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This is a necessary condition for stability; but it is not always a su\ufb03cient con- dition, as Suydam\u2019s criterion is derived from consideration of local-mode behavior only. Newcomb derived the necessary and su\ufb03cient conditions for the stability of a cylindrical plasma. His twelve theorems are described in (ref.[11]). 8.3d Tokamak Con\ufb01guration In this case the longitudinal magnetic \ufb01eld Bs is much larger than the poloidal magnetic \ufb01eld B\u3b8. The plasma region is r \u2264 a and the vacuum region is a \u2264 r \u2264 b and an ideal conducting wall is at r = b. It is assumed that ka\ufffd 1, kb\ufffd 1. The function \u3b6 in (8.90) for WV is \u3b6 = i (mBa + kaBs) 1\u2212 (a/b)2m \u3ber(a) am bm ( bm rm \u2212 r m bm ) (from (8.91)), and WV becomes WV = \u3c0 2\u3bc0 (mBa + kaBs)2 m \u3be2r (a)\u3bb, \u3bb \u2261 1 + (a/b)2m 1\u2212 (a/b)2m . From the periodic condition for a torus, it follows that 2\u3c0n k = \u22122\u3c0R (n is an integer) so that (k ·B) is given by a(k ·B) = mBa + kaBs = mBa ( 1\u2212 nqa m ) in terms of the safety factor. The Wa term in (8.92) is reduced to k2B2s \u2212 ( m a )2 B2a = ( kBs + m a Ba )2 \u2212 2m a Ba ( kBs + m a Ba ) = ( nBa a )2(( 1\u2212 nqa m )2 \u2212 2 ( 1\u2212 nqa m )) . Accordingly, the energy integral becomes Wp +WV = \u3c0 2\u3bc0 B2a\u3be 2 r (a) (( 1\u2212 nqa m )2 (1 +m\u3bb)\u2212 2 ( 1\u2212 nqa m )) + \u3c0 2\u3bc0 \u222b ( f ( d\u3ber dr )2 + g\u3be2r ) dr. (8.98) The 1st term of (8.98) is negative when 1\u2212 2 1 +m\u3bb < nqa m < 1. (8.99) The assumption nqa/m \u223c 1 corresponds to ka \u223c mBa/Bs. As Ba/Bs \ufffd 1, this is consistent with the assumption ka \ufffd 1. When m = 1, (m2 \u2212 1)/m2 in the 2nd term of (8.95) for g is zero. The magnitude of g is of the order of k2r2, which is very small since kr \ufffd 1. The term in f(d\u3ber/dr)2 can be very small if \u3ber is nearly constant. Accordingly the contribution of the integral term in (8.98) is negligible. When m = 1 and a2/b2 < nqa < 1, the energy integral becomes negative (W < 0). The mode m = 1 is unstable in the region speci\ufb01ed by (8.99) irrespective of the current distribution. The Kruskal-Shafranov condition for the mode m = 1 derived from the sharp-boundary con\ufb01guration is also applicable to the di\ufb00use-boundary plasma. The growth rate \u3b32 = \u2212\u3c92 is \u3b32 \ufffd \u2212W\u222b (\u3c1m0|\u3be|2/2)dr = 1 \u3008\u3c1m0\u3009 B2a \u3bc0a2 ( 2(1\u2212 nqa)\u2212 2(1\u2212 nqa) 2 1\u2212 a2/b2 ) , (8.100) 102 8.3 Instabilities of a Cylindrical Plasma 103 Fig.8.9 The relation of the growth rate \u3b3 and nqa for kink instability (\u22122W/(\u3c0\u3be2aB2a/\u3bc0) = \u3b32a2(\u3008\u3c1m0\u3009\u3bc0/B2a)). After (ref.[12]). \u3008\u3c1m0\u3009 = \u222b \u3c1m0|\u3be|22\u3c0rdr \u3c0a2\u3be2r (a) . The maximum growth rate is \u3b32max \u223c (1\u2212 a2/b2)B2a/(\u3bc0\u3008\u3c1\u3009a2). When m = 1, (m2 \u2212 1)/m2 in the 2nd term of (8.95) for g is large, and g \u223c 1. Accordingly, the contribution of the integral term to Wp must be checked. The region g < 0 is given by \u3c71 < \u3c7 < \u3c72, when \u3c7 \u2261 \u2212krBz/B\u3b8 = nq(r) and \u3c71,2 = m\u2212 2 m(m2 \u2212 1)k 2r2 ± 2k 2r2 m(m2 \u2212 1) ( 1\u2212 m 2(m2 \u2212 1) 2k2r2 \u3bc0rp \u2032 0 B2\u3b8 )1/2 . (8.101) Since kr \ufffd 1, the region g < 0 is narrow and close to the singular point nq(r) = m and the contribution of the integral term to Wp can be neglected. Therefore if nqa/m is in the range given by (8.99), the plasma is unstable due to the displacement \u3ber(a) of the plasma boundary. When the current distribution is j(r) = j0 exp(\u2212\u3ba2r2/a2) and the conducting wall is at in\ufb01nity (b = \u221e), \u3b32 can be calculated from (8.100), using the solution of Euler\u2019s equation; and the dependence of \u3b32 on qa can be estimated. The result is shown in \ufb01g.8.9. When the value of nqa/m is outside the region given by (8.99), the e\ufb00ect of the displacement of the plasma boundary is not great and the contribution of the integral term in Wp is dominant. However, the growth rate \u3b32 is k2r2 times as small as that given by (8.100), as is clear from consideration of (8.101). 