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equilibrium equation (6.9), 2 a2 \u222b a 0 ( \u3bc0p0 + B2z 2 ) rdr = ( \u3bc0p0 + B2z +B2\u3b8 2 ) r=a we can convert the necessary condition for stability to a2B2\u3b8 (a) > 4\u3bc0 \u222b a 0 rp0dr i.e., \u3b2\u3b8 \u2261 ( 2\u3bc0 B2\u3b8 ) 2\u3c0 \u3c0a2 \u222b p0rdr < 1. (8.112) Next let us study the stability of the mode m = 0 in the reversed \ufb01eld pinch. It is assumed that Bz reverses at r = r1. The substitution \u3ber(r) = \u3bbr 0 \u2264 r \u2264 r1 \u2212 \u3b5, \u3ber(r) = 0 r > r1 + \u3b5 105 106 8 Magnetohydrodynamic Instabilities into (8.82) yields Wp = \u3c0 2\u3bc0 \u3bb2 \u222b r1 0 rdr ( 4B2z \u2212 2B\u3b8 d dr (rB\u3b8) + k2r2B2z ) . Using the equilibrium equation (6.8), we obtain the necessary condition for stability: \u3bc\u221210 r 2 1B 2 \u3b8 (r1) > 8 \u222b r1 0 rp0dr \u2212 4r21p0(r1). If p0(r1) \u223c 0, we have the condition \u3b2\u3b8 < 1 2 . (8.113) 8.4 Hain-Lu¨st Magnetohydrodynamic Equation When the displacement \u3be is denoted by \u3be(r, \u3b8, z, t) = \u3be(r) exp i(m\u3b8 + kz \u2212 \u3c9t) and the equilibrium magnetic \ufb01eld B0 is expressed by B(r) = (0, B\u3b8(r), Bz(r)) the (r, \u3b8, z) components of magnetohydrodynamic equation of motion are given by \u2212\u3bc0\u3c1m\u3c92\u3ber = ddr ( \u3bc0\u3b3p(\u2207 · \u3be) +B2 1 r d dr (r\u3ber) + iD(\u3be\u3b8Bz \u2212 \u3bezB\u3b8) ) \u2212 ( F 2 + r d dr ( B\u3b8 r )2) \u3ber \u2212 2ikB\u3b8 r (\u3be\u3b8Bz \u2212 \u3bezB\u3b8), (8.114) \u2212\u3bc0\u3c1m\u3c92\u3be\u3b8 = im r \u3b3\u3bc0p(\u2207 · \u3be) + iDBz 1 r d dr (r\u3ber) + 2ik B\u3b8Bz r \u3ber \u2212H2Bz(\u3be\u3b8Bz \u2212 \u3bezB\u3b8), (8.115) \u2212\u3bc0\u3c1m\u3c92\u3bez = ik\u3b3\u3bc0p(\u2207 · \u3be)\u2212 iDB\u3b8 1 r d dr (r\u3ber)\u2212 2ikB 2 \u3b8 r \u3ber +H2B\u3b8(\u3be\u3b8Bz \u2212 \u3bezB\u3b8) (8.116) where F = m r B\u3b8 + kBz = (k ·B), D = m r Bz \u2212 kB\u3b8, H2 = ( m r )2 + k2, \u2207 · \u3be = 1 r d dr (r\u3ber) + im r \u3be\u3b8 + ik\u3bez. When \u3be\u3b8, \u3bez are eliminated by (8.115),(8.116), we \ufb01nd d dr ( (\u3bc0\u3c1m\u3c92 \u2212 F 2) \u394 (\u3bc0\u3c1m\u3c92(\u3b3\u3bc0p+B2)\u2212 \u3b3\u3bc0pF 2)1 r d dr (r\u3ber) ) + [ \u3bc0\u3c1m\u3c9 2 \u2212 F 2 \u2212 2B\u3b8 ddr ( B\u3b8 r ) \u2212 4k 2 \u394 B2\u3b8 r2 (\u3bc0\u3c1m\u3c92B2 \u2212 \u3b3\u3bc0pF 2) +r d dr ( 2kB\u3b8 \u394r2 ( m r Bz \u2212 kB\u3b8 ) (\u3bc0\u3c1m\u3c92(\u3b3\u3bc0p+B2)\u2212 \u3b3\u3bc0pF 2) )] \u3ber = 0 (8.117) 106 8.5 Ballooning Instability 107 Fig.8.11 Ballooning mode where \u394 is \u394 = \u3bc20\u3c1 2 m\u3c9 4 \u2212 \u3bc0\u3c1m\u3c92H2(\u3b3\u3bc0p+B2) + \u3b3\u3bc0pH2F 2. This equation was derived by Hain-Lu¨st (ref.