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Drift kinetic theory of alpha transport by tokamak perturbations

Part of: Focus on Fusion Invited Contributions from the 18th European Fusion Theory Conference

Published online by Cambridge University Press:  08 March 2021

Elizabeth A. Tolman Open the ORCID record for Elizabeth A. Tolman [Opens in a new window]  and
Peter J. Catto Open the ORCID record for Peter J. Catto [Opens in a new window]
Elizabeth A. Tolman*
Affiliation:
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA02139, USA Institute for Advanced Study, Princeton, NJ08540, USA
Peter J. Catto
Affiliation:
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

Upcoming tokamak experiments fuelled with deuterium and tritium are expected to have large alpha particle populations. Such experiments motivate new attention to the theory of alpha particle confinement and transport. A key topic is the interaction of alpha particles with perturbations to the tokamak fields, including those from ripple and magnetohydrodynamic modes like Alfvén eigenmodes. These perturbations can transport alphas, leading to changed localization of alpha heating, loss of alpha power and damage to device walls. Alpha interaction with these perturbations is often studied with single-particle theory. In contrast, we derive a drift kinetic theory to calculate the alpha heat flux resulting from arbitrary perturbation frequency and periodicity (provided these can be studied drift kinetically). Novel features of the theory include the retention of a large effective collision frequency resulting from the resonant alpha collisional boundary layer, correlated interactions over many poloidal transits and finite orbit effects. Heat fluxes are considered for the example cases of ripple and the toroidal Alfvén eigenmode (TAE). The ripple heat flux is small. The TAE heat flux is significant and scales with the square of the perturbation amplitude, allowing the derivation of constraints on mode amplitude for avoidance of significant alpha depletion. A simple saturation condition suggests that TAEs in one upcoming experiment will not cause significant alpha transport via the mechanisms in this theory. However, saturation above the level suggested by the simple condition, but within numerical and experimental experience, which could be accompanied by the onset of stochasticity, could cause significant transport.

Keywords

fusion plasmaplasma wavesplasma instabilities
Type
Research Article
Information
Journal of Plasma Physics , Volume 87 , Issue 2 , April 2021 , 855870201
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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References

Berk, H. L., Breizman, B. N., Fitzpatrick, J. & Wong, H. V. 1995 Line broadened quasi-linear burst model [fusion plasma]. Nucl. Fusion 35 (12), 1661.CrossRefGoogle Scholar
Berk, H. L., Breizman, B. N. & Ye, H. 1992 Scenarios for the nonlinear evolution of alpha-particle-induced Alfvén wave instability. Phys. Rev. Lett. 68 (24), 3563.CrossRefGoogle ScholarPubMed
Betti, R. & Freidberg, J. P. 1992 Stability of Alfvén gap modes in burning plasmas. Phys. Fluids B 4 (6), 1465–1474.CrossRefGoogle Scholar
Betti, R. & Freidberg, J. P. 1993 Destabilization of the internal kink by energetic-circulating ions. Phys. Rev. Lett. 70 (22), 3428.CrossRefGoogle ScholarPubMed
Bhatnagar, P. L., Gross, E. P. & Krook, M. 1954 A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94 (3), 511.CrossRefGoogle Scholar
