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surface passing X points is the separatix surface. \u3bas = 2Zx Rmax \u2212Rx = AZx R \u3b2p0 \u2261 p(R, 0)\u2212 pb B2z (Rx, 0)/2\u3bc0 = a a+ (R2/R2x)b . When A and \u3bas are speci\ufb01ed, \u3b2p0 is \ufb01xed. To avoid this inadequateness, Weening (ref.[2]) added an additional particular solution r2 ln(r2/R2\u3b1)\u2212 r2 to Solovev solution; that is, \u3c8 = b+ d 2 ( 1\u2212 r 2 R2x ) z2 + a+ (R2/R2x)(b+ d) 8R2 ( (r2 \u2212R2)2\u2212(R2 \u2212R2x)2 ) (6.17) \u2212d 4 ( r2 ln r2 R2x \u2212 (r2 \u2212R2x) ) . When the plasma boundary is chosen to be the separatrix \u3c8(r, z) = 0, the aspect ratioA, elongation ratio\u3bas, and central poloidal beta \u3b2p0 are Z2x R2x = \u22121 2 ( a b+ d + R2 R2x )( 1\u2212 R 2 x R2 ) R2max R2 = ( 2\u2212 R 2 x R2 ) + 2d[x ln x/(x\u2212 1)\u2212 1] a+ (R2/R2x)(b+ d) , x \u2261 R 2 max R2x 1 A = Rmax/R \u2212Rx/R 2 , \u3bas = AZx R \u3b2p0 = a a+ (R2/R2x)(b+ d) [ 1 + 2d(ln(R2/R2x)\u2212 (1\u2212R2/R2x)) (a+ (R2/R2x)(b+ d))(1\u2212R2x/R2) ] . The magnetic surface \u3c8, the magnetic \ufb01eld B and the pressure p in translationally symmetric system (\u2202/\u2202z = 0) are given by \u3c8 = Az(r, \u3b8), Br = 1 r \u2202\u3c8 \u2202\u3b8 , B\u3b8 = \u2212\u2202\u3c8 \u2202r , Bz = \u3bc0 2\u3c0 I(\u3c8), p = p(\u3c8). 56 6.3 Tokamak Equilibrium 57 Fig.6.3 Toroidal coordinates. The equilibrium equation is reduced to 1 r \u2202 \u2202r ( r \u2202\u3c8 \u2202r ) + 1 r2 \u22022\u3c8 \u2202\u3b82 + \u3bc0 \u2202p(\u3c8) \u2202\u3c8 + \u3bc20 8\u3c02 \u2202I2(\u3c8) \u2202\u3c8 = 0, j = 1 2\u3c0 I \u2032B + p\u2032ez, \u394\u3c8 + \u3bc0jz = 0. It is possible to drive the similar equilibrium equation in the case of helically symmetric system. 6.3 Tokamak Equilibrium (ref.[3]) The equilibrium equation for an axially symmetric system is given by (6.14). The 2nd and 3rd terms of the left-hand side of the equation are zero outside the plasma region. Let us use toroidal coordinates (b, \u3c9, \u3d5) (\ufb01g.6.3). The relations between these to cylindrical coordinates (r, \u3d5, z) are r = R0 sinh b cosh b\u2212 cos\u3c9 , z = R0 sin\u3c9 cosh b\u2212 cos\u3c9 . The curves b = b0 are circles of radius a = R0 sinh b0, centered at r = R0 coth b0, z = 0. The curves \u3c9 =const. are also circles with the center at r = 0, a = R0 cos\u3c9/ sin\u3c9. When the magnetic-surface function \u3c8 is replaced by F , according to \u3c8 = F (b, \u3c9) 21/2(cosh b\u2212 cos\u3c9)1/2 the function F satis\ufb01es \u22022F \u2202b2 \u2212 coth b\u2202F \u2202b + \u22022F \u2202\u3c92 + 1 4 F = 0 outside the plasma region. When F is expanded as F = \u3a3gn(b) cosn\u3c9, the coe\ufb03cient gn satis\ufb01es d2gn db2 \u2212 coth bdgn db \u2212 ( n2 \u2212 1 4 ) gn = 0. 