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The transport equation of particles is \u2202 \u2202t n(r, t) +\u2207 · (n(r, t)V (r, t)) = 0 (7.1) provided processes of the ionization of neutrals and the recombination of ions are negligible (see ch.5.1). The particle \ufb02ux \u393 = nV is given by n(r, t)V (r, t) = \u2212D(r, t)\u2207n(r, t) in many cases, where D is di\ufb00usion coe\ufb03cient. (Additional terms may be necessary in more general cases.) Di\ufb00usion coe\ufb03cient D and particle con\ufb01nement time \u3c4p are related by the di\ufb00usion equation of the plasma density n as follows: \u2207 · (D\u2207n(r, t)) = \u2202 \u2202t n(r, t). Substitution of n(r, t) = n(r) exp(\u2212t/\u3c4p) in di\ufb00usion equation yields \u2207 · (D\u2207n(r)) = \u2212 1 \u3c4p n(r). When D is constant and the plasma column is a cylinder of radius a, the di\ufb00usion equation is reduced to 1 r \u2202 \u2202r ( r \u2202n \u2202r ) + 1 D\u3c4p n = 0. The solution satisfying the boundary condition n(a) = 0 is n = n0J0 ( 2.4r a ) exp ( \u2212 t \u3c4p ) and the particle con\ufb01nement time is \u3c4p = a2 2.42D = a2 5.8D , (7.2) where J0 is the zeroth-order Bessel function. The relationship (7.2) between the particle con\ufb01ne- ment time \u3c4p and D holds generally, with only a slight modi\ufb01cation of the numerical factor. This formula is frequently used to obtain the di\ufb00usion coe\ufb03cient from the observed values of the plasma radius and particle con\ufb01nement time. 70 7 Di\ufb00usion of Plasma, Con\ufb01nement Time 71 The equation of energy balance is given by (A.19), which will be derived in appendix A, as follows: \u2202 \u2202t ( 3 2 n\u3baT ) +\u2207 · ( 3 2 \u3baTnv ) +\u2207 · q = Q\u2212 p\u2207 · v \u2212 \u2211 ij \u3a0ij \u2202vi \u2202xj . (7.3) The \ufb01rst term in the right-hand side is the heat generation due to particle collisions per unit volume per unit time, the second term is the work done by pressure and the third term is viscous heating. The \ufb01rst term in the left-hand side is the time derivative of the thermal energy per unit volume, the second term is convective energy loss and the third term is conductive energy loss. Denoting the thermal conductivity by \u3baT, the thermal \ufb02ux due to heat conduction may be expressed by q = \u2212\u3baT\u2207(\u3baT ). If the convective loss is neglected and the heat sources in the right-hand side of (7.3) is zero, we \ufb01nd that \u2202 \u2202t ( 3 2 n\u3baT ) \u2212\u2207 · \u3baT\u2207(\u3baT ) = 0. In the case of n = const., this equation reduces to \u2202 \u2202t ( 3 2 \u3baT ) = \u2207 · ( \u3baT n \u2207(\u3baT ) ) . When the thermal di\ufb00usion coe\ufb03cient \u3c7T is de\ufb01ned by \u3c7T = \u3baT n , the same equation on \u3baT is obtained as (7.1). In the case of \u3c7T = const., the solution is \u3baT = \u3baT0J0 ( 2.4 a r ) exp ( \u2212 t \u3c4E ) , \u3c4E = a2 5.8(2/3)\u3c7T . (7.4) The term \u3c4E is called energy