Hostname: page-component-5cf477f64f-tx7qf Total loading time: 0 Render date: 2025-04-03T09:34:50.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear physics and energetic particle transport features of the beam–plasma instability

Part of: Complex Plasma Phenomena

Published online by Cambridge University Press:  17 August 2015

Nakia Carlevaro ,
Matteo V. Falessi ,
Giovanni Montani  and
Fulvio Zonca
Nakia Carlevaro*
Affiliation:
ENEA for EUROfusion, Via E. Fermi, 45 (00044) Frascati (Roma), Italy
Matteo V. Falessi
Affiliation:
Department of Physics, ‘RomaTre’ University, Via della Vasca Navale, 84 (00146) Roma, Italy
Giovanni Montani
Affiliation:
ENEA for EUROfusion, Via E. Fermi, 45 (00044) Frascati (Roma), Italy Department of Physics, ‘Sapienza’ University of Rome, P.le Aldo Moro, 5 (00185) Roma, Italy
Fulvio Zonca
Affiliation:
ENEA for EUROfusion, Via E. Fermi, 45 (00044) Frascati (Roma), Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study transport features of a one-dimensional beam–plasma system in the presence of multiple resonances. As a model description of the general problem of a warm energetic particle beam, we assume $n$ cold supra-thermal beams and investigate the self-consistent evolution in the presence of the complete spectrum of nearly degenerate Langmuir modes. A qualitative transport estimation is obtained by computing the Lagrangian Coherent Structures of the system on given temporal scales. This leads to the splitting of the phase space into regions where the local transport processes are relatively faster. The general theoretical framework is applied to the case of the nonlinear dynamics of two cold beams, for which numerical simulation results are illustrated and analysed.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 5 , October 2015 , 495810515
Copyright
© Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Al’Tshul’, L. M. & Karpman, V. I. 1966 Theory of nonlinear oscillations in a collisionless plasma. Sov. Phys. JETP 22 (2), 361–369.Google Scholar
Antoni, M., Elskens, Y. & Escande, D. F. 1998 Explicit reduction of n-body dynamics to self-consistent particle–wave interaction. Phys. Plasmas 5 (4), 841–852.Google Scholar
Antoniazzi, A., DeNinno, G., Fanelli, D., Guarino, A. & Ruffo, S. 2005 Wave–particle interaction: from plasma physics to the free-electron laser. J. Phys.: Conf. Ser. 7, 143–153.Google Scholar
Antoniazzi, A., Elskens, Y., Fanelli, D. & Ruffo, S. 2006 Statistical mechanics and Vlasov equation allow for a simplified hamiltonian description of single-pass free electron laser saturated dynamics. Eur. Phys. J. B 50 (4), 603–611.Google Scholar
Antoniazzi, A., Johal, R. S., Fanelli, D. & Ruffo, S. 2008 On the origin of quasi-stationary states in models of wave particle interaction. Commun. Nonlinear Sci Numer. Simul. 13, 2–10.Google Scholar
Berk, H. L. & Breizman, B. N. 1990a Saturation of a single mode driven by an energetic injected beam. i. Plasma wave problem. Phys. Fluids B 2 (9), 2226–2234.Google Scholar
Berk, H. L. & Breizman, B. N. 1990b Saturation of a single mode driven by an energetic injected beam. ii. Electrostatic ‘universal’ destabilization mechanism. Phys. Fluids B 2 (9), 2235–2245.Google Scholar
Berk, H. L. & Breizman, B. N. 1990c Saturation of a single mode driven by an energetic injected beam. iii. Alfvén wave problem. Phys. Fluids B 2 (9), 2246–2252.Google Scholar
Berk, H. L., Breizman, B. N., Fitzpatrick, J., Pekker, M. S., Wong, H. V. & Wong, K. L. 1996a Nonlinear response of driven systems in weak turbulence theory. Phys. Plasmas 3 (5), 1827–1838.Google Scholar
