Influence of Poloidal Rotation of Plasma on PeelingBallooning Modes *


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1 Applied Physics Frontier Volume, 1, PP.1 Influence of Poloidal Rotation of Plasma on PeelingBallooning Modes * Yiyu Xiong 1, Shaoyong Chen 1, Changjian Tang 1, Jie Huang 1, Yang Luo 1 1. Key Laboratory of High Energy Density Physics and Technology, Ministry of Education, Sichuan University, Chengdu 15, China Abstract Influence of the poloidal rotation of the plasma on PeelingBallooning modes (PB modes) is studied with BOUT++ code. The results show that the poloidal rotation destabilizes PB modes because of the effect of KelvinHelmholtz instability when the poloidal rotation is large enough and the shear is small, and the shear of the poloidal rotation can stabilize highn PB modes when the shear is increased. Keywords: Plasma; PoloidalRotation; PeelingBallooning Mode 1 INTRODUCTION Edge localized modes (ELMs) are magnetohydrodynamic (MHD) instabilities occurring in fusion plasmas in high confinement regime (Hmode) of tokamaks. PB modes which are driven by gradients of pressure and current in the pedestal are widely accepted as the instability triggering large ELMs [13]. The effect of the toroidal rotation of the plasma on ELM and PB modes has been studied. For eample, results from JTU showed that the toroidal rotation can reduce ELM size and increase ELM frequency []. The ELITE code [5,] shows that highn ideal PB modes are stabilized by the shear of the toroidal rotation, while lown PB modes are destabilized. Results from BOUT++ in 1 showed nonideal physics effects, such as diamagnetic drift, EB drift, resistivity, and anomalous electron viscosity, are important to PB modes [7]. Results from BOUT++ in 1 showed the shear of the toroidal rotation can stabilize highn PB modes, and the KelvinHelmholtz instability which is caused by the shear of the toroidal rotation destabilizes PB modes []. The above two articles have not taken into account the influence of the poloidal rotation on PB modes. In the absence of an eternal driving source, the poloidal rotation of the plasma in the Tokamak will be damped out because of magnetic pumping [9], so the influence of the poloidal rotation of the plasma on PB modes or ELM is ignored generally. There is study showing that the electron cyclotron wave can drive the poloidal rotation of the plasma [9]. Meanwhile, eperiments have found that the momentum of the injection can cause the poloidal rotation of the plasma when the supersonic molecular beam is injected along the tangential direction [1]. So it is necessary to figure out the influence of the poloidal rotation of the plasma on PB modes and ELM. In this paper, we will study the influence of the poloidal rotation of the plasma on PB modes with BOUT++ code, and the nonideal physics effects will be considered in our simulations. SIMULATION EQUATIONS In our simulations, we assume that the poloidal rotation of the plasma is eisted without eternal driving source, so there isn t driving source term in threefield reduced MHD equations. Threefield equations describe the evolution of vorticity, pressure, and parallel vector potential with time. The flute perturbation model, k / k 1is used in these * Supported by Chinese National Fusion Project for ITER Grant No 13GB17, and by the National Natural Science Foundation of China Grant No
2 equations t 1 + v + v + v + v = B J + b k P (1) pol 1 E,dia t A p 1 + pol p 1 + E,dia p P + 1 p 1 = t v v v v () η η v (3) 1 H = pola 1 A 1 A 1 μ 1 1 = 1 p1 () B Zen = b v (5) pol J = J 1 () A 1 μ (7) dia 1 In these equations, v pol is poloidal rotation velocity. b b b, b = A 1b B, and k b b is equilibrium magnetic field line curvature vector. ϕ 1 is perturbed electrostatic potential, Φ dia is electrostatic potential which can drive EB drift velocity. v E,dia is EB drift velocity which can balance ion diamagnetic drift velocity. η is resistivity, η H is hyperresistivity. In fact, In Eq. (5) b b E,dia D pol but ion diamagnetic drift velocity v D balances EB drift velocity v E,dia, so b v pol. To meet the conditions of incompressible fluid, the poloidal rotation velocity is written artificially v = v v v, v pol =CBp () Bp is poloidal magnetic field. Referring to the curve of poloidal rotation velocity in the Ref. [11], we define S. Here, ais sep ais C C 1 tanh C is normalized radial coordinate with poloidal magnetic flues, Ψ ais and Ψ sep are poloidal magnetic flues at the magnetic ais and separatri, respectively. controls the location of the shear of poloidal rotation. C controls the amplitude of the poloidal rotation, C S controls the shear of the poloidal rotation. 3 SIMULATION RESULTS We solve the threefield reduced MHD equations with BOUT++ code. In our simulations, the major radius on magnetic ais is Rais=3.9m. The magnetic field on ais is B =1.9T. In order to study the influence of the poloidal rotation and the shear of the poloidal rotation respectively, we give two different profiles of poloidal rotation velocity as shown in Fig. 1 and Fig (a) (b) FIG. 1. C S =, PROFILES OF THE POLOIDAL ROTATION VELOCITY FOR DIFFERENT C. (A), (B), (C) CORRESPOND TO C = , 1.5 1, , RESPECTIVELY (c)
