Abstract
Global gyrokinetic simulations of reversed shear Alfven eigenmodes (RSAE) excited by energetic particles (EP) in the DIII-D discharge #159,243 find that self-generated zonal flows dominate the RSAE nonlinear saturation. Effects of Coulomb collisions are negligible since the effective EP collision frequency is much smaller than the RSAE linear growth rate. Nonlinear interactions of multiple unstable RSAEs lower the saturation amplitude and resulting EP transport, and nonlinearly generate toroidal Alfven eigenmodes (TAE) and ellipticity induced Alfven eigenmodes (EAE). In both single mode and multiple modes simulations, the initial RSAE saturation amplitude and associated EP transport are an order of magnitude higher than the experimental observations, but quickly diminish within 0.1 ms after nonlinear saturation due to the formation of coherent phase space structures that flatten the EP distribution function at resonances. Results from these simulations of the RSAE only show that additional dissipation such as background micro-turbulence is needed to sustain the quasi-steady state RSAE amplitude and EP transport observed in this DIII-D experiment.
Similar content being viewed by others
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
References
M.E. Austin, J. Lohr, Electron cyclotron emission radiometer upgrade on the DIII-D tokamak. Rev. Sci. Inst. 74, 1457 (2003)
E.M. Bass, R.E. Waltz, Gyrokinetic simulations of mesoscale energetic particle-driven alfvénic turbulent transport embedded in microturbulence. Phys. Plasmas 17, 112319 (2010)
H.L. Berk, B.N. Breizman, Saturation of a single mode driven by an energetic injected beam. III Alfvén wave problem. Phys. Fluids B 2, 2246 (1990)
H.L. Berk, B.N. Breizman, H. Ye, Scenarios for the nonlinear evolution of alpha-particle-induced Alfvén wave instability. Phys. Rev. Lett. 68, 3563 (1992)
A. Biancalani et al., Gyrokinetic investigation of Alfvén instabilities in the presence of turbulence. Plasma Phys. Control. Fusion 63, 065009 (2021)
B.N. Breizman, S.E. Sharapov, Major minority: energetic particles in fusion plasmas. Plasma Phys. Control. Fusion 53, 054001 (2011)
A.J. Brizard, T.S. Hahm, Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421 (2007)
G. Brochard, R. Dumont, H. Lütjens, X. Garbet, T. Nicolas, P. Maget, Nonlinear dynamics of the fishbone-induced alpha transport on ITER. Nucl. Fusion 60, 126019 (2020)
G. Brochard et al., Verification and validation of linear gyrokinetic and kinetic-MHD simulations for internal kink instability in DIII-D tokamak. Nucl. Fusion 62, 036021 (2022)
J. Candy, R.E. Waltz, An Eulerian Gyrokinetic-Maxwell Solver. J. Comput. Phys. 186, 545 (2003)
Y. Chen, S.E. Parker, A Δf particle method for gyrokinetic simulations with kinetic electrons and electromagnetic perturbations. J. Comput. Phys. 189, 463 (2003)
L. Chen, F. Zonca, Physics of Alfvén waves and energetic particles in burning plasmas. Rev. Mod. Phys. 88, 015008 (2016)
Y. Chen, T. Munsat, S.E. Parker, W.W. Heidbrink, M.A. Van Zeeland, B.J. Tobias, C.W. Domier, Gyrokinetic simulations of reverse shear Alfvén eigenmodes in DIII-D plasmas. Phys. Plasm. 20, 012109 (2013)
Y. Chen, G.Y. Fu, C. Collins, S. Taimourzadeh, S.E. Parker, Zonal structure effect on the nonlinear saturation of reverse shear Alfven eigenmodes. Phys. Plasm. 25, 032304 (2018)
