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Second-order nonlinear gyrokinetic theory: from the particle to the gyrocentre

Part of: Focus on Fusion

Published online by Cambridge University Press:  05 June 2018

Natalia Tronko  and
Cristel Chandre
Natalia Tronko*
Affiliation:
Max-Planck Institute for Plasma Physics, 85748 Garching, Germany TU Munich, Mathematics Center, 85747 Garching, Germany
Cristel Chandre
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
*
†Email address for correspondence: [email protected]
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Abstract

A gyrokinetic reduction is based on a specific ordering of the different small parameters characterizing the background magnetic field and the fluctuating electromagnetic fields. In this tutorial, we consider the following ordering of the small parameters: $\unicode[STIX]{x1D716}_{B}=\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6FF}}^{2}$ where $\unicode[STIX]{x1D716}_{B}$ is the small parameter associated with spatial inhomogeneities of the background magnetic field and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6FF}}$ characterizes the small amplitude of the fluctuating fields. In particular, we do not make any assumption on the amplitude of the background magnetic field. Given this choice of ordering, we describe a self-contained and systematic derivation which is particularly well suited for the gyrokinetic reduction, following a two-step procedure. We follow the approach developed in Sugama (Phys. Plasmas, vol. 7, 2000, p. 466): In a first step, using a translation in velocity, we embed the transformation performed on the symplectic part of the gyrocentre reduction in the guiding-centre one. In a second step, using a canonical Lie transform, we eliminate the gyroangle dependence from the Hamiltonian. As a consequence, we explicitly derive the fully electromagnetic gyrokinetic equations at the second order in $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6FF}}$.

Keywords

fusion plasmaplasma nonlinear phenomenaplasma simulation
Type
Tutorial
Information
Journal of Plasma Physics , Volume 84 , Issue 3 , June 2018 , 925840301
Copyright
© EUROfusion Consortium Research Institutions 2018 

