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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   i1   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 )]ckm 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/ kmax , 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




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