23P143-B
Collisional Kinetic-Fluid Closure Model for Zonal Flows
ゾーナルフローに対する衝突性運動論的流体クロージャーモデル
Hideo Sugama and Osamu Yamagishi
洲鎌英雄, 山岸統
National Institute for Fusion Science / Graduate University for Advanced Studies
322-6 Oroshi-cho, Toki, Gifu 509-5292, Japan
核融合科学研究所・総合研究大学院大学 〒509-5392 岐阜県土岐市下石町322-6
A novel kinetic-fluid closure model is presented, which describes collisional damping of zonal flows in theion temperature gradient turbulence in tokamaks. The closure relations representing the parallel heat fluxes
are derived from the Laplace transform of the solution to the bounce-averaged collisional ion gyrokinetic
equation for the radial wave number vector corresponding to the zonal-flow component. Approximate
expressions for the closure relations are also obtained, which are suitable for numerical simulation.
)
1. Introduction[ "n (gk # , n 0 u||k # , "p||k # , "p#k # ](1)
Zonal flows are intensively investigated in the= d 3v "f k(g# )[1, v|| , mv||2 , mv#2 /2] .
fusion research as an attractive mechanism forrealizing a good plasma confinement [1]. An
Consequently, we obtain the perturbed gyrocenter
accurate theoretical description of zonal-flow
density equation,)
evolution is a key issue for correctly predicting' n 0 u||k $ *"#n (gk$+ B% &)!
,the turbulent transport of fusion plasmas. In fact,"t( B +
unless the residual zonal flow [2] is properlyctreated in a gyrofluid model, the gyrofluid(2)= - i 2 k $ % (b . &B)[#p||k $ + #p$k $simulation cannot reproduce the same turbulenteBtransport as given by the gyrokinetic simulation.
+ n 0e/ k $ e -b / 2 (2 - b /2)] + N 0k $ ,In order for a set of gyrofluid fluid equations tothe parallel momentum balance equation,
describe the residual zonal-flow level given by' &p||k # * &p#k #"u||k #Rosenbluth and Hinton [1], a novel collisionlessm n0+ B$ %)b$ %B,+"tB( B +kinetic-fluid closure model of zonal flows in!
tokamaks was presented by Sugama et al.[3],mc(3)= - i 2 k#$ (b . %B)(q||k # + q#k # + 4 p0 u||k # )which differs from the zonal-flow closure model
eBby Beer and Hammett [4]. It is confirmed fromnethe fluid simulations of zonal-flow damping [5]- n 0eb$ %(/ k # e -b / 2 ) + 0 / k # e -b / 2b$ %B + N1k # ,2Bthat the collisionless kinetic-fluid closure model
theperturbed
parallelpressureequation,of zonal flows [3] correctly reproduces the"#p||k $residual zonal flow level predicted by the kinetic+ B% &[(q||k $ + 3p0 u||k $ ) /B]"ttheory, in which effects of the noncircular!
2tokamak cross section can be included. Behaviors+ (q$k $ + p0 u||k $ )b% &B(4)of zonal flows are also influenced by collisions
B[6]. In fact, the gyrokinetic simulation of the ion
c= ' i 2 k$% (b ( &B)[m(#r||,||k $ + #r||,$k $ )temperature gradient (ITG) turbulence shows thateB
the turbulent transport level significantly depends+ p0e) k $ e 'b / 2 (4 ' b /2)] + N 2||k $ ,on the collision frequency through the collisional
damping of zonal flows [7]. In the present work,and the perturbed perpendicular pressure equation,"#p$k $
we extend the closure model to take account of+ B% &[(q$k $ + p0 u||k $ ) /B]
collisional effects on zonal flows.
"t!

$

2. Kinetic-Fluid Equations
In the same way as in [3], we take the velocity
moments of the electrostatic gyrokinetic equation
for the perturbed gyrocenter distribution function"f k(g# ) in order to obtain the equations which governtime evolution of the fluid variables defined by!

1(q + p0 u||k $ )b% &B(5)
B $k $
c= ' i 2 k $ % (b ( &B)[m(#r||,$k $ + #r$,$k $ )eB+ p0e) k $ e 'b / 2 (3 ' 3b /2 + b 2 /8)] + N 2$k $ ,'where

N 0k " , N1k " , N 2||k " , and!
!

!

