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A gyrokinetic model for the plasma periphery of tokamak devices

Part of: Focus on Fusion

Published online by Cambridge University Press:  16 March 2020

B. J. Frei Open the ORCID record for B. J. Frei [Opens in a new window] ,
R. Jorge Open the ORCID record for R. Jorge [Opens in a new window]  and
P. Ricci
B. J. Frei*
Affiliation:
École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC), CH-1015Lausanne, Switzerland
R. Jorge
Affiliation:
École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC), CH-1015Lausanne, Switzerland Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa, 1049-001Lisboa, Portugal
P. Ricci
Affiliation:
École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC), CH-1015Lausanne, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

A gyrokinetic model is presented that can properly describe large and small amplitude electromagnetic fluctuations occurring on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and the arbitrarily large deviations from thermal equilibrium that are present in the plasma periphery of tokamak devices. The formulation of the gyrokinetic model is based on a second-order accurate description of the single charged particle dynamics, derived from Lie perturbation theory, where the fast particle gyromotion is decoupled from the slow drifts assuming that the ratio of the ion sound Larmor radius to the perpendicular equilibrium pressure scale length is small. The collective behaviour of the plasma is obtained by a gyrokinetic Boltzmann equation that describes the evolution of the gyroaveraged distribution function. The collisional effects are included by a nonlinear gyrokinetic Dougherty collision operator. The gyrokinetic model is then developed into a set of coupled fluid equations referred to as the gyrokinetic moment hierarchy. To obtain this hierarchy, the gyroaveraged distribution function is expanded onto a Hermite–Laguerre velocity-space polynomial basis. Then, the gyrokinetic equation is projected onto the same basis yielding the spatial and temporal evolution of the Hermite–Laguerre expansion coefficients. A closed set of fluid equations for the lowest-order coefficients is presented. The Hermite–Laguerre projection is performed accurately at arbitrary perpendicular wavenumber values. Finally, the self-consistent evolution of the electromagnetic fields is described by a set of gyrokinetic Maxwell equations derived from a variational principle where the velocity integrals are explicitly evaluated.

Keywords

fusion plasmaplasma confinementplasma dynamics
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 2 , April 2020 , 905860205
Copyright
© Cambridge University Press 2020

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References

Abel, I. G., Barnes, M., Cowley, S. C., Dorland, W. & Schekochihin, A. A. 2008 Linearized model Fokker–Planck collision operators for gyrokinetic simulations. I. Theory. Phys. Plasmas 15 (12), 122509.CrossRefGoogle Scholar
Abel, I. G. & Hallenbert, A. 2018 Multiscale modelling for tokamak pedestals. J. Plasma Phys. 84 (2), 745840202.CrossRefGoogle Scholar
Abramowitz, M. 1974 Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover Publications, Incorporated.Google Scholar
Adkins, T. & Schekochihin, A. A. 2018 A solvable model of Vlasov-kinetic plasma turbulence in Fourier–Hermite phase space. J. Plasma Phys. 84 (1), 905840107.CrossRefGoogle Scholar
Agostini, M., Terry, J. L., Scarin, P. & Zweben, S. J. 2011 Edge turbulence in different density regimes in Alcator C-Mod experiment. Nucl. Fusion 51 (5), 053020.CrossRefGoogle Scholar
Artun, M. & Tang, W. M. 1994 Nonlinear electromagnetic gyrokinetic equations for rotating axisymmetric plasmas. Phys. Plasmas 1 (8), 2682.CrossRefGoogle Scholar
Banos, A. 1967 The guiding centre approximation in lowest order. J. Plasma Phys. 1 (03), 305.CrossRefGoogle Scholar
Barnes, M., Abel, I. G., Dorland, W., Ernst, D. R., Hammett, G. W., Ricci, P., Rogers, B. N., Schekochihin, A. A. & Tatsuno, T. 2009 Linearized model Fokker–Planck collision operators for gyrokinetic simulations. II. Numerical implementation and tests. Phys. Plasmas 16 (7), 072107.CrossRefGoogle Scholar
Batishchev, O. V., Krasheninnikov, S. I., Catto, P. J., Batishcheva, A. A., Sigmar, D. J., Xu, X. Q., Byers, J. A., Rognlien, T. D., Cohen, R. H., Shoucri, M. M. et al. 1997 Kinetic effects in tokamak scrape-off layer plasmas. Phys. Plasmas 4 (5), 1672.CrossRefGoogle Scholar
Batishcheva, A. A., Batishchev, O. V., Shoucri, M. M., Krasheninnikov, S. I., Catto, P. J., Shkarofsky, I. P. & Sigmar, D. J. 1996 A kinetic model of transient effects in tokamak edge plasmas. Phys. Plasmas 3 (5), 1634.CrossRefGoogle Scholar
Battaglia, D. J., Burrell, K. H., Chang, C. S., Ku, S., DeGrassie, J. S. & Grierson, B. A. 2014 Kinetic neoclassical transport in the H-mode pedestal. Phys. Plasmas 21 (7), 072508.CrossRefGoogle Scholar
Beer, M. A. & Hammett, G. W. 1996 Toroidal gyrofluid equations for simulations of tokamak turbulence. Phys. Plasmas 3 (11), 4046.CrossRefGoogle Scholar
Belli, E. A. & Candy, J. 2012 Full linearized Fokker–Planck collisions in neoclassical transport simulations. Plasma Phys. Control. Fusion 54 (1), 015015.CrossRefGoogle Scholar
Bernstein, I. B. & Catto, P. J. 1985 Generalized gyrokinetics. Phys. Fluids 28 (5), 1342.CrossRefGoogle Scholar
Beurskens, M. N. A., Osborne, T. H., Schneider, P. A., Wolfrum, E., Frassinetti, L., Groebner, R., Lomas, P., Nunes, I., Saarelma, S., Scannell, R. et al. 2011 H-mode pedestal scaling in DIII-D, ASDEX Upgrade, and JET. Phys. Plasmas 18 (5), 056120.CrossRefGoogle Scholar