8.3e Reversed Field Pinch (ref.[13]) The characteristics of the Reversed \ufb01eld pinch is that Ba and Bs are of the same order of magnitude, so that the approximation based upon ka\ufffd 1 or Ba \ufffd Bs can no longer be used. As is clear from the expression (8.82) for Wp, the plasma is stable if 2\u3bc0jz B\u3b8 r = 2 B\u3b8 r2 d dr (rB\u3b8) = 1 r3 d dr (rB\u3b8)2 < 0 (8.102) is satis\ufb01ed everywhere. This is a su\ufb03cient condition; however, it can never be satis\ufb01ed in real cases. When the expression (8.95) for g is rewritten in terms of P \u2261 rBz/B\u3b8 (2\u3c0P is the pitch of the magnetic-\ufb01eld lines), we \ufb01nd g = 2(kr)2\u3bc0 m2 + (kr)2 dp0 dr + B2\u3b8/r (m2 + (kr)2)2 (kP +m) × ( kP ((m2 + k2r2)2 \u2212 (m2 \u2212 k2r2)) +m((m2 + k2r2)2 \u2212 (m2 + 3k2r2)) ) . (8.103) When m = 1, g becomes g = 2(kr)2\u3bc0 1 + (kr)2 dp0 dr + (kr)2B2\u3b8/r (1 + (kr)2)2 (kP + 1)(kP (3 + k2r2) + (k2r2 \u2212 1)). (8.104) 103 104 8 Magnetohydrodynamic Instabilities Fig.8.10 Dependence of pitch P (r) on r, and the region gj(P, r) < 0. The displacements \u3ber(r) of the unstable modes are also shown. Parts a-d are for k < 0. The 2nd term in (8.104) is quadratic with respect to P and has the minimum value g(r) > 2\u3bc0 dp0 dr k2r2 1 + k2r2 \u2212 4B 2 \u3b8 r k2r2 (1 + k2r2)2(3 + k2r2) . The condition g(r) > 0 is reduced to r\u3bc0 B2\u3b8 dp0 dr > 2 (1 + k2r2)(3 + k2r2) (8.105) (dp0/dr must be positive). Accordingly if the equilibrium solution is found to satisfy the condition (8.105) near the plasma center and also to satisfy the condition (8.102) at the plasma boundary, the positive contribution of the integral term may dominate the negative contribution from the plasma boundary and this equilibrium con\ufb01guration may be stable. Let us consider the 2nd term of (8.104): gj = k2rB2\u3b8 (1 + k2r2)2 (kP + 1)(kP (3 + k2r2) + (k2r2 \u2212 1)). (8.106) This term is positive when kP < \u22121 or kP > (1\u2212 k2r2)/(3 + k2r2). (8.107) The point kP = \u22121 is a singular point. The region gj < 0 is shown in the P, r diagrams of \ufb01g.8.10a-d for given k (< 0). rs is a singular point. Several typical examples of P (r) are shown in the \ufb01gure. It is clear that \u3ber(r) shown in (b) and (c) makes W negative. The example (d) is the case where the longitudinal magnetic \ufb01eld Bz is reversed at r = r1. If k (> 0) is chosen so that kP (0) < 1/3 and if the singular point rs satisfying kP (rs) = \u22121 is smaller than r = b (i.e., rs does not lie on the conducting wall r = b), the plasma is unstable for the displacement \u3ber(r) shown in \ufb01g.8.10d\u2032. The necessary condition for the stability of the reverse-\ufb01eld con\ufb01guration is \u2212P (b) < 3P (0). (8.108) 104 8.3 Instabilities of a Cylindrical Plasma 105 This means that B\u3b8 cannot be very small compared with Bz and that the value of the reversed Bz at the wall cannot be too large. When m = 1, (8.82) for Wp yields the su\ufb03cient condition of stability: 2\u3bc0jz B\u3b8 r < B2\u3b8 r2 (1 + kP )2. (8.109) The most dangerous mode is k = \u22121/P (a). With the assumption B\u3b8 > 0, the stability condition becomes \u3bc0jz < 1 2 B\u3b8 r ( \u2212P (r) P (a) + 1 )2 . (8.110) Accordingly if the condition\u2223\u2223\u2223\u2223P (r)P (a) \u2223\u2223\u2223\u2223 > 1 (8.111) is satis\ufb01ed at small r and jz is negative at r near the boundary, this con\ufb01guration may be stable. Let us consider limitations on the beta ratio from the standpoint of stability. For this purpose the dangerous mode kP (a) = \u22121 is examined, using (8.82) for Wp. The substitution \u3ber(r) = \u3be 0 \u2264 r \u2264 a\u2212 \u3b5, \u3ber(r) = 0 r \u2265 a+ \u3b5 into (8.82) yields Wp = \u3c0 2\u3bc0 \u3be2 \u222b a 0 dr r ( \u22122B\u3b8 ddr (rB\u3b8) + (krBz +mB\u3b8) 2 + (krBz \u2212mB\u3b8)2 m2 + k2r2 ) . When m = 1, then Wp is Wp = \u3c0 2\u3bc0 \u3be2 \u222b a 0 dr r ( \u22122B\u3b8 ddr (rB\u3b8) + 2k 2r2B2z + 2B 2 \u3b8 \u2212 k2r2(krBz \u2212B\u3b8)2 1 + k2r2 ) . Since the last term in the integrand is always negative, the integration of the other three terms must be positive, i.e., \u3c0 2\u3bc0 \u3be2 ( \u2212B2\u3b8 (a) + 2k2 \u222b a 0 rB2zdr ) > 0. Using kP (a) = \u22121 and the