Characterization and Nonlinear Interaction of Shear Alfvén Waves in the Presence of Strong Tearing Modes in Tokamak Plasmas

Transcription

1 1 EX/P7-7 Characterization and Nonlinear Interaction of Shear Alfvén Waves in the Presence of Strong Tearing Modes in Tokamak Plasmas W. Chen 1, Z. Qiu, X. T. Ding 1, X. S. Xie, L. M. Yu 1, X. Q. Ji 1, J. X. Li 1, Y. G. Li 1, J. Q. Dong 1, Z. B. Shi 1, Y. P. Zhang 1, J. Y. Cao 1, X. M. Song 1, S. D. Song 1, X. Xu 1, Q. W. Yang 1, Yi. Liu 1, L. W. Yan 1, Y. H. Xu 1 and X. R. Duan 1 1 Southwestern Institute of Physics, P.O. Box 43 Chengdu 6141, China Institute for Fusion Theory and Simulation, Zhejiang Univ., Hangzhou 317, China Corresponding Author: chenw@swip.ac.cn Abstract: Experimental observation and theoretical analysis are presented for the nonlinear mode couplings between shear Alfvén wave and magnetic island. In the sub-alfvén frequency range, two kind axi-symmetry magnetic activities with n= have been observed during NBI on HL-A. One kind has been identified, and belongs to EGAM. Another kind is found for the first time, its frequency lies in the range of TAE frequency. The Fourier bicoherence analysis suggests these axi-symmetry modes are generated by the nonlinear mode coupling via the decay process between Alfvén eigenmodes and low-frequency MHD modes. The experimental results indicate that the nonlinear mode coupling is one of mechanisms of the energy cascade in energetic-particle turbulence or Alfvén turbulence. Shear Alfvén waves (SAWs) are very common in magnetized plasmas both in space and laboratory[1][]. In present-day fusion and future burning plasmas, SAWs are easily excited by fast particles and energetic alpha particles (E=3.5MeV) produced by nuclear fusion reactions. SAWs can cause the loss and redistribution of energetic particles, and consequently, degrade plasma performance or damage plasma wall[3]. The physics of SAWs is an intriguing but complex area of research[4]. Meanwhile, it is important of s- tudying axi-symmetry mode and nonlinear mode coupling. The axi-symmetry mode such as zonal flows(zfs) and geodesic acoustic mode (GAM) is associated with the plasma turbulence and confinement[5]. The nonlinear mode coupling can produce coherent mode structures which can provide overlap of wave-particle resonances in the minor radius, and transfer wave energy across different spatial scale[6]. The role of nonlinear mode coupling is generally important in determining the mode excitation, saturation or damping[7][8]. The nonlinear interaction also affects energetic particle redistribution/transport or plasma confinement[9]. The phenomena can occur commonly in the space and Lab plasma, e.g. creating of the kinetic Alfvén waves which are very efficient in the energy and momentum exchange with plasma particles, heating plasma and accelerating particles in various space plasmas[1][11], couplings of various Alfvén eigenmodes (AEs) and energeticparticle modes (EPMs) in Lab plasmas[1][13][14]. The SAWs can interact nonlinearly, and lone wave packets can suffer decay instabilities including the parametric and modulational those[15][16], where the pump decays into a daughter SAW, ion acoustic wave or