C.Z. Cheng, M.S. Chance, Low-n shear Alfvén spectra in axisymmetric toroidal plasmas. Phys. Fluids 29, 3695–3701 (1986)
J. Cheng, W. Zhang, Z. Lin, D. Li, C. Dong, J. Cao, Nonlinear co-existence of beta-induced Alfvén eigenmodes and beta-induced Alfvén-acoustic Eigenmodes. Phys Plasm 24, 092516 (2017)
G.J. Choi, P. Liu, X.S. Wei, J.H. Nicolau, G. Dong, W.L. Zhang, Z. Lin, W.W. Heidbrink, T.S. Hahm, Gyrokinetic simulation of low-frequency Alfvénic modes in DIII-D tokamak. Nucl. Fusion 61, 066007 (2021)
M.D.J. Cole, A. Biancalani, A. Bottino, R. Kleiber, A. Könies, A. Mishchenko, Toroidal Alfvén Eigenmodes with nonlinear gyrokinetic and fluid hybrid models. Phys. Plasmas 24, 022508 (2017)
C.S. Collins et al., Observation of critical-gradient behavior in Alfvén-eigenmode-induced fast-ion transport. Phys. Rev. Lett. 116, 095001 (2016)
W. Deng, Z. Lin, I. Holod, Z. Wang, Y. Xiao, H. Zhang, Linear properties of reversed shear Alfvén eigenmodes in the DIII-D tokamak. Nucl. Fusion 52, 043006 (2012a)
W. Deng, Z. Lin, I. Holod, Gyrokinetic simulation model for kinetic magnetohydrodynamic processes in magnetized plasmas. Nucl. Fusion 52, 023005 (2012b)
G. Dong, J. Bao, A. Bhattacharjee, A. Brizard, Z. Lin, P. Porazik, Gyrokinetic particle simulations of the effects of compressional magnetic perturbations on Drift-Alfvenic instabilities in tokamaks. Phys. Plas. 24, 081205 (2017)
V.N. Duarte, H.L. Berk, N.N. Gorelenkov, W.W. Heidbrink, G.J. Kramer, R. Nazikian, D.C. Pace, M. Podestà, B.J. Tobias, M.A. Van Zeeland, Prediction of nonlinear evolution character of energetic-particle-driven instabilities. Nucl. Fusion 57, 054001 (2017)
A. Fasoli, B.N. Breizman, D. Borba, R.F. Heeter, M.S. Pekker, S.E. Sharapov, Nonlinear splitting of fast particle driven waves in a plasma: observation and theory. Phys. Rev. Lett. 81, 5564 (1998)
A. Fasoli et al., Chapter 5: physics of energetic Ions. Nucl. Fusion 47, S264 (2007)
G.Y. Fu, W. Park, Nonlinear hybrid simulation of the toroidicity-induced Alfvén Eigenmode. Phys. Rev. Lett. 74, 1594 (1995)
N.N. Gorelenkov, V.N. Duarte, Microturbulence-mediated route for energetic ion transport and Alfvénic mode amplitude oscillations in tokamaks. Phys. Lett. A 386, 126944 (2021)
N.N. Gorelenkov, S.D. Pinches, K. Toi, Energetic particle physics in fusion research in preparation for burning plasma experiments. Nucl. Fusion 54, 125001 (2014)
W.W. Heidbrink, Basic physics of Alfvén instabilities driven by energetic particles in toroidally confined plasmas. Phys. Plasmas 15, 055501 (2008)
W.W. Heidbrink, R.B. White, Mechanisms of energetic-particle transport in magnetically confined plasmas. Phys. Plasmas 27, 030901 (2020)
W.W. Heidbrink, J.M. Park, M. Murakami, C.C. Petty, C. Holcomb, M.A. Van Zeeland, Evidence for fast-ion transport by microturbulence. Phys. Rev. Lett. 103, 175001 (2009)
W.W. Heidbrink, C.S. Collins, M. Podestà, G.J. Kramer, D.C. Pace, C.C. Petty, L. Stagner, M.A. Van Zeeland, R.B. White, Y.B. Zhu, Fast-Ion transport by Alfvén eigenmodes above a critical gradient threshold. Phys. Plasmas 24, 056109 (2017)
H. Hezaveh, Y. Todo, Z. Qu, B. Breizman, M. Hole, Simulation of convective transport during frequency chirping of a TAE using the MEGA code. Nucl. Fusion 62, 036025 (2022)
