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force F s, which is very small for a perturbation where (k ·B) = 0, and the plasma becomes unstable. This type of instability is called resistive instability. 9.1 Tearing Instability Let us consider a slab model in which the zeroth-order magnetic \ufb01eld B0 depends on only x and B is given as follows; B0 = B0y(x)ey +B0z(x)ez. (9.4) From Ohm\u2019s law (9.1) we \ufb01nd \u2202B \u2202t = \u2212\u2207×E = \u2207× ((V ×B)\u2212 \u3b7j) = \u2207× (V ×B) + \u3b7 \u3bc0 \u394B (9.5) where \u3b7 is assumed to be constant. It is assumed that the plasma is incompressible. Since the growth rate of the resistive instability is small compared with the MHD characteristic rate (inverse of Alfven transit time) and the movement is slower than the sound velocity, the assumption of incompressibility is justi\ufb01ed and it follows that \u2207 · V = 0. (9.6) The magnetic \ufb01eld B always satis\ufb01es \u2207 ·B = 0. (9.7) The equation of motion is \u3c1m dV dt = 1 \u3bc0 (\u2207×B)×B \u2212\u2207p 114 9.1 Tearing Instability 115 = 1 \u3bc0 ( (B0 · \u2207)B1 + (B1 · \u2207)B0 \u2212 \u2207B 2 2 ) \u2212\u2207p. (9.8) Let us consider the perturbation expressed by f1(r, t) = f1(x) exp (i(kyy + kzz) + \u3b3t). Then (9.5) reduces to \u3b3B1x = i(k ·B)Vx + \u3b7 \u3bc0 ( \u22022 \u2202x2 \u2212 k2 ) B1x (9.9) where k2 = k2y+k 2 z . The \ufb01rst term in the right-hand side of (9.8) becomes (B0 ·\u2207)B1 = i(k·B0)B1. The rotation of (9.8) is \u3bc0\u3c1m\u3b3\u2207× V = \u2207× ( i(k ·B0)B1 + ( B1x \u2202 \u2202x ) B0 ) . (9.10) Equations (9.6),(9.7) reduce to \u2202B1x \u2202x + ikyB1y + ikzB1z = 0, (9.11) \u2202Vx \u2202x + ikyVy + ikzVz = 0. (9.12) Multiply ky and z component of (9.10) and multiply kz and the y component and take the di\ufb00erence. Use the relations of (9.11) and (9.12); then we \ufb01nd \u3bc0\u3c1m\u3b3 ( \u22022 \u2202x2 \u2212 k2 ) Vx = i(k ·B0) ( \u22022 \u2202x2 \u2212 k2 ) B1x \u2212 i(k ·B0)\u2032\u2032B1x (9.13) where \u2032 is di\ufb00erentiation in x. Ohm\u2019s law and the equation of motion are reduced to (9.9) and (9.13) (ref.[1]). It must be noti\ufb01ed that the zeroth-order magnetic \ufb01eld B0 appears only in the form of (k ·B0). When we introduce a function F (x) \u2261 (k ·B0) (9.14) the location of F (x) = 0 is the position where resistive instabilities are likely occurred. We choose this position to be x = 0 (see \ufb01g.9.1). F (x) is equal to (k · B0) \ufffd (k · B0)\u2032x near x = 0. As is clear from eqs.(9.9) and (9.13), B1x is an even function and Vx is an odd function near x = 0. The term |\u394B1x| \u223c |\u3bc0kyj1z| can be large only in the region |x| < \u3b5. Since the growth rate of resistive instability is much smaller than MHD growth rate, the left-hand side of the equation of motion (9.13) can be neglected in the region |x| > \u3b5 and we have d2B1x dx2 \u2212 k2B1x = F \u2032\u2032 F B1x, |x| > \u3b5. (9.15) The solution in the region x > 0 