2 EX/P7-7 ZFs[17][18][19]. In this letter, we will report the excitation and generation of the two kind axi-symmetry magnetic activities in the presence of strong tearing modes (TMs) during NBI on HL-A, and present that there exists the intense nonlinear interactions between AEs and low-frequency MHD modes, and develop a theory to explain this experimental phenomenon. HL-A is a medium-size tokamak with major/minor radius R/a = 1.65m/.4m. The experiments discussed here were performed in deuterium plasmas with plasma current I p 15 17kA, toroidal field B t T, and safety factor q a at the plasma edge. The line averaged density was detected by a hydrogen cyanide interferometer. The polodial mode-number m is measured using a set of seven Mirnov probes localized in the high field side (HFS) and eleven ones localized in the low field side (LFS). But the toroidal mode-number n is measured using a set of ten Mirnov probes localized in the LFS of the vessel. The neutral beam is tangentially co-injected with a tangency radius of 1.35 m. The acceleration voltage E b and port-through power P NBI of the neutral beam were typically 45kV and 1.M W, respectively. The energetic-ions produced by the NBI destabilize AEs and EPMs on HL-A. In a toroidal plasma, various geometric effects become important in the SAW dynamics, and they can act to open some gaps in the SAW continuum, and discrete Alfvén modes (so-called AEs) which reside in these gaps[]. AEs are excited easily by EPs due to the absence of continuum damping in gaps. A well-known discrete mode is the toroidal Alfvén eigenmode(tae) which are a bound eigenstate existing in the gap opened up due to poloidal symmetry breaking, and the center frequency of TAE-gap is given approximately by ω T AE v A /qr, and the location r of the TAE structure is determined approximately by the condition q(r ) = (m + 1)/n. Another well-known example about discrete modes is the beta-induced Alfvén eigenmode(bae). The BAE is a low frequency mode with parallel wave number k = (n m/q)/r = and frequency ω BAE = (T i /m i ) 1/ (7/4 + T e /T i ) 1/ /R = β 1/ i (7/4 + T e /T i ) 1/ v A /R, which is due to the plasma finite beta effect under the geodesic curvature, and usually believed to be electromagnetic oscillation, and driven by fast particles or pressure gradients. MMfM(kHz) 4 3 ne(1 19 m 3 ) Ip(kA)/5 P NBI (MW) BAE with n=1 EGAM with n= BAE1 with n=-1 TM with n=1 P ECRH (MW) timem(ms) fm(khz) Ip(kA)/5 ne(1 19 m 3 ) P NBI (MW) TAE with n=1 TAE1 with n=-1 n= TM with n= timem(ms) FIG. 1: Typical discharges with axi-symmetry magnetic activities on HL-A. Plasma current, Ip; electron density, ne; ECRH power, P ECRH and NBI power, P NBI. D pattern is spectrogram of Mirnov signal. Lelt col.(shot I, Bt=1.4T) with BAEs (HF, n=±1) and TMs (LF, n=1); Right col.(shot II, Bt=1.39T) with TAEs (HF, n=±1) and TMs (LF, n=1).

3 EX/P7-7 3 FIG. : Toroidal mode numbers of MHD instabilities at different frequency-regions which are obtained by the narrow-band filtering of toroidal Mirnov signals. Lelt col. from shot I; Right col. from shot II. The characterization of SAWs in the presence of strong TMs have been observed on HL-A. In FIG.1 some coherent MHD fluctuations are visible at t 67 77ms and f 5 35kHz for shot I, but for shot II, at t 6 65ms and f 9 16kHz. In shot I and shot II, the frequencies of TMs with m/n=/1 are respectively f 3kHz and f 1kHz. In detail the toroidal mode numbers of MHD instabilities at different frequency-regions are shown in FIG.. It is found the subsequent generation of Alfve nic modes including the co-/counter-propagating AEs and n= axi-symmetry magnetic activities. Meanwhile, the magnetic fluctuation spectrogram indicates that the n= modes are accompanied by strong TM and AEs, and their frequencies comply with fn= mode = fae ft M and fn= mode = fae1 + ft M. The BAEs and EGAM had been investigated in the previous works[1][]. Here we focus on TAEs in the presence of strong TMs. In the toroidal geometry, the linear shear Alfve n continuum and radial eigenfunctions can be obtained by solving the Eq.(1)[3], (Lm + Lm )ϕm + (L1m,m+1 )ϕm+1 + L1m,m 1 ϕm 1 = (Lm+1 + Lm+1 )ϕm+1 + L1m+1,m ϕm = (Lm 1 + Lm 1 )ϕm 1 + L1m 1,m ϕm (1) = (ω km )r r mr (ω km ), L1m,m+1 = ω { r r(ε+ ) r m(m±1) (ε ) where Lm = r r m mα m [ε+(r ) ]m r } km { r r r km±1 r (ε+ )km±1 m[ε+(r ) ] r km±1 } q ( r r ), Lm = r ω 4ε r r + mr [4ε(ε+ )ω + k r α/q ], and all other symbols are standard. Eq.(1) is from the first two terms of eq.(), noting VA = B /(4πρ), B = B (1 (r/r ) cos θ) and

4 EX/P7-7 4 FIG. 3: Plasma density and q-profiles. The solid curves from the inversion of HCN measurements (ne-profile) and TSC simulation (q-profile), and the dot curves are fitting results. k = i l = (nq m)/(qr ), while the translational invariance of parallel mode structure localized at each rational surface is assumed. The Alfvén continuum and radial mode structures have be obtained by employing experimental profiles of ne and q (see FIG.3) in shot II. It is found from FIG.4 that the TAEs with n=1-3 localize in the core plasma, and their frequencies are f 13 15kHz which agree very well with the experimental observations. FIG. 4: Alfvén continuum (a-c) and radial eigenfunctions (e-g) of TAEs with different toroidal mode numbers. Note that the dot fitting curves in the fig.3 are employed in the calculation.