[14]). The solution of (8.117) gives \u3ber(r) in the region of 0 < r < a. The equations for the vacuum region a < r < aw (aw is the radius of wall) are \u2207×B1 = 0, \u2207 ·B1 = 0 so that we \ufb01nd B1 = \u2207\u3c8, \ufffd\u3c8 = 0 and \u3c8 = (bIm(kr) + cKm(kr)) exp(im\u3b8 + ikz), B1r = \u2202\u3c8 \u2202r = ( bI \u2032m(kr) + cK \u2032 m(kr) ) exp(im\u3b8 + ikz). (8.118) B1r in the plasma region is given by B1r = i(k ·B)\u3ber = iF\u3ber and the boundary conditions at r = a are B1r(a) = iF\u3ber(a), (8.119) B\u20321r(a) = i(F \u2032\u3ber(a) + F\u3be\u2032r(a)), (8.120) and the coe\ufb03cients b,c can be \ufb01xed. To deal with this equation as an eigenvalue problem, boundary conditions must be imposed on \u3ber; one is that \u3ber \u221d rm\u22121 at r = 0, and the other is that the radial component of the perturbed magnetic \ufb01eld at the perfect conducting wall B1r(aw) = 0. After \ufb01nding suitable \u3c92 to satisfy these conditions, the growth rate \u3b32 \u2261 \u2212\u3c92 is obtained (ref.[15]). 8.5 Ballooning Instability In interchange instability, the parallel component k\u2016 = (k ·B)/B of the propagation vector is zero and an average minimum-B condition may stabilize such an instability. Suydam\u2019s condition and the local-mode stability condition of toroidal-system are involved in perturbations with k\u2016 = 0. In this section we will study perturbations where k\u2016 = 0 but |k\u2016/k\u22a5| \ufffd 1. Although the interchange instability is stabilized by an average minimum-B con\ufb01guration, it is possible that the perturbation 107 108 8 Magnetohydrodynamic Instabilities with k\u2016 = 0 can grow locally in the bad region of the average minimum-B \ufb01eld. This type of instability is called the ballooning mode (see \ufb01g.8.11). The energy integral \u3b4W is given by \u3b4W = 1 2\u3bc0 \u222b ((\u2207× (\u3be ×B0))2 \u2212 (\u3be × (\u2207×B0)) · \u2207 × (\u3be ×B0) +\u3b3\u3bc0p0(\u2207 · \u3be)2 + \u3bc0(\u2207 · \u3be)(\u3be · \u2207p0))dr. Let us consider the case that \u3be can be expressed by \u3be = B0 ×\u2207\u3c6 B20 , (8.121) where \u3c6 is considered to be the time integral of the scalar electrostatic potential of the perturbed electric \ufb01eld. Because of \u3be ×B0 = \u2207\u22a5\u3c6 the energy integral is reduced to \u3b4W = 1 2\u3bc0 \u222b ( (\u2207×\u2207\u22a5\u3c6)2 \u2212 ( (B0 ×\u2207\u22a5\u3c6)× \u3bc0j0 B20 ) \u2207×\u2207\u22a5\u3c6 +\u3b3\u3bc0p0(\u2207 · \u3be)2 + \u3bc0(\u2207 · \u3be)(\u3be · \u2207p0) ) dr. \u2207 · \u3be is given by \u2207 · \u3be = \u2207 · ( B0 ×\u2207\u3c6 B20 ) = \u2207\u3c6 · \u2207 × ( B0 B20 ) = \u2207\u3c6 · (( \u2207 1 B2 ) ×B + 1 B2 \u2207×B ) . The 2nd term in ( ) is negligible compared with the 1st term in the low beta case. By means of \u2207p0 = j0 ×B0, \u3b4W is expressed by \u3b4W = 1 2\u3bc0 \u222b (\u2207×\u2207\u22a5\u3c6)2 + \u3bc0\u2207p0 · (\u2207\u22a5\u3c6×B0) B20 ( B0 · \u2207 ×\u2207\u22a5\u3c6 B20 ) \u2212\u3bc0(j0 ·B0) B20 \u2207\u22a5\u3c6 · \u2207 ×\u2207\u22a5\u3c6+ \u3b3\u3bc0p0 ( \u2207 ( 1 B20 ) · (B0 ×\u2207\u22a5\u3c6) )2 + \u3bc0\u2207p0 · (B0 ×\u2207\u22a5\u3c6) B20 ( \u2207 ( 1 B20 ) · (B0 ×\u2207\u22a5\u3c6) ) dr. Let us use z coordinate as a length along a \ufb01eld line, r as radial coordinate of magnetic surfaces and \u3b8 as poloidal angle in the perpendicular direction to \ufb01eld lines. The r, \u3b8, z components of \u2207p0, B, and \u2207\u3c6 are approximately given by \u2207p0 = (p\u20320, 0, 0), B = (0, B\u3b8(r), B0(1\u2212 rR\u22121c (z))), \u2207\u3c6 = (\u2202\u3c6/\u2202r, \u2202\u3c6/r\u2202\u3b8, \u2202\u3c6/\u2202z), \u3c6(r, \u3b8, z) = \u3c6(r, z)Re(exp im\u3b8). Rc(z) is the radius of curvature of the line of magnetic force: 1 Rc(z) = 1 R0 ( \u2212w + cos 2\u3c0 z L ) . 