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A drift-kinetic analytical model for scrape-off layer plasma dynamics at arbitrary collisionality

Part of: Focus on Fusion

Published online by Cambridge University Press:  20 November 2017

R. Jorge Open the ORCID record for R. Jorge [Opens in a new window] ,
P. Ricci  and
N. F. Loureiro
R. Jorge*
Affiliation:
École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC), CH-1015 Lausanne, Switzerland Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
P. Ricci
Affiliation:
École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC), CH-1015 Lausanne, Switzerland
N. F. Loureiro
Affiliation:
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: rogerio.jorge@epfl.ch
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Abstract

A drift-kinetic model to describe the plasma dynamics in the scrape-off layer region of tokamak devices at arbitrary collisionality is derived. Our formulation is based on a gyroaveraged Lagrangian description of the charged particle motion, and the corresponding drift-kinetic Boltzmann equation that includes a full Coulomb collision operator. Using a Hermite–Laguerre velocity space decomposition of the gyroaveraged distribution function, a set of equations to evolve the coefficients of the expansion is presented. By evaluating explicitly the moments of the Coulomb collision operator, distribution functions arbitrarily far from equilibrium can be studied at arbitrary collisionalities. A fluid closure in the high-collisionality limit is presented, and the corresponding fluid equations are compared with previously derived fluid models.

Keywords

fusion plasmaplasma confinementplasma nonlinear phenomena
Type
Research Article
Information
Journal of Plasma Physics , Volume 83 , Issue 6 , December 2017 , 905830606
Copyright
© Cambridge University Press 2017 

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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. Prog. Phys. 76 (11), 116201.CrossRefGoogle ScholarPubMed
Abramowicz, M., Czerny, B., Lasota, J. & Szuszkiewicz, E. 1988 Slim accretion disks. Astrophys. J. 332, 646.CrossRefGoogle Scholar
Abramowitz, M., Stegun, I. & Miller, D. 1965 Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics Series No. 55, vol. 32. United States Department of Commerce, National Institute of Standards and Technology (NBS).Google Scholar
Agostini, M., Terry, J., Scarin, P. & Zweben, S. 2011 Edge turbulence in different density regimes in Alcator C-Mod experiment. Nucl. Fusion 51 (5), 053020.CrossRefGoogle Scholar
Armstrong, T. 1967 Numerical Studies of the Nonlinear Vlasov Equation. Phys. Fluids 10 (6), 1269.CrossRefGoogle Scholar
Balescu, R. 1988 Transport Processes in Plasmas. North-Holland.Google 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 (5/2), 1672–1680.Google 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.Google Scholar
Beer, M. & Hammett, G. 1996 Toroidal gyrofluid equations for simulations of tokamak turbulence. Phys. Plasmas 3 (1996), 4046.Google Scholar
Braginskii, S. I. 1965 Transport Processes in a Plasma. Consultants Bureau.Google Scholar
Brizard, A. 1992 Nonlinear gyrofluid description of turbulent magnetized plasmas. Phys. Fluids B 4 (1992), 1213.CrossRefGoogle Scholar