5 5 EX/P7-7 A novel result, which is nonlinear mode couplings among TM, AEs and n= mode, has been observed on HL-A. For studying the nonlinear mode coupling, the squared bicoherence[4] is given by ˆb (f 1, f ) = ˆB XY Z (f 1, f ) / < X(f 1 )Y (f ) >< Z(f 3 ) > with the Fourier bispectral ˆBXY Z (f 1, f ) =< X(f 1 )Y (f )Z (f 3 ) >, f 3 = f 1 ± f and < ˆb (f 1, f ) < 1. Where X(f), Y (f) and Z(f) are the Fourier transform of the time traces of x(t), y(t) and z(t), respectively. The symbol <> denotes the ensemble average over many realizations. It is convenient to represent the contribution of the nonlinear coupling from multiple modes to one mode with the summed squared bicoherence, which is defined as Σb XY Z = Σ f=f 1 ±f ˆb (f 1, f )/N(f). Where N(f) is the numbers of realizations. Fig.5 shows the squared bicoherence and summed squared bicoherence of a poloidal Mirnov signal. It is found the nonlinear interaction between the fundamental n AE = 1 (or n AE = 1 ) AE with f AE and n = 1 TM with f T M at each different time. The following matching conditions are satisfied among these modes, i.e. n T M + n AE = n AE/n= mode and f T M + f AE = f AE/n= mode for TM and AEs. Moreover, the n = ±1 AEs interact with TM further and create a multitude of AEs/n= modes, and f AE/n= mode = (k 1) f T M +f AE, n AE/n= mode = (k 1) n T M + n AE, where k is positive integer. If only considering the fundamental frequencies of TM, AEs and n= modes, their nonlinear interaction relations are f AE f AE1 = f T M, f AE + f AE1 = f n= mode, and other expressions are f AE = f n= mode ± f T M (i.e., n = mode + T M AE) or f n= mode = f AE ± f T M (i.e., AE + T M n = mode). Need to point out here that the direction of wave energy transfer is unknown, and it needs to be assessed. The auto-bicoherence of magnetic fluctuation is shown that f = k f T M and f ± f 1 = k f T M are strong coupling lines. It suggests that the nonlinear mode coupling process between TM and AEs is similar to the coupling of drift-wave turbulence and GAM [5][5] as well as TAE and EPM [1]. FIG. 5: Squared bicoherence ˆb Bθ (f Bθ Bθ 1, f ), self-power spectrum and summed squared bicoherence Σˆb Bθ (f) of Mirnov signal for shot I (top row) and shot II (bottom row). Bθ Bθ The Alfvén continuum structures can be modified by the interaction with low-frequency