is B1x = e\u2212kx (\u222b x \u2212\u221e e2k\u3be d\u3be \u222b \u3be \u221e (F \u2032\u2032/F )B1xe\u2212k\u3b7 d\u3b7 +A ) and the solution in the region x < 0 is B1x = ekx (\u222b x \u221e e\u22122k\u3be d\u3be \u222b \u3be \u221e (F \u2032\u2032/F )B1xek\u3b7 d\u3b7 +B ) . Let us de\ufb01ne \u394\u2032 as the di\ufb00erence between B\u20321x(+\u3b5) at x = +\u3b5 and B\u20321x(\u2212\u3b5) at x = \u2212\u3b5 as follows; \u394\u2032 = B\u20321x(+\u3b5)\u2212B\u20321x(\u2212\u3b5) B1x(0) . (9.16) 115 116 9 Resistive Instability Fig.9.1 Zeroth-order magnetic con\ufb01guration and magnetic islands due to tearing instability. Pro\ufb01les of B1x and Vx are also shown. Then the value of \u394\u2032 obtained from the solutions in the region |x| > \u3b5 is given by \u394\u2032 = \u22122k \u2212 1 B1x(0) (\u222b \u2212\u3b5 \u2212\u221e + \u222b \u221e \u3b5 ) exp(\u2212k|x|)(F \u2032\u2032/F )B1x dx. (9.17) For a trial function of F (x) = Fsx/Ls (|x| < Ls), F (x) = Fsx/|x| (x > |Ls|) we can solve (9.15) and \u394\u2032 is reduced to \u394\u2032 = ( 2\u3b1 Ls ) e\u22122\u3b1 + (1\u2212 2\u3b1) e\u22122\u3b1 \u2212 (1\u2212 2\u3b1) \u2248 2 Ls ( 1 \u3b1 \u2212 \u3b1 ) Here \u3b1 \u2261 kLs was used and Ls is shear length de\ufb01ned by Ls = (F/F \u2032)x=0. For more general cases of F (x), B1x(x) has logarithmic singularity at x = 0, since F \u2032\u2032/F \u221d 1/x generally. Referene [2] describes the method to avoid di\ufb03culties arising from the corresponding logarithmic singularity. Equations (9.9) and (9.13) in the region |x| < \u3b5 reduce to \u22022B1x \u2202x2 \u2212 ( k2 + \u3b3\u3bc0 \u3b7 ) B1x = \u2212i\u3bc0 \u3b7 F \u2032xVx, (9.18) \u22022Vx \u2202x2 \u2212 ( k2 + (F \u2032)2 \u3c1m\u3b7\u3b3 x2 ) Vx = i ( F \u2032x 1 \u3c1m\u3b7 \u2212 F \u2032\u2032 \u3bc0\u3c1m\u3b3 ) B1x. (9.19) The value of \u394\u2032 obtained from the solution in the region |x| < \u3b5 is given from (9.18) as follows; \u394\u2032 ×B1x(0) = \u2202B1x(+\u3b5) \u2202x \u2212 \u2202B1x(\u2212\u3b5) \u2202x = \u3bc0 \u3b7 \u222b \u3b5 \u2212\u3b5 (( \u3b3 + \u3b7 \u3bc0 k2 ) B1x \u2212 iF \u2032xVx ) dx. (9.20) 116 9.1 Tearing Instability 117 The value \u394\u2032 of (9.20) must be equal to the value of \u394\u2032 of (9.17). This requirement gives the eigenvalue \u3b3 and the growth rate of this resistive instability can be obtained (ref.[1]). However we try to reduce the growth rate in qualitative manner in this section. In the region |x| < \u3b5, it is possible to write \u22022B1x \u2202x2 \u223c \u394 \u2032B1x \u3b5 . It is assumed that the three terms of (9.9), namely the term of induced electric \ufb01eld (the left-hand side), the V ×B term (the \ufb01rst term in the right-hand side) and Ohm\u2019s term (the second term) are the same order: \u3b3B1x \u223c \u3b7 \u3bc0 \u394\u2032B1x \u3b5 , (9.21) \u3b3B1x \u223c iF \u2032\u3b5Vx. (9.22) Then (9.21) yields \u3b3 \u223c \u3b7 \u3bc0 \u394\u2032 \u3b5 . (9.23) Accordingly \u394\u2032 > 0 (9.24) is