Plasma and Fusion Research

Volume 16, 1403087 (2021)

Regular Articles

On the Effect of Beating during Nonlinear Frequency Chirping
Andreas BIERWAGE, Roscoe B. WHITE1) and Vinícius N. DUARTE1)
National Institutes for Quantum and Radiological Science and Technology (QST), Rokkasho Fusion Institute, Aomori 039-3212, and Naka Fusion Institute, Ibaraki 311-0193, Japan
Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA
(Received 22 March 2021 / Accepted 19 May 2021 / Published 12 July 2021)


Spectroscopic analyses of energetic particle (EP) driven bursts of MHD fluctuations in magnetically confined plasmas often exhibit chirps that occur simultaneously in groups of two or more. While the superposition of oscillations at multiple frequencies necessarily causes beating in the signal acquired by a localized external probe, self-consistent hybrid simulations of chirping EP modes in a JT-60U tokamak plasma have demonstrated the possibility of global beating, where the mode's electromagnetic field vanishes globally between beats and reappears with opposite phase [Bierwage et al., Nucl. Fusion 57, 016036 (2017)]. This implies that there can be a single coherent field mode that oscillates at multiple frequencies simultaneously when it is resonantly driven by multiple density waves in EP phase space. Conversely, this means that the EP density waves are mutually coupled and interfere with each other via the jointly driven field, a mechanism ignored in some theories of chirping. In this thesis-style treatise, we study the role of field pulsations in general and beating in particular using the Hamiltonian guiding center orbit-following code ORBIT with a reduced wave-particle interaction model in realistic geometry. Beating is found to drive the evolution of EP phase space structures. A key mechanism is the pulsation of effective phase space islands combined with the alternation of their effective O- and X-points due to phase jumps between each beat. Observations: (1) Beating causes density wave fronts to advance radially in a pulsed manner and the resulting chirps become staircase-like. (2) The pulsations facilitate convective transfer of material between neighboring layers of phase space density waves. On the one hand, this may inhibit the early detachment of solitary phase space vortices. On the other hand, it facilitates the accumulation of hole and clump fragments into larger structures. (3) Long-range chirping is observed when massive holes or clumps detach and drift away from the turbulent belt around the seed resonance. It is remarkable that the detached vortices remain robust and, on average, maintain their concentric nested layers while being visibly perturbed by the field's continued beating.


tokamak, fast particle, Hamiltonian guiding center simulation, phase space dynamics, chirping

