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A low-frequency variational model for energetic particle effects in the pressure-coupling scheme

Published online by Cambridge University Press:  03 July 2018

Alexander R. D. Close*
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK
Joshua W. Burby
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Cesare Tronci
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK
*
Email address for correspondence: alexander.rd.close@gmail.com

Abstract

Energetic particle effects in magnetic confinement fusion devices are commonly studied by hybrid kinetic-fluid simulation codes whose underlying continuum evolution equations often lack the correct energy balance. While two different kinetic-fluid coupling options are available (current coupling and pressure coupling), this paper applies the Euler–Poincaré variational approach to formulate a new conservative hybrid model in the pressure-coupling scheme. In our case the kinetics of the energetic particles are described by guiding center theory. The interplay between the Lagrangian fluid paths and phase space particle trajectories reflects an intricate variational structure which can be approached by letting the four-dimensional guiding center trajectories evolve in the full six-dimensional phase space. Then, the redundant perpendicular velocity is integrated out to recover a four-dimensional description. A second equivalent variational approach is also reported, which involves the use of phase space Lagrangians. Not only do these variational structures confer on the new model a correct energy balance, but also they produce a cross-helicity invariant which is lost in the other pressure-coupling schemes reported in the literature.

Type
Research Article
Copyright
© Cambridge University Press 2018 

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References

Belova, E. V., Denton, R. E. & Chan, A. A. 1997 Hybrid simulations of the effects of energetic particles on low-frequency MHD waves. Comput. Phys. 136 (2), 324336.CrossRefGoogle Scholar
Belova, E. V., Gorelenkov, N. N., Fredrickson, E. D., Tritz, K. & Crocker, N. 2015 Coupling of neutral-beam-driven compressional Alfvén eigenmodes to kinetic Alfvén waves in NSTX tokamak and energy channeling. Phys. Rev. Lett. 115, 015001.CrossRefGoogle ScholarPubMed
Belova, E. V. & Park, W. 1999 3D hybrid and MHD/particle simulations of field-reversed configurations. In NIFS-PROC–41, Japan (ed. Yoshimura, S. & Seiichi, G.), NIFS-PROC–41, pp. 8187. National Inst. for Fusion Science, Nagoya, Japan.Google Scholar
Briguglio, S., Vlad, G. & Zonca, F. 1998 Hybrid magnetohydrodynamic-particle simulation of linear and nonlinear evolution of Alfvén modes in tokamaks. Phys. Plasmas 5 (9), 32873301.CrossRefGoogle Scholar
Briguglio, S., Vlad, G., Zonca, F. & Kar, C. 1995 Hybrid magnetohydrodynamic-gyrokinetic simulation of toroidal Alfvén modes. Phys. Plasmas 10 (2), 37113723.CrossRefGoogle Scholar
Brizard, A. 1994 Eulerian action principles for linearized reduced dynamical equations. Phys. Plasmas 1 (8), 24602472.CrossRefGoogle Scholar
Brizard, A. J. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79 (2), 421468.CrossRefGoogle Scholar
Brizard, A. J. & Tronci, C. 2016 Variational principles for the guiding-center Vlasov–Maxwell equations. Phys. Plasmas 23 (6), 062107.CrossRefGoogle Scholar
Burby, J. & Tronci, C. 2017 Variational approach to low-frequency kinetic-MHD in the current-coupling scheme. Plasma Phys. Control. Fusion 59 (4), 045013.CrossRefGoogle Scholar
Burby, J. W.2015 Chasing Hamiltonian structure in gyrokinetic theory. PhD thesis, Princeton University.Google Scholar
Burby, J. W., Brizard, A., Morrison, P. & Qin, H. 2015 Hamiltonian gyrokinetic Vlasov–Maxwell system. Phys. Lett. A 379 (36), 20732077.CrossRefGoogle Scholar