Borba, D. & Kerner, W. 1999 CASTOR-K: stability analysis of Alfvén eigenmodes in the presence of energetic ions in tokamaks. J. Comput. Phys. 153 (1), 101–138.CrossRefGoogle Scholar
Breizman, B. N. & Sharapov, S. E. 1995 Energetic particle drive for toroidicity-induced Alfvén eigenmodes and kinetic toroidicity-induced Alfvén eigenmodes in a low-shear tokamak. Plasma Phys. Control. Fusion 37 (10), 1057.CrossRefGoogle Scholar
Calvo, I., Parra, F. I., Velasco, J. L. & Alonso, J. A. 2017 The effect of tangential drifts on neoclassical transport in stellarators close to omnigeneity. Plasma Phys. Control. Fusion 59 (5), 055014.CrossRefGoogle Scholar
Catto, P. J. 2018 Ripple modifications to alpha transport in tokamaks. J. Plasma Phys. 84 (5), 905840508.CrossRefGoogle Scholar
Catto, P. J. 2019 a Collisional alpha transport in a weakly non-quasisymmetric stellarator magnetic field. J. Plasma Phys. 85 (2), 905850213.CrossRefGoogle Scholar
Catto, P. J. 2019 b Collisional alpha transport in a weakly rippled magnetic field. J. Plasma Phys. 85 (2), 905850203.CrossRefGoogle Scholar
Catto, P. J. 2020 Collisional effects on resonant particles in quasilinear theory. J. Plasma Phys. 86 (3), 815860302.CrossRefGoogle Scholar
Chen, L. & Zonca, F. 2012 Nonlinear excitations of zonal structures by toroidal Alfvén eigenmodes. Phys. Rev. Lett. 109 (14), 145002.CrossRefGoogle ScholarPubMed
Cheng, C. Z. & Chance, M. S. 1986 Low-n shear Alfvén spectra in axisymmetric toroidal plasmas. Phys. Fluids 29 (11), 3695–3701.CrossRefGoogle Scholar
Collins, C. S., Heidbrink, W. W., Austin, M. E., Kramer, G. J., Pace, D. C., Petty, C. C., Stagner, L., Van Zeeland, M. A., White, R. B., Zhu, Y. B., et al. 2016 Observation of critical-gradient behavior in Alfvén-eigenmode-induced fast-ion transport. Phys. Rev. Lett. 116 (9), 095001.CrossRefGoogle ScholarPubMed
Collins, C. S., Heidbrink, W. W., Podestà, M., White, R. B., Kramer, G. J., Pace, D. C., Petty, C. C., Stagner, L., Van Zeeland, M. A., Zhu, Y. B., et al. 2017 Phase-space dependent critical gradient behavior of fast-ion transport due to Alfvén eigenmodes. Nucl. Fusion 57 (8), 086005.CrossRefGoogle Scholar
Cordey, J. G. 1976 Effects of particle trapping on the slowing-down of fast ions in a toroidal plasma. Nucl. Fusion 16 (3), 499.CrossRefGoogle Scholar
Creely, A. J., et al. 2020 Overview of the SPARC tokamak. J. Plasma Phys. 86 (5), 865860502.CrossRefGoogle Scholar
Duarte, V. N., Gorelenkov, N. N., White, R. B. & Berk, H. L. 2019 Collisional resonance function in discrete-resonance quasilinear plasma systems. Phys. Plasmas 26 (12), 120701.CrossRefGoogle Scholar
Fasoli, A., Borba, D., Gormezano, C., Heeter, R., Jaun, A., Jacquinot, J., Kerner, W., King, Q., Lister, J. B., Sharapov, S., et al. 1997 Alfvén eigenmode experiments in tokamaks and stellarators. Plasma Phys. Control. Fusion 39 (12B), B287.CrossRefGoogle Scholar
Fitzgerald, M., Sharapov, S. E., Rodrigues, P. & Borba, D. 2016 Predictive nonlinear studies of TAE-induced alpha-particle transport in the $Q=10$ ITER baseline scenario. Nucl. Fusion 56 (11), 112010.CrossRefGoogle Scholar
Fu, G. Y. & Cheng, C. Z. 1992 Excitation of high-$n$ toroidicity-induced shear Alfvén eigenmodes by energetic particles and fusion alpha particles in tokamaks. Phys. Fluids B 4 (11), 3722–3734.CrossRefGoogle Scholar
Fu, G. Y. & Park, W. 1995 Nonlinear hybrid simulation of the toroidicity-induced Alfvén eigenmode. Phys. Rev. Lett. 74 (9), 1594.CrossRefGoogle ScholarPubMed