57 58 6 Equilibrium Fig.6.4 The coordinates r, z and \u3c1, \u3c9\u2032 There are two independent solutions:( n2 \u2212 1 4 ) gn = sinh b d db Qn\u22121/2(cosh b), ( n2 \u2212 1 4 ) fn = sinh b d db Pn\u22121/2(cosh b). P\u3bd(x) and Q\u3bd(x) are Legendre functions. If the ratio of the plasma radius to the major radius a/R0 is small, i.e., when eb0 \ufffd 1, then gn and fn are given by g0 = eb/2, g1 = \u221212e \u2212b/2, f0 = 2 \u3c0 eb/2(b+ ln 4\u2212 2), f1 = 23\u3c0e 3b/2. If we take terms up to cos\u3c9, F and \u3c8 are F = c0g0 + d0f0 + 2(c1g1 + d1f1) cos\u3c9, \u3c8 = F 21/2(cosh b\u2212 cos\u3c9)1/2 \u2248 e \u2212b/2(1 + e\u2212b cos\u3c9)F. Use the coordinates \u3c1, \u3c9\u2032 shown in \ufb01g.6.4. These are related to the cylindrical and toroidal coordinates as follows: r = R0 + \u3c1 cos\u3c9\u2032 = R0 sinh b cosh b\u2212 cos\u3c9 z = \u3c1 sin\u3c9 \u2032 = R0 sin\u3c9 cosh b\u2212 cos\u3c9 . When b is large, the relations are \u3c9\u2032 = \u3c9, \u3c1 2R0 \u2248 e\u2212b. Accordingly the magnetic surface \u3c8 is expressed by \u3c8 =c0 + 2 \u3c0 d0(b+ ln 4\u2212 2) + [( c0 + 2 \u3c0 d0(b+ ln 4\u2212 2) ) e\u2212b + ( 4 3\u3c0 d1e b \u2212 c1e\u2212b )] cos\u3c9 =d\u20320 ( ln 8R \u3c1 \u2212 2 ) + ( d\u20320 2R ( ln 8R \u3c1 \u2212 1 ) \u3c1+ h1 \u3c1 + h2\u3c1 ) cos\u3c9. In terms of \u3c8, the magnetic-\ufb01eld components are given by rBr = \u2212\u2202\u3c8 \u2202z , rBz = \u2202\u3c8 \u2202r , rB\u3c1 = \u2212 \u2202\u3c8 \u3c1\u2202\u3c9\u2032 , rB\u3c9\u2032 = \u2202\u3c8 \u2202\u3c1 . 58 6.3 Tokamak Equilibrium 59 From the relation \u2212d \u2032 0 \u3c1 = rB\u3c9\u2032 \u2248 R\u2212\u3bc0Ip2\u3c0\u3c1 , the parameter d\u20320 can be taken as d\u20320 = \u3bc0IpR/2\u3c0. Here Ip is the total plasma current in the \u3d5 direction. The expression of the magnetic surface is reduced to \u3c8 = \u3bc0IpR 2\u3c0 ( ln 8R \u3c1 \u2212 2 ) + ( \u3bc0Ip 4\u3c0 ( ln 8R \u3c1 \u2212 1 ) \u3c1+ h1 \u3c1 + h2\u3c1 ) cos\u3c9\u2032 (6.18) where R0 has been replaced by R. In the case of a/R \ufffd 1, the equation of pressure equilibrium (6.9) is \u3008p\u3009 \u2212 pa = 12\u3bc0 ((B 2 \u3d5v)a + (B 2 r +B 2 z )a \u2212 \u3008B2\u3d5\u3009). Here \u3008 \u3009 indicates the volume average and pa is the plasma pressure at the plasma boundary. The value of B2r +B 2 z is equal to B 2 \u3c9\u2032 . The ratio of \u3008p\u3009 to \u3008B2\u3c9\u2032\u3009/2\u3bc0 is called the poloidal beta ratio \u3b2p. When pa = 0, \u3b2p is \u3b2p = 1 + B2\u3d5v \u2212 \u3008B2\u3d5\u3009 B2\u3c9\u2032 \u2248 1 + 2B\u3d5v B2\u3c9\u2032 \u3008B\u3d5v \u2212B\u3d5\u3009. B\u3d5 and B\u3d5v are the toroidal magnetic \ufb01elds in the plasma and the vacuum toroidal \ufb01elds respec- tively. When B\u3d5 is smaller than B\u3d5v, the plasma is diamagnetic, \u3b2p > 1. When B\u3d5 is larger than B\u3d5v, the plasma is paramagnetic, \u3b2p < 1. When the plasma current \ufb02ows along a line of magnetic force, the current produces the poloidal magnetic \ufb01eld B\u3c9\u2032 and a poloidal component of the plasma current appears and induces an additional toroidal magnetic \ufb01eld. This is the origin of the paramagnetism. When the function (6.18) is used, the magnetic \ufb01eld is given by B\u3c9\u2032 = 1 r \u2202\u3c8 \u2202\u3c1 = \u2212\u3bc0Ip 