Burby, J. W. & Sengupta, W. 2018 Hamiltonian structure of the guiding center plasma model. Phys. Plasmas 25 (2), 020703.CrossRefGoogle Scholar
Cary, J. R. & Brizard, A. J. 2009 Hamiltonian theory of guiding-center motion. Rev. Mod. Phys. 81 (2), 693738.CrossRefGoogle Scholar
Cendra, H., Holm, D. D., Hoyle, M. J. W. & Marsden, J. E. 1998 The Maxwell–Vlasov equations in Euler-Poincaré form. J. Math. Phys. 39 (6), 31383157.CrossRefGoogle Scholar
Chen, L., White, R. B. & Rosenbluth, M. N. 1984 Excitation of internal kink modes by trapped energetic beam ions. Phys. Rev. Lett. 52 (13), 12221225.CrossRefGoogle Scholar
Cheng, B., Süli, E. & Tronci, C. 2017 Existence of global weak solutions to a hybrid Vlasov-MHD model for plasma dynamics. Proc. Lond. Math. Soc. 3, 143.Google Scholar
Cheng, C. Z. 1991 A kinetic-magnetohydrodynamic model for low-frequency phenomena. J. Geophys. Res. 96 (A12), 2115921171.CrossRefGoogle Scholar
Coppi, B. & Porcelli, F. 1986 Theoretical model of fishbone oscillations in magnetically confined plasmas. Phys. Rev. Lett. 57 (18), 22722275.CrossRefGoogle ScholarPubMed
Dewar, R. 1972 A Lagrangian theory for nonlinear wave packets in a collisionless plasma. J. Plasma Phys. 7 (2), 267284.CrossRefGoogle Scholar
Evstatiev, E. G. 2014 Application of the phase space action principle to finite-size particle plasma simulations in the drift-kinetic approximation. Comput. Phys. Commun. 185, 2851.CrossRefGoogle Scholar
Fu, G. Y. & Park, W. 1995 Nonlinear hybrid simulation of the toroidicity-induced Alfvén Eigenmode. Phys. Rev. Lett. 74 (9), 15941596.CrossRefGoogle ScholarPubMed
Fu, G. Y., Park, W., Strauss, H. R., Breslau, J., Chen, J., Jardin, S. & Sugiyama, L. E. 2006 Global hybrid simulations of energetic particle effects on the $n=1$ mode in tokamaks: Internal kink and fishbone instability. Phys. Plasmas 13 (5), 052517.CrossRefGoogle Scholar
Holm, D. D., Marsden, J. E. & Ratiu, T. S. 1998 The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Maths 137, 181.CrossRefGoogle Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. S. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123 (1–2), 1116.CrossRefGoogle Scholar
Holm, D. D., Schmah, T. & Stoica, C. 2009 Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions. Oxford University Press.Google Scholar
Holm, D. D. & Tronci, C. 2012 Euler-Poincaré formulation of hybrid plasma models. Commun. Math. Sci. 10 (1), 191222.CrossRefGoogle Scholar
Hou, Y., Zhu, P., Kim, C. C., Hu, Z., Zou, Z., Wang, Z. & NIMROD Team 2018 NIMROD calculations of energetic particle driven toroidal Alfvén eigenmodes. Phys. Plasmas 25 (1), 012501.Google Scholar
Ilgisonis, V. I. & Lakhin, V. P. 1999 Lagrangean structure of hydrodynamic plasma models and conservation laws. Plasma Phys. Rep. 25 (1), 5869.Google Scholar
Kaufman, A. N. 1986 The electric dipole of a guiding center and the plasma momentum density. Phys. Fluids 29 (5), 17361737.CrossRefGoogle Scholar
Kim, C. C. 2008 Preliminary simulations of FLR effects on RFP tearing modes. J. Fusion Energ. 27 (1), 6164.CrossRefGoogle Scholar
Kim, C. C., Sovinec, C. R., Parker, S. E. & the NIMROD team 2004 Hybrid kinetic-MHD simulations in general geometry. Comput. Phys. 164, 448455.CrossRefGoogle Scholar
Kim, C. C. & the NIMROD team 2008 Impact of velocity space distribution on hybrid kinetic-magnetohydrodynamic simulation of the $(1,1)$ mode. Phys. Plasmas 15, 072507.CrossRefGoogle Scholar
Kraus, M., Kormann, K., Morrison, P. J. & Sonnendrücker, E. 2017 GEMPIC: Geometric electromagnetic particle-in-cell methods. J. Plasma Phys. 83 (4), 905830401.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1976 Mechanics. Elsevier & Butterworth-Heinemann.Google Scholar
Littlejohn, R. G. 1983 Variational principles of guiding centre motion. J. Plasma Phys. 29 (1), 111125.CrossRefGoogle Scholar
Low, F. E. 1958 A Lagrangian formulation of the Boltzmann-Vlasov equation for plasmas. Proc. R. Soc. Lond. A 248, 282287.CrossRefGoogle Scholar
