22F05
Analytical Formua of offset toroidal rotation in NTV
新古典トロイダル粘性によるオフセットトロイダル回転の解析式
Mitsuru Kikuchi, Mitsuru Honda
菊池 満, 本多 充
Japan Atomic Energy Agency, Fusion Research and Development Directorate, Naka Fusion Institute801-1, Mukoyama, Naka, Ibaraki 311-0193, Japan
日本原子力研究開発機構、那珂核融合研究所 核融合研究開発部門〒311-0193 茨城県那珂市向山801-1
An analytic formula of offset toroidal rotation due to NTV is derived for e-ion-impurity plasma
assuming negligible impurity viscous forces in parallel and toroidal directions. The formula is expressed inthe radial coordinate ρ=(φ/φa)1/2a to see parametric dependence.

1. Introduction
Tokamak has toroidal symmetry and symmetry
breaking via application of non-axisymmetric field
produces dissipation of toroidal momentum by
NTV [1] and produces offset toroidal rotation [2].
For toroidal rotation measurement, impurity(typically carbon) charge exchange recombination
spectroscopy (CXRS) is used. In this case, we
expect some difference of toroidal rotation between
ion and impurity [3].
An analytic formula of offset toroidal rotation isderived assuming negligible impurity viscous forcesin parallel and toroidal directions [4], [5]. In thistalk, we show these expressions in the radial
coordinate ρ=(φ/φa)1/2a, where φ and a are toroidal
flux and minor radius, respectively, to show its
importance in high βp regime, which is relevant forsteady state operation.
2. Analytical form of offset ion toroidal rotation
The 0-th order ion force balance is give by
0 = eZ i n i (E + ui × B) - ∇Pi . The radial componentof above equation can be obtained by taking innerproduct with tangent vector ∂x/∂ψ and using theidentity ∂x/∂ψ⋅∇ψ=1 in the flux coordinates (ψ, θ,ζ) [6], where x is position vector as follows,€ dΦ
1 dPi u i ⋅ ∇ζ = −+ + qu i ⋅ ∇θ dψ eZ i n i dψ (1)Here, q=dΦ/dψ is safety factor. This equation isexpressed by using Bθ=R-1dψ/dρ, uiζ=Rui⋅∇ζ,
uiθ=Rui⋅∇θ as follows,€uiζ = −€1 Pi' 
Φ' + + quiθBθ eZ i n i (2)Here, ' denotes derivative with respect to ρ.
This equation implies that role of pressure gradientand radial electric field in the flow balance becomes
more important in high βp regime, where Bθ is€relatively small.
Tokamak is an axisymmetric system and theparticle diffusion is ambipolar and its radial electricfield -dΦ/dρ is free, so that it needs to be
determined from the toroidal momentum diffusion
equation. When symmetric breaking occurs by theapplication of non-axisymmetric magnetic field,
particle diffusion is no longer ambipolar. And theradial electric field is determined so that
non-ambipolar flux becomes zero. Shaing [7]derived non-ambipolar flux=0 condition as follows,

Φ' +1 Pi'λ= − 2 Ti'eZ i n ieZ i λ1(3)This leads to following formula for offset
toroidal rotation uiζ0 ,€uiζ 0 =λ2 Ti'+ quiθ 0eZ i λ1 Bθ

(4)Here uiθ0 is a residual poloidal rotation mainly
driven by thermal force dTi/dr and is given by Kim€[3] as follows,uiθ 0 = −K1F(B ⋅ ∇θ )Ti'eZ i B 2 Bθ(5)

Substitution of (5) to (4) gives following formulaof offset toroidal rotation of bulk ion,€ λqK F(B ⋅ ∇θ )  Ti'uiζ 0 =  2 − 1eZ i B 2eZ i λ1
 Bθ

€(5)

In the cylindrical approximation, q=rBt/RBθ,
F=RBt, <B2>=Bt2. So, following approximate
analytical formula is obtained considering
λ2/λ1=3.54 as given in [8].

uiζ 0 =

3.54 − K1 dTi
eZ i Bθ dr(6)The approximate cylindrical expression in [5] is
not correct. Since dTi/dr is negative, offset toroidal€rotation in NTV is counter to the plasma current.
The value of K1 in collisionless plasma is ~1.1 nearcenter and 0.5 at r/R~0.3 as seen in Fig.2 of [3]. So,
this changes theoretical coefficient in 1/ν regime
kc=uiζ0/[(dTi/dr)/eZiBθ]
becomes
closertoexperimental observation including this residual
poloidal flow correction.
Since the measurement of roroidal rotation
corresponds to impurity toroidal rotation, there isfinite difference between ion and impurity parallel
flow as follows,

Bu// I = Bu// i −1.5K 2 Bζ
Ti'eZ i Bθ(7)

The local toroidal flow uaζ=Rua⋅∇ζ of ion (a=i)
and impurity (a=I) is given by,€uaζ =€Bζ Bu// aB2
B2  1 P' − 1− ζ2  Φ' + a eZ a n a B  Bθ (9)In the large aspect ratio cylindrical plasma, offset
toroidal rotation measurement corresponds tofollowing impurity offset toroidal rotation.

uiζ 0 =€(8)

We obtain following form of offset troidal
rotation for impurity species.
2 λqK F(B ⋅ ∇θ ) 1.5K 2 Bζ  Ti'uiζ 0 =  2 − 1−eZ i B 2eZ i B 2  Bθ
eZ i λ1
B2  1  P 'P' −1− ζ2   I − i B  Bθ  eZ I n I eZ i n i €If we plot these value in Garofalo’s paper [2],measurement becomes closer as shown below.

3.54 − K1 −1.5K 2 dTi
eZ i Bθdr(10)

Fig. 1 Offset toroidal rotation in DIII-D [2] andcorrection of theoretical value in this talk.

Acknowledgments
One of the authors acknowledges Prof. Shaingfor useful conversation.
References[1] W. Zhu, S. A. Sabbagh et al., Phys. Rev. Lett.
96, 225002 (2006).
[2] A. M. Garofalo et al., Phys. Rev. Lett. 101,195005 (2008).
[3] Y.B. Kim, P. Diamond, R.J. Groebner, Phys.
Fluids B3, 2050(1991).
[4] M. Kikuchi, “On the Offset Toroidal Rotationin Neoclassical Toroidal Viscosity”, 1st
APTWG Intern. Conf. June 14-17, 2011, NIFS,
Japan, PLI-1.
[5] M. Kikuchi, “On offset toroidal rotation inNTV”, Proc. 38th EPS Conf. on Plasma Phys.
(2011) P4.115.
[6] M. Kikuchi, Frontier in Fusion Research Physics and Fusion, (Springer Verlag, London,2011).
[7] K.C. Shaing, Phys. Plasmas10,1443(2003).
[8] K.C. Shaing, S. Sabbagh, M. Peng, Phys.
Plasmas 14(2007)024501.