N 2"k " arethenonlinear terms defined in [3]. The right-hand sidesof (2)-(4) contain the third-order fluid variables (orparallel heat fluxes),[q||k " , q"k " ](6)% u ( p)(B $2||k #(s)* . (9)"'$u||k # (t = 0) + u||k(p)B#Bp'&*)
Here, " j ( p)( j = 1,2) are defined by{}[ "1 ( p), "2 ( p)] = (mi /n 0Ti ) # d 3v G( p)(v|| /B)[1, µ ] (10)with G( p) being the solution of!
and the fourth-order fluid variables(11)
G( p) " p "1Cii (G( p)) = (v|| /B) f iM .
!
[ "r||,||k # , "r||,#k # , "r#,#k # ]For installing the closure relations into numerical!(7)= d 3v "f k(g# )m[v||4 , v||2v#2 /2, v#4 /4 ] .
simulation codes, it is useful to express " j ( p)!
!
In order to construct a closed system of( j = 1,2) approximately as
kinetic-fluid equations, we need closure relations, !
(12)" j ( p) = " j p # /(1+ p # ) ,which express the higher-order fluid variables inwhere " j = lim p # $ " j ( p) ( j = 1,2) !are defined in! (6)-(7) in terms of the lower-order variables in (1).
[3] and " represents the typical time scale of the!
collisional damping of the zonal-flow potential. The
3. Closure Model!
approximate expressions in (12) are shown toHere, we consider the ion temperature gradient! enable the closure relations to be written in the form
(ITG) turbulence for typical perpendicular wave!of the first-order differential equations in time.
numbers of fluctuations are given by k" # ti $ 1 ,where " ti denotes the ion thermal gyroradius. The
4. Summary
ITG turbulence are governed by the ion gyrokinetic
In this paper, the new kinetic-fluid model isequation for "f ik(g#) , the relation !of the perturbed
derived, which describes behaviors of zonal flows
in the ITG turbulence for the weakly collisional! electron density to the electrostatic potential"n ek = n 0e $ k % $ k
Te and the quasineutrality
case. The collisional effects are included in theclosure relations for the long-time evolution part ofcondition!
"n ik # = "n ek # . We now derive the closure
the parallel heat fluxes. The approximate
relations for the high-order fluid variables in theexpressions for the closure model are also shown
fluid equations (2)-(5) for ions instead of the ion!
and numerical simulation using them remains as a
gyrokinetic equation to accurately describe thefuture task to confirm their validity.
!
zonal-flow behaviors in the ITG turbulence. The
collionless case was already treated in [3]. We
Acknowledgments
assume the weakly collisional case, in which theThis work is supported in part by the Japanesecollisional effects on the short-time behaviors of theMinistry of Education, Culture, Sports, Science,
fluid variables are neglected. Then, the sameand Technology, Grant No. 21560871 and in partclosure relations as (61)-(62) in [3] can be used forby the NIFS Collaborative Research Program,the shot-time evolution parts of the parallel heat
NIFS10KNTT003.
(s)fluxes (q||k, q(s)"k " ) and the fourth-order variables"References("r||,||k # , "r||,#k # , "r#,#k # ) . The long-time evolution= % d 3v #f k(g" )v|| [(mv||2 $ 3T), (mv "2 /2 $ T)] ,$#(##)(l )parts of the parallel heat fluxes (q||k, q(l"k) " ) are"! derived from the solution of the bounce-averaged
collisional ion gyrokinetic equation for the radial!

wave number vector k "!= kr #r corresponding to
the zonal-flow component. The resultant closure
relations are given in terms of the Laplace
#(l )
(l )
transform [q||k( p), q(l"k) " ( p)] = $ [q||k(t), q(l"k) " (t)]""0"exp(# pt)dt as(l!
)
q||k( p) = #2q(l"k) " ( p)"= 2n 0Ti [U1k " ( p)B # U 2k " ( p)B 2 ] ,where U jk " ( p)( j = 1,2) are given by!

!

{U jk " ( p) = #1 ( p) $ B $2!
!
!

$1} # ( p)j(8)[1] P. H. Diamond, S.-I. Itoh, K. Itoh, and T. S. Hahm:
Plasma Phys. Controlled Fusion 47 (2005) R35.
[2] M. N. Rosenbluth and F. L. Hinton: Phys. Rev. Lett.
80 (1998) 724.
[3] H. Sugama, T.-H. Watanabe, and W. Horton: Phys.
Plasmas 14 (2007) 022502.
[4] M. A. Beer and G. W. Hammett: Proceedings of theJoint Varenna-Lausanne International Workshop on
Theory of Fusion Plasmas, Varenna, 1998, edited byJ. W. Connor et al. (Societa Italiana de Fisca,Bologna, Italy, 1999), p. 19.
[5] O. Yamagishi: Plasma Conference 2011, 23P141-B.
[6] F. L. Hinton and M. N. Rosenbluth: Plasma Phys.
Control. Fusion 41 (1999) A653.
[7] Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang, and P.
H. Diamond: Phys. Rev. Lett. 83 (1999) 3645.