Marsden, J. E., Patrick, G. W. & Shkoller, S. 1998 Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199, 351395.CrossRefGoogle Scholar
Marsden, J. E. & Ratiu, T. S. 1998 Introduction to Mechanics and Symmetry. Springer.Google Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70 (2), 467521.CrossRefGoogle Scholar
Morrison, P. J. & Greene, J. M. 1980 Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics. Phys. Rev. Lett. 45, 790794.CrossRefGoogle Scholar
Morrison, P. J., Tassi, E. & Tronci, C. 2014 Energy stability analysis for a hybrid fluid-kinetic plasma model. In Nonlinear Physical Systems (ed. Kirillov, O. & Pelinovsky, D.), chap. 14, pp. 311329. Wiley-Blackwell.CrossRefGoogle Scholar
Newcomb, W. A. 1962 Lagrangian and Hamiltonian methods in magnetohydrodynamics. Nuclear Fusion: Supplement, part 2, 451463.Google Scholar
Park, W., Belova, E. V., Fu, G. Y., Tang, X. Z., Strauss, H. R. & Sugiyama, L. E. 1999 Plasma simulation studies using multilevel physics models. Phys. Plasmas 6 (5), 17961803.CrossRefGoogle Scholar
Park, W., Parker, S., Biglari, H., Chance, M., Chen, L., Cheng, C. Z., Hahm, T. S., Lee, W. W., Kulsrud, R., Monticello, D. et al. 1992 Three-dimensional hybrid gyrokinetic-magnetohydrodynamics simulation. Phys. Fluids B4 (7), 20332037.CrossRefGoogle Scholar
Pei, Y., Xiang, N., Hu, Y., Todo, Y., Li, G., Shen, W. & Xu, L. 2017 Kinetic-MHD hybrid simulation of fishbone modes excited by fast ions on the experimental advanced superconducting tokamak (EAST). Phys. Plasmas 24, 032507.CrossRefGoogle Scholar
Squire, J., Qin, H. & Tang, W. M. 2013 The Hamiltonian structure and Euler-Poincaré formulation of the Vlasov–Maxwell and gyrokinetic system. Phys. Plasmas 20, 022501.CrossRefGoogle Scholar
Takahashi, R., Brennan, D. P. & Kim, C. C. 2009a A detailed study of kinetic effects of energetic particles on resistive MHD linear stability. Nucl. Fusion 49, 065032.CrossRefGoogle Scholar
Takahashi, R., Brennan, D. P. & Kim, C. C. 2009b Kinetic effects of energetic particles on resistive MHD stability. Phys. Rev. Lett. 102, 135001.CrossRefGoogle Scholar
Thyagaraja, A. & McClements, K. G. 2009 Plasma physics in noninertial frames. Phys. Plasmas 16, 092506.CrossRefGoogle Scholar
Todo, Y. 2006 Properties of energetic-particle continuum modes destabilized by energetic ions with beam-like velocity distributions. Phys. Plasmas 13 (8), 082503.CrossRefGoogle Scholar
Todo, Y., Sato, T., Hayashi, T., Watanabe, K., Horiuchi, R., Takamaru, H., Watanabe, T. H. & Kageyama, A. 1996 Vlasov-MHD and particle-MHD simulations of the toroidal Alfvén eigenmode. In Proc. 16th Int. Conf. Plasma Phys. Control. Nucl. Fusion Res., pp. 423430. International Atomic Energy Agency (IAEA).Google Scholar
Todo, Y., Sato, T., Watanabe, K., Watanabe, T. H. & Horiuchi, R. 1995 Magnetohydrodynamic Vlasov simulation of the toroidal Alfvén eigenmode. Phys. Plasmas 2 (7), 27112716.CrossRefGoogle Scholar
Tronci, C. 2010 Hamiltonian approach to hybrid plasma models. J. Phys. A: Math. Theor. 43, 375501.CrossRefGoogle Scholar
Tronci, C. 2013 A Lagrangian kinetic model for collisionless magnetic reconnection. Plasma Phys. Control. Fusion 55 (3), 035001.CrossRefGoogle Scholar
Tronci, C. 2016 From liquid crystal models to the guiding-center theory of magnetized plasmas. Ann. Phys. 371, 323337.CrossRefGoogle Scholar
Tronci, C. & Camporeale, E. 2015 Neutral Vlasov kinetic theory of magnetized plasmas. Phys. Plasmas 22 (2), 020704.CrossRefGoogle Scholar
Tronci, C., Tassi, E., Camporeale, E. & Morrison, P. J. 2014 Hybrid Vlasov-MHD models: Hamiltonian versus non-Hamiltonian. Plasma Phys. Control. Fusion 56 (9), 095008.CrossRefGoogle Scholar
Tronci, C., Tassi, E. & Morrison, P. J. 2015 Energy-Casimir stability of hybrid Vlasov-MHD models. J. Phys. A 48 (18), 185501.Google Scholar
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