DOI: 10.1585/pfr.16.1403087


  • [1] L. Chen and F. Zonca, Physics of Alfvén waves and energetic particles in burning plasmas, Rev. Mod. Phys. 88, 15008 (2016).
  • [2] Y. Todo, Introduction to the interaction between energetic particles and Alfvén eigenmodes in toroidal plasmas, Rev. Mod. Plasma Phys. 3, 1 (2019).
  • [3] W.W. Heidbrink and R.B. White, Mechanisms of energetic energetic-particle transport in magnetically confined plasmas, Phys. Plasmas 27, 030901 (2020).
  • [4] F. Zonca, L. Chen, S. Briguglio, G. Fogaccia, G. Vlad and X. Wang, Nonlinear dynamics of phase space zonal structures and energetic particle physics in fusion plasmas, New J. Phys. 17(1), 013052 (2015).
  • [5] G. Meng, N.N. Gorelenkov, V.N. Duarte, H.L. Berk, R.B. White and X.G. Wang, Resonance frequency broadening of wave-particle interaction in tokamaks due to Alfvénic eigenmode, Nucl. Fusion 58(8), 082017 (2018).
  • [6] C.T. Hsu, C.Z. Cheng, P. Helander, D.J. Sigmar and R. White, Particle dynamics in chirped-frequency fluctuations, Phys. Rev. Lett. 72(16), 122506 (1994).
  • [7] R.L. Dewar, Saturation of kinetic plasma instabilities by particle trapping, Phys. Fluids 16(3), 431 (1973).
  • [8] F. Zonca and L. Chen, Destabilization of energetic particle modes by ICRF-induced fast minority ion tails on TFTR, In JAERI-Conf 2000-004 (6th IAEA TCM on EP in Magnetic Confinement Systems, 1999, Naka, Japan), pages 52-56. JAERI, 2000.
  • [9] W.W. Heidbrink, M.A Van Zeeland, M.E. Austin, N.A. Crocker, X. Du, G.R. McKee and D.A. Spong, Stability of beta-induced Alfvén eigenmodes (BAE) in DIII-D, Nucl. Fusion 61(6), 066031 (2021).
  • [10] H.L. Berk, B.N. Breizman and N.V. Petviashvili, Spontaneous hole-clump pair creation in weakly unstable plasmas, Phys. Lett. A 234, 213 (1997), Erratum: Phys. Lett. A 238, 408 (1998).
  • [11] F. Zonca, Private communication concerning a theory of auroral chorus generation, Article in preparation.
  • [12] A. Bierwage, K. Shinohara, Y. Todo, N. Aiba, M. Ishikawa, G. Matsunaga, M. Takechi and M. Yagi, Self-consistent long-time simulation of chirping and beating energetic particle modes in JT-60U plasmas, Nucl. Fusion 57(1), 016036 (2017).
  • [13] R.B. White, V.N. Duarte, N.N. Gorelenkov, E.D. Fredrickson and M. Podesta, Phase-space dynamics of Alfvén mode chirping, Phys. Plasmas 27, 052108 (2020).
  • [14] H.K. Berk and K.V. Roberts, Nonlinear study of Vlasov's equation for a special class of distribution functions, Phys. Fluids 10, 1595 (1967).
  • [15] P. Khain and L. Friedland, A waterbag theory of autoresonant Bernstein-Greene-Kruskal modes, Phys. Plasmas 14(8), 082110 (2007).
  • [16] H. Hezaveh, Z.S. Qu, M.J. Hole and R.L. Dewar, Theoretical description of chirping waves using phase-space waterbags, Plasma Phys. Control. Fusion. 63(6), 065008 (2021).
  • [17] R.B. White and M.S. Chance, Hamiltonian guiding center drift orbit calculation for plasmas of arbitrary cross section, Phys. Fluids 27(10), 2455 (1984).
  • [18] Y. Chen, R.B. White, G.-Y. Fu and R. Nazikian, Numerical study of the nonlinear evolution of toroidicity-induced Alfvén eigenmodes, Phys. Plasmas 6, 226 (1999).
  • [19] R.B. White, The Theory of Toroidally Confined Plasmas, (Imperial College Press, London, 3rd edition, 2014).
  • [20] K.V. Roberts and H.L. Berk, Nonlinear evolution of a two-stream instability, Phys. Rev. Lett. 19(6), 297 (1967).
  • [21] T.P. Armstrong, Numerical studies of the nonlinear Vlasov equation, Phys. Fluids 10(6), 1269 (1967).
  • [22] G. Wang, H.L. Berk, B.N. Breizman and L.-J. Zheng, Frequency chirping in the Alfvén continuum, Nucl. Fusion 58(8), 082014 (2018).
  • [23] I.B. Bernstein, J.M. Greene and M.D. Kruskal, Exact non-linear plasma oscillations, Phys. Rev. 108(3), 546 (1957).
  • [24] H.L. Berk, B.N. Breizman, J. Candy, M. Pekker and N.V. Petviashvili, Spontaneous hole-clump pair creation, Phys. Plasmas 6, 3102 (1999).
  • [25] B.N. Breizman, Nonlinear travelling waves in energetic particle phase space, Nucl. Fusion 50(8), 084014 (2010).
  • [26] R.M. Nyqvist, M.K. Lilley and B.N. Breizman, Adiabatic description of long range frequency sweeping, Nucl. Fusion 52(9), 094020 (2012).