con\ufb01nement time. 7.1 Collisional Di\ufb00usion (Classical Di\ufb00usion) 7.1a Magnetohydrodynamic Treatment A magnetohydrodynamic treatment is applicable to di\ufb00usion phenomena when the electron-to- ion collision frequency is large and the mean free path is shorter than the connection length of the inside regions of good curvature and the outside region of bad curvature of the torus; i.e., vTe \u3bdei <\u223c 2\u3c0R \u3b9 , \u3bdei >\u223c \u3bdp \u2261 1 R \u3b9 2\u3c0 vTe = 1 R \u3b9 2\u3c0 ( \u3baTe me )1/2 where vTe is electron thermal velocity and \u3bdei is electron to ion collision frequency. From Ohm\u2019s law (5.28) E + v ×B \u2212 1 en \u2207pi = \u3b7j, the motion of plasma across the lines of magnetic force is expressed by nv\u22a5 = 1 B (( nE \u2212 \u3baTi e \u2207n ) × b ) \u2212 me\u3bdei e2 \u2207p B2 71 72 7 Di\ufb00usion of Plasma, Con\ufb01nement Time Fig.7.1 Electric \ufb01eld in a plasma con\ufb01ned in a toroidal \ufb01eld. The symbols \u2297 and \ufffd here show the direction of the P\ufb01rsch-Schlu¨ter current. = 1 B (( nE \u2212 \u3baTi e \u2207n ) × b ) \u2212 (\u3c1\u3a9e)2\u3bdei ( 1 + Ti Te ) \u2207n (7.5) where \u3c1\u3a9e = vTe/\u3a9e, vTe = (\u3baTe/me)1/2 and \u3b7 = me\u3bdei/e2ne (see sec.2.8). If the \ufb01rst term in the right-hand side can be neglected, the particle di\ufb00usion coe\ufb03cient D is given by D = (\u3c1\u3a9e)2\u3bdei ( 1 + Ti Te ) . (7.6) The classical di\ufb00usion coe\ufb03cient Dei is de\ufb01ned by Dei \u2261 (\u3c1\u3a9e)2\u3bdei = n\u3baTe \u3c3\u22a5B2 = \u3b2e\u3b7\u2016 \u3bc0 , (7.7) where \u3c3\u22a5 = nee2/(me\u3bdei), \u3b7\u2016 = 1/2\u3c3\u22a5. However the \ufb01rst term of the right-hand side of (7.5) is not always negligible. In toroidal con- \ufb01guration, the charge separation due to the toroidal drift is not completely cancelled along the magnetic \ufb01eld lines due to the \ufb01nite resistivity and an electric \ufb01eld E arises (see \ufb01g.7.1). Therefore the E×b term in (7.5) contributes to the di\ufb00usion. Let us consider this term. From the equilibrium equation, the diamagnetic current j\u22a5 = b B ×\u2207p, j\u22a5 = \u2223\u2223\u2223\u2223 1B \u2202p\u2202r \u2223\u2223\u2223\u2223 \ufb02ows in the plasma. From \u2207 · j = 0, we \ufb01nd \u2207 · j\u2016 = \u2212\u2207 · j\u22a5. By means of the equation B = B0(1\u2212 (r/R) cos \u3b8), the P\ufb01rsch-Schlu¨ter current j\u2016 is given by (refer to (6.43)) j\u2016 = 2 2\u3c0 \u3b9 1 B0 \u2202p \u2202r cos \u3b8. (7.8) If the electric conductivity along the magnetic lines of force is \u3c3\u2016, the parallel electric \ufb01eld is E\u2016 = j\u2016/\u3c3\u2016. As is clear from \ufb01g.7.1, the relation E\u3b8 E\u2016 \u2248 B0 B\u3b8 holds. From B\u3b8/B0 \u2248 (r/R)(\u3b9/2\u3c0), the \u3b8 component of the electric \ufb01eld is given by E\u3b8 = B0 B\u3b8 E\u2016 = R r 2\u3c0 \u3b9 1 \u3c3\u2016 j\u2016 = 2 \u3c3\u2016 R r ( 2\u3c0 \u3b9 )2 1 B0 \u2202p \u2202r