3 Growth Rate (/ a ) (a) (d) (b) FIG.. C = , PROFILES OF THE POLOIDAL ROTATION VELOCITY FOR DIFFERENT C S. (A), (B), (C), (D), (E) CORRESPOND TO C S =, 1, 15,, 5, RESPECTIVELY. In Fig. 1, C S = is invariant, the shear of the poloidal rotation is small and can be ignored, but the velocity gradient of the poloidal rotation will change with the amplitude of the poloidal rotation. In Fig., C = is invariant, the shear of the poloidal rotation becomes large when C S is increased. In Fig. 3 and Fig., when the poloidal rotation is zero, the simulation result is consistent with the result in the Ref. [7]. In Fig. 3, the results show that the poloidal rotation increases the growth rates of PB modes when the poloidal rotation is large enough and the shear of the poloidal rotation can be considered to be ignored. In Fig., the results show the growth rates of highn PB modes is reduced when the shear of the poloidal rotation is increased (e) without poloidal rotation poloidal rotation C = poloidal rotation C =1.5 1 poloidal rotation C = FIG. 3. C S =, GROWTH RATES OF PB MODE VERSUS TOROIDAL MODE NUMBER N. THE GROWTH RATES ARE NORMALIZED TO THE ALFVEN FREQUENCY Ω Α. THE CURVES OF THE DIFFERENT COLORS CORRESPOND TO DIFFERENT C Toroidal Mode Number(n) (c) 1 1
4 Growth Rate (/ a ) without poloidal rotation poloidal rotation Cs= poloidal rotation Cs=1 poloidal rotation Cs=15 poloidal rotation Cs= poloidal rotation Cs=5 FIG.. C = , GROWTH RATES OF PB MODE VERSUS TOROIDAL MODE NUMBER N. THE GROWTH RATES ARE NORMALIZED TO THE ALFVEN FREQUENCY Ω Α. THE CURVES OF THE DIFFERENT COLORS CORRESPOND TO THE DIFFERENT SHEAR PARAMETER C S. SUMMARY The current results show the poloidal rotation of the plasma has a great influence on PB modes. When the poloidal rotation of the plasma is large enough and the shear is small, the poloidal rotation of the plasma destabilizes PB modes. When the shear of the poloidal rotation becomes large, the poloidal rotation of the plasma can stabilize highn PB modes. In our simulations, when C S = is invariant, the velocity gradient of the poloidal rotation will change with the amplitude of the poloidal rotation, and the velocity gradient will lead to KelvinHelmholtz instability. The destabilizing effect of the poloidal rotation on PB modes is caused by the KelvinHelmholtz instability. The quantitative relationship between the eternal driving source and the poloidal rotation of the plasma and the selfconsistent association with the plasma parameters are still not clear, so in this paper the formula of the poloidal rotation velocity has not yet reflected the selfconsistent relationship between the poloidal rotation and the actual driving source as well as the plasma parameters. However, we can change the amplitude of the poloidal rotation and the shear of the poloidal rotation and the location of the shear by this formula. So our study on the influence of the poloidal rotation of the plasma on PB modes is based on the assumption that the poloidal rotation of the plasma is eisted in the Tokamak. In fact, the poloidal rotation in our paper is probable to be derived as mentioned in the section Introduction. The selfconsistent relationship between the poloidal rotation and the actual driving source as well as the plasma parameters will be considered in a followup study. ACKNOWLEDGEMENTS The authors wish to thank X. Q. Xu, B.D. Dudson, and M.V. Umansky, for their contributions to BOUT++ framework. The authors also wish to thank C. J. Tang and S. Y. Chen for fruitful discussions. This work was supported by Chinese National Fusion Project for ITER Grant No. 13GB17, and by the National Natural Science Foundation of China Grant No REFERENCES [1] Connor J W, Hastie R J, Wilson H R and Miller R L. Magnetohydrodynamic stability of tokamak edge plasmas [J]. Physics of Plasmas, 199, 5: [] Snyder P B, Wilson H R, Ferron J R, et al. Edge localized modes and the pedestal: A model based on coupled peeling ballooning modes [J]. Physics of Plasmas,, 9: Toroidal Mode Number(n) [3] Snyder P B, Wilson H R, and Xu X Q. Progress in the peelingballooning model of edge localized modes: Numerical studies of nonlinear dynamicsa [J]. Physics of Plasmas, 5,
5 [] Oyama N, Sakamoto Y, Isayama A, et al. Energy loss for grassy ELMs and effects of plasma rotation on the ELM characteristics in JTU [J]. Nuclear Fusion, 5, 5: 711 [5] Wilson H R, Cowley S C, Kirk A, et al. Magnetohydrodynamic stability of the Hmode transport barrier as a model for edge localized modes: an overview [J]. Plasma Physics and Controlled Fusion,, : A71A [] Snyder P B, Burrell K H, Wilson H R, et al. Stability and dynamics of the edge pedestal in the low collisionality regime: physics mechanisms for steadystate ELMfree operation [J]. Nuclear Fusion, 7, 7: 91 9 [7] Xu X Q, Dudson B, Snyder P B, et al. Nonlinear Simulations of PeelingBallooning Modes with Anomalous Electron Viscosity and their Role in Edge Localized Mode Crashes [J]. Physical Review Letters, 1, [] Xi P W, Xu X Q, Wang X G, et al. Influence of equilibrium shear flow on peelingballooning instability and edge localized mode crash [J]. Physics of Plasmas, 1, [9] 刘才根, 钱尚介, 万华明. 电子回旋波驱动的托卡马克芯部等离子体极向旋转 [J]. 物理学报,199,7:1515 [1] 陈程远. HLA 装置上超声分子束注入的发展及其相关物理研究 [D]. 博士学位论文 : 核工业西南物理研究院,1 [11] Hinton F L, Kim J, Kim Y B, et al. Poloidal Rotation near the Edge of a Tokamak Plasma in H Mode [J]. Physical Review Letters, 199, 7: 11 Authors 1 Yiyu Xiong(1991), master student, major research interests: plasma physics theory
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