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Study of electron temperature gradient instability in toroidal plasmas with negative magnetic shear Jian Guang-De Dong Jia-Qi Key words electron temperature gradient instability critical gradient negative magnetic shear In this paper, the electron temperature gradient (ETG) instability and corresponding turbulent transport in toroidal plasmas with negative magnetic shear is studied using the integral eigenvalue [1] equations . The full electron kinetics is considered and the behaviours of the modes and the transport in the parameter regimes close to the instability threshold are emphasized. The fitting formulas of the critical gradient, for negative magnetic shear, are given. 1 Integral eigenmode equations An axis symmetric toroidal geometry with circular flux surfaces is employed. The pressure gradient parameter is not independent of and may be expressed with other parameters as d q e 2 Rq 2 (1 i1 e ) . (1) dr n In the local approximation, the magnetic shear effects are neglected and the parallel wave vector k // const is used. We can straightforwardly derive the following two coupled eigenmode equations [2] which are the reduced versions, k 2 2 ˆ ˆ 1 i 2e P0 (k ) P1 A// (k ' ) 0 , (2) 2 pe ˆ 1 2 ˆ e P1 (k ) k e P2 A// (k ' ) 0 , (3) 2 where 2 e 1 k // vte (k , k ) , P0 i 2 *e d D1 D0 0 (4) ae 2 2 ae 0 - 2 k // vte e 3 k // vte (k , k ) , P1 *e d 0 D1 D0 (5) ae 2 2 ae 0 - ae 2 2 P2 i 2 *e d D1 0 1 k // vte D e 5 k // vte e (k , k ) , (6) 0 a 2 2a e 2 2 a e 0 - e 2 ae D1 e i exp[ (k // vte / 2 ae ) 2 ] /( ae (1 ae )) . （7） All the physical quantities, in the above formulas, have been explained in detail in Ref.[2]. 2 Numerical results The parameters for the numerical results given here are s 1 , q 1.5 , i 3 and ˆ d s e / pe 0 unless otherwise stated. These parameters are typical for tokamak plasmas in 2 2 confinement regions. From the numerical results, it is explic itly demonstrated that the maximum growth rate is an offset linear function of e , m ax ns*e ( e ecns ) , (8) or of R / LTe , m ax ns *e n R / LTe ( R / LTe ) cns , (9) where the threshold gradient c ns e or ( R / LTe ) cns and the proportionality factors ns all depend on the plasma parameters, such as i , n , s and q . ˆ Project supported by the National Natural Science Foundation of China (Grant No. 10135020). 1 We consider the nominal negative magnetic shear s 1 . The obtained critical gradient ˆ parameters ec ns and ( R / LTe )cns are shown in Fig.1 for i 3 (the line with solid squares), 2 (the line with open circles), 1 (the line with open squares), 1/3 (the line with solid circles). Here, the data points are the numerical results and the curves are from the fitting functions, 6.549 n2 1.199 n 1.018 , i 1 / 3, 4.089 n 2.831 n 0.6331, i 1, 2 ecns (10) 2.760 n 7.175 n 0.3237 , i 2, 2 2.274 2 10 .81 0.1970 , i 3, n n and R / LTe cns ecns / n , (11) with e given by Eq. (10). From Eq. (10) it is easy to find that d e / d n 0 in the parameter c ns cns regime studied here. We may also find that R / L Te cns decreases with n for n n0 . Here n0 0.4, 0.32,0.3 and 0.25 for i 1/3, 1, 2 and 3, respectively. Again, a higher i is in favour of raising the threshold temperature gradient of ETG modes for 0.1 n 0.6 . In addition, higher ETGs seem needed to drive the modes unstable in plasmas with peaked density profiles. In comparison with Ref. [3], we see that the threshold temperature gradient for a negative magnetic shear ec ns is higher than the value for a positive magnetic shear of the same magnitude when the other parameters are also the same. For example, the critical R / LTe values are 10 and 12 for s 1 ˆ and s 1 , respectively, when i 3 . For i 1 these values are approximately 5 and 6 when ˆ n 0.2 . FIG.1 Critical gradient parameters ec ns (a) and ( R / LTe )cns (b) versus n