Berk, H. L., Breizman, B. N., Fitzpatrick, J. & Wong, H. V. 1995a Line broadened quasi-linear burst model [fusion plasma]. Nucl. Fusion 35 (12), 1661–1668.Google Scholar
Berk, H. L., Breizman, B. N. & Pekker, M. 1995b Numerical simulation of bump-on-tail instability with source and sink. Phys. Plasmas 2 (8), 3007–3016.CrossRefGoogle Scholar
Berk, H. L., Breizman, B. N. & Pekker, M. 1996b Nonlinear dynamics of a driven mode near marginal stability. Phys. Rev. Lett. 76 (8), 1256–1259.CrossRefGoogle ScholarPubMed
Berk, H. L., Breizman, B. N. & Pekker, M. S. 1997 Nonlinear theory of kinetic instabilities near threshold. Plasma Phys. Rep. 23 (9), 778–788.Google Scholar
Borgogno, D., Grasso, D., Pegoraro, F. & Schep, T. J. 2008 Stable and unstable invariant manifolds in a partially chaotic magnetic configuration generated by nonlinear reconnection. Phys. Plasmas 15 (10), 102308.Google Scholar
Borgogno, D., Grasso, D., Pegoraro, F. & Schep, T. J. 2011a Barriers in the transition to global chaos in collisionless magnetic reconnection. i. Ridges of the finite time Lyapunov exponent field. Phys. Plasmas 18 (10), 102307.Google Scholar
Borgogno, D., Grasso, D., Pegoraro, F. & Schep, T. J. 2011b Barriers in the transition to global chaos in collisionless magnetic reconnection. ii. Field line spectroscopy. Phys. Plasmas 18 (10), 102308.Google Scholar
Breizman, B. 2011 Nonlinear consequences of energetic particle instabilities. Fusion Sci. Technol. 59 (3), 549–560.Google Scholar
Breizman, B. N., Berk, H. L., Pekker, M. S., Porcelli, F., Stupakov, G. V. & Wong, K. L. 1997 Critical nonlinear phenomena for kinetic instabilities near threshold. Phys. Plasmas 4 (5), 1559–1568.Google Scholar
Breizman, B. N., Berk, H. L. & Ye, H. 1993 Collective transport of alpha particles due to Alfvén wave instability. Phys. Fluids B 5 (9), 3217–3226.Google Scholar
Breizman, B. N. & Sharapov, S. E. 2011 Major minority: energetic particles in fusion plasmas. Plasma Phys. Control. Fusion 53 (5), 054001.Google Scholar
Briguglio, S., Wang, X., Zonca, F., Vlad, G., Fogaccia, G., Di Troia, C. & Fusco, V. 2014 Analysis of the nonlinear behavior of shear-Alfvén modes in tokamaks based on hamiltonian mapping techniques. Phys. Plasmas 21 (11), 112301.Google Scholar
Carlevaro, N., Fanelli, D., Garbet, X., Ghendrih, P., Montani, G. & Pettini, M. 2014 Beam–plasma instability and fast particles: the Lynden–Bell approach. Plasma Phys. Control. Fusion 56 (3), 035013.Google Scholar
Chen, L. & Zonca, F. 2007 Theory of Alfvén waves and energetic particle physics in burning plasmas. Nucl. Fusion 47 (10), S727–S734.Google Scholar
Chen, L. & Zonca, F. 2013 On nonlinear physics of shear Alfvén waves. Phys. Plasmas 20 (5), 055402.Google Scholar
Chen, L. & Zonca, F. 2015 Physics of Alfvén waves and energetic particles in burning plasmas. Rev. Mod. Phys. (submitted).Google Scholar
Chian, A., Rempel, E. L., Aulanier, G., Schmieder, B., Shadden, S. C., Welsch, B. T. & Yeates, A. R. 2014 Detection of coherent structures in photospheric turbulent flows. Astrophys. J. 786 (1), 51.Google Scholar
Chirikov, B. V. 1979 A universal instability of many-dimensional oscillator systems. Phys. Rep. 52 (5), 263–379.Google Scholar
Deneef, P. 1975 Two waves on a beam–plasma system. Phys. Fluids 18, 1209–1212.Google Scholar
Elskens, Y. & Escande, D. F. 2003 Microscopic Dynamics of Plasmas Chaos. Taylor & Francis.CrossRefGoogle Scholar
Esarey, E., Sprangle, P., Krall, J. & Ting, A. 1996 Overview of plasma-based accelerator concepts. IEEE Trans. Plasma Sci. 24 (2), 252–288.Google Scholar