Bleuel, J., Endler, M., Niedermeyer, H., Schubert, M. & Thomsen, H. 2002 The spatial structure of edge fluctuations in the Wendelstein 7-AS stellarator. New J. Phys. 4 (1), 38.CrossRefGoogle Scholar
Bottino, A. & Sonnendrucker, E. 2015 Monte Carlo particle-in-cell methods for the simulation of the Vlasov–Maxwell gyrokinetic equations. J. Plasma Phys. 81 (5), 435810501.CrossRefGoogle Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Courier Corporation.Google Scholar
Braginskii, S. I. 1965 Transport processes in a plasma. Rev. Plasma Phys. 1, 205.Google Scholar
Brizard, A. J. 1989 Nonlinear gyrokinetic Maxwell–Vlasov equations using magnetic co-ordinates. J. Plasma Phys. 41 (3), 541.CrossRefGoogle Scholar
Brizard, A. J. 1992 Nonlinear gyrofluid description of turbulent magnetized plasmas. Phys. Fluids B 4 (5), 1213.CrossRefGoogle Scholar
Brizard, A. J. 1995 Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks. Phys. Plasmas 2 (2), 459.CrossRefGoogle Scholar
Brizard, A. J. 2000 New variational principle for the Vlasov–Maxwell equations. Phys. Rev. Lett. 84 (25), 5768.CrossRefGoogle ScholarPubMed
Brizard, A. J. 2010 Exact energy conservation laws for full and truncated nonlinear gyrokinetic equations. Phys. Plasmas 17 (4), 042303.CrossRefGoogle Scholar
Brizard, A. J. 2013 Beyond linear gyrocenter polarization in gyrokinetic theory. Phys. Plasmas 20 (9), 0922309.CrossRefGoogle Scholar
Brizard, A. J. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79 (2), 421.CrossRefGoogle Scholar
Brizard, A. J. & Mishchenko, A. 2009 Guiding-center recursive Vlasov and Lie-transform methods in plasma physics. J. Plasma Phys. 75 (3), 675.CrossRefGoogle Scholar
Brizard, A. J. & Tronko, N. 2011 Exact momentum conservation laws for the gyrokinetic Vlasov–Poisson equations. Phys. Plasmas 18 (8), 082307.CrossRefGoogle Scholar
Brower, D. L., Peebles, W. A. & Luhmann, N. C. 1987 The spectrum, spatial distribution and scaling of microturbulence in the text tokamak. Nucl. Fusion 27 (12), 2055.CrossRefGoogle Scholar
Burrell, K. H., Doyle, E. J., Gohil, P., Groebner, R. J., Kim, J., La Haye, R. J., Lao, L. L., Moyer, R. A., Osborne, T. H. & Peebles, W. A. 1994 Role of the radial electric field in the transition from L (low) mode to H (high) mode to VH (very high) mode in the DIII-D tokamak. Phys. Plasmas 1 (5), 1536.CrossRefGoogle Scholar
Cary, J. R. 1981 Lie transform perturbation theory for Hamiltonian systems. Phys. Rep. 79 (2), 129.Google Scholar
Cary, J. R. & Brizard, A. J. 2009 Hamiltonian theory of guiding-center motion. Rev. Mod. Phys. 81 (2), 693.CrossRefGoogle Scholar
Cary, J. R. & Littlejohn, R. G. 1983 Noncanonical Hamiltonian mechanics and its application to magnetic field line flow. Ann. Phys. 151 (1), 1.CrossRefGoogle Scholar
Catto, P. J. 1978 Linearized gyro-kinetics. Plasma Phys. 20 (7), 719.CrossRefGoogle Scholar
Catto, P. J. & Simakov, A. N. 2004 A drift ordered short mean free path description for magnetized plasma allowing strong spatial anisotropy. Phys. Plasmas 11 (1), 90.CrossRefGoogle Scholar
Catto, P. J. & Tsang, K. T. 1977 Linearized gyro-kinetic equation with collisions. Phys. Fluids 20 (3), 396.CrossRefGoogle Scholar
Chang, C. S., Ku, S., Tynan, G. R., Hager, R., Churchill, R. M., Cziegler, I., Greenwald, M., Hubbard, A. E. & Hughes, J. W. 2017 Fast low-to-high confinement mode bifurcation dynamics in a tokamak edge plasma gyrokinetic simulation. Phys. Rev. Lett. 118 (17), 1.CrossRefGoogle Scholar
Chew, G. F., Goldberger, M. L. & Low, F. E. 1956 The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. Proc. R. Soc. Lond. A 236 (1204), 112.Google Scholar
Cohen, R. H. & Xu, X. Q. 2008 Progress in kinetic simulation of edge plasmas. Contrib. Plasma Phys. 48 (1), 212.CrossRefGoogle Scholar
Colyer, G. J., Schekochihin, A. A., Parra, F. I., Roach, C. M., Barnes, M. A., Ghim, Y. C. & Dorland, W. 2017 Collisionality scaling of the electron heat flux in ETG turbulence. Plasma Phys. Control. Fusion 59 (5), 055002.CrossRefGoogle Scholar
Connor, J. W. & Wilson, H. R. 2000 A review of theories of the L-H transition. Plasma Phys. Control. Fusion 1 (1), R1.CrossRefGoogle Scholar
Correa-Restrepo, D. & Pfirsch, D. 2005 The electromagnetic gauge in the variational formulation of kinetic and other theories. J. Plasma Phys. 71 (4), 503.CrossRefGoogle Scholar
Dannert, T. & Jenko, F. 2005 Gyrokinetic simulation of collisionless trapped-electron mode turbulence. Phys. Plasmas 12 (7), 072309.CrossRefGoogle Scholar
Dickinson, D., Roach, C. M., Saarelma, S., Scannell, R., Kirk, A. & Wilson, H. R. 2012 Kinetic instabilities that limit beta in the edge of a tokamak plasma: a picture of an H-mode pedestal. Phys. Rev. Lett. 108 (13), 1.CrossRefGoogle ScholarPubMed
Dimits, A. M. 2010 Gyrokinetic equations in an extended ordering. Phys. Plasmas 17 (5), 055901.CrossRefGoogle Scholar
Dimits, A. M. 2012 Gyrokinetic equations for strong-gradient regions. Phys. Plasmas 19 (2), 022504.CrossRefGoogle Scholar
Dimits, A. M., LoDestro, L. L. & Dubin, D. H. E. 1992 Gyroaveraged equations for both the gyrokinetic and drift-kinetic regimes. Phys. Fluids B 4 (1), 274.CrossRefGoogle Scholar
D’Ippolito, D. A., Myra, J. R. & Krasheninnikov, S. I. 2002 Cross-field blob transport in tokamak scrape-off-layer plasmas. Phys. Plasmas 9 (1), 222.CrossRefGoogle Scholar
Dorland, W. & Hammett, G. W. 1993 Gyrofluid turbulence models with kinetic effects. Phys. Fluids B 5 (3), 812.CrossRefGoogle Scholar
Dorland, W., Jenko, F., Kotschenreuther, M. & Rogers, B. N. 2000 Electron temperature gradient turbulence. Phys. Rev. Lett. 85 (26), 5579.CrossRefGoogle ScholarPubMed
Dougherty, J. P. 1964 Model Fokker–Planck equation for a plasma and its solution. Phys. Fluids 7 (11), 1788.CrossRefGoogle Scholar