Camporeale, E., Delzanno, G. L., Bergen, B. K. & Moulton, J. D. 2016 On the velocity space discretization for the Vlasov–Poisson system: comparison between implicit Hermite spectral and Particle-in-Cell methods. Comput. Phys. Commun. 198, 47–58.CrossRefGoogle Scholar
Carralero, D., Birkenmeier, G., Müller, H., Manz, P., Demarne, P., Müller, S., Reimold, F., Stroth, U., Wischmeier, M. & Wolfrum, E. 2014 An experimental investigation of the high density transition of the scrape-off layer transport in ASDEX Upgrade. Nucl. Fusion 54 (12), 123005.Google Scholar
Carreras, B. 2005 Plasma edge cross-field transport: experiment and theory. J. Nucl. Mater. 337–339 (1–3), 315–321.Google Scholar
Cary, J. & Brizard, A. 2009 Hamiltonian theory of guiding-center motion. Rev. Mod. Phys. 81 (2), 693–738.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), 90–102.Google Scholar
Chang, C., Ku, S., Tynan, G., Hager, R., Churchill, R., Cziegler, I., Greenwald, M., Hubbard, A. & Hughes, J. 2017 Fast low-to-high confinement mode bifurcation dynamics in a tokamak edge plasma gyrokinetic simulation. Phys. Rev. Lett. 118 (17), 175001.Google Scholar
Chapman, S. 1962 The Mathematical Theory of Non-Uniform Gases.Google Scholar
D’Ippolito, D., Myra, J. & Krasheninnikov, S. 2002 Cross-field blob transport in tokamak scrape-off-layer plasmas. Phys. Plasmas 9 (1), 222.Google Scholar
D’Ippolito, D., Myra, J. & Zweben, S. 2011 Convective transport by intermittent blob-filaments: comparison of theory and experiment. Phys. Plasmas 18 (6), 060501.Google Scholar
Dorland, W. & Hammett, G. W. 1993 Gyrofluid turbulence models with kinetic effects. Phys. Fluids B-Plasma Phys. 5 (1993), 812–835.CrossRefGoogle Scholar
Dubin, D., Krommes, J. & Oberman, C. 1983 Nonlinear gyrokinetic equations. Phys. Fluids 26 (12), 3524.CrossRefGoogle Scholar
Dudson, B., Umansky, M., Xu, X., Snyder, P. & Wilson, H. 2009 BOUT++: a framework for parallel plasma fluid simulations. Comput. Phys. Commun. 180 (9), 1467–1480.Google Scholar
Easy, L., Militello, F., Omotani, J., Dudson, B., Havlíčková, 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.Google 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), 1307–1339.Google Scholar
Erents, S., Chankin, A., Matthews, G. & Stangeby, P. 2000 Parallel flow in the JET scrape-off layer. Plasma Phys. Control. Fusion 42 (8), 905–915.Google Scholar
Garcia, O., Horacek, J. & Pitts, R. A. 2015 Intermittent fluctuations in the TCV scrape-off layer. Nucl. Fusion 55 (6), 062002.Google Scholar
Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2 (4), 331–407.CrossRefGoogle Scholar
Grant, F. C. & Feix, M. C. 1967 Fourier–Hermite solutions of the Vlasov equations in the linearized limit. Phys. Fluids 10 (4), 696–702.Google Scholar
Grošelj, D., Cerri, S. S., Navarro, A. B., Willmott, C., Told, D., Loureiro, N. F., Califano, F. & Jenko, F 2017 Fully kinetic versus reduced-kinetic modeling of collisionless plasma turbulence. Astrophys. J. 847 (1), 28.Google 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), 43012.Google Scholar
Hahm, T. 1988 Nonlinear gyrokinetic equations for tokamak microturbulence. Phys. Fluids 31 (9), 2670–2673.Google Scholar
Hahm, T. S., Wang, L. & Madsen, J. 2009 Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence. Phys. Plasmas 16 (2), 022305.Google Scholar
Halpern, F., Ricci, P., Jolliet, S., Loizu, J., Morales, J., Mosetto, A., Musil, F., Riva, F., Tran, T. & Wersal, C. 2016 The GBS code for tokamak scrape-off layer simulations. J. Comput. Phys. 315, 388–408.Google Scholar