108 8.5 Ballooning Instability 109 When Rc(z) < 0, the curvature is said to be good. If the con\ufb01guration is average minimum-B, w and R0 must be 1 > w > 0 and R0 > 0. Since B\u3b8/B0, r/R0, r/L are all small quantities, we \ufb01nd \u2207\u22a5\u3c6 = \u2207\u3c6\u2212\u2207\u2016\u3c6 \u2248 Re ( \u2202\u3c6 \u2202r , im r \u3c6, 0 ) , \u2207× (\u2207\u22a5\u3c6) \u2248 Re ( \u2212im r \u2202\u3c6 \u2202z , \u22022\u3c6 \u2202z\u2202r , 0 ) , B0 ×\u2207\u22a5\u3c6 \u2248 Re (\u2212im r B0\u3c6,B0 \u2202\u3c6 \u2202r , 0 ) and \u3b4W is reduced to \u3b4W = 1 2\u3bc0 \u222b m2 r2 (( \u2202\u3c6(r, z) \u2202z )2 \u2212 \u3b2 rpRc(z) (\u3c6(r, z))2 ) 2\u3c0rdrdz where \u2212p0/p\u20320 = rp and \u3b2 = p0/(B20/2\u3bc0). The 2nd term contributes to stability in the region Rc(z) < 0 and contributes to instability in the region of Rc(z) > 0. Euler\u2019s equation is given by d2\u3c6 dz2 + \u3b2 rpRc(z) \u3c6 = 0. (8.122) Rc is nearly equal to B/|\u2207B|. Equation (8.122) is a Mathieu di\ufb00erential equation, whose eigenvalue is w = F (\u3b2L2/2\u3c02rpR0). Since F (x) = x/4, x\ufffd 1, F (x) = 1\u2212 x\u22121/2, x\ufffd 1, we \ufb01nd the approximate relation \u3b2c \u223c 4w(1 + 3w)(1\u2212 w)2 2\u3c02rpR0 L2 . Since w is of the order of rp/2R0 and the connection length is L \u2248 2\u3c0R0(2\u3c0/\u3b9) (\u3b9 being the rotational transform angle), the critical beta ratio \u3b2c is \u3b2c \u223c ( \u3b9 2\u3c0 )2 (rp R ) . (8.123) If \u3b2 is smaller than the critical beta ratio \u3b2c, then \u3b4W > 0, and the plasma is stable. The stability condition for the ballooning mode in the shearless case is given by (ref.[16]). \u3b2 < \u3b2c. In the con\ufb01guration with magnetic shear, more rigorous treatment is necessary. According to the analysis (ref.[17]-[19]) for ballooning modes with large toroidal mode number n\ufffd 1 and m\u2212nq \u223c 0 (see appendix B), the stable region in the shear parameter S and the measure of pressure gradient \u3b1 of ballooning mode is shown in \ufb01g.8.12. The shear parameter S is de\ufb01ned by S = r q dq dr 109 110 8 Magnetohydrodynamic Instabilities Fig.8.12 The maximum stable pressure gradient \u3b1 as a function of the shear parameter S of ballooning mode. The dotted line is the stability boundary obtained by imposing a more restricted boundary condition on the perturbation (ref.[17]). where q is the safety factor (q \u2261 2\u3c0/\u3b9: \u3b9 rotational transform angle) and the measure of pressure gradient \u3b1 is de\ufb01ned by \u3b1 = \u2212 q 2R B2/2\u3bc0 dp dr . The straight-line approximation of the maximum pressure gradient in the range of large positive shear (S > 0.8) is \u3b1 \u223c 0.6S as is shown in \ufb01g.8.12. Since \u3b2 = 1 B20/2\u3bc0 1 \u3c0a2 \u222b a 0 p2\u3c0rdr = \u2212 1 B20/2\u3bc0 1 a2 \u222b a 0 dp dr