Fu, G. Y. & Van Dam, J. W. 1989 Excitation of the toroidicity-induced shear Alfvén eigenmode by fusion alpha particles in an ignited tokamak. Phys. Fluids B 1 (10), 1949–1952.CrossRefGoogle Scholar
Fülöp, T., Lisak, M., Kolesnichenko, Y. I. & Anderson, D. 1996 Finite orbit width stabilizing effect on toroidal Alfvén eigenmodes excited by passing and trapped energetic ions. Plasma Phys. Control. Fusion 38 (6), 811.CrossRefGoogle Scholar
Goldston, R. J., White, R. B. & Boozer, A. H. 1981 Confinement of high-energy trapped particles in tokamaks. Phys. Rev. Lett. 47 (9), 647.CrossRefGoogle Scholar
Hazeltine, R. D. 1973 Recursive derivation of drift-kinetic equation. Plasma Phys. 15 (1), 77.CrossRefGoogle Scholar
Heidbrink, W. W. 2002 Alpha particle physics in a tokamak burning plasma experiment. Phys. Plasmas 9 (5), 2113–2119.CrossRefGoogle 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
Heidbrink, W. W. & White, R. B. 2020 Mechanisms of energetic-particle transport in magnetically confined plasmas. Phys. Plasmas 27 (3), 030901.CrossRefGoogle Scholar
Helander, P. & Sigmar, D. J. 2005 Collisional Transport in Magnetized Plasmas, vol. 4. Cambridge University Press.Google Scholar
Hirvijoki, E., Asunta, O., Koskela, T., Kurki-Suonio, T., Miettunen, J., Sipilä, S., Snicker, A. & Äkäslompolo, S. 2014 ASCOT: solving the kinetic equation of minority particle species in tokamak plasmas. Comput. Phys. Commun. 185 (4), 1310–1321.CrossRefGoogle Scholar
Hirvijoki, E., Snicker, A., Korpilo, T., Lauber, P., Poli, E., Schneller, M. & Kurki-Suonio, T. 2012 Alfvén eigenmodes and neoclassical tearing modes for orbit-following implementations. Comput. Phys. Commun. 183 (12), 2589–2593.CrossRefGoogle Scholar
Ikeda, K. 2007 Progress in the ITER physics basis. Nucl. Fusion 47 (6), E01.CrossRefGoogle Scholar
Kim, K., Park, J. K. & Boozer, A. H. 2013 Numerical verification of bounce-harmonic resonances in neoclassical toroidal viscosity for tokamaks. Phys. Rev. Lett. 110 (18), 185004.CrossRefGoogle ScholarPubMed
Kurki-Suonio, T., Asunta, O., Hellsten, T., Hynönen, V., Johnson, T., Koskela, T., Lönnroth, J., Parail, V., Roccella, M., Saibene, G., et al. 2009 Ascot simulations of fast ion power loads to the plasma-facing components in ITER. Nucl. Fusion 49 (9), 095001.CrossRefGoogle Scholar
La Haye, R. J. 2006 Neoclassical tearing modes and their control. Phys. Plasmas 13 (5), 055501.CrossRefGoogle Scholar
Linsker, R. & Boozer, A. H. 1982 Banana drift transport in tokamaks with ripple. Phys. Fluids 25 (1), 143–147.CrossRefGoogle Scholar
Logan, N. C., Park, J. K., Kim, K., Wang, Z. & Berkery, J. W. 2013 Neoclassical toroidal viscosity in perturbed equilibria with general tokamak geometry. Phys. Plasmas 20 (12), 122507.CrossRefGoogle Scholar
Mynick, H. E. 1986 Generalized banana-drift transport. Nucl. Fusion 26 (4), 491.CrossRefGoogle Scholar
Nagaoka, K., Isobe, M., Toi, K., Shimizu, A., Fujisawa, A., Ohshima, S., Nakano, H., Osakabe, M., Todo, Y., Akiyama, T., et al. 2008 Radial transport characteristics of fast ions due to energetic-particle modes inside the last closed-flux surface in the compact helical system. Phys. Rev. Lett. 100 (6), 065005.CrossRefGoogle ScholarPubMed
Nazikian, R., Fu, G. Y., Batha, S. H., Bell, M. G., Bell, R. E., Budny, R. V., Bush, C. E., Chang, Z., Chen, Y., Cheng, C. Z., et al. 1997 Alpha-particle-driven toroidal Alfvén eigenmodes in the Tokamak Fusion Test Reactor. Phys. Rev. Lett. 78 (15), 2976.CrossRefGoogle Scholar