Doyle, E. J., Groebner, R. J., Burrell, K. H., Gohil, P., Lehecka, T., Luhmann, N. C., Matsumoto, H., Osborne, T. H., Peebles, W. A. & Philipona, R. 1991 Modifications in turbulence and edge electric fields at the L-H transition in the DIII-D tokamak. Phys. Fluids B 3 (8), 2300.CrossRefGoogle Scholar
Doyle, E. J., Houlberg, W. A., Kamada, Y., Mukhovatov, V., Osborne, T. H., Polevoi, A., Bateman, G., Connor, J. W., Cordey, J. G., Fujita, T. et al. 2007 Chapter 2: plasma confinement and transport. Nucl. Fusion 47 (6), S18.Google Scholar
Dragt, A. J. & Finn, J. M. 1976 Lie series and invariant functions for analytic symplectic maps. J. Math. Phys. 17 (12), 2215.CrossRefGoogle Scholar
Dubin, D., Krommes, J. & Oberman, C. 1983 Nonlinear gyrokinetic equations. Phys. Fluids 26 (12), 3524.CrossRefGoogle Scholar
Dudson, B. D., Umansky, M. V., Xu, X. Q., Snyder, P. B. & Wilson, H. R. 2009 BOUT++: a framework for parallel plasma fluid simulations. Comput. Phys. Commun. 180 (9), 1467.CrossRefGoogle Scholar
Easy, L., Militello, F., Omotani, J., Dudson, B., Havlíĉková, E., Tamain, P., Naulin, V. & Nielsen, A. H. 2014 Three dimensional simulations of plasma filaments in the scrape off layer: a comparison with models of reduced dimensionality. Phys. Plasmas 21 (12), 122515.CrossRefGoogle Scholar
Elmore, S., Allan, S. Y., Kirk, A., Fishpool, G., Harrison, J., Tamain, P., Kočan, M., Gaffka, R., Stephen, R. & Bradley, J. W. 2012 Upstream and divertor ion temperature measurements on MAST by retarding field energy analyser. Plasma Phys. Control. Fusion 54 (6), 065001.CrossRefGoogle Scholar
Endler, M., Niedermeyer, H., Giannone, L., Kolzhauer, E., Rudyj, A., Theimer, G. & Tsois, N. 1995 Measurements and modelling of electrostatic fluctuations in the scrape-off layer of ASDEX. Nucl. Fusion 35 (11), 1307.CrossRefGoogle Scholar
Erents, S. K., Chankin, A. V., Matthews, G. F. & Stangeby, P. C. 2000 Parallel flow in the JET scrape-off layer. Plasma Phys. Control. Fusion 42 (8), 905.CrossRefGoogle Scholar
Estève, D., Garbet, X., Sarazin, Y., Grandgirard, V., Cartier-Michaud, T., Dif-Pradalier, G., Ghendrih, P., Latu, G. & Norscini, C. 2015 A multi-species collisional operator for full-F gyrokinetics. Phys. Plasmas 22 (12), 122506.CrossRefGoogle Scholar
Fonck, R. J., Cosby, G., Durst, R. D., Paul, S. F., Bretz, N., Scott, S., Synakowski, E. & Taylor, G. 1993 Long-wavelength density turbulence in the TFTR tokamak. Physica 70 (24), 3736.Google ScholarPubMed
Frenet, F. 1852 Sur les courbes à double courbure. Journal de Mathématiques Pures et Appliquées 17, 437.Google Scholar
Frieman, E. A. & Chen, L. 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25 (3), 502.CrossRefGoogle Scholar
Furnish, G., Horton, W., Kishimoto, Y., LeBrun, M. & Tajima, T. 1999 Global gyrokinetic simulation of tokamak transport. Phys. Plasmas 6 (4), 1227.CrossRefGoogle Scholar
Garcia, O. E., Horacek, J. & Pitts, R. A. 2015 Intermittent fluctuations in the TCV scrape-off layer. Nucl. Fusion 55 (6), 062002.CrossRefGoogle Scholar
Garcia, O. E., Horacek, J., Pitts, R. A., Nielsen, A. H., Fundamenski, W., Naulin, V. & Juul Rasmussen, J. 2007a Fluctuations and transport in the TCV scrape-off layer. Nucl. Fusion 47 (7), 667.CrossRefGoogle Scholar
Garcia, O. E., Pitts, R. A., Horacek, J., Madsen, J., Naulin, V., Nielsen, A. H. & Rasmussen, J. J. 2007b Collisionality dependent transport in TCV SOL plasmas. Plasma Phys. Control. Fusion 49 (12B), B47.CrossRefGoogle Scholar
Gillis, J. & Weiss, G. 1960 Products of Laguerre polynomials. Maths Comput. 14 (69), 60.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. Comput. Phys. 230 (18), 7053.CrossRefGoogle Scholar
Gohil, P., Burrell, K. H., Doyle, E. J., Groebner, R. J., Kim, J. & Seraydarian, R. P. 1994 The phenomenology of the L-H transition in the DIII-D tokamak. Nucl. Fusion 34 (8), 1057.CrossRefGoogle Scholar
Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence. 2: Strong alfvenic turbulence. Astrophys. J. 438, 763.CrossRefGoogle Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 2007 Table of Integrals, Series, and Products, 7th edn.Elsevier/Academic Press.Google Scholar
Graessle, D. E., Prager, S. C. & Dexter, R. N. 1991 Q Dependence of magnetic turbulence in a tokamak. Phys. Fluids B 3 (9), 2626.CrossRefGoogle Scholar
Grant, F. C. & Feix, M. C. 1967 Fourier–Hermite solutions of the vlasov equations in the linearized limit. Phys. Fluids 10 (4), 696.CrossRefGoogle Scholar
Green, B. J., Teams, ITER International Team & Participant 2003 Plasma physics and controlled fusion related content ITER: burning plasma physics experiment. Plasma Phys. Control. Fusion 45 (5), 687.CrossRefGoogle Scholar
Grulke, O., Terry, J. L., Cziegler, I., Labombard, B. & Garcia, O. E. 2014 Experimental investigation of the parallel structure of fluctuations in the scrape-off layer of Alcator C-Mod. Nucl. Fusion 54 (4), 043012.CrossRefGoogle Scholar
Guttenfelder, W. & Candy, J. 2011 Resolving electron scale turbulence in spherical tokamaks with flow shear. Phys. Plasmas 18 (2), 022506.CrossRefGoogle Scholar
Guttenfelder, W., Peterson, J. L., Candy, J., Kaye, S. M., Ren, Y., Bell, R. E., Hammett, G. W., LeBlanc, B. P., Mikkelsen, D. R. & Nevins, W. M. 2013 Progress in simulating turbulent electron thermal transport in NSTX. Nucl. Fusion 53 (9), 093022.CrossRefGoogle Scholar
Hahm, T. S. 1988 Nonlinear gyrokinetic equations for tokamak microturbulence. Phys. Fluids 31 (9), 2670.CrossRefGoogle Scholar
Hahm, T. S. 1996 Nonlinear gyrokinetic equations for turbulence in core transport barriers. Phys. Plasmas 3 (12), 4658.CrossRefGoogle Scholar