for s 1 . ˆ The coefficients in Eq. (10), depending on the parameter i , may be expressed by polynomials. In this way, Eqs. (10) and (11) may be re-written as ecns c1ns ( i ) n2 c 2ns ( i ) n c3ns ( i ) , (12) and c ns ( ) R / LTe cns c1ns ( i ) n c 2ns ( i ) 3 i , n where c1ns 0.794 i2 4.18 i 7.74 c 2ns 0.603 i2 6.47 i 3.21 . c 3ns 0.117 i2 0.695 i 1.23 2 The proportionality factor ns ( n ) in Eq. (8) is given in Fig. 2, where the data points are from the numerical results and the symbols have the same meanings as those in Fig. 1. Here, the curves are from the fitting functions, 1.45 n2 0.288 n 0.0374 , i 1 / 3, 0.289 n 0.125 n 0.00567 , i 1, 2 ns ( n ) (13) 0.0518 n 0.101 n 0.000795 , i 2, 2 0.0203 2 0.0968 0.0025 , i 3, n n The definition of ns in Eq. (9) leads to the essentially linear fitting functions in Eq. (13) for i 2 and 3. The same procedure as for Eq. (10) may be carried out for the coefficients in Eq. (13) and results in ns ( n , i ) g1ns ( i ) n2 g 2ns ( i ) n g 3ns ( i ) , (14) where g 1ns ( i ) 0.306 i3 1.92 i2 3.85 i 2.53 g 2ns ( i ) 0.148 i3 0.880 i2 1.58 i 0.722 . g ( i ) 0.0168 0.0979 0.171 i 0.0841 ns 3 i 3 i 2 Fig.2 Proportionality factor versus n . ns Again, the maximum growth rate for a negative magnetic shear may be expressed as follows and used in future studies, m ax [ g 1ns ( i ) n2 g 2ns ( i ) n g 3ns ( i )]*e R / LTe [c1ns ( i ) n c 2ns ( i ) c 3ns ( i ) / n ]. (15) 3 Conclusions and discussion Substituting Eq. (14) into Eq. (9), for negative magnetic shear, we have, m ax [ g 1ns ( i ) n2 g 2ns ( i ) n g 3ns ( i )]ckm axTe /(eBLn )R / LTe ( R / LTe ) cns , (16) m ax where c is the speed of light, k is the poloidal wave vector corresponding to the maximum growth rate m ax , B and e are the magnetic field and the magnitude of electron charge, respectively. Considering that the finite effects are not significant for nowadays tokamak plasmas, as an approximation, we may assume that ~ c / pe ~ 1/ kmax , the collisionless skin depth, and make an estimation for the electron thermal transport coefficient, e ~ DB [ g 1ns ( i ) n2 g 2ns ( i ) n g 3ns ( i )] c 1 / LTe (1 / LTe ) cns G ns (s, q) ˆ (17) pe for LT Lcns , with DB cTe / eB and G ns (s, q) being the Bohm diffusion coefficient and an order Te ˆ unity function of the rest parameters, respectively. 3 In summary, the ETG-driven instability in toroidal plasmas in the parameter regime close to the stability boundary has been studied with gyro-kinetic theory in this work. The emphasis is placed on the calculations of the maximum growth rate and threshold temperature gradient of the modes. Direct calculation of critical gradient is performed. The scaling of the maximum growth rate with ETG is given. The scaling of the critical gradient with the temperature ratio (Te / Ti ) and toroidicity ( n / R) is presented. The algebraic fitting formulas for the maximum growth rate and for the critical gradient are created from the numerical data. In addition, the critical temperature ˆ gradient (TG) in plasmas with negative magnetic shear s is higher than that with positive s ˆ ˆ when the other parameters and the magnitude of s are the same. This is also qualitatively in agreement with the experimental observations. The theoretical results for critical TG, obtained in this work, lie in the range of experimental results. References 1 Dong J Q, Horton W and Kim J. Toroidal kinetic i mode study in high–temperature plasmas. Phys. Fluids 1992 B 4 1867 2 Jian Guang-De， Dong Jia-Qi. Study of electron temperature gradient instability in toroidal plasmas with negative magnetic shear.2004 Chinese Physics Vol.13 No.6 3 Jian Guang-De，Dong Jia-Qi. Particle simulation method for the electron temperature gradient instability in toroidal plasmas. Acta Phys. Sin. 2003 V ol.52 No.7 1656 4

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