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= ky n\u2dck n0 \u3baTe eB \u3b3k \u2212 \u3c9kri \u3c9\u2217k = ky n\u2dck n0 \u3baTe eB (\u2212iAk exp i\u3b1k). 77 78 7 Di\ufb00usion of Plasma, Con\ufb01nement Time Then the di\ufb00usion coe\ufb03cient may be obtained from (7.34) as follows: D = 1 \u3bann0 Re(n\u2dckV\u2dc\u2212kx = (\u2211 k ky\u3b3k \u3ban\u3c9 \u2217 k \u2223\u2223\u2223\u2223 n\u2dckn0 \u2223\u2223\u2223\u22232 ) \u3baTe eB = (\u2211 k ky \u3ban Ak sin\u3b1k \u2223\u2223\u2223\u2223 n\u2dckn0 \u2223\u2223\u2223\u22232 ) \u3baTe eB . (7.38) The anomalous di\ufb00usion coe\ufb03cient due to \ufb02uctuation loss increases with time (from (7.35) and (7.38)) and eventually the term with the maximum growth rate \u3b3k > 0 becomes dominant. However, the amplitude |n\u2dck| will saturate due to nonlinear e\ufb00ects; the saturated amplitude will be of the order of |n\u2dck| \u2248 |\u2207n0|\u394x \u2248 \u3ban kx n0, where \u394x is the correlation length of the \ufb02uctuation in the direction of x and the inverse of the propagation constant kx in the x direction. Then (7.35) becomes D = \u3b3k \u3ba2n \u2223\u2223\u2223\u2223 n\u2dckn0 \u2223\u2223\u2223\u22232 \u2248 \u3b3kk2x \u2248 (\u394x)2\u3b3k \u2248 (\u394x) 2 \u3c4c , (7.39) where \u3c4c is the autocorrelation time of the \ufb02uctuation and is nearly equal to the inverse of \u3b3k in the saturation stage of the \ufb02uctuation. When the nondimensional coe\ufb03cient inside the parentheses in (7.38) is at its maximum of 1/16, we have the Bohm di\ufb00usion coe\ufb03cient DB = 1 16 \u3baTe eB . (7.40) It appears that (7.40) give the largest possible di\ufb00usion coe\ufb03cient. When the density and potential \ufb02uctuations n\u2dck, \u3c6\u2dck are measured, V\u2dc k can be calculated, and the estimated outward particle \ufb02ux \u393 and di\ufb00usion coe\ufb03cient D can be compared to the values obtained by experiment. As the relation of n\u2dck and \u3c6\u2dck is given by (7.37), the phase di\ufb00erence will indicate whether \u3c9k is real (oscillatory mode) or \u3b3k > 0 (growing mode), so that this equation is very useful in interpreting experimental results. Let us take an example of the \ufb02uctuation driven by ion temperature gradient drift instability (refer to sec.8.6). The mode is described by \u3c6(r, \u3b8, z) = \u2211 \u3c6mn(r) exp(\u2212im\u3b8 + inz/R). The growth rate of the \ufb02uctuation has the maximum at around k\u3b8 = (\u2212i/r)\u2202/\u2202\u3b8 = \u2212m/r of (ref.[4],[5]) |k\u3b8| = m r \u223c \u3b1\u3b8 \u3c1i , \u3b1\u3b8 = 0.7 \u223c 0.8. Then the correlation length \u394\u3b8 in \u3b8 direction is \u394\u3b8 \u223c \u3c1i/\u3b1\u3b8 (\u3c1i is ion Larmor radius). The propagation constant k\u2016 along the line of magnetic force near the rational surface q(rm) = m/n is k\u2016 = \u2212ib · \u2207 = B\u3b8 B (\u2212m r ) + Bt B ( n R ) \u2248 1 R ( n\u2212 m q(r) ) = m rR rq\u2032 q2 (r \u2212 rm) = s Rq k\u3b8(r \u2212 rm) where q(r) \u2261 (r/R)(Bt/B\u3b8) is the safety factor (B\u3b8 and Bt are poloidal and toroidal \ufb01elds, respectively) and s is the shear parameter (refer to Section 8.3c) s \u2261 rq\u2032/q. |k\u2016| is larger than 78 7.3 Fluctuation Loss, Bohm, Gyro-Bohm Di\ufb00usion, and Stationary · · · 79 Fig.7.4 In the upper \ufb01gure, the radial width of eigenmode \u394r is larger than the radial separation of the rational surfaces \u394rm. A semi-global eigenmode structure \u394rg takes place due to the mode couplings. In the lower \ufb01gure, the radial width of eigenmode \u394r is smaller than the radial separation of the rational surfaces \u394rm. The modes with the radial width \u394r are independent of each other. the inverse of the connection length qR of torus and is less than the inverse of, say, the pressure gradient scale Lp, that is 1 qR < |k\u2016| < 1 Lp . The radial width \u394r = |r \u2212 rm| of the mode near the rational surface r = rm is roughly expected to be \u394r = |r\u2212 rm| = (Rq/s)(k\u2016/k\u3b8) = (\u3c1i/s\u3b1\u3b8) \u223c O(\u3c1i/s). The more accurate radial width of the eigenmode of ion temperature gradient driven drift turbulence is given by (ref.