  • [27] R.M. Nyqvist and B.N. Breizman, Modeling of long range frequency sweeping for energetic particle modes, Phys. Plasmas 20(4), 042106 (2013).
  • [28] H. Hezaveh, Z.S. Qu, B. Layden and M.J. Hole, Impact of energetic particle orbits on long range frequency chirping of BGK modes, Nucl. Fusion 57(12), 126010 (2017).
  • [29] H. Hezaveh, Z.S. Qu, B.N. Breizman and M.J. Hole, Long range frequency chirping of Alfvén eigenmodes, Nucl. Fusion 60(5), 056014 (2020).
  • [30] L. Chen, Theory of magnetohydrodynamic instabilities excited by energetic particles in tokamaks, Phys. Plasmas 1(5), 1519 (1994).
  • [31] A. Bierwage and K. Shinohara, Orbit-based analysis of resonant excitations of Alfvén waves in tokamaks, Phys. Plasmas 21(11), 112116 (2014).
  • [32] A. Fasoli, D. Borba, C. Gormezano, R. Heeter, A. Jaun, J. Jacquinot, W. Kerner, Q. King, J.B. Lister, S. Sharapov, D. Start and L. Villard, Alfvén eigenmode experiments in tokamaks and stellarators, Plasma Phys. Control. Fusion, 39, B287 (1997).
  • [33] A. Fasoli, B.N. Breizman, D. Borba, R.F. Heeter, M.S. Pekker and S.E. Sharapov, Nonlinear splitting of fast particle driven waves in a plasma: Observation and theory, Phys. Rev. Lett. 81(25), 5564 (1998).
  • [34] Y. Todo, M.A. Van Zeeland, A. Bierwage, W.W. Heidbrink and M.E. Austin, Validation of comprehensive magnetohydrodynamic hybrid simulations for Alfvén eigenmode induced energetic particle transport in DIII-D plasmas, Nucl. Fusion 55(7), 073020 (2015).
  • [35] Y. Todo, M.A. Van Zeeland and W.W. Heidbrink, Fast ion profile stiffness due to the resonance overlap of multiple Alfvén eigenmodes, Nucl. Fusion 56(11), 112008 (2016).
  • [36] A. Bierwage, K. Shinohara, Y. Todo, N. Aiba,M. Ishikawa, G. Matsunaga, M. Takechi and M. Yagi, Simulations tackle abrupt massive migrations of energetic beam ions in a tokamak plasma, Nature Comms. 9, 3282 (2018).
  • [37] R.B. White, V.N. Duarte, N.N. Gorelenkov, E.D. Fredrickson, M. Podesta and H.L. Berk,Modeling of chirping toroidal Alfvén eigenmodes in NSTX, Phys. Plasmas 26, 092103 (2019).
  • [38] S.D. Pinches, L.C. Appel, J. Candy, S.E. Sharapov, H.L. Berk, D. Borba, B.N. Breizman, T.C. Hender, K.I. Hopcraft, G.T.A. Huysmans and W. Kerner, The HAGIS self-consistent nonlinear wave-particle interaction model, Comp. Phys. Comm. 111, 133 (1998).
  • [39] S.D. Pinches, H.L. Berk, M.P. Gryaznevich, S.E. Sharapov and JET-EFDA Contributors, Spectroscopic determination of the internal amplitude of frequency sweeping TAE, Plasma Phys. Control. Fusion 46(7), S47 (2004).
  • [40] Y. Todo, H.L. Berk and B.N. Breizman, Simulation of intermittent beam ion loss in a Tokamak Fusion Test Reactor experiment, Phys. Plasmas 10(7), 2888 (2003).
  • [41] S. Nishimura, Y. Todo, D.A. Spong, Y. Suzuki and N. Nakajima, Simulation study of Alfvén-eigenmode-induced energetic ion transport in LHD, Plasma Fusion Res. 8(7),2403090 (2013).
  • [42] A. Bierwage and K. Shinohara, Orbit-based analysis of nonlinear energetic ion dynamics in tokamaks. I. Effective mode number profile and resonant frequency tracking, Phys. Plasmas 23(4), 042511 (2016).
  • [43] A. Bierwage and K. Shinohara, Orbit-based analysis of nonlinear energetic ion dynamics in tokamaks. II. Mechanisms for rapid chirping and convective amplification, Phys. Plasmas 23(4), 042512 (2016).
  • [44] T. Wang, Z. Qiu, F. Zonca, S. Briguglio and G. Vlad, Dynamics of reversed shear Alfvén eigenmode and energetic particles during current ramp-up, Nucl. Fusion 60, 126032 (2020).
  • [45] F. Zonca and L. Chen, Resonant damping of toroidicity-induced shear-Alfvén eigenmodes in tokamaks, Phys. Rev. Lett. 68(5), 592 (1992).
  • [46] J.R. Cary and A.J. Brizard, Hamiltonian theory of guiding-center motion, Rev. Mod. Phys. 81, 693 (2009).
  • [47] R.B. White and A. Bierwage, Particle resonances in toroidal fusion devices, Phys. Plasmas 28(3), 032507 (2021).
  • [48] C.T. Hsu and D.J. Sigmar, Alpha-particle losses from toroidicity-induced Alfvén eigenmodes. Part I: Phase-space topology of energetic particle orbits in tokamak plasma, Phys. Fluids B: Plasma Physics 4, 1492 (1992).
  • [49] R.F. Heeter, A.F. Fasoli and S.E. Sharapov, Chaotic regime of Alfvén eigenmode wave-particle interaction, Phys. Rev. Lett. 85(15), 3177 (2000).