2\u3c0\u3c1 + ( \u3bc0Ip 4\u3c0R ln 8R \u3c1 + 1 R ( h2 \u2212 h1 \u3c12 )) cos\u3c9\u2032, B\u3c1 = \u2212 1 r\u3c1 \u2202\u3c8 \u2202\u3c9\u2032 = ( \u3bc0Ip 4\u3c0R ( ln 8R \u3c1 \u2212 1 ) + 1 R ( h2 + h1 \u3c12 )) sin\u3c9\u2032. \u23ab\u23aa\u23aa\u23aa\u23ac\u23aa\u23aa\u23aa\u23ad (6.19) The cross section of the magnetic surface is the form of \u3c8(\u3c1, \u3c9\u2032) = \u3c80(\u3c1) + \u3c81 cos\u3c9\u2032. When \u394 = \u2212\u3c81/\u3c8\u20320 is much smaller than \u3c1, the cross section is a circle whose center is displaced by an amount (see \ufb01g.6.5) \u394(\u3c1) = \u3c12 2R ( ln 8R \u3c1 \u2212 1 ) + 2\u3c0 \u3bc0IpR (h1 + h2\u3c12). Let us consider the parameters h1 and h2. As will be shown in sec.6.4, the poloidal component B\u3c9\u2032 of the magnetic \ufb01eld at the plasma surface (r = a) must be B\u3c9\u2032(a, \u3c9\u2032) = Ba ( 1 + a R \u39bcos\u3c9\u2032 ) (6.20) at equilibrium. a is the plasma radius and \u39b = \u3b2p + li 2 \u2212 1 (6.21) 59 60 6 Equilibrium Fig.6.5 Displacement of the plasma column. \u3c80(\u3c1\u2032) = \u3c80(\u3c1)\u2212 \u3c8\u20320(\u3c1)\u394 cos\u3c9, \u3c1\u2032 = \u3c1\u2212\u394cos\u3c9. and \u3b2p is the poloidal beta ratio \u3b2p = p (B2a/2\u3bc0) (6.22) and li is li = \u222b B2\u3c9\u2032\u3c1d\u3c1d\u3c9 \u2032 \u3c0a2B2a . (6.23) The parameters h1 and h2 must be chosen to satisfy B\u3c1 = 0 and B\u3c9\u2032 = Ba(1+ (a/R)\u39b cos\u3c9\u2032) at the plasma boundary, i.e., h1 = \u3bc0Ip 4\u3c0 a2 ( \u39b+ 1 2 ) , h2 = \u2212\u3bc0Ip4\u3c0 ( ln 8R a + \u39b\u2212 1 2 ) . (6.24) Accordingly \u3c8 is given by \u3c8 = \u3bc0IpR 2\u3c0 ( ln 8R \u3c1 \u2212 2 ) \u2212 \u3bc0Ip 4\u3c0 ( ln \u3c1 a + ( \u39b+ 1 2 )( 1\u2212 a 2 \u3c12 )) \u3c1 cos\u3c9\u2032. (6.25) The term h2\u3c1 cos\u3c9\u2032 in \u3c8 brings in the homogeneous vertical \ufb01eld Bz = h2 R , which is to say that we must impose a vertical external \ufb01eld. When we write \u3c8e = h2\u3c1 cos\u3c9\u2032 so that \u3c8 is the sum of two terms, \u3c8 = \u3c8p + \u3c8e, \u3c8e and \u3c8p are expressed by \u3c8e = \u2212\u3bc0Ip4\u3c0 ( ln 8R a + \u39b\u2212 1 2 ) \u3c1 cos\u3c9\u2032 (6.26) \u3c8p = \u3bc0IpR 2\u3c0 ( ln 8R \u3c1 \u2212 2 ) + \u3bc0Ip 4\u3c0 (( ln 8R \u3c1 \u2212 1 ) \u3c1+ a2 \u3c1 ( \u39b + 1 2 )) cos\u3c9\u2032. (6.27) These formulas show that a uniform magnetic \ufb01eld in the z direction, B\u22a5 = \u2212\u3bc0Ip4\u3c0R ( ln 8R a +\u39b\u2212 1 2 ) , (6.28) must be applied in order to maintain a toroidal plasma in equilibrium (\ufb01g.6.6). This vertical \ufb01eld weakens the inside poloidal \ufb01eld and strengthens the outside poloidal \ufb01eld. The amount of B\u22a5 (6.28) for the equilibrium can be derived more intuitively. The hoop force by which the current ring of a plasma tends to expand is given by Fh = \u2212 \u2202 \u2202R LpI 2 p 2 \u2223\u2223\u2223\u2223 LpIp=const. = 1 2 I2p \u2202Lp \u2202R , 60 6.3 Tokamak Equilibrium 61 Fig.6.6 Poloidal magnetic \ufb01eld due to the combined plasma current and vertical magnetic \ufb01eld.