Park, J. K., Boozer, A. H. & Menard, J. E. 2009 Nonambipolar transport by trapped particles in tokamaks. Phys. Rev. Lett. 102 (6), 065002.CrossRefGoogle ScholarPubMed
Parra, F. I. & Catto, P. J. 2010 Transport of momentum in full $f$ gyrokinetics. Phys. Plasmas 17 (5), 056106.CrossRefGoogle Scholar
Poli, E., García-Muñoz, M., Fahrbach, H.-U., Günter, S. & ASDEX Upgrade Team 2008 Observation and modeling of fast trapped ion losses due to neoclassical tearing modes. Phys. Plasmas 15 (3), 032501.CrossRefGoogle Scholar
Rodrigues, P., Figueiredo, A. C. A., Borba, D., Coelho, R., Fazendeiro, L., Ferreira, J., Loureiro, N. F., Nabais, F., Pinches, S. D., Polevoi, A. R., et al. 2016 Sensitivity of alpha-particle-driven Alfvén eigenmodes to q-profile variation in ITER scenarios. Nucl. Fusion 56 (11), 112006.CrossRefGoogle Scholar
Rodrigues, P., Figueiredo, A., Ferreira, J., Coelho, R., Nabais, F., Borba, D., Loureiro, N. F., Oliver, H. J. C. & Sharapov, S. E. 2015 Systematic linear-stability assessment of Alfvén eigenmodes in the presence of fusion $\alpha$-particles for ITER-like equilibria. Nucl. Fusion 55 (8), 083003.CrossRefGoogle Scholar
Rodriguez-Fernandez, P., Howard, N. T., Greenwald, M. J., Creely, A. J., Hughes, J. W., Wright, J. C., Holland, C., Lin, Y., Sciortino, F. & The SPARC Team 2020 Predictions of core plasma performance for the SPARC tokamak. J. Plasma Phys. 86 (5), 865860503.CrossRefGoogle Scholar
Sanchis, L., Garcia-Munoz, M., Snicker, A., Ryan, D. A., Zarzoso, D., Chen, L., Galdon-Quiroga, J., Nocente, M., Rivero-Rodriguez, J. F., Rodriguez-Ramos, M., et al. 2018 Characterisation of the fast-ion edge resonant transport layer induced by 3D perturbative fields in the ASDEX Upgrade tokamak through full orbit simulations. Plasma Phys. Control. Fusion 61 (1), 014038.CrossRefGoogle Scholar
Scott, S. D., Kramer, G. J., Tolman, E. A., Snicker, A., Varje, J., Särkimäki, K., Wright, J. C. & Rodriguez-Fernandez, P. 2020 Fast-ion physics in SPARC. J. Plasma Phys. 86 (5).CrossRefGoogle Scholar
Shaing, K.-C. 2015 Superbanana and superbanana plateau transport in finite aspect ratio tokamaks with broken symmetry. J. Plasma Phys. 81 (2).CrossRefGoogle Scholar
Sharapov, S. E., Borba, D., Fasoli, A., Kerner, W., Eriksson, L. G., Heeter, R. F., Huysmans, G. T. A. & Mantsinen, M. J. 1999 Stability of alpha particle driven Alfvén eigenmodes in high performance JET DT plasmas. Nucl. Fusion 39 (3), 373.CrossRefGoogle Scholar
Sigmar, D. J., Hsu, C. T., White, R. & Cheng, C. Z. 1992 Alpha-particle losses from toroidicity-induced Alfvén eigenmodes. Part II: Monte Carlo simulations and anomalous alpha-loss processes. Phys. Fluids B 4 (6), 1506–1516.CrossRefGoogle Scholar
Slaby, C., Könies, A., Kleiber, R. & García-Regaña, J. M. 2018 Effects of collisions on the saturation dynamics of TAEs in tokamaks and stellarators. Nucl. Fusion 58 (8), 082018.CrossRefGoogle Scholar
Snicker, A., Hirvijoki, E. & Kurki-Suonio, T. 2013 Power loads to ITER first wall structures due to fusion alphas in a non-axisymmetric magnetic field including the presence of MHD modes. Nucl. Fusion 53 (9), 093028.CrossRefGoogle Scholar
Snicker, A., Kilpeläinen, J., Jacobsen, A. S., Garcia Munoz, M., Sanchis-Sanchez, L. & ASDEX Upgrade Team 2019 The combined effect of neoclassical tearing modes and ELM control coils on fast ions: validation in AUG and extrapolation for ITER. In 27th IAEA Fusion Energy Conference (FEC 2018). International Atomic Energy Agency.Google Scholar