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References

Abel, I. G., Plunk, G. G., Wang, E., Barnes, M., Cowley, S. C., Dorland, W. & Schekochihin, A. A. 2013 Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows. Rep. Progr. Phys. 76, 116201.CrossRefGoogle ScholarPubMed
Bottino, A. & Sonnendrücker, E. 2015 Monte Carlo particle-in-cell methods for the simulation of the Vlasov–Maxwell gyrokinetic equations. J. Plasma Phys. 81, 435810501.Google Scholar
Brizard, A. J. 1989 Nonlinear gyrokinetic Maxwell–Vlasov equations using magnetic co-ordinates. J. Plasma Phys. 41, 541.Google Scholar
Brizard, A. J. 2000 New variational principle for the Vlasov–Maxwell equations. Phys. Rev. Lett. 84, 5768.Google Scholar
Brizard, A. J. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421.CrossRefGoogle Scholar
Burby, J. W., Brizard, A. J., Morrison, P. J. & Qin, H. 2015 Hamiltonian gyrokinetic Vlasov–Maxwell system. Phys. Lett. A 379, 2073.Google Scholar
Burby, J. W., Squire, J. & Qin, H 2013 Automation of the guiding center expansion. Phys. Plasmas 20, 072105.Google Scholar
Candy, J. & Waltz, R. E. 2003 An Eulearian gyrokinetic-Maxwell solver. J. Comput. Phys. 186, 545.Google Scholar
Cary, J. R. 1981 Lie transform perturbation theory for Hamiltonian systems. Phys. Rep. 79, 129.CrossRefGoogle Scholar
Cary, J. R. & Brizard, A. J. 2009 Hamiltonian theory of guiding center motion. Rev. Mod. Phys. 81, 693.Google Scholar
Casati, A., Gerbaud, T., Hennequin, P., Bourdelle, C., Candy, J., Clairet, F., Garbet, X., Grandgirard, V., Gürcan, Ö. D., Heuraux, S. et al. 2009 Turbulence in the tore supra tokamak: measurements and validation of nonlinear simulations. Phys. Rev. Lett. 102 (16), 165005–165005.Google Scholar
Catto, P. J. 1978 Linearized gyro-kinetics. Plasma Phys. Control. Fusion 20 (7), 719–722.Google Scholar
Catto, P. J. 1981 Generalized gytokonetics. Plasma Phys. Control. Fusion 23 (7), 639–650.Google Scholar
Frieman, E. A. & Chen, L. 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25, 502.Google Scholar
Garbet, X., Idomura, Y., Villard, L. & Watanabe, T. H. 2010 Gyrokinetic simulations of turbulent transport. Nuclear Fusion 50, 043002.CrossRefGoogle Scholar
Goerler, T., Lapillonne, X., Brunner, S., Dannert, T., Jenko, F., Merz, F. & Told, D. 2011 The global version of the gyrokinetic turbulence code GENE. J. Computat. Phys. 230, 7053–7071.Google Scholar
Grandgirard, V., Abiteboul, J., Bigot, J., Cartier-Michaud, J., Crouseillese, N., Dif-Pradalier, G., Ehrlacher, C., Esteve, D., Garbet, X., Ghendrih, P. et al. 2016 A 5d gyrokinetic full-f global semi-Lagrangian code for flux-driven ion turbulence simulations. Comput. Phys. Commun. 207, 35–68.CrossRefGoogle Scholar
Grebogi, C., Kaufman, A. N. & Littlejohn, R. G. 1979 Hamiltonian theory of ponderomotive effects of an electromagnetic wave in a nonuniform magnetic field. Phys. Rev. Lett. 43, 22.Google Scholar
Hahm, T. S. 1988 Nonlinear gyrokinetic equations for tokamak microturbulence. Phys. Fluids 31, 2670.CrossRefGoogle Scholar
Jolliet, S., Bottino, A., Angelino, P., Hatzky, R., Tran, T. M., Mcmillan, B. F., Sauter, O., Appert, K., Idomura, Y. & Villard, L. 2007 A global collisionless PIC code in magnetic coordinates. Comput. Phys. Commun. 177, 409.Google Scholar
Krause, T. B., Apte, A. & Morrison, P. J. 2007 A unified approach to the darwin approximation. Phys. Plasmas 14, 102112.Google Scholar
Littlejohn, R. G. 1979 A guiding center Hamiltonian: a new approach. J. Math. Phys. 20, 2445.Google Scholar
Littlejohn, R. G. 1981 Hamiltonian formulation of guiding center motion. Phys. Fluids 29, 1730.Google Scholar
Littlejohn, R. G. 1983 Variational principles of guiding centre motion. J. Plasma Phys. 29, 111.Google Scholar
Parra, F. I. & Calvo, I 2011 Phase-space Lagrangian derivation of electrostatic gyrokinetics in general geometry. Plasma Phys. Control. Fusion 53, 045001.CrossRefGoogle Scholar
Parra, F. I., Calvo, I., Burby, J. W. & Qin, H. 2014 Equivalence of two independent calculations of the higher order guiding center Lagrangian. Phys. Plasmas 21, 104506.Google Scholar
Peeters, A. G., Camenen, Y., Casson, F. J., Hornsby, W. A., Snodin, A. P., Strintzi, D. & Szepesi, G. 2009 The nonlinear gyro-kinetic flux tube code gkw. Comput. Phys. Commun. 180 (12), 2650–2672.Google Scholar
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. 182, 310.Google Scholar
Squire, J., Qin, H., Tang, W. M. & Chandre, C. 2013 The Hamiltonian structure and Euler–Poincaré formulation of the Vlasov–Maxwell and gyrokinetic systems. Phys. Plasmas 20, 022501.Google Scholar
Sugama, H. 2000 Gyrokinetic field theory. Phys. Plasmas 7, 466.Google Scholar
Tronko, N., Bottino, A., Chandre, C. & Sonnendrücker, E. 2017 Hierarchy of second order gyrokinetic Hamiltonian models for particle-in-cell codes. Plasma Phys. Control. Fusion 59, 064008.CrossRefGoogle Scholar
Tronko, N., Bottino, A. & Sonnendrücker, E. 2016 Second order gyrokinetic theory for particle-in-cell codes. Phys. Plasmas 23, 082505.Google Scholar
Tronko, N. & Brizard, A. J. 2015 Lagrangian and Hamiltonian constraints for guiding center Hamiltonian theories. Phys. Plasmas 22, 112507.CrossRefGoogle Scholar
Wersal, C., Bottino, A., Angelino, P. & Scott, B. D. 2012 Fluid moments and spectral diagnostics in global particle-in-cell simulations. J. Phys. Conf. Ser. 401, 012025.Google Scholar