cos \u3b8. (7.9) Accordingly (7.5) is reduced to nVr = \u2212nE\u3b8 B \u2212 (\u3c1\u3a9e)2\u3bdei ( 1 + Ti Te ) \u2202n \u2202r 72 7.1 Collisional Di\ufb00usion (Classical Di\ufb00usion) 73 Fig.7.2 Magnetic surface (dotted line) and drift surfaces (solid lines). = \u2212 ( R r · 2 ( 2\u3c0 \u3b9 )2 n\u3baTe \u3c3\u2016B20 cos \u3b8 ( 1 + r R cos \u3b8 ) + n\u3baTe \u3c3\u22a5B20 ( 1 + r R cos \u3b8 )2) × ( 1 + Ti Te ) \u2202n \u2202r . Noting that the area of a surface element is dependent of \u3b8, and taking the average of nVr over \u3b8, we \ufb01nd that \u3008nVr\u3009 = 12\u3c0 \u222b 2\u3c0 0 nVr ( 1 + r R cos \u3b8 ) d\u3b8 = \u2212 n\u3baTe \u3c3\u22a5B20 ( 1 + Ti Te )( 1 + 2\u3c3\u22a5 \u3c3\u2016 ( 2\u3c0 \u3b9 )2) \u2202n \u2202r . (7.10) Using the relation \u3c3\u22a5 = \u3c3\u2016/2, we obtain the di\ufb00usion coe\ufb03cient of a toroidal plasma: DP.S. = nTe \u3c3\u22a5B20 ( 1 + Ti Te )( 1 + ( 2\u3c0 \u3b9 )2) . (7.11) This di\ufb00usion coe\ufb03cient is (1 + (2\u3c0/\u3b9)2) times as large as the di\ufb00usion coe\ufb03cient of (7.2). This value is called P\ufb01rsch-Schlu¨ter factor (ref.[1]). When the rotational tranform angle \u3b9/2\u3c0 is about 0.3, P\ufb01rsch-Schlu¨ter factor is about 10. 7.1b A Particle Model The classical di\ufb00usion coe\ufb03cient of electrons Dei = (\u3c1\u3a9e)2\u3bdei is that for electrons which move in a random walk with a step length equal to the Larmor radius. Let us consider a toroidal plasma. For rotational transform angle \u3b9, the displacement \u394 of the electron drift surface from the magnetic surface is (see \ufb01g.7.2) \u394 \u2248 ±\u3c1\u3a9e 2\u3c0 \u3b9 . (7.12) The ± signs depend on that the direction of electron motion is parallel or antiparallel to the magnetic \ufb01eld (see sec.3.5). As an electron can be transferred from one drift surface to the other by collision, the step length across the magnetic \ufb01eld is \u394 = ( 2\u3c0 \u3b9 ) \u3c1\u3a9e. (7.13) 73 74 7 Di\ufb00usion of Plasma, Con\ufb01nement Time Consequently, the di\ufb00usion coe\ufb03cient is given by DP.S. = \u3942\u3bdei = ( 2\u3c0 \u3b9 )2 (\u3c1\u3a9e)2\u3bdei, (7.14) thus the P\ufb01rsch-Schlu¨ter factor has been reduced (|2\u3c0/\u3b9| \ufffd 1 is assumed). 7.2 Neoclassical Di\ufb00usion of Electrons in Tokamak The magnitude B of the magnetic \ufb01eld of a tokamak is given by B = RB0 R(1 + \ufffdt cos \u3b8) = B0(1\u2212 \ufffdt cos \u3b8), (7.15) where \ufffdt = r R . (7.16) Consequently, when the perpendicular component v\u22a5 of a electron velocity is much larger than the parallel component v\u2016, i.e., when( v\u22a5 v )2 > R R+ r , that is, v\u22a5 v\u2016 > 1 \ufffd 1/2 t , (7.17) the electron is trapped outside of the torus, where the magnetic \ufb01eld is weak. Such an electron drifts in a banana orbit (see \ufb01g.3.9). In order to complete a circuit of the banana orbit, the e\ufb00ective collision time \u3c4e\ufb00 = 1/\u3bde\ufb00 of the trapped electron must