  • [50] F. Zonca, S. Briguglio, L. Chen, G. Fogaccia and G. Vlad, Collective effects and self-consistent energetic particle dynamics in advanced tokamaks, In Proc. 19th Fusion Energy Conference, Lyon, France, 14-19 Oct. 2002, pages IAEACSP-19/CD/TH/4-4, Vienna, 2003. IAEA.
  • [51] R.B. White, R.J. Goldston, K. McGuire, A.H. Boozer, D.A. Monticello and W. Park, Theory of mode-induced beam particle loss in tokamaks, Phys. Fluids 26, 2958 (1983).
  • [52] M.J. Hole and L.C. Appel, A modulation model for mode splitting of magnetic perturbations in the Mega Ampere Spherical Tokamak, Plasma Phys. Control. Fusion 51, 045002 (2009).
  • [53] W. Chen, M. Jiang, Y. Xu, P.W. Shi, L.M. Yu, X.T. Ding, Z.B. Shi, X.Q. Ji, D.L. Yu, Y.G. Li, Z.C. Yang, W.L. Zhong, Z.Y. Qiu, J.Q. Li, J.Q. Dong, Q.W. Yang, Yi. Liu, L.W. Yan, M. Xu and X.R. Duan, Experimental observation of multi-scale interactions among kink/tearing modes and high-frequency fluctuations in the HL-2A core NBI plasmas, Nucl. Fusion 57(11), 114003 (2017).
  • [54] J. Zhu, Z.W. Maa, S. Wang and W. Zhang, Nonlinear dynamics of toroidal Alfvén eigenmodes in the presence of tearing modes, Nucl. Fusion 58, 046019 (2018).
  • [55] I. Barth and L. Friedland, Quantum phenomena in a chirped parametric anharmonic oscillator, Phys. Rev. Lett. 113(4), 040403 (2014).
  • [56] V.I. Veksler, A new method of acceleration of relativistic particles, J. Phys. Acad. Sci. U.S.S.R. 9, 153 (1945).
  • [57] E.M. McMillan, The Synchrotron—a proposed high energy particle accelerator, Phys. Rev. 68(1), 143 (1945).
  • [58] B. Chapman, R.O. Dendy, S.C. Chapman, K.G. McClements, G.S. Yun, S.G. Thatipamula and M.H. Kim, Nonlinear wave interactions generate high-harmonic cyclotron emission from fusion-born protons during a KSTAR ELM crash, Nucl. Fusion 58(9), 096027 (2018).
  • [59] G.J. Kramer, B.J. Tobias, A. Turnbull and E.M. Bass, Enhanced radial energy transport induced by radially curved Alfvén eigenmode wavefronts, Nucl. Fusion 59(9), 094001 (2019).
  • [60] G. Meng, Ph. Lauber, Z. Lu and X. Wang, Effects of the non-perturbative mode structure on energetic particle transport, Nucl. Fusion 60, 056017 (2020).
  • [61] F. Zonca, L. Chen and R.A. Santoro, Kinetic theory of low-frequency Alfvén modes in tokamaks, Plasma Phys. Control. Fusion 38, 2011 (1996).
  • [62] F. Zonca, L. Chen, R.A. Santoro and J.Q. Dong, Existence of discrete modes in an unstable shear Alfvén continuous spectrum, Plasma Phys. Control. Fusion 40, 2009 (1998).
  • [63] Ph. Lauber, S. Günter, A. Könies and S.D. Pinches, LIGKA: A linear gyrokinetic code for the description of background kinetic and fast particle effects on the MHD stability in tokamaks, J. Comp. Phys. 226, 447 (2007).
  • [64] A. Bierwage and Ph. Lauber, Gyrokinetic analysis of low-n shear Alfvén and ion sound waves spectra in a high-beta tokamak plasma, Nucl. Fusion 57(11), 116063 (2017).
  • [65] A. Könies, C. Slaby, R. Kleiber, T. Fehér, M. Borchardt and A. Mishchenko, The MHD continuum with a radial electric field, Phys. Plasmas 27(12), 122511 (2020).
  • [66] R.B. White, V.N. Duarte, N.N. Gorelenkov, A. Bierwage, E.D. Fredrickson, M. Podesta and H.L. Berk, Alfvén mode chirping, In Proc. 28th Fusion Energy Conference, Virtual Event, 10-15 May 2021, pages TH/P1-2, Vienna, 2021. IAEA.
  • [67] M.K. Lilley and R.M. Nyquist, Formation of phase space holes and clumps, Phys. Rev. Lett. 112(15), 155002 (2014).
  • [68] A. Bierwage, C. Di Troia, S. Briguglio and G. Vlad, Orbit-based representation of equilibrium distribution functions for low-noise initialization of kinetic simulations of toroidal plasmas, Comp. Phys. Comm. 183, 1107 (2012).
  • [69] Y. Li, J. Razavilar and K.J.R. Liu, A high-resolution technique for multidimensional NMR spectroscopy, IEEE Trans. Biomed. Eng. 45(1), 78 (1998).
  • [70] C. Slaby, S. Äkäslompolo, M. Borchardt, J. Geiger, R. Kleiber, A. Könies, S. Bozhenkov, C. Brandt, A. Dinklage, M. Dreval, O. Ford, G. Fuchert, D. Hartmann, M. Hirsch, U. Höfel, Z. Huang, P. McNeely, N. Pablant, K. Rahbarnia, N. Rust, J. Schilling, A. von Stechow, H. Thomsen and the Wendelstein 7-X team, Investigation of mode activity in NBI-heated experiments of Wendelstein 7-X, Nucl. Fusion 60(11), 112004 (2020).