23P141-B
Zonal flow damping simulation in non-circular tokamaks by a fluid closure
model based on collisionless gyrokinetics
無衝突運動論的クロージャー流体モデルに基づく、非円形断面トカマクにおけるゾーナル流減衰シミュレーション
Osamu Yamagishi1), Hideo Sugama1)2)
山岸統, 洲鎌英雄
1

National Institute for Fusion Science / 2 Graduate University for Advanced Studies
322-6 Oroshi-cho, Toki, Gifu 509-5292 Japan1
核融合科学研究所/ 2総合研究大学院大学〒509-5392 岐阜県土岐市下石町322-6
Simulation study of zonal flow (ZF) damping is done by using a fluid model based on the collisionlessgyrokinetics [H. Sugama et al., Phys. Plasmas 14, 022502 (2007)]. The residual zonal flow level isdiscussed in tokamaks with various shapes of the magnetic surfaces, whose MHD equilibria are obtained
numerically by VMEC code. For circular toakamak, it is shown that our fluid model gives a residual level
predicted by Rosenbluth-Hinton (R-H) formula [M. N. Rosenbluth et al., Phys. Rev. Lett. 80, 724 (1998)].
For non-circular tokamaks, the ellipticity is an increasing effect for ZF. On the other hand, the triangularity
is more weak effect. These simulation results are also compared with some ZF theories. Effects of finite
initial parallel flow are also considered. It is shown that the toroidal component of initial parallel flow is
not damped while the poloidal component is damped as R-H formula shows, and resulting perpendicularflow can be kept with high level. The quasi-linear results are also shown by considering the linear ion
temperature gradient modes and trapped electron modes as the nonlinear sources for the fluid variables.

1. Intoroduction
It is known that the zonal flow (ZF) componentof E x B flow plays an important role for thedetermination of the anomalous transport levelvia the turbulence suppression. This means thatthe zonal flow itself should be treated accurately.
The fluid model has an advantage that it needs
much smaller computation resources than those thekinetic model needs. However, it has come torealize unfortunately that the zonal flow is nottreated accurately in the classical fluid models.
Hammett and Perkins (H-P) [1] improved the fluid
model to include the effect of kinetic Landau
damping. This introduces the transient response of
the potential in the short time (so-called GAMoscillation), and fluid simulation studies with thismodel have been developed [2,3]. However, it waspointed out that even with H-P model, the fluid
model cannot reproduce the residual ZF level
predicted by the kinetic equations [4]. To improvethe discripancy, an attempt was made [5]. In
consideration of these results, a new fluid closure
model is developed to reproduce the kinetic ZF
level in the long time limit, based on thecollisionless gyrokinetics [6].
In this work, we use the model of Ref.[6] for thesimulation study of the zonal flow damping. Theresidual ZF level is discussed in tokamaks withvarious shapes of the magnetic surfaces, whose
MHD equilibria are obtained numerically by anMHD code, VMEC [7]. For circular toakamak, it isshown that our fluid model gives a residual level
predicted by Rosenbluth-Hinton (R-H) formula [4].
For non-circular tokamaks, the ellipticity is foundto have an increasing effect on ZF. On the otherhand, the triangularity is more weak effect. Thesesimulation results are also compared with some ZFtheories. Effects of initial parallel flow are also
investigated. It is shown that when resultant vectorof initial parallel flow and perpendicular E x B flow
is in the toroidal direction, the flow, potential,density, are not damped at all.
2. The fluid model
The model fluid equations are given in eqs.
(48)-(51) of Ref.[6] with quasi-neutrality condition.
Four fluid variables n, u//, p//, and p are solved.
The closure is introduced to the parallel andperpendicular energy flux along the field line, q//
and q. These are divided into two parts, short andlong time terms,
q//=q// (s)+q// (l),q=q (s)+q (l).
(1)
The short time term q(s) is assumed to be H-P model,
while the long time term q(l) is given by eq.(57) of[6]. The typical simulation results in the circular
tokamak are shown in Fig.1. If the energy flux q is
dropped entirely, the potential is oscillating and notconverged. If only the short time closure isconsidered (H-P model), the initially given potential
is GAM-damped and it becomes negligibly small in
the long time limit. In the case that both q(s) and q(l)are included, the finite level of residual zonal flow
is obtained, for which we can confirm that theresidual level reproduce the R-H result.

Fig.1: q=0(dot), q=q(s) (dash) and q=q(s)+q(l) (solid)

3. Shaping effects on the residual zonal flow
Here the effects of the ellipticity and triangularityof the magnetic surfaces on the residual zonal flow
are investigated. Here a/R0=1/3 and q=1+22 are
fixed, and a surface  and ~0.2 is considered.
The radial wave number is assumed to be krai=0.01.
The R-H formula, f( ∞ )/f(0)=1/(1+1.6q2/1/2), is
obtained for the circular tokamaks so that it does
not change as the ellipticity or triangularity changes(dashed lines). On the other hand, the simulation
result (marks) increases with the ellipticity. A
formula from Ref.[6] in the low kr limit is also
shown (solid lines), which coincides with thesimulation results very well. The effect oftriangularity on the residual ZF level is found to beweak. These tendencies are roughly consistent withtheories [8,9] (dotted, dot-dashed lines).

Fig.2: Residual potential as a function of the ellipticity(left) and triangularity (right).

4. Effects of finite initial parallel flow
Effects of finite initial parallel flow have notbeen discussed so much in the past. Here wediscuss finite u//(0) cases. In fig.3, three cases are
considered: (i)(0)=1 and u//(0)=0, (ii)(0)=0 andu//(0)= -iKvt2/B, and (iii) (0)=1 and u//(0)= -iK
vt2/B, where K=kBt/p. In the case (i), (0)=1 at
t=0 is damped as usual, and the residual potential
becomes the R-H level. On the other hand, we
assume only finite u//(0)/vt= -iKvt/B in case (ii).
Here this u//(0) corresponds to the first term ofeq.(A7) of [6], in which case total flow (u//b+u) isin the toroidal direction. Since we consider linearsystems, a solution is linear combination of caseswith different initial value. Thus the case (iii) isexpected to be the sum of case (i) and case (ii), andwe see that the potential  at t=0 is not damped atall. This is expected because in this case, balance
equation eq.(66) of [6] is satisfied for krai=0.01<<1,which forces , n and u// to be fixed in time
simultaneously without nonlinear source. This is
one example, and we can choose function form ofu//(0) freely. The effects of quasi-linear density orflows, derived from linear gyrokinetic modes (ITG,TEMs, etc.) will be discussed.

Fig.3: Residual potential with finite u//(t=0).
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