Hammett, G., Dorland, W. & Perkins, F. 1992 Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics. Phys. Fluids B 4 (7), 2052–2061.Google Scholar
Hammett, G. W., Beer, M. A., Dorland, W. D., Cowley, S. C. & Smith, S. A. 1993 Developments in the gyrofluid approach to Tokamak turbulence simulations. Plasma Phys. Control. Fusion 35 (8), 973–985.CrossRefGoogle Scholar
Hatch, D. R., Jenko, F., Navarro, A. B., Bratanov, V., Terry, P. W. & Pueschel, M. J. 2016 Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra. New J. Phys. 18 (7), 075018.CrossRefGoogle Scholar
Hazeltine, R. D. 1998 Transport theory in the collisionless limit. Phys. Plasmas 5 (9), 3282.CrossRefGoogle Scholar
Hazeltine, R. D. & Meiss, J. D. 2003 Plasma Confinement. Dover.Google Scholar
Helander, P. & Sigmar, D. 2005 Collisional Transport in Magnetized Plasmas, Cambridge Monographs on Plasma Physics. Cambridge University Press.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.Google Scholar
Hidalgo, C., Gonçalves, B., Pedrosa, M., Castellano, J., Erents, K., Fraguas, L., Hron, M., Jimenez, J., Matthews, G., van Milligen, B. et al. 2002 Empirical similarity in the probability density function of turbulent transport in the edge plasma region in fusion plasmas. Plasma Phys. Control. Fusion 44, 1557–1564.Google Scholar
Hinton, F. & Hazeltine, R. D. 1976 Theory of plasma transport in toroidal confinements systems. Rev. Mod. Phys. 48 (2), 239–308.CrossRefGoogle Scholar
Hirvijoki, E., Lingam, M., Pfefferlé, D., Comisso, L., Candy, J. & Bhattacharjee, A. 2016 Fluid moments of the nonlinear Landau collision operator. Phys. Plasmas 23 (8), 1–10.Google Scholar
Jackson, J. 1998 Classical electrodynamics, 3rd Edition. Am. J. Phys. 841 (1999), 159.Google Scholar
Ji, J.-Y. & Held, E. D. 2006 Exact linearized Coulomb collision operator in the moment expansion. Phys. Plasmas 13 (10), 102103.Google Scholar
Ji, J.-Y. & Held, E. D. 2008 Landau collision operators and general moment equations for an electron-ion plasma. Phys. Plasmas 15 (10), 102101.Google Scholar
Ji, J.-Y. & Held, E. D. 2009 Full Coulomb collision operator in the moment expansion. Phys. Plasmas 16 (10), 102108.Google Scholar
Ji, J.-Y. & Held, E. D. 2010 Analytical solution of the kinetic equation for a uniform plasma in a magnetic field. Phys. Rev. E 82 (1), 1–5.Google Scholar
Jorge, R., Ricci, P., Halpern, F., Loureiro, N. F. & Silva, C. 2016 Plasma turbulence in the scrape-off layer of the ISTTOK tokamak. Phys. Plasmas 23 (10), 102511.Google Scholar
Kočan, M., Gunn, J., Carpentier-Chouchana, S., Herrmann, A., Kirk, A., Komm, M., Müller, H., Pascal, J., Pitts, R., Rohde, V. et al. 2011 Measurements of ion energies in the tokamak plasma boundary. J. Nucl. Mater. 415 (1 suppl), S1133–S1138.CrossRefGoogle Scholar
Krommes, J. A. 2013 The physics of the second-order gyrokinetic magnetohydrodynamic Hamiltonian: $\unicode[STIX]{x1D707}$ conservation, Galilean invariance, and ponderomotive potential. Phys. Plasmas 20 (12), 124501.Google Scholar
LaBombard, B., Boivin, R., Greenwald, M., Hughes, J., Lipschultz, B., Mossessian, D., Pitcher, C., Terry, J. & Zweben, S. 2001 Particle transport in the scrape-off layer and its relationship to discharge density limit in Alcator C-Mod. Phys. Plasmas 8 (5 II), 2107–2117.CrossRefGoogle Scholar