Hahm, T. S., Wang, L. & Madsen, J. 2009 Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence. Phys. Plasmas 16 (2), 022305.CrossRefGoogle Scholar
Hakim, A., Shi, E. L., Abel, I. G. & Hammett, G. W.2016 Scrape-Off layer turbulence in tokamaks simulated with a continuum gyrokinetic code. Preprint, arXiv:1610.09056.Google Scholar
Halpern, F. D., Jolliet, S., Loizu, J., Mosetto, A. & Ricci, P. 2013 Ideal ballooning modes in the tokamak scrape-off layer. Phys. Plasmas 20 (5), 052306.CrossRefGoogle Scholar
Halpern, F. D., Ricci, P., Jolliet, S., Loizu, J., Morales, J., Mosetto, A., Musil, F., Riva, F., Tran, T. M. & Wersal, C. 2016 The GBS code for tokamak scrape-off layer simulations. J. Comput. Phys. 315, 388.CrossRefGoogle Scholar
Halpern, F. D., Ricci, P., Jolliet, S., Loizu, J. & Mosetto, A. 2014 Theory of the scrape-off layer width in inner-wall limited tokamak plasmas. Nucl. Fusion 54 (4), 043003.CrossRefGoogle Scholar
Hammett, G. W., Beer, M. A., Dorland, W., Cowley, S. C. & Smith, S. A. 1993 Developments in the gyrofluid approach to Tokamak turbulence simulations. Plasma Phys. Control. Fusion 35 (8), 973.CrossRefGoogle Scholar
Hammett, G. W., Dorland, W. & Perkins, F. W. 1992 Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics. Phys. Fluids B 4 (7), 2052.CrossRefGoogle Scholar
Hammett, G. W. & Perkins, F. W. 1990 Fluid moment models for Landau damping with application to the ion-temperature-gradient instability. Phys. Rev. Lett. 64 (25), 3019.CrossRefGoogle ScholarPubMed
Hatch, D. R., Terry, P. W., Jenko, F., Merz, F. & Nevins, W. M. 2011 Saturation of gyrokinetic turbulence through damped eigenmodes. Phys. Rev. Lett. 106 (11), 1.CrossRefGoogle ScholarPubMed
Hazeltine, R. D. 1973 Recursive derivation of drift-kinetic equation. Plasma Phys. 15 (1), 77.CrossRefGoogle Scholar
Hazeltine, R. D. & Meiss, J. D. 2003 Plasma Confinement. Courier Corporation.Google Scholar
Heikkinen, J. A., Henriksson, S., Janhunen, S., Kiviniemi, T. P. & Ogando, F. 2006 Gyrokinetic simulation of particle and heat transport in the presence of wide orbits and strong profile variations in the edge plasma. Contrib. Plasma Phys. 46 (7–9), 490.CrossRefGoogle Scholar
Held, M., Wiesenberger, M. & Kendl, A.2019 Padé-based arbitrary wavelength polarization closures for full-f gyro-kinetic and -fluid models. Preprint, arXiv:1907.13439.Google Scholar
Held, M., Wiesenberger, M., Madsen, J. & Kendl, A. 2016 The influence of temperature dynamics and dynamic finite ion Larmor radius effects on seeded high amplitude plasma blobs. Nucl. Fusion 56 (12), 126005.CrossRefGoogle Scholar
Hirvijoki, E., Brizard, A. J. & Pfefferlé, D. 2017 Differential formulation of the gyrokinetic landau operator. J. Plasma Phys. 83 (1), 595830102.CrossRefGoogle Scholar
Hubbard, A. E. E. 2000 Physics and scaling of the H-mode pedestal. Plasma Phys. Control. Fusion 42 (5A), A15.CrossRefGoogle Scholar
Idomura, Y., Urano, H., Aiba, N. & Tokuda, S. 2009 Study of ion turbulent transport and profile formations using global gyrokinetic full-f Vlasov simulation. Nucl. Fusion 49 (6), 065029.CrossRefGoogle Scholar
Jackson, J. D. 2012 Classical Electrodynamics. Wiley.Google Scholar
Jenko, F., Dorland, W., Kotschenreuther, M. & Rogers, B. N. 2000 Electron temperature gradient driven turbulence. Phys. Plasmas 7 (5), 1904.CrossRefGoogle Scholar
Jorge, R., Frei, B. J. & Ricci, P. 2019a Nonlinear gyrokinetic Coulomb collision operator. J. Plasma Phys. 85 (6), 905850604.CrossRefGoogle Scholar
Jorge, R., Ricci, P., Brunner, S., Gamba, S., Konovets, V., Loureiro, N. F., Perrone, L. M. & Teixeira, N. 2019b Linear theory of electron-plasma waves at arbitrary collisionality. J. Plasma Phys. 85 (2), 905850211.CrossRefGoogle Scholar
Jorge, R., Ricci, P. & Loureiro, N. F. 2017 A full-F drift-kinetic analytical model for SOL plasma dynamics at arbitrary collisionality. J. Plasma Phys. 83 (6), 905830606.CrossRefGoogle Scholar
Jorge, R., Ricci, P. & Loureiro, N. F. 2018 Theory of the drift-wave instability at arbitrary collisionality. Phys. Rev. Lett. 121 (16), 165001.CrossRefGoogle ScholarPubMed
Kaufman, A. N. 1986 The electric-dipole of a guiding center and the plasma momentum density. Phys. Fluids 29 (1986), 1736.CrossRefGoogle Scholar
Kirk, A., Counsell, G. F., Arends, E., Meyer, H., Taylor, D., Valovic, M., Walsh, M., Wilson, H. et al. 2004 H-mode pedestal characteristics on mast. Plasma Phys. Control. Fusion 46 (5A), A187.CrossRefGoogle Scholar
Kirk, A., Wilson, H. R., Akers, R., Conway, N. J., Counsell, G. F., Cowley, S. C., Dowling, J., Dudson, B., Field, A., Lott, F. et al. 2005 Structure of ELMs in MAST and the implications for energy deposition. Plasma Phys. Control. Fusion 47 (2), 315.CrossRefGoogle Scholar
Kocan, M., Gunn, J. P., Pascal, J. Y., Bonhomme, G., Fenzi, C., Gauthier, E. & Segui, J. L. 2008 Edge ion-to-electron temperature ratio in the Tore Supra tokamak. Plasma Phys. Control. Fusion 50 (12), 125009.CrossRefGoogle Scholar
Krause, T. B., Apte, A. & Morrison, P. J. 2007 A unified approach to the Darwin approximation. Phys. Plasmas 14 (10), 102112.CrossRefGoogle Scholar
Krommes, J. A. 2012 The gyrokinetic description of microturbulence in magnetized plasmas. Annu. Rev. Fluid Mech. 44 (1), 175.CrossRefGoogle Scholar
Kruskal, M. D. 1965 Elementary orbit and drift theory. In Plasma Physics: Lectures presented at the seminar on plasma physics, organized by and held at the International Centre for Theoretical Physics, Trieste, from October 5–31, 1964, p. 67. International Atomic Energy Agency.Google Scholar
Ku, S., Chang, C. S. & Diamond, P. H. 2009 Full-f gyrokinetic particle simulation of centrally heated global ITG turbulence from magnetic axis to edge pedestal top in a realistic tokamak geometry. Nucl. Fusion 49 (11), 115021.CrossRefGoogle Scholar
