24D06
Dynamic Response of Gradient and Turbulent Transport in Resistive BallooningTurbulence
抵抗性バルーニング乱流における勾配・輸送の動的応答	 
Satoru Sugita1, Kimitaka Itoh2,3, Sanae-I. Itoh1,3, Masatoshi Yagi1,3,4, Guillaume Fuhr5, Peter Beyer5,
and Sadruddin Benkadda5
杉田 暁1, 伊藤公孝2,3，伊藤早苗1,3，矢木雅敏1,3,4, Fuhr Guillaume5, Beyer Peter5,Benkadda Sadruddhin5
1

Research Institute for Applied Mechanics, Kyushu University
6-1, Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan1
九州大学 応用力学研究所	  〒816-8580 福岡県春日市春日公園6-1
2
National Institute for Fusion Science, National Institutes of Natural Science322-6, Oroshi-cho, Toki, Gifu 509-5292, Japan2
自然科学研究機構 核融合科学研究所	  〒509-5292 岐阜県土岐市下石町322-6
3
Itoh Research Center for Plasma Turbulence, Kyushu University
6-1, Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan3
九州大学 伊藤極限プラズマ研究連携センター	  〒816-8580 福岡県春日市春日公園6-1
4
Japan Atomic Energy Agency
2-166, Oaza-obuchi-aza-omotedate, Rokkasho, Kamikita, Aomori 039-3212, Japan4
日本原子力研究開発機構	  〒039-3212 青森県上北郡六ヶ所村大字尾駮字表舘2-166
5International Institute for Fusion Science, CNRS-Université de Provence
Case 321, Centre de Saint Jérôme, 13397 Marseille Cedex20, France
The flux-driven nonlinear simulation of resistive ballooning mode with tokamak edge geometry isperformed. In the situation with the poloidal rotation is artificially suppressed, the radial propagation ofpulses of pressure gradient is observed. It is found that the fluctuation with higher mode number propagateswith the pulse. Statistical values of the pulses propagation are evaluated. The scale length and propagation
velocity are compared with theoretical estimations and it gives good agreement.

1. Introduction
Recent experimental study reveals thatfluctuations with the long-range radial correlation
exist in fusion experimental device and their
spatiotemporal structure is characterized by theballistic radial propagation [1]. Ballistic
propagation is considered to play an importantrole in determining the turbulent transport infusion experimental devices such as ITER andDEMO. Ballistic behaviors and dynamics havebeen studied widely [2]. However, the statisticalcharacteristics have not been discussed enough.
In this presentation, we report statisticalcharacteristics of ballistic radial propagation ofpulses of pressure gradient using the global
nonlinear simulation. They are compared withtheoretical estimation and give good agreement.
2. Model and Simulation Conditions
Thetwo-field
reduced-RBM(ResistiveBallooning Mode) equations are solved toreproduce the plasma edge turbulence in thetokamak geometry [3]. The following equations areemployed for the normalized electrostatic potential! and pressure p .
(1)(2)
The notations and symbols are explained as follows.
!|| and ! " are gradient parallel andperpendicular to the magnetic field line.
[ f , g] = (!f / !x ) (!g / !y) " (!f / !y) (!g / !x ) is
the Poisson bracket. The curvature operator G isintroduced. µ indicates the collisional viscosity,! || and ! ! are parallel and perpendicularcollisional heat diffusivities.

= 2! ( L p / R0 ) is

the parameter that is related to a curvature
coefficient, where ! = 5 / 3 is the adiabatic
index, L p is the scale length of the pressure
gradient and R0 is the major radius at thereference surface. In this RBM model, thediamagnetic velocity is neglected in comparison
with the ExB velocity, and the parallel current isevaluated by using a simplified electrostatic Ohm’slaw, !||0 j|| = !"||" , where !||0 is a reference valueof the parallel resistivity and j|| is the parallel
current. The system is normalized using the(2resistive interchange time ! int= R0 L p / 2cs02

thetime,resistiveballooning) inlength! = m n ! L / (! int B ) in the perpendicular2bal