Escande, D. F. & Elskens, Y.2008 Self-consistency vanishes in the plateau regime of the bump-on-tail instability. arXiv:0807.1839.Google Scholar
Evstatiev, E. G., Horton, W. & Morrison, P. J. 2003 Multiwave model for plasma–wave interaction. Phys. Plasmas 10 (10), 4090–4094.Google Scholar
Falessi, M. V., Pegoraro, F. & Schep, T. J. 2015 Lagrangian coherent structures and plasma transport processes. J. Plasma Phys. 81 (5), 495810505.Google Scholar
Farina, D., Casagrande, F., Colombo, U. & Pozzoli, R. 1994 Hamiltonian analysis of the transition to the high-gain regime in a compton free-electron-laser amplifier. Phys. Rev. E 49 (2), 1603–1609.Google Scholar
Farina, D. & Pozzoli, R. 2004 Large-amplitude oscillations and chaos in a hamiltonian plasma system with many degrees of freedom. Phys. Rev. E 70 (3), 036407.Google Scholar
Fasoli, A., Gormenzano, C., Berk, H. L., Breizman, B., Briguglio, S., Darrow, D. S., Gorelenkov, N., Heidbrink, W. W., Jaun, A., Konovalov, S. V., Nazikian, R., Noterdaeme, J.-M., Sharapov, S., Shinohara, K., Testa, D., Tobita, K., Todo, Y., Vlad, G. & Zonca, F. 2007 Chapter 5: physics of energetic ions. Nucl. Fusion 47 (6), S264–S284.Google Scholar
Firpo, M.-C. & Elskens, Y. 2000 Phase transition in the collisionless damping regime for wave–particle interaction. Phys. Rev. Lett. 84 (15), 3318–3321.Google Scholar
Garth, C., Gerhardt, F., Tricoche, X. & Hagen, H. 2007 Efficient computation and visualization of coherent structures in fluid flow applications. IEEE Trans. Vis. Comput. Graphics 13 (6), 1464–1471.Google Scholar
Ghantous, K., Berk, H. L. & Gorelenkov, N. N. 2014 Comparing the line broadened quasilinear model to Vlasov code. Phys. Plasmas 21 (3), 032119.Google Scholar
Ghantous, K., Gorelenkov, N. N., Berk, H. L., Heidbrink, W. W. & Van Zeeland, M. A. 2012 1.5d quasilinear model and its application on beams interacting with Alfvén eigenmodes in diii-d. Phys. Plasmas 19 (9), 092511.Google Scholar
Gorelenkov, N. N., Pinches, S. D. & Toi, K. 2014 Energetic particle physics in fusion research in preparation for burning plasma experiments. Nucl. Fusion 54 (12), 125001.Google Scholar
Haller, G. 2001 Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149 (4), 248–277.Google Scholar
Haller, G. 2011 A variational theory of hyperbolic lagrangian coherent structures. Physica D 240 (7), 574–598.Google Scholar
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47 (1), 137–162.Google Scholar
Heidbrink, W. W. 2008 Basic physics of Alfvén instabilities driven by energetic particles in toroidally confined plasmas. Phys. Plasmas 15 (5), 055501.CrossRefGoogle Scholar
Kasten, J., Petz, C., Hotz, I., Hege, H.-C., Noack, B. R. & Tadmor, G. 2010 Lagrangian feature extraction of the cylinder wake. Phys. Fluids 22 (9), 091108.Google Scholar
Keinigs, R. & Jones, M. E. 1987 Two-dimensional dynamics of the plasma wakefield accelerator. Phys. Fluids 30 (1), 252–263.Google Scholar
Klimontovich, Yu. L. 1967 The Statistical Theory of Non-Equilibrium Processes in a Plasma. MIT Press.Google Scholar
Krafft, C. & Volokitin, A. 2014 Hamiltonian models for resonant wave–particle interaction processes in magnetized and inhomogeneous plasmas. Eur. Phys. J. D 68 (12), 370.Google Scholar
Lacina, J., Krlín, L. & Körbel, S. 1976 Effect of beam density and of higher harmonics on beam–plasma interaction. Plasma Phys. 18, 471–483.Google Scholar
Lauber, P. 2013 Super-thermal particles in hot plasmas – kinetic models, numerical solution strategies, and comparison to tokamak experiments. Phys. Rep. 533 (2), 33–68.Google Scholar