LaBombard, B., Boivin, R. L., Greenwald, M., Hughes, J., Lipschultz, B., Mossessian, D., Pitcher, C. S., Terry, J. L. & Zweben, S. J. 2001 Particle transport in the scrape-off layer and its relationship to discharge density limit in Alcator C-Mod. Phys. Plasmas 8 (5), 2107.CrossRefGoogle Scholar
LaBombard, B., Hughes, J. W., Mossessian, D., Greenwald, M., Lipschultz, B., Terry, J. L., Team & the Alcator C-Mod 2005 Evidence for electromagnetic fluid drift turbulence controlling the edge plasma state in the Alcator C-Mod tokamak. Nucl. Fusion 45 (12), 1658.CrossRefGoogle Scholar
Lanthaler, S., Graves, J. P., Pfefferlé, D. & Cooper, W. A. 2019 Guiding-centre theory for kinetic-mhd modes in strongly flowing plasmas. Plasma Phys. Control. Fusion 61, 074006.CrossRefGoogle Scholar
Lanti, E., Ohana, N., Tronko, N., Hayward-Schneider, T., Bottino, A., McMillan, B. F., Mishchenko, A., Scheinberg, A., Biancalani, A., Angelino, P. et al. 2019 ORB5: a global electromagnetic gyrokinetic code using the PIC approach in toroidal geometry. Preprint, arXiv:1905.01906.CrossRefGoogle Scholar
Lee, W. W. 1983 Gyrokinetic approach in particle simulation. Phys. Fluids 26 (2), 556.CrossRefGoogle Scholar
Lee, W. W. & Kolesnikov, A. 2009 On higher order corrections to gyrokinetic Vlasov–Poisson equations in the long wavelength limit. Phys. Plasmas 16 (12), 044506.Google Scholar
Levinson, J., Beall, J. M., Powers, E. J. & Bengtson, R. D. 1984 Space/time statistics of the turbulence in a tokamak edge plasma. Nucl. Fusion 24 (5), 527.CrossRefGoogle Scholar
Li, B. & Ernst, D. R. 2011 Gyrokinetic Fokker–Planck collision operator. Phys. Rev. Lett. 106 (19), 1.CrossRefGoogle ScholarPubMed
Littlejohn, R. G. 1979 A guiding center Hamiltonian: a new approach. J. Math. Phys. 20 (12), 2445.CrossRefGoogle Scholar
Littlejohn, R. G. 1981 Hamiltonian formulation of guiding center motion. Phys. Fluids 24 (9), 1730.CrossRefGoogle Scholar
Littlejohn, R. G. 1982 Hamiltonian perturbation theory in noncanonical coordinates. J. Math. Phys. 23 (5), 742.CrossRefGoogle Scholar
Littlejohn, R. G. 1988 Phase anholonomy in the classical adiabatic motion of charged particles. Phys. Rev. A 38 (12), 6034.CrossRefGoogle ScholarPubMed
Lönnroth, J. S., Bateman, G., Bécoulet, M., Beyer, P., Corrigan, G., Figarella, C., Fundamenski, W., Garcia, O. E., Garbet, X., Huysmans, G. et al. 2006 Integrated ELM modelling. Contrib. Plasma Phys. 46 (7–9), 726.CrossRefGoogle Scholar
Loureiro, N. F., Dorland, W., Fazendeiro, L., Kanekar, A., Mallet, A., Vilelas, M. S. & Zocco, A. 2016 Viriato: a Fourier–Hermite spectral code for strongly magnetized fluid-kinetic plasma dynamics. Comput. Phys. Commun. 206, 45.CrossRefGoogle Scholar
Madsen, J. 2010 Second order guiding-center Vlasov–Maxwell equations. Phys. Plasmas 17 (8), 082107.CrossRefGoogle Scholar
Madsen, J. 2013 Full-F gyrofluid model. Phys. Plasmas 20 (7), 072301.CrossRefGoogle Scholar
Mandell, N. R., Dorland, W. & Landreman, M. 2018 Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics. J. Plasma Phys. 84 (01), 905840108.CrossRefGoogle Scholar
Mandell, N. R., Hakim, A., Hammett, G. W. & Francisquez, M.2019 Electromagnetic full-$f$ gyrokinetics in the tokamak edge with discontinuous galerkin methods. Preprint, arXiv:1908.05653.Google Scholar
Martin, Y. R. & Takizuka, T. 2008 Power requirement for accessing the H-mode in ITER. J. Phys.: Conf. Ser. 123 (1), 012033.Google Scholar
Mazzucato, E., Smith, D. R., Bell, R. E., Kaye, S. M., Hosea, J. C., Leblanc, B. P., Wilson, J. R., Ryan, P. M., Domier, C. W., Luhmann, N. C. et al. 2008 Short-scale turbulent fluctuations driven by the electron-temperature gradient in the national spherical torus experiment. Phys. Rev. Lett. 101 (7), 075001.CrossRefGoogle ScholarPubMed
Militello, F. & Omotani, J. T. 2016 Scrape off layer profiles interpreted with filament dynamics. Nucl. Fusion 56 (10), 104004.CrossRefGoogle Scholar
Militello, F., Tamain, P., Fundamenski, W., Kirk, A., Naulin, V., Nielsen, A. H. et al. 2013 Experimental and numerical characterization of the turbulence in the scrape-off layer of mast. Plasma Phys. Control. Fusion 55 (2), 025005.CrossRefGoogle Scholar
Miyato, N. & Scott, B. D. 2011 Fluid moments in the reduced model for plasmas with large flow velocity. Plasma Fusion Res. 6 (2011), 1.CrossRefGoogle Scholar
Miyato, N., Scott, B. D., Strintzi, D. & Tokuda, S. 2009 A modification of the guiding-centre fundamental 1-form with strong E $\times$ B flow. J. Phys. Soc. Japan 78 (10), 1.CrossRefGoogle Scholar
Miyato, N., Scott, B. D. & Yagi, M. 2013 On the gyrokinetic model in long wavelength regime. Plasma Phys. Control. Fusion 55 (7), 074011.CrossRefGoogle Scholar
Mosetto, A., Halpern, F. D., Jolliet, S., Loizu, J. & Ricci, P. 2013 Turbulent regimes in the tokamak scrape-off layer Turbulent regimes in the tokamak scrape-off layer. Phys. Plasmas 20 (9), 092308.CrossRefGoogle Scholar
Mosetto, A., Halpern, F. D., Jolliet, S., Loizu, J. & Ricci, P. 2015 Finite ion temperature effects on scrape-off layer turbulence. Phys. Plasmas 22 (1), 012308.CrossRefGoogle Scholar
Naulin, V. 2007 Turbulent transport and the plasma edge. J. Nucl. Mater. 363–365 (1–3), 24.CrossRefGoogle Scholar
Neiser, T. F., Jenko, F., Carter, T. A., Schmitz, L., Told, D., Merlo, G., Crandall, P. C., Mckee, G. & Yan, Z.2018 Gyrokinetic GENE simulations of DIII-D near-edge L-mode plasmas. Preprint, arXiv:1808.06607.Google Scholar
Nespoli, F., Furno, I., Labit, B., Ricci, P., Avino, F., Halpern, F. D., Musil, F. & Riva, F. 2017 Blob properties in full-turbulence simulations of the TCV scrape-off layer. Plasma Phys. Control. Fusion 59 (5), 055009.CrossRefGoogle Scholar