Labombard, B., Hughes, J., Mossessian, D., Greenwald, M., Lipschultz, B., Terry, J.& the Alcator C-Mod Team 2005 Evidence for electromagnetic fluid drift turbulence controlling the edge plasma state in the Alcator C-Mod tokamak. Nucl. Fusion 45 (12), 1658.Google Scholar
Lee, W. 1983 Gyrokinetic approach in particle simulation. Phys. Fluids 26 (2), 556.Google Scholar
Leonard, A. W. 2014 Edge-localized-modes in tokamaks. Phys. Plasmas 21 (9), 090501.CrossRefGoogle Scholar
Liang, Y., Koslowski, H. R., Thomas, P. R., Nardon, E., Alper, B., Andrew, P., Andrew, Y., Arnoux, G., Baranov, Y., Bécoulet, M. et al. 2007 Active control of type-I edge-localized modes with n=1 perturbation fields in the JET tokamak. Phys. Rev. Lett. 98 (26), 265004.Google Scholar
Littlejohn, R. G. 1983 Variational principles of guiding centre motion. J. Plasma Phys. 29 (1), 111–125.CrossRefGoogle Scholar
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), 726–738.Google 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, 45–63.Google Scholar
Loureiro, N. F., Schekochihin, A. A. & Zocco, A. 2013 Fast collisionless reconnection and electron heating in strongly magnetized plasmas. Phys. Rev. Lett. 111 (2), 025002.Google Scholar
Madsen, J. 2013 Full-F gyrofluid model. Phys. Plasmas 20 (7), 072301.Google Scholar
Madsen, J., Naulin, V., Nielsen, A. H. & Rasmussen, J. J. 2016 Collisional transport across the magnetic field in drift-fluid models. Phys. Plasmas 23 (3), 032306.Google Scholar
Mandell, N. R., Dorland, W. & Landreman, M.2017 Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics. arXiv:1708.04029.CrossRefGoogle Scholar
Martin, Y., Takizuka, T.& Group, the ITPA CDBM H-mode Threshold Data 2008 Power requirement for accessing the H-mode in ITER. J. Phys.: Conf. Ser. 123, 012033.Google Scholar
Mikhailovskii, A. & Tsypin, V. 1971 Transport equations and gradient instabilities in a high pressure collisional plasma. Plasma Phys. 13 (9), 785–798.Google Scholar
Mosetto, A., Halpern, F., Jolliet, S., Loizu, J. & Ricci, P. 2015 Finite ion temperature effects on scrape-off layer turbulence. Phys. Plasmas 22 (1), 0–17.Google Scholar
Myra, J., Davis, W., D’Ippolito, D., Labombard, B., Russell, D., Terry, J. & Zweben, S. 2013 Edge sheared flows and the dynamics of blob-filaments. Nucl. Fusion 53 (7), 073013.CrossRefGoogle Scholar
Nespoli, F., Furno, I., Labit, B., Ricci, P., Avino, F., Halpern, F., Musil, F. & Riva, F. 2017 Blob properties in full-turbulence simulations of the TCV scrape-off layer. Plasma Phys. Control. Fusion 59 (5), 055009.Google Scholar
Ng, C. S., Bhattacharjee, A. & Skiff, F. 1999 Kinetic eigenmodes and discrete spectrum of plasma oscillations in a weakly collisional plasma. Phys. Rev. Lett. 83 (10), 1974–1977.Google Scholar
Omotani, J., Dudson, B., Havlíčková, E. & Umansky, M. 2015 Non-local parallel transport in BOUT++. J. Nucl. Mater. 463, 769–772.Google Scholar
Parker, J. T.2016 Gyrokinetic simulations of fusion plasmas using a spectral velocity space representation. arXiv:1603.04727.Google Scholar
Parker, J. T. & Dellar, P. J. 2015 Fourier–Hermite spectral representation for the Vlasov–Poisson system in the weakly collisional limit. J. Plasma Phys. 2 (2011), 1–36.Google Scholar
Paškauskas, R. & De Ninno, G. 2009 Lyapunov stability of Vlasov equilibria using Fourier–Hermite modes. Phys. Rev. E 80 (3), 036402.CrossRefGoogle ScholarPubMed