Lekien, F. & Ross, S. D. 2010 The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds. Chaos 20 (1), 017505.Google Scholar
Leoncini, X. & Zaslavsky, G. M. 2002 Jets, stickiness, and anomalous transport. Phys. Rev. E 65 (4), 046216.Google Scholar
Lesur, M. & Idomura, Y. 2012 Nonlinear categorization of the energetic-beam-driven instability with drag and diffusion. Nucl. Fusion 52 (9), 094004.Google Scholar
Lesur, M., Idomura, Y. & Garbet, X. 2009 Fully nonlinear features of the energetic beam-driven instability. Phys. Plasmas 16 (9), 092305.Google Scholar
Lesur, M., Idomura, Y., Shinohara, K., Garbet, X.& JT-60 Team 2010 Spectroscopic determination of kinetic parameters for frequency sweeping Alfvén eigenmodes. Phys. Plasmas 17 (12), 122311.Google Scholar
Levin, M. B., Lyubarskiǐ, M. G., Onishchenko, I. N., Shapiro, V. D. & Shevchenko, V. I. 1972 Contribution to the nonlinear theory of kinetic instability of an electron beam in plasma. Sov. Phys. JETP 35 (5), 898–901.Google Scholar
Lifshitz, E. M. & Pitaevskii, L. P. 1976 Course of Theoretical Physics, Volume 10: Physical Kinetics. Butterworth-Heinemann.Google Scholar
Lilley, M. K. & Breizman, B. N. 2012 Convective transport of fast particles in dissipative plasmas near an instability threshold. Nucl. Fusion 52 (9), 094002.Google Scholar
Lilley, M. K. & Nyqvist, R. M. 2014 Formation of phase space holes and clumps. Phys. Rev. Lett. 112 (15), 155002.Google Scholar
Litos, M., Adli, E., An, W., Clarke, C. I., Clayton, C. E., Corde, S., Delahaye, J. P., England, R. J., Fisher, A. S., Frederico, J., Gessner, S., Green, S. Z., Hogan, M. J., Joshi, C., Lu, W., Marsh, K. A., Mori, W. B., Muggli, P., Vafaei-Najafabadi, N., Walz, D., White, G., Wu, Z., Yakimenko, V. & Yocky, G. 2014 High-efficiency acceleration of an electron beam in a plasma wakefield accelerator. Nature 515, 92–95.Google Scholar
Malhotra, N. & Wiggins, S. 1998 Geometric structures, lobe dynamics, and Lagrangian transport in flows with aperiodic time-dependence, with applications to Rossby wave flow. J. Nonlinear Sci. 8 (4), 401–456.Google Scholar
Matsiborko, N. G., Onishchenko, I. N., Shapiro, V. D. & Shevchenko, V. I. 1972 On non-linear theory of instability of a mono-energetic electron beam in plasma. Plasma Phys. 14 (6), 591–600.Google Scholar
Mynick, H. E. & Kaufman, A. N. 1978 Soluble theory of nonlinear beam–plasma interaction. Phys. Fluids 21, 653–663.Google Scholar
O’Neil, T. M. & Malmberg, J. H. 1968 Transition of the dispersion roots from beam-type to Landau-type solutions. Phys. Fluids 11 (8), 1754–1760.Google Scholar
O’Neil, T. M., Winfrey, J. H. & Malmberg, J. H. 1971 Nonlinear interaction of a small cold beam and a plasma. Phys. Fluids 14 (6), 1204–1212.CrossRefGoogle Scholar
Onishchenko, I. N., Linetskii, A. R., Matsiborko, N. G., Shapiro, V. D. & Shevchenko, V. I. 1970 Contribution to the nonlinear theory of excitation of a monochromatic plasma wave by an electron beam. JETP Lett. 12 (8), 281–285.Google Scholar
Pascucci, V., Tricoche, X., Hagen, H. & Tierny, J. 2010 Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications. Springer.Google Scholar
Peacock, T. & Haller, G. 2013 Lagrangian coherent structures: the hidden skeleton of fluid flows. Phys. Today 66 (2), 41–47.Google Scholar
Pechhacker, R. & Tsiklauri, D. 2014 Three-dimensional particle-in-cell simulation of electron acceleration by langmuir waves in an inhomogeneous plasma. Phys. Plasmas 21 (1), 012903.Google Scholar
Pinches, S. D., Chapman, I. T., Lauber, P. W., Oliver, H. J. C., Sharapov, S. E., Shinohara, K. & Tani, K. 2015 Energetic ions in iter plasmas. Phys. Plasmas 22 (2), 021807.Google Scholar