Sorbom, B. N., Ball, J., Palmer, T. R., Mangiarotti, F. J., Sierchio, J. M., Bonoli, P., Kasten, C., Sutherland, D. A., Barnard, H. S., Haakonsen, C. B., et al. 2015 ARC: a compact, high-field, fusion nuclear science facility and demonstration power plant with demountable magnets. Fusion Engng Des. 100, 378–405.CrossRefGoogle Scholar
Spong, D. A., Carreras, B. A. & Hedrick, C. L. 1994 Nonlinear evolution of the toroidal Alfvén instability using a gyrofluid model. Phys. Plasmas 1 (5), 1503–1510.CrossRefGoogle Scholar
Su, C. H. & Oberman, C. 1968 Collisional damping of a plasma echo. Phys. Rev. Lett. 20 (9), 427.CrossRefGoogle Scholar
Tani, K., Azumi, M., Kishimoto, H. & Tamura, S. 1981 Effect of toroidal field ripple on fast ion behavior in a tokamak. J. Phys. Soc. Japan 50 (5), 1726–1737.CrossRefGoogle Scholar
Todo, Y. 2019 Introduction to the interaction between energetic particles and Alfvén eigenmodes in toroidal plasmas. Rev. Mod. Plasma Phys. 3 (1), 1.CrossRefGoogle Scholar
Todo, Y., Berk, H. L. & Breizman, B. N. 2003 Simulation of intermittent beam ion loss in a Tokamak Fusion Test Reactor experiment. Phys. Plasmas 10 (7), 2888–2902.CrossRefGoogle Scholar
Tolman, E. A., Loureiro, N. F., Rodrigues, P., Hughes, J. W. & Marmar, E. S. 2019 Dependence of alpha-particle-driven Alfvén eigenmode linear stability on device magnetic field strength and consequences for next-generation tokamaks. Nucl. Fusion 59 (4), 046020.CrossRefGoogle Scholar
Van Zeeland, M. A., Kramer, G. J., Austin, M. E., Boivin, R. L., Heidbrink, W. W., Makowski, M. A., McKee, G. R., Nazikian, R., Solomon, W. M. & Wang, G. 2006 Radial structure of Alfvén eigenmodes in the DIII-D tokamak through electron-cyclotron-emission measurements. Phys. Rev. Lett. 97 (13), 135001.CrossRefGoogle ScholarPubMed
Varje, J., Särkimäki, K., Kontula, J., Ollus, P., Kurki-Suonio, T., Snicker, A., Hirvijoki, E. & Äkäslompolo, S. 2019 High-performance orbit-following code ASCOT5 for Monte Carlo simulations in fusion plasmas. arXiv:1908.02482.Google Scholar
Wang, X. & Briguglio, S. 2016 Saturation of single toroidal number Alfvén modes. New J. Phys. 18 (8), 085009.CrossRefGoogle Scholar
White, R. B. 2013 The Theory of Toroidally Confined Plasmas. World Scientific Publishing Company.Google Scholar
White, R. B. & Boozer, A. H. 1995 Rapid guiding center calculations. Phys. Plasmas 2 (8), 2915–2919.CrossRefGoogle Scholar
White, R. B., Duarte, V. N., Gorelenkov, N. N. & Meng, G. 2019 Collisional enhancement of energetic particle Alfvénic resonance width in tokamaks. Phys. Plasmas 26 (3), 032508.CrossRefGoogle Scholar
White, R. B., Fredrickson, E., Darrow, D., Zarnstorff, M., Wilson, R., Zweben, S., Hill, K., Chen, Y. & Fu, G. 1995 Toroidal Alfvén eigenmode-induced ripple trapping. Phys. Plasmas 2 (8), 2871–2873.CrossRefGoogle Scholar
White, R. B., Gorelenkov, N., Heidbrink, W. W. & Van Zeeland, M. A. 2010 Particle distribution modification by low amplitude modes. Plasma Phys. Control. Fusion 52 (4), 045012.CrossRefGoogle Scholar
Wu, Y. & White, R. B. 1994 Self-consistent study of the alpha-particle-driven toroidicity-induced Alfvén eigenmode. Phys. Plasmas 1 (8), 2733–2740.CrossRefGoogle Scholar
Zhou, M. & White, R. 2016 Collisional dependence of Alfvén mode saturation in tokamaks. Plasma Phys. Control. Fusion 58 (12), 125006.CrossRefGoogle Scholar
Zonca, F., Romanelli, F., Vlad, G. & Kar, C. 1995 Nonlinear saturation of toroidal Alfvén eigenmodes. Phys. Rev. Lett. 74 (5), 698.CrossRefGoogle ScholarPubMed