Northrop, T. G. 1963 The Adabatic Motion of Charged Particles. Interscience Publishers.Google Scholar
Omotani, J. T., Dudson, B. D., Havlìĉková, E. & Umansky, M. 2015 Non-local parallel transport in BOUT++. J. Nucl. Mater. 463, 769.CrossRefGoogle Scholar
Ono, M., Kaye, S. M., Peng, Y.-K. M., Barnes, G., Blanchard, W., Carter, M. D., Chrzanowski, J., Dudek, L., Ewig, R., Gates, D. et al. 2000 Exploration of spherical torus physics in the NSTX device. Nucl. Fusion 40 (3Y), 557.CrossRefGoogle Scholar
Pan, Q. & Ernst, D. R. 2019 Gyrokinetic Landau collision operator in conservative form. Phys. Rev. E 99 (2), 023201.Google ScholarPubMed
Pan, Q., Told, D. & Jenko, F. 2016 Fully nonlinear $\unicode[STIX]{x1D6FF}$f gyrokinetics for scrape-off layer parallel transport. Phys. Plasmas 23 (10), 102302.CrossRefGoogle Scholar
Pan, Q., Told, D., Shi, E., Hammett, G. W. & Jenko, F. 2018 Full-f version of GENE for turbulence in open-field-line systems. Phys. Plasmas 062303 (25), 1.Google Scholar
Parra, F. I. & Calvo, I. 2014 Corrigendum: phase-space Lagrangian derivation of electrostatic gyrokinetics in general geometry (2011 Plasma Phys. Control. Fusion 53 045001). Plasma Phys. Control. Fusion 56 (9), 099501.CrossRefGoogle Scholar
Parra, F. I. & Catto, P. J. 2008 Limitations of gyrokinetics on transport time scales. Plasma Phys. Control. Fusion 50 (6), 065014.CrossRefGoogle Scholar
Paruta, P., Ricci, P., Fiva, F., Wersal, C., Beadle, C. & Frei, B. 2018 Simulation of plasma turbulence in the periphery of diverted tokamak by using the GBS code. Phys. Plasmas 25 (11), 112301.CrossRefGoogle Scholar
Petty, C. C., Luce, T. C., DeBoo, J. C., Waltz, R. E., Baker, D. R. & Wade, M. R. 1998 Scaling of heat transport with beta in the DIII-D tokamak. Nucl. Fusion 38 (8), 1183.CrossRefGoogle Scholar
Pezzi, O., Valentini, F., Servidio, S. & Camporeale, E. 2019 Fourier–Hermite decomposition of the collisional Vlasov–Maxwell system: implications for the velocity-space cascade. Plasma Phys. Control. Fusion 61 (5), 054005.CrossRefGoogle Scholar
Pfirsch, D. & Morrison, P. J. 1985 Local conservation laws for the Maxwell–Vlasov and collisionless kinetic guiding-center theories. Phys. Rev. A 32 (3), 1714.CrossRefGoogle ScholarPubMed
Pitts, R. A., Alberti, S., Blanchard, P., Horacek, J., Reimerdes, H. & Stangeby, P. C. 2003 ELM driven divertor target currents on TCV. Nucl. Fusion 43 (10), 1145.CrossRefGoogle Scholar
Pitts, R. A., Horacek, J., Fundamenski, W., Garcia, O. E., Nielsen, A. H., Wischmeier, M., Naulin, V. & Juul Rasmussen, J. 2007 Parallel SOL flow on TCV. J. Nucl. Mater. 363, 505.CrossRefGoogle Scholar
Qin, H., Cohen, R. H., Nevins, W. M. & Xu, X. Q. 2006 General gyrokinetic equations for edge plasmas. Contrib. Plasma Phys. 46 (7–9), 477.CrossRefGoogle Scholar
Qin, H., Cohen, R. H., Nevins, W. M. & Xu, X. Q. 2007 Geometric gyrokinetic theory for edge plasmas. Phys. Plasmas 14 (5), 056110.CrossRefGoogle Scholar
Qin, H. & Tang, W. M. 2004 Pullback transformations in gyrokinetic theory. Phys. Plasmas 11 (3), 1052.CrossRefGoogle Scholar
Qin, H., Tang, W. M. & Rewoldt, G. 1998 Gyrokinetic theory for arbitrary wavelength electromagnetic modes in tokamaks. Phys. Plasmas 5 (4), 1035.CrossRefGoogle Scholar
Qin, H., Tang, W. M., Rewoldt, G. & Lee, W. W. 2000 On the gyrokinetic equilibrium. Phys. Plasmas 7 (3), 991.CrossRefGoogle Scholar
Ribeiro, T. T. & Scott, B. 2008 Gyrofluid turbulence studies of the effect of the poloidal position of an axisymmetric Debye sheath. Plasma Phys. Control. Fusion 50 (5), 055007.CrossRefGoogle Scholar
Ricci, P. 2015 Simulation of the scrape-off layer region of tokamak devices. J. Plasma Phys. 81 (2), 435810202.CrossRefGoogle Scholar
Ricci, P., Halpern, F. D., Jolliet, S., Loizu, J., Mosetto, A., Fasoli, A., Furno, I. & Theiler, C. 2012 Simulation of plasma turbulence in scrape-off layer conditions: the GBS code, simulation results and code validation. Plasma Phys. Control. Fusion 54 (12), 124047.CrossRefGoogle Scholar
Ritz, C. P. P., Brower, D. L., Rhodes, T. L., Bengtson, R. D., Levinson, S. J., Luhmann, N. C., Peebles, W. a. & Powers, E. J. 1987 Characterization of tokamak edge turbulence by far-infrared laser scattering and Langmuir probes. Nucl. Fusion 27 (7), 1125.CrossRefGoogle Scholar
Rogers, B. N. & Dorland, W. 2005 Noncurvature-driven modes in a transport barrier. Phys. Plasmas 12 (6), 1.CrossRefGoogle Scholar
Rogers, B. N. & Drake, J. F. 1997 Enhancement of turbulence in tokamaks by magnetic fluctuations. Phys. Rev. Lett. 79 (2), 229.CrossRefGoogle Scholar
Rogers, B. N., Drake, J. F. & Zeiler, A. 1998 Phase space of tokamak edge turbulence, the L-H transition, and the formation of the edge pedestal. Phys. Rev. Lett. 81 (20), 4396.CrossRefGoogle Scholar
Rogers, B. N. & Ricci, P. 2010 Low-frequency turbulence in a linear magnetized plasma. Phys. Rev. Lett. 104 (22), 225002.CrossRefGoogle Scholar
Rogers, B. N., Zhu, B. & Francisquez, M. 2018 Gyrokinetic theory of slab universal modes and the non-existence of the gradient drift coupling (GDC) instability. Phys. Plasmas 25 (5), 052115.CrossRefGoogle Scholar
Rossel, J. X., Moret, J. M., Coda, S., Sauter, O., Goodman, T. P., Felici, F., Testa, D. & Martin, Y. 2012 Edge-localized mode control by electron cyclotron waves in a tokamak plasma. Nucl. Fusion 52 (3), 032004.CrossRefGoogle Scholar
Saibene, G., Oyama, N., Lönnroth, J., Andrew, Y., De La Luna, E., Giroud, C., Huysmans, G. T. A., Kamada, Y., Kempenaars, M. A. H., Loarte, A. et al. 2007 The H-mode pedestal, ELMs and TF ripple effects in JT-60U/JET dimensionless identity experiments. Nucl. Fusion 47 (8), 969.CrossRefGoogle Scholar
