\u2212 V B
2
\u3b7
. (9.2)
When \u3b7 tends to zero, this force becomes in\ufb01nite and prevents the deviation of the plasma from
the line of magnetic force. When the magnitude B of magnetic \ufb01eld is small, this force does not
become large, even if \u3b7 is small, and the plasma can deviate from the line of magnetic force. When
we consider a perturbation with the propagation vector k, only the parallel (to k) component of
the zeroth-order magnetic \ufb01eld B a\ufb00ects the perturbation, as will be shown later. Even if shear
exists, we can choose a propagation vector k perpendicular to the magnetic \ufb01eld B:
(k ·B) = 0. (9.3)
Accordingly, if there is any force F dr driving the perturbation, this driving force may easily exceed
the