Schneller, M., Lauber, P., Bilato, R., García-Muñoz, M., Brüdgam, M., Günter, S.& the ASDEX Upgrade Team 2013 Multi-mode Alfvénic fast particle transport and losses: numerical versus experimental observation. Nucl. Fusion 53 (12), 123003.Google Scholar
Schneller, M., Lauber, P., Brüdgam, M., Pinches, S. D. & Günter, S. 2012 Double-resonant fast particle–wave interaction. Nucl. Fusion 52 (10), 103019.Google Scholar
Senatore, C. & Ross, S. D. 2011 Detection and characterization of transport barriers in complex flows via ridge extraction of the finite time Lyapunov exponent field. Intl J. Numer. Meth. Engng 86 (9), 1163–1174.Google Scholar
Shadden, S. C., Lekien, F. & Marsden, J. E. 2005 Definition and properties of lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D 212 (3–4), 271–304.Google Scholar
Shapiro, V. D. 1963 Nonlinear theory of the interaction of a monoenergetic beam with a plasma. Sov. Phys. JETP 17 (2), 416–423.Google Scholar
Shapiro, V. D. & Shevchenko, V. I. 1971 Contribution to the nonlinear theory of relaxation of a ‘monoenergetic’ beam in a plasma. Sov. Phys. JETP 33 (3), 555–561.Google Scholar
Shoucri, M. 2010 Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas. Nova Science.Google Scholar
Smith, G. R. & Pereira, N. R. 1978 Phase-locked particle motion in a large-amplitude plasma wave. Phys. Fluids 21 (12), 2253–2262.Google Scholar
Tang, W., Mathur, M., Haller, G., Hahn, D. C. & Ruggiero, F. H. 2010 Lagrangian coherent structures near a subtropical jet stream. J. Atmos. Sci. 67 (7), 2307–2319.Google Scholar
Tennyson, J. L., Meiss, J. D. & Morrison, P. J. 1994 Self-consistent chaos in the beam–plasma instability. Physica D 71 (1–2), 1–17.Google Scholar
Thompson, J. R. 1971 Nonlinear evolution of collisionless electron beam–plasma systems. Phys. Fluids 14 (7), 1532–1541.Google Scholar
Vlad, G., Briguglio, S., Fogaccia, G., Zonca, F., Fusco, V. & Wang, X. 2013 Electron fishbone simulations in tokamak equilibria using XHMGC. Nucl. Fusion 53 (8), 083008.Google Scholar
Volokitin, A. & Krafft, C. 2012 Velocity diffusion in plasma waves excited by electron beams. Plasma Phys. Control. Fusion 54 (8), 085002.Google Scholar
Voth, G. A., Haller, G. & Gollub, J. P. 2002 Experimental measurements of stretching fields in fluid mixing. Phys. Rev. Lett. 88 (25), 254501.Google Scholar
Wang, X., Briguglio, S., Chen, L., Di Troia, C., Fogaccia, G., Vlad, G. & Zonca, F. 2011 An extended hybrid magnetohydrodynamics gyrokinetic model for numerical simulation of shear Alfvén waves in burning plasmas. Phys. Plasmas 18 (5), 052504.Google Scholar
Wang, X., Briguglio, S., Chen, L., Di Troia, C., Fogaccia, G., Vlad, G. & Zonca, F. 2012 Nonlinear dynamics of beta-induced Alfvén eigenmode driven by energetic particles. Phys. Rev. E 86 (4), 045401(R).Google Scholar
Wong, H. V. & Berk, H. L. 1998 Growth and saturation of toroidal Alfvén eigenmode modes destabilized by ion cyclotron range of frequency produced tails. Phys. Plasmas 5 (7), 2781–2796.Google Scholar
Zaslavsky, A., Krafft, C., Gorbunov, L. & Volokitin, A. 2008 Wave–particle interaction at double resonance. Phys. Rev. E 77 (5), 056407.Google Scholar
Zonca, F., Chen, L., Briguglio, S., Fogaccia, G., Milovanov, A. V., Qiu, Z., Vlad, G. & Wang, X. 2015a Energetic particles and multi-scale dynamics in fusion plasmas. Plasma Phys. Control. Fusion 57 (1), 014024.Google Scholar
Zonca, F., Chen, L., Briguglio, S., Fogaccia, G., Vlad, G. & Wang, X. 2015b Nonlinear dynamics of phase space zonal structures and energetic particle physics in fusion plasmas. New J. Phys. 17 (1), 013052.Google Scholar