2i 0 ||0 s

20

length, and magnetic shear length Ls in the(parallel length, where cs0 = p0 / ( n0 mi )1/2)

is thereference sound speed. p0 , n0 , and B0 arereference values of the pressure , density andmagnetic field, respectively. mi is the ion mass.
The model equations are solved numerically by afinite difference method in the radial direction andby a Fourier expansion in the poloidal and toroidal
directions. The maximum radial grid number is 104,
the maximum toroidal mode number is 32, and themaximum poloidal mode number is 112. To save
calculation time, only one in four toroidal modes isincluded in the simulations. In the ( m, n ) space,the ( n = 0, m > 0 ) modes are suppressed to keep
the neoclassical terms consistent (especially for the(1, 0) mode), where m is the poloidal mode
number and n is the toroidal mode number,respectively. By assuming the safety factor q ( r )is monotonically increasing, the simulation domain
covers a region between q = 2.5 and q = 3.5 .
The regions q < 2.5 and q > 3.5 are used for a
dumping buffer. In the dumping buffer region, thedissipation rate is set to be large. In the simulation
region, the ion viscosity µ and the diffusivity! ! are set to be 0.93. The parallel diffusivity is! || = 0.5 and= 0.04 . A Gaussian typepressure source is introduced in the innerbuffer region, i.e., q ! 2.5 . We vary theamplitude of the pressure source S ( r ) in thesimulations. The simulations are performed in
the situation that the poloidal rotation velocity
is always suppressed artificially.

3. Simulation Results
Flux-driven Nonlinear simulations for edge
turbulence are performed. It is found that thecoexistence of the radial propagation of the pulse ofpressure gradient and the appearance/disappearanceof the radial global structure of heat flux innonlinearly saturation phase. Here, we focus on thepulse propagation. The pulses of pressure gradientpropagate radially outward and inward. It is foundthat the propagation front is a boundary of thefluctuation with higher poloidal mode number andthat with lower poloidal mode number. When thepulse of pressure gradient propagates outward, thepropagation of fluctuation with higher poloidal
mode number is also observed. The details of pulse
propagation are statistically analyzed for threedifferent input energy sources S . The scale lengthof the pulse l , averaged velocity of the pulse v ,and effective diffusion coefficient Deff areevaluated, where the scale length is calculated fromthe curvature of the peak of the auto-correlation andthe effective diffusion coefficient is evaluated by
Deff = ! / "p , where ! indicates heat flux
andindicates spatial and temporal average.

The spatial scale length l is compared with the1/2( Deff / ! )theoretical estimation[4], where !

is the linear growth rate and the order of magnitudeagrees. The velocity of pulse v is compared withthe estimation

1/2( Deff! )[4] and the order andtendency agree.
4. Summary
The flux-driven nonlinear simulation of resistive
ballooning mode with tokamak edge geometry isperformed. In the situation with poloidal rotation is
artificially suppressed, the radial propagation ofpulses of pressure gradient is observed. It is foundthat the fluctuation with higher mode number
propagates with the pulse. Statistical values of thepulses propagation are evaluated and comparedwith theoretical estimations and it gives goodagreement.
Acknowledgments
This work is partially supported by the
Grant-in-Aid for Scientific Research (S) of JSPS
(21224014), Grant-in-Aid for JSPS Fellows(22-4534), collaboration programs of RIAM,Kyushu Univ., NIFS/NINS under the project ofFormation of International Network, and Collage
Doctoral Franco-Japonais.
References[1] S. Inagaki, et al.,: Phys. Rev. Lett. 107 (2011)
115001.
[2] X. Garbet, et al.,: Phys. Plasmas 14 (2007) 122305.
[3] P. Beyer, et al.,: Plamsa Phys. Control. Fusion. 49(2007) 507.
[4] Ö. D. Gürcan, et al.,: Phys. Plasmas 12 (2005)
032303.