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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