6 EX/P7-7 6 magnetic islands which can be considered as cause of nonaxisymmetric distortion of the plasma equilibrium[6]. The field line bending and Alfvén continuum distortion induced by strong TMs maybe provide a large contribution for the nonlinearity. The nonlinear coupling between AMs (ω, k ) and magnetic island (ω is, k is ), can be interpreted within the theoretical framework of modulational instability. The Alfvénic modes driven by energetic particles, may couple to the magnetic island, and generate a series of modes, with the frequency being ω q = ω + ω is and k q = k + k is. Note that these modes, may not satisfy the eigenmode dispersion relation of Alfvén eigenmode, and are thus, quasi-modes. The governing nonlinear equation describing quasi-mode generation by the beating of AE and magnetic island, can be derived from MHD equations. Note that AEs (TAE and RSAE as two examples) are typically localized away from the rational surfaces where resistivity plays crucial role on magnetic island, in our calculation only ideal region contribution of magnetic island is taken into account. Meanwhile, we will ignore J or ω here, i.e., we assume AE and magnetic island are given, and investigate only the nonlinear interaction between them. Here, J and ω are, respectively, equilibrium current and diamagnetic drift frequency, corresponding to the linear drive of tearing mode and Alfvén modes. Due to the frequency separation between AE and magnetic island, i.e. ω is ω A, magnetic island can be treated as a distortion of equilibrium here, and we obtain +( v ) iω c v A c v A δφ + B 1 B δa δφ + δ B is 1 B δa =. () The first two terms of equation () are, respectively, the inertial and field line bending terms, the balance of which gives the SAWs. The third and fourth terms are, respectively, Reynold s and Maxwell s stresses due to nonlinear beating of TAE and TM [8, 7]. For both TAE and TM in ideal region, to the lowest order, we have δe =, thus, δa = ic/ω l δφ. v = (c/b ) b δφ is is the electric drift induced by magnetic island. After some algebra, we have, [ 1 ω ω l + 1 ] va δφ = c ( 1 k,isk ) b δφis B va ω ω δφ is = c 1 (1 k,isk )(k B va ω r,is k θ,t AE k r,t AE k θ,is ) ω is (k k,is )δφ is δφ. For the nonlinear coupling between TM and AE, with 1 k,is k /(ω ω is ), k r,is k θ,t AE k r,t AE k θ,is, and k k,is, the coupling coefficient is finite, i.e., there is finite generation of n is ± n T AE TAE by an n is TM and an n T AE TAE. Namely, in the presence of an m/n=/1 magnetic island, there are a series of Alfvénic modes with toroidal mode numbers being n=...,,1,,-1,-,..., and the frequency between two Alfvénic modes with neighboring n matches the the frequency of the TM with n=1. In summary, the experimental observation and theoretical analysis of a nonlinear interaction between AEs and TMs are presented in a laboratory plasma. The AEs are driven (3)

7 7 EX/P7-7 by energetic particles. The AEs then are modulated by the strong TMs, resulting in the subsequent generation of Alfvénic sidebands including the co-/counter-propagating AEs and n= axi-symmetry magnetic activities. The relevant phenomena are typical examples with respect to multi-scale nonlinear interactions. The n= axi-symmetry modes maybe have a significant effect on plasma transport in the vicinity of the magnetic island[9], and also have a profound regulatory effect on the turbulence around magnetic island[3]. The experimental results indicate that the couplings possibly induce the energy transfer among TMs, AEs and n= modes, and this could be one of mechanisms of the energy cascade in the Alfvén turbulence, and the n= modes may be an energy channeling between different scales, such as macro-, meso- and micro-scale. Specifically, the couplings of SAWs and TMs may be AEs converted to the region of shear Alfvén continuum, thereby facilitating the mode saturation and dissipation. The new findings give a deep insight into the underlying physics mechanism for the excitation of the LF Alfvénic/acoustic fluctuation and ZFs. The authors (C.W. and Q.Z.) are very grateful to the HL-A Group, and thank Prof. L. Chen and Dr. F. Zonca for valuable discussions. This work is supported in part by the ITER-CN under Grants No. 13GB141, 13GB144 and 13GB164. References [1] H. Alfvén Nature 15, 45 (194). [] N. F. Cramer, The physics of Alfvén waves, WILEY-VCH Verlag Berlin GmbH, Berlin (Federal Republic of Germany), 1. [3] A. Fasoli, et al. Nucl. Fusion 47, s64 (7). [4] L. Chen and F. Zonca, Phys. Plasmas, 554 (1). [5] P. H. Diamond et al. Plasma Phys. Control. Fusion 47, R35 (5). [6] B. N. Breizman and S.E. Sharapov, Plasma Phys. Control. Fusion 53, 541 (11). [7] L. Chen et al. Plasma Phys. Control. Fusion 4, 183 (1998). [8] L. Chen and F. Zonca, Nucl. Fusion 47, s77 (7). [9] G. Vlad et al. Nucl. Fusion 49, 754 (9). [1] Y. M. Voitenko, J. Plasma Physics 6, 497 (1998). [11] J. S. Zhao, D. J. Wu and J. Y. Lu, Phys. Plasmas 18, 393 (11). [1] N. A. Crocker et al. Phys. Rev. Lett. 97, 45 (6). [13] N. A. Crocker et al. Phys. Plasmas 16, 4513 (9). [14] M. Podestá et al. Nucl. Fusion 51, 6335 (11). [15] S. Spangler et al. Phys. Plasmas 4, 846 (1997). [16] J. V. Hollweg, J. Geophys. Res. 99, 3431 (1994).

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