Pitts, R., Alberti, S., Blanchard, P., Horacek, J., Reimerdes, H. & Stangeby, P. 2003 ELM driven divertor target currents on TCV. Nucl. Fusion 43 (10), 1145–1166.CrossRefGoogle Scholar
Qin, H., Cohen, R., Nevins, W. & Xu, X. 2007 Geometric gyrokinetic theory for edge plasmas. Phys. Plasmas 14 (5), 056110.Google Scholar
Ribeiro, 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.Google Scholar
Ritz, C., Brower, D., Rhodes, T., Bengtson, R., Levinson, S., Luhmann, N., Peebles, W. & Powers, E. 1987 Characterization of tokamak edge turbulence by far-infrared laser scattering and Langmuir probes. Nucl. Fusion 27 (7), 1125–1134.CrossRefGoogle Scholar
Rossel, J., Moret, J., Coda, S., Sauter, O., Goodman, T., Felici, F., Testa, D. & Martin, Y. 2012 Edge-localized mode control by electron cyclotron waves in a tokamak plasma. Nucl. Fusion 52 (3), 2–6.Google Scholar
Schekochihin, A. A., Parker, J. T., Highcock, E. G., Dellar, P. J., Dorland, W. & Hammett, G. 2016 Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence. J. Plasma Phys. 82 (02), 905820212.Google Scholar
Serianni, G., Agostini, M., Antoni, V., Cavazzana, R., Martines, E., Sattin, F., Scarin, P., Spada, E., Spolaore, M., Vianello, N. et al. 2007 Coherent structures and transport properties in magnetized plasmas. Plasma Phys. Control. Fusion 49 (12B), B267–B280.Google Scholar
Shi, E., Hakim, A. & Hammett, G. 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 a straight open-field-line plasma. J. Plasma Phys. 83 (03), 905830304.Google Scholar
Silva, C., Figueiredo, H., Duarte, P. & Fernandes, H. 2011 Poloidal asymmetries in the ISTTOK edge plasma. J. Nucl. Mater. 415 (1), S455–S458.CrossRefGoogle Scholar
Snyder, P. & Hammett, G. 2001 A Landau fluid model for electromagnetic plasma microturbulence. Phys. Plasmas 8 (7), 3199–3216.Google Scholar
Stangeby, P. 2000 The Plasma Boundary of Magnetic Fusion Devices, Series in Plasma Physics, vol. 43. CRC Press.Google Scholar
Sugama, H., Watanabe, T.-H. & Horton, W. 2001 Collisionless kinetic-fluid closure and its application to the three-mode ion temperature gradient driven system. Phys. Plasmas 8 (6), 2617–2628.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. 390–391 (1), 347–350.Google Scholar
Tassi, E. 2016 Hamiltonian reduced fluid model for plasmas with temperature and heat flux anisotropies. Theor. Math. Phys. 188 (3), 1377–1393.Google Scholar
Terry, J. L., Zweben, S. J., Umansky, M. V., Cziegler, I., Grulke, O., Labombard, B. & Stotler, D. P. 2009 Spatial structure of scrape-off-layer filaments near the midplane and X-point regions of Alcator-C-Mod. J. Nucl. Mater. 390–391 (1), 339–342.Google Scholar
Tskhakaya, D. 2012 On recent massively parallelized PIC simulations of the SOL. Contrib. Plasma Phys. 52 (5–6), 490–499.CrossRefGoogle Scholar
Tskhakaya, D., Subba, F., Bonnin, X., Coster, D. P., Fundamenski, W. & Pitts, R. A. 2008 On kinetic effects during parallel transport in the SOL. Contrib. Plasma Phys. 48 (1–3), 89–93.Google Scholar
Wootton, A., Carreras, B., Matsumoto, H., Mcguire, K., Peebles, W., Ritz, C., Terry, P. & Zweben, S. 1990 Fluctuations and anomalous transport in tokamaks. Phys. Fluids B 2 (1990), 2879.Google Scholar
Xu, G., Naulin, V., Fundamenski, W., Hidalgo, C., Alonso, J., Silva, C., Gonçalves, B., Nielsen, A., Juul Rasmussen, J., Krasheninnikov, S. et al. 2009 Blob/hole formation and zonal-flow generation in the edge plasma of the JET tokamak. Nucl. Fusion 49 (9), 092002.Google Scholar
Xu, X