the condition of instability. In order to get the value of \u3b3, the evaluation of \u3b5 is necessary. Equation (9.13) reduces to \u3bc0\u3c1m\u3b3 (\u2212Vx \u3b52 ) \u223c iF \u2032\u3b5\u394 \u2032B1x \u3b5 . (9.25) If the terms Vx, B1x, \u3b3 are eliminated by (9.21), (9.22) and (9.25), we \ufb01nd \u3b55 \u223c ( \u3b7 \u3bc0a2 )2 (\u394\u2032a) \u3c1m\u3bc0 (F \u2032a)2 a5, \u3b5 a \u223c (( \u3c4A \u3c4R )2 (\u394\u2032a) ( B0 F \u2032a2 )2)1/5 \u223c S\u22122/5(\u394\u2032a)1/5 ( B0 (k ·B0)\u2032a2 )2/5 (9.26) where the physical quantities \u3c4R = \u3bc0a 2 \u3b7 , \u3c4A = a B0/(\u3bc0\u3c1m)1/2 are the resistive di\ufb00usion time and Alfve´n transit time respectively. A non-dimensional factor S = \u3c4R/\u3c4A is magnetic Reynolds number and a is a typical plasma size. Accordingly the growth rate \u3b3 is given by \u3b3 = \u3b7 \u3bc0a2 a \u3b5 (\u394\u2032a) = (\u394\u2032a)4/5 \u3c4 3/5 R \u3c4 2/5 A ( (k ·B0)\u2032a2 B0 )2/5 = (\u394\u2032a)4/5 S3/5 ( (k ·B0)\u2032a2 B0 )2/5 1 \u3c4A . (9.27) Since this mode is likely break up the plasma into a set of magnetic islands as is shown in \ufb01g.9.1, this mode is called tearing instability (ref.[1]). 117 118 9 Resistive Instability The foregoing discussion has been based on the slab model. Let us consider this mode in a toroidal plasma. The poloidal and the toroidal components of the propagation vector k are m/r and \u2212n/R respectively. Accordingly there are correspondences of ky \u2194 m/r, and kz \u2194 \u2212n/R, and (k ·B0) = m r B\u3b8 \u2212 n R Bz = n r B\u3b8 ( m n \u2212 q ) , q \u2261 r R Bz B\u3b8 . Therefore weak positions for tearing instability are given by (k ·B0) = 0 and these are rational surfaces satisfying q(rs) = m/n. The shear is given by (k ·B0)\u2032 = \u2212n r B\u3b8 dq dr , (k ·B0)\u2032r2s B0 = \u2212n ( rs R ) q\u2032rs q . The tearing mode is closely related to the internal disruption in tokamak and plays important role as is described in sec.16.3. It has been assumed that the speci\ufb01c resistivity \u3b7 and the mass density \u3c1m are uniform and there is no gravitation (acceleration) g = 0. If \u3b7 depends on x, the resistive term in (9.5) becomes \u2207×(\u3b7\u2207×B)/\u3bc0. When there is temperature gradient (\u3b7\u2032 = 0), rippling mode with short wavelength (kLs \ufffd 1) may appear in the smaller-resistivity-side (high-temperature-side) of x = 0 position. When there is gravitation, the term \u3c1g is added to the equation of motion (9.8). If the direction of g is opposit to \u2207\u3c1m (g is toward low-density-side), gravitational interchange mode may appear (ref.[1]). 9.2 Resistive Drift Instability A \ufb01nite density and temperature gradient always exists at a plasma boundary. Con\ufb01gurations including a gradient may be unstable under certain conditions. Let us consider a slab model. The directon of the uniform magnetic \ufb01eld is taken in the z direction and B0 = (0, 0, B0). The x axis is taken in the direction of