F.L. Hinton, R.D. Hazeltine, Theory of plasma transport in toroidal confinement systems. Rev. Mod. Phys. 48, 239 (1976)
W. Horton, Drift waves and transport. Rev. Mod. Phys. 71, 735 (1999)
S. Jolliet, A. Bottino, P. Angelino, R. Hatzky, T.M. Tran, B.F. Mcmillan, O. Sauter, K. Appert, Y. Idomura, L. Villard, A global collisionless PIC code in magnetic coordinates. Comp. Phys. Communicat. 177, 409 (2007)
J. Lang, G.-Y. Fu, Y. Chen, Nonlinear simulation of toroidal Alfvén eigenmode with source and sink. Phys. Plasmas 17, 042309 (2010)
P. Lauber, Super-thermal particles in hot plasmas—kinetic models, numerical solution strategies, and comparison to tokamak experiments. Phys. Rep. 533, 33 (2013)
Z. Lin, W.M. Tang, W.W. Lee, Gyrokinetic particle simulation of neoclassical transport. Phys. Plasmas 2, 14 (1995)
Z. Lin, T.S. Hahm, W.W. Lee, W.M. Tang, R.B. White, Turbulent transport reduction by zonal flows: massively parallel simulations. Science 281, 1835 (1998)
Z. Lin, S. Ethier, T.S. Hahm, W.M. Tang, Size scaling of turbulent transport in magnetically confined plasmas. Phys. Rev. Lett. 88, 195004 (2002)
P. Liu, X. Wei, Z. Lin, G. Brochard, G.J. Choi, W.W. Heidbrink, J.H. Nicolau, G.R. McKee, Regulation of Alfvén eigenmodes by microturbulence in fusion plasmas. Phys. Rev. Lett. 128, 185001 (2022)
J. McClenaghan, Z. Lin, I. Holod, W. Deng, Z. Wang, Verification of gyrokinetic particle simulation of current-driven instability in fusion plasmas i internal kink mode. Phys. Plasmas 21, 122519 (2014)
A. Mishchenko, A. Könies, R. Kleiber, M. Cole, Pullback transformation in gyrokinetic electromagnetic simulations. Phys. Plasmas 21, 092110 (2014)
J.H. Nicolau, G. Choi, J. Fu, P. Liu, X. Wei, Z. Lin, Global Gyrokinetic simulation with kinetic electron for collisionless damping of zonal flow in stellarators. Nucl. Fusion 61, 126041 (2021)
S.E. Parker, W.W. Lee, A fully nonlinear characteristic method for gyrokinetic simulation. Phys. Fluids B 5, 77 (1993)
S.D. Pinches et al., The HAGIS self-consistent nonlinear wave-particle interaction model. Comp. Phys. Communicat. 111, 133 (1998)
D.A. Spong, B.A. Carreras, C.L. Hedrick, Nonlinear evolution of the toroidal Alfvén instability using a gyrofluid model. Phys. Plasmas 1, 1503 (1994)
D.A. Spong, E.M. Bass, W. Deng, W.W. Heidbrink, Z. Lin, B. Tobias, M.A. Van Zeeland, M.E. Austin, C.W. Domier, N.C. Luhmann, Verification and Validation of linear gyrokinetic simulation of Alfvén eigenmodes in the DIII-D tokamak. Phys. Plasmas 19, 082511 (2012)
S. Taimourzadeh et al., Verification and validation of integrated simulation of energetic particles in fusion plasmas. Nucl. Fusion 59, 066006 (2019)
B. Tobias et al., Commissioning of electron cyclotron emission imaging instrument on the DIII-D tokamak and first data. Rev. Sci. Inst. 81, 10D928 (2010)
Y. Todo, Introduction to the Interaction between energetic particles and Alfven eigenmodes in toroidal plasmas. Rev. Mod. Plasma Phys. 3, 1 (2019)
Y. Todo, T. Sato, Linear and nonlinear particle-magnetohydrodynamic simulations of the toroidal Alfvén eigenmode. Phys. Plasmas 5, 1321 (1998)
Y. Todo, H.L. Berk, B.N. Breizman, Simulation of intermittent beam ion loss in a tokamak fusion test reactor experiment. Phys. Plasmas 10, 2888 (2003)
Y. Todo, H.L. Berk, B.N. Breizman, Nonlinear magnetohydrodynamic effects on Alfvén eigenmode evolution and zonal flow generation. Nucl. Fusion 50, 084016 (2010)
