23P049-P
Neoclassical Toroidal Viscosity Calculations in Tokamaks using a delta-f
Simulation and Their Verificationsdelta-f法を用いたトカマクにおける新古典トロイダル粘性計算法の検証
Shinsuke Satake Hideo Sugama, Ryutaro Kanno, Jong-Kyu Park1
佐竹真介, 洲鎌英雄，菅野龍太郎，J.-K. Park
National Institute for Fusion Science322-6 Oroshi, Toki, Gifu 509-5292, Japan
核融合科学研究所 〒509-5292 岐阜県土岐市下石町322-6
1Princeton Plasma Physics Laboratory
Princeton, New Jersey 08543, USA
Neoclassical toroidal viscosity, which arises in tokamaks when non-axisymmetric magnetic perturbation isapplied, is known to affect the toroidal rotation of tokamak plasma even if the perturbation amplitude isquite weak. FORTEC-3D, a neoclassical transport code using the δf Monte Carlo method, is utilized toevaluate the neoclassical toroidal viscosity caused by weak magnetic perturbations. The calculation method
is applied to investigate the dependence of neoclassical toroidal viscosity on the radial electric fieldstrength. It is shown that the neoclassical toroidal viscosity is sensitive to the radial electric field of which
amplitude is about the same order as that is expected from the force balance relation and vanishes when theforce balance relation is satisfied.

1. Introduction
The control of the toroidal rotation is an
important issue in tokamaks in order to improve thestability of the contained plasmas. Recent studies
have shown that the non-axisymmetry as small as
δB/B~10can induce significant damping intoroidal rotation which otherwise would be muchbetter maintained. Such small perturbation can beapplied for the purpose of stabilizing ordestabilizing edge localized modes[1], or is inherent
in tokamaks because of the error in coil systems.
One of the main physics which affect the toroidal
rotation of the plasmas is the neoclassical toroidal
viscosity (NTV), which arises when the toroidal
symmetry of the magnetic field is broken. The NTV
torque has been observed and studied in many
tokamak experiments[2–4] and with a numbers oftheories[5-6]. However, theoretical prediction ofNTV is nontrivial because it involves complicateddiffusion process in the perturbed magnetic field,
which is also affected by collisions and radial
electric field. Conventional theories have been
therefore developed in each limited regime tosimplify or ignore other processes; the effects bypassing particles are ignored, effects by trapped
particles are studied in many different collisionality
regimes, the finite drift width of particle orbits are
neglected, and so on. In order to give a moregeneral and quantitatively reliable method toestimate the NTVs in weakly perturbed tokamaks,
we have adapted FORTEC-3D code[7] for direct
numerical simulation of NTV. Since it is a first

application of the δf transport code to neoclassicalviscosity calculation, we have been carefully
verifying its applicability step by step. Previous
benchmarks[8, 9] have demonstrated that it can beapplied to the wide range of collision frequency andthat the calculation results agree very well with thecombined NTV formula by Park[10] in the limit ofzero radial electric field.
In this paper, as the second step of theverification of the code, the NTV dependence on theradial electric field (is investigated. It is shown that
the NTV in low collisionality plasma is very sensitive toof which amplitude is about the same order as that isexpected from the force balance relation. Whensatisfies the force balance, the NTV is found to vanishcompletely, as expected from the neoclassical theory.

2. Numerical method
FORTEC-3D solves the drift-kinetic equation forthe perturbed distribution functionin the 5-D
phase space [7]:
∥∙

∙,

1

where
is the local Maxwellian, andandare the test-particle and field-particle part ofthe linearized collision terms respectively. The
magnetic field is expressed by Fourier spectrum in
Boozercoordinates
asB r, θ, ζ
1
∑ ,cos. Utilizing the, cos
field expression, the NTV in perturbed tokamak canbe expressed as [8]〈⋅⋅ P〉,〈sin3e-04〉, 2,

depends on the plasma collisionality ∗ and ,→ 0 limit. To avoid
which is ≓ 1.17 in the ∗ ,the complexity and for future benchmark withanalytic theories which sometimes uses anapproximation in , we used a constantprofile.
The collisionality is banana regime, ∗ 0.06 .
Since the simulation starts from zero toroidal flow
and the flow seldom develops during a simulation
profile which satisfies the force
run, thebalance can be obtained from Eq. (3) by settingln /0. Each simulation is carried out
between 2.0 times of its
by multiplying theforce-balance value.
4. Simulation results
Figure 1 shows the radial profile of NTV density
for several amplitude of. It has a peak at theresonant surface. One can see that NTV completely
profile satisfies the force
vanishes when the1.0 case). Though Eq. (3) is derived
balance (assuming the toroidal symmetry, the relation stillholds true when a weak non-axisymmetricperturbation is applied.
The NTV dependence on theamplitude is0shown in Fig. 2. NTV has the maximum atand becomes almost constant for large | |. It isand∙ ζ profiles of aexpected that thesteady-state in perturbed tokamak plasmas are close
to the force balance solution if there is not strong
torque input (NBI) or non-ambipolar flux. The
simulation result anticipates that the NTV is almost
zero if the force balance is completely satisfied,
while it is also sensitive to the deviation from thebalanced state. It has nearly a linear dependence,〈 ⋅ ⋅ P〉 ∝ |when theforce banalce | ,deviation is small.

NTV density [N/m2]

1e-04
5e-05
0e+00
-5e-05-1e-04
0.4

0.45

0.5

0.55

0.6

0.65

0.7

r/a.

Fig.1. NTV profile for each amplitude of
3e-04
2e-04
2

where χ is the poloidal flux and Φ is the. The factor
electrostatic potential,

2e-04

NTV density [N/m ]where δP ≡/2
and 〈⋯ 〉∥means the flux-surface average. Eq. (2) isnumerically evaluated in the code from thequasi-steady state solution of.
3. Simulation condition
We used the same simple tokamak geometry as isused in Ref. 8. Single (7, 3)-mode perturbation
0.01 /
is applied, which has a resonant,rational flux surface q 7/3 at r/a 0.487 .
, we utilize the force balance
Concerning
relation (for ions) in tokamaks[10],∙ ζ,
3Er x-2.0
Er x-1.0
Er x-0.5
Er x 0.0
Er x 0.5
Er x 1.0
Er x 2.0

2e-04

r/a=0.485
r/a=0.531
r/a=0.652

2e-04
1e-04
5e-05
0e+00
-5e-05-1e-04-2 -1.5 -1 -0.5 0 0.5 1
Er amplitude

1.522.5

Fig.2. Dependence of NTV on . The horizontal axis
amplitude.
represents the magnification factor of theAcknowledgments
This study is supported by JSPS Grant-in-Aid forYoung Scientists (B), No. 23760810 and NIFS
collaborative Research Programs 11KNST014.
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