[5][6]) \u394r = \u3c1i ( qR sLp )1/2 ( \u3b3k \u3c9kr )1/2 . The radial separation length \u394rm of the adjacent rational surface rm and rm+1 is q\u2032\u394rm = q(rm+1)\u2212 q(rm) = m+ 1 n \u2212 m n = 1 n , \u394rm = 1 nq\u2032 = m/n rq\u2032 r m \u223c 1 sk\u3b8 . When the mode width \u394r is larger than the radial separation of the rational surface \u394rm, the di\ufb00erent modes are overlapped and the toroidal mode coupling takes place (see \ufb01g.7.4). The half width \u394rg of the envelope of coupled modes is estimated to be (ref.[6]-[8]) \u394rg = ( \u3c1iLp s )1/2 . The radial correlation length becomes the large value of \u394rg (\u394rg/\u394r \u223c (Lp/\u3c1i)1/2) and the radial propagation constant becomes kr \u223c 1/\u394rg. In this case, the di\ufb00usion coe\ufb03cient D is D = (\u394rg)2\u3b3k \u223c \u3c1iLp s \u3c9\u2217k \u223c T eB \u3b1\u3b8 s . where \u3c9\u2217k is the drift frequency (sec.8.6, sec.9.2). This coe\ufb03cient is of the Bohm type. When the mode width \u394r is less than \u394rm (weak shear case), there is no coupling between di\ufb00erent modes and the radial correlation length is \u394r = \u3c1i ( qR sLp )1/2 . 79 80 7 Di\ufb00usion of Plasma, Con\ufb01nement Time Fig.7.5 Magnetic surface \u3c8 = const. and electric-\ufb01eld equipotential \u3c6 = const. The plasma moves along the equipotential surfaces by virtue of E ×B. The di\ufb00usion coe\ufb03cient D in this case is D \u223c (\u394r)2\u3c9\u2217k \u223c \u3c12i ( qR sLp )( k\u3b8T eBLp ) \u223c T eB \u3c1i Lp ( \u3b1\u3b8qR sLp ) \u221d T eB \u3c1i Lp . (7.41) This is called gyro-Bohm type di\ufb00usion coe\ufb03cient. It may be expected that the transport in toroidal systems becomes small in the weak shear region of negative shear con\ufb01guration near the minimum q position (refer to sec.16.7). Stationary Convective Loss Next, let us consider stationary convective losses across the magnetic \ufb02ux. Even if \ufb02uctuations in the density and electric \ufb01eld are not observed at a \ufb01xed position, it is possible that the plasma can move across the magnetic \ufb01eld and continuously escape. When a stationary electric \ufb01eld exists and the equipotential surfaces do not coincide with the magnetic surfaces \u3c6 = const., the E ×B drift is normal to the electric \ufb01eld E, which itself is normal to the equipotential surface. Consequently the plasma drifts along the equipotential surfaces (see \ufb01g.7.5) which cross the magnetic surfaces. The resultant loss is called stationary convective loss. The particle \ufb02ux is given by \u393k = n0 Ey B . (7.42) The losses due to di\ufb00usion by binary collision are proportional to B\u22122; but \ufb02uctuation or convective losses are proportional to B\u22121. Even if the magnetic \ufb01eld is increased, the loss due to \ufb02uctuations does not decrease rapidly. 7.4 Loss by Magnetic Fluctuation When the magnetic \ufb01eld in a plasma \ufb02uctuates, the lines of magnetic force will wander radially. Denote the radial shift of the \ufb01eld line by \u394r and the radial component of magnetic \ufb02uctuation \u3b4B by \u3b4Br respectively. Then we \ufb01nd \u394r = \u222b L 0 brdl, where br = \u3b4Br/B and l is the length along the line of magnetic force. The ensemble average of (\u394r)2 is given by \u3008(\u394r)2\u3009 = \u2329\u222b L 0 br dl \u222b L 0 br dl\u2032 \u232a = \u2329\u222b L 0 dl \u222b L 0 dl\u2032 br(l) br(l\u2032) \u232a = \u2329\u222b L 0 dl \u222b L\u2212l \u2212l ds br(l) br(l + s) \u232a \u2248 L \u2329 b2r \u232a lcorr, where lcorr is lcorr = \u2329\u222b\u221e \u2212\u221e br(l) br(l + s) ds \u232a \u3008b2r\u3009 . 80 References 81 If electrons run along the lines of magnetic force with the velocity vTe, the di\ufb00usion coe\ufb03cient De of electrons becomes (ref.[9]) De = \u3008(\u394r)2\u3009 \u394t = L \u394t \u3008b2r\u3009lcorr = vTelcorr \u2329( \u3b4Br B )2\u232a . (7.43) We may take lcorr \u223c R in the case of tokamak and lcorr \u223c a in the case of reverse \ufb01eld pinch (RFP, refer sec.17.1). References [1] D. P\ufb01rsh and A. Schlu¨ter: MPI/PA/7/62, Max-Planck Institute fu¨r Physik und Astrophysik Mu¨nchen (1962). [2] A. A. Galeev and R. Z. Sagdeev: Sov. Phys. JETP 26, 233 (1968). [3] B. B. Kadomtsev and O. P. Pogutse: Nucl. Fusion 11, 67 (1971). F. L. Hinton and R. D. Hazeltine: Rev. Modern Phys. 48 239 (1976). [4] W. Horton: Phys. Rev. Lett. 37, 1269 (1976). [5] S. Hamaguchi and W. Horton: Phys. Fluids B4, 319 (1992). [6] W. Horton, Jr., R. Esres, H. Kwak, and D.-I. Choi: Phys. Fluids 21, 1366 (1978). [7] F. Romanelli and F. Zonca: Phys. Fluids B5, 4081 (1993). J. Y. Kim and M. Wakatani: Phys. Rev. Lett. 73, 2200 (1994). [8] Y. Kishimoto,