M.A. Van Zeeland et al., Alfvén eigenmodes and fast ion transport in negative triangularity DIII-D plasmas. Nucl. Fusion 59, 086028 (2019)
J. Varela, D.A. Spong, L. Garcia, Analysis of Alfvén eigenmode destabilization by energetic particles in large helical device using a landau-closure model. Nucl. Fusion 57, 046018 (2017)
G. Vlad, S. Briguglio, G. Fogaccia, V. Fusco, C. Di Troia, E. Giovannozzi, X. Wang, F. Zonca, Single- n versus multiple- n simulations of Alfvénic modes. Nucl. Fusion 58, 082020 (2018)
Z. Wang, Z. Lin, I. Holod, W.W. Heidbrink, B. Tobias, M. Van Zeeland, M.E. Austin, Radial localization of toroidicity-induced alfvén eigenmodes. Phys. Rev. Lett. 111, 145003 (2013)
H.Y. Wang, I. Holod, Z. Lin, J. Bao, J.Y. Fu, P.F. Liu, J.H. Nicolau, D. Spong, Y. Xiao, Global gyrokinetic particle simulations of microturbulence in W7-X and LHD stellarators. Phys. Plasmas 27, 082305 (2020)
X.Q. Wang et al., Nonlinear simulations of energetic particle-driven instabilities interacting with Alfvén continuum during frequency chirping. Plasma Phys. Control. Fusion 63, 015004 (2021)
H. Wang, P. Liu, Z. Lin, W. Zhang, Linear gyrokinetic simulations of reversed shear Alfvén eigenmodes and ion temperature gradient modes in DIII-D tokamak. Plasma Sci. Technol. 23, 015101 (2021)
R.B. White, The theory of toroidally confined plasmas (Published By Imperial College Press And Distributed By World Scientific Publishing Co, USK, 2006)
Y. Xiao, Z. Lin, Turbulent transport of trapped-electron modes in collisionless plasmas. Phys. Rev. Lett. 103, 085004 (2009)
W. Zhang, Z. Lin, L. Chen, Transport of energetic particles by microturbulence in magnetized plasmas. Phys. Rev. Lett. 101, 095001 (2008)
H.S. Zhang, Z. Lin, I. Holod, Nonlinear frequency oscillation of Alfvén eigenmodes in fusion plasmas. Phys. Rev. Lett. 109, 025001 (2012)
H.S. Zhang, Z. Lin, W. Deng, I. Holod, Z.X. Wang, Y. Xiao, W.L. Zhang, Nonlinear dynamics of beta-induced Alfvén eigenmode in tokamak. Phys Plasmas 20, 012510 (2013)
F. Zonca, L. Chen, Theory on excitations of drift Alfvén waves by energetic particles. II the general fishbone-like dispersion relation. Phys. Plasmas 21, 072121 (2014)
F. Zonca, L. Chen, S. Briguglio, G. Fogaccia, G. Vlad, X. Wang, Nonlinear dynamics of phase space zonal structures and energetic particle physics in fusion plasmas. New J. Phys. 17, 013052 (2015)
Acknowledgements
The authors would like to thank W. W. Heidbrink and G. R. McKee for providing experimental data and for useful discussions. This work was supported by DOE SciDAC ISEP and INCITE, used computing resources at ORNL (DOE Contract DE-AC05-00OR22725) and NERSC (DOE Contract DE-AC02-05CH11231), and experimental data from DIII-D National Fusion Facility (DOE Contract DE-FC02-04ER54698). This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
Funding
U.S. Department of Energy, DE-AC05-00OR22725, Zhihong Lin, DE-AC02-05CH11231, Zhihong Lin.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Liu, P., Wei, X., Lin, Z. et al. Nonlinear gyrokinetic simulations of reversed shear Alfven eigenmodes in DIII-D tokamak. Rev. Mod. Plasma Phys. 7, 15 (2023). https://doi.org/10.1007/s41614-023-00117-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s41614-023-00117-4