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Plunk, G. G., Quataert, E. & Tatsuno, T. 2008 Gyrokinetic turbulence: a nonlinear route to dissipation through phase space. Plasma Phys. Control. Fusion 50 (12), 124024.CrossRefGoogle 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. Ser. 182 (1), 310.CrossRefGoogle Scholar
Schekochihin, A. A., Parker, J. T., Highcock, E. G. & Dellar, P. J. 2016 Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence. J. Plasma Phys. 82 (2), 905820212.CrossRefGoogle Scholar
Schirmer, J., Conway, G. D., Zohm, H., Suttrop, W., Team & the ASDEX Upgrade 2006 The radial electric field and its associated shear in the ASDEX upgrade tokamak. Nucl. Fusion 46 (9), S780.CrossRefGoogle Scholar
Scott, B. D. 1997 Three-dimensional computation of drift Alfvén turbulence. Plasma Phys. Control. Fusion 39 (10), 1635.CrossRefGoogle Scholar
Scott, B. D. 2003 Computation of electromagnetic turbulence and anomalous transport mechanisms in tokamak plasmas. Plasma Phys. Control. Fusion 45 (12A), A385.CrossRefGoogle Scholar
Scott, B. D. 2007 Tokamak edge turbulence: background theory and computation. Plasma Phys. Control. Fusion 49 (7), S25.CrossRefGoogle Scholar
Scott, B. D. 2010 Derivation via free energy conservation constraints of gyrofluid equations with finite-gyroradius electromagnetic nonlinearities. Phys. Plasmas 17 (10), 102306.CrossRefGoogle Scholar
Shats, M. G., Xia, H. & Punzmann, H. 2005 Spectral condensation of turbulence in plasmas and fluids and its role in low-to-high phase transitions in toroidal plasma. Phys. Rev. E 71 (4), 1.CrossRefGoogle ScholarPubMed
Shi, E. L., Hakim, A. H. & Hammett, G. W. 2015 A gyrokinetic one-dimensional scrape-off layer model of an edge-localized mode heat pulse. Phys. Plasmas 22 (2), 022504.CrossRefGoogle Scholar
Shi, E. L., Hammett, G. W., Stoltzfus-Dueck, T. & Hakim, A. 2017 Gyrokinetic continuum simulation of turbulence in open-field-line plasmas. J. Plasma Phys. 83 (2017), 1.CrossRefGoogle Scholar
Shimada, M., Campbell, D. J., Mukhovatov, V., Fujiwara, M., Kirneva, N., Lackner, K., Nagami, M., Pustovitov, V. D., Uckan, N., Wesley, J. et al. 2007 Chapter 1: overview and summary. Nucl. Fusion 47 (6), S1.CrossRefGoogle Scholar
Sigmar, D. J., Batishcheva, A. A., Batishchev, O. V., Krasheninnikov, S. I. & Catto, P. J. 1996 Kinetic models of ELMs burst. Contrib. Plasma Phys. 36 (2–3), 230.CrossRefGoogle Scholar
Simakov, A. N. & Catto, P. J. 2005 Drift kinetic equation exact through second order in gyroradius expansion. Phys. Plasmas 12 (1), 1.CrossRefGoogle Scholar
Snyder, P. B., Aiba, N., Beurskens, M., Groebner, R. J., Horton, L. D., Hubbard, A. E., Hughes, J. W., Huysmans, G. T. A., Kamada, Y., Kirk, A. et al. 2009a Pedestal stability comparison and iter pedestal prediction. Nucl. Fusion 49 (8), 085035.CrossRefGoogle Scholar
Snyder, P. B. B., Groebner, R. J. J., Hughes, J. W. W., Osborne, T. H. H., Beurskens, M., Leonard, A. W. W., Wilson, H. R. R. & Xu, X. Q. Q. 2011 A first-principles predictive model of the pedestal height and width: development, testing and ITER optimization with the EPED model. Nucl. Fusion 51 (10), 103016.CrossRefGoogle Scholar
Snyder, P. B., Groebner, R. J., Leonard, A. W., Osborne, T. H. & Wilson, H. R. 2009b Development and validation of a predictive model for the pedestal height. Phys. Plasmas 16 (5), 056118.CrossRefGoogle Scholar
Snyder, P. B. & Hammett, G. W. 2001 A Landau fluid model for electromagnetic plasma microturbulence. Phys. Plasmas 8 (7), 3199.CrossRefGoogle Scholar
Snyder, P. B., Osborne, T. H., Burrell, K. H., Groebner, R. J., Leonard, A. W., Nazikian, R., Orlov, D. M., Schmitz, O., Wade, M. R. & Wilson, H. R. 2012 The EPED pedestal model and edge localized mode-suppressed regimes: studies of quiescent H-mode and development of a model for edge localized mode suppression via resonant magnetic perturbations. Phys. Plasmas 19 (5), 056115.CrossRefGoogle Scholar
Snyder, P. B., Wilson, H. R., Ferron, J. R., Lao, L. L., Leonard, A. W., Osborne, T. H., Turnbull, A. D., Mossessian, D., Murakami, M. & Xu, X. Q. 2002 Edge localized modes and the pedestal: a model based on coupled peeling-ballooning modes. Phys. Plasmas 9 (5), 2037.CrossRefGoogle Scholar
Sötckel, J., Badalec, J., Duran, I., Hron, M., Horacek, J., Jakubka, K., Kryska, L., Petrzilka, J., Zacek, F. & Heller, M. V. P. 1999 Plasma physics and controlled fusion magnetic and electrostatic fluctuations in the CASTOR tokamak. Plasma Phys. Control. Fusion 41 (3A), A577.CrossRefGoogle 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 (2), 022501.CrossRefGoogle Scholar
Stoneking, M. R., Hokin, S. A., Prager, S. C., Fiksel, G., Ji, H. & Den Hartog, D. J. 1994 Particle transport due to magnetic fluctuations. Phys. Rev. Lett. 73 (4), 549.CrossRefGoogle ScholarPubMed
Strintzi, D. & Scott, B. 2004 Nonlocal nonlinear electrostatic gyrofluid equations. Phys. Plasmas 11 (12), 5452.CrossRefGoogle Scholar
Sugama, H. 2000 Gyrokinetic field theory. Phys. Plasmas 7 (2000), 466.CrossRefGoogle Scholar
Sugama, H. & Nishimura, S. 2008 Moment-equation methods for calculating neoclassical transport coefficients in general toroidal plasmas. Phys. Plasmas 15 (4), 042502.CrossRefGoogle Scholar
Sugama, H., Nunami, M., Nakata, M. & Watanabe, T. H. 2017 Conservation laws for collisional and turbulent transport processes in toroidal plasmas with large mean flows. Phys. Plasmas 24 (2), 020701.CrossRefGoogle Scholar
Sugama, H., Watanabe, T. H. & Nunami, M. 2009 Linearized model collision operators for multiple ion species plasmas and gyrokinetic entropy balance equations. Phys. Plasmas 16 (11), 112503.CrossRefGoogle Scholar
Sugama, H., Watanabe, T. H. & Nunami, M. 2014 Extended gyrokinetic field theory for time-dependent magnetic confinement fields. Phys. Plasmas 21 (1), 012515.CrossRefGoogle Scholar
Sugama, H., Watanabe, T. H. & Nunami, M. 2015 Effects of collisions on conservation laws in gyrokinetic field theory. Phys. Plasmas 22 (8), 082306.CrossRefGoogle Scholar
Tamain, P., Ghendrih, P., Tsitrone, E., Sarazin, Y., Garbet, X., Grandgirard, V., Gunn, J., Serre, E., Ciraolo, G. & Chiavassa, G. 2009 3D modelling of edge parallel flow asymmetries. J. Nucl. Mater. 390 (1), 347.CrossRefGoogle Scholar
Tatsuno, T., Dorland, W., Schekochihin, A. A., Plunk, G. G., Barnes, M., Cowley, S. C. & Howes, G. G. 2009 Nonlinear phase mixing and phase-space cascade of entropy in gyrokinetic plasma turbulence. Phys. Rev. Lett. 103 (1), 1.CrossRefGoogle ScholarPubMed
Thomsen, H., Endler, M., Bleuel, J., Chankin, A. V., Erents, S. K. & Matthews, G. F. 2002 Parallel correlation measurements in the scrape-off layer of the Joint European Torus. Phys. Plasmas 9 (4), 1233.CrossRefGoogle Scholar
Tronko, N., Bottino, A., Chandre, C. & Sonnendruecker, E. 2017a Hierarchy of second order gyrokinetic Hamiltonian models for particle-in-cell codes. Plasma Phys. Control. Fusion 59 (6), 064008.CrossRefGoogle Scholar
Tronko, N., Bottino, A., Görler, T., Sonnendrücker, E., Told, D. & Villard, L. 2017b Verification of Gyrokinetic codes: theoretical background and applications. Phys. Plasmas 24 (5), 056115.CrossRefGoogle Scholar
Tronko, N., Bottino, A. & Sonnendrücker, E. 2016 Second order gyrokinetic theory for particle-in-cell codes. Phys. Plasmas 23 (8), 082505.CrossRefGoogle Scholar
Tskhakaya, D. 2012 On recent massively parallelized PIC simulations of the SOL. Contrib. Plasma Phys. 52 (5–6), 490.CrossRefGoogle Scholar
Wagner, F. 2007 A quarter-century of H-mode studies. Plasma Phys. Control. Fusion 49 (12B), B1.CrossRefGoogle Scholar
Wagner, F. 2009 The physics basis of ITER confinement. AIP Conf. Proc. 1095 (1), 31.Google Scholar
Waltz, R. E., Kerbel, G. D. & Milovich, J. 1994 Toroidal gyro-Landau fluid model turbulence simulations in a nonlinear ballooning mode representation with radial modes. Phys. Plasmas 4 (10), 3138.Google Scholar
Wang, L. & Hahm, T. S. 2010 Nonlinear gyrokinetic theory with polarization drift. Phys. Plasmas 17 (8), 082304.Google Scholar
Wang, W. H., He, Y. X., Gao, Z., Zeng, L., Zhang, G. P., Xie, L. F., Yang, X. Z., Feng, C. H., Wang, L., Xiao, Q. et al. 2004 Edge plasma electrostatic fluctuation and anomalous transport characteristics in the Sino-united spherical tokamak. Plasma Phys. Control. Fusion 47 (1), 1.CrossRefGoogle Scholar
Wimmel, H. K. 1984 Kinetic guiding-center equations for the theory of drift instabilities and anomalous transport. Phys. Scr. 29 (2), 141.Google Scholar
Winslow, D. L., Bengtson, R. D., Richards, B. & Craven, W. A. 1997 Determination of field line location and safety factor in TEXT-U. Rev. Sci. Instrum. 68 (1), 396.CrossRefGoogle Scholar
Wolf, R. C. 2002 Internal transport barriers in tokamak plasmas. Plasma Phys. Control. Fusion 45 (1), R1.CrossRefGoogle Scholar
Wong, M. W. 1998 The Weyl Transform. Springer.Google Scholar
Wootton, A. J., Carreras, B. A., Matsumoto, H., McGuire, K., Peebles, W. A., Ritz, C. P., Terry, P. W. & Zweben, S. J. 1990 Fluctuations and anomalous transport in tokamaks. Phys. Fluids B 2 (12), 2879.CrossRefGoogle Scholar
Xia, H., Shats, M. G. & Punzmann, H. 2006 Strong E$\times$B shear flows in the transport-barrier region in H-mode plasma. Phys. Rev. Lett. 97 (25), 1.CrossRefGoogle ScholarPubMed
Xu, G. S., Naulin, V., Fundamenski, W., Hidalgo, C., Alonso, J. A., Silva, C., Goncalves, B., Nielsen, A. H., Juul Rasmussen, J., Krasheninnikov, S. I. et al. 2009 Blob/hole formation and zonal-flow generation in the edge plasma of the JET tokamak. Nucl. Fusion 49 (9), 092002.CrossRefGoogle Scholar
Xu, G. S., Wan, B. N., Song, M. & Li, J. 2003 Direct measurement of poloidal long-wavelength E $\times$ B flows in the HT-7 Tokamak. Phys. Rev. Lett. 91 (12), 1.CrossRefGoogle Scholar
Xu, X. Q., Xiong, Z., Dorr, M. R., Hittinger, J. A., Bodi, K., Candy, J., Cohen, B. I., Cohen, R. H., Colella, P. & Kerbel, G. D. 2007 Edge gyrokinetic theory and continuum simulations. Nucl. Fusion 47 (8), 809.CrossRefGoogle Scholar
Zeiler, A., Biskamp, D., Drake, J. F. & Rogers, B. N. 1998 Transition from resistive ballooning to $\unicode[STIX]{x1D708}_{i}$ driven turbulence in tokamaks. Phys. Plasmas 5 (7), 2654.CrossRefGoogle Scholar
Zeiler, a., Drake, J. F. & Rogers, B. 1997 Nonlinear reduced Braginskii equations with ion thermal dynamics in toroidal plasma. Phys. Plasmas 4 (6), 2134.CrossRefGoogle Scholar
Zocco, A., Loureiro, N. F., Dickinson, D., Numata, R. & Roach, C. M. 2015 Kinetic microtearing modes and reconnecting modes in strongly magnetised slab plasmas. Plasma Phys. Control. Fusion 57 (6), 065008.CrossRefGoogle Scholar
Zocco, A. & Schekochihin, A. A. 2011 Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection, and electron heating in low-beta plasmas. Phys. Plasmas 18 (10), 102309.CrossRefGoogle Scholar
Zohm, H. 1996 Edge localized modes (ELMs). Plasma Phys. Control. Fusion 38 (2), 105.CrossRefGoogle Scholar
Zweben, S. J., Boedo, J. A., Grulke, O., Hidalgo, C., LaBombard, B., Maqueda, R. J., Scarin, P. & Terry, J. L. 2007 Edge turbulence measurements in toroidal fusion devices. Plasma Phys. Control. Fusion 49 (7), S1.CrossRefGoogle Scholar
Zweben, S. J., Davis, W. M., Kaye, S. M., Myra, J. R., Bell, R. E., Leblanc, B. P., Maqueda, R. J., Munsat, T., Sabbagh, S. A., Sechrest, Y. et al. 2015 Edge and SOL turbulence and blob variations over a large database in NSTX. Nucl. Fusion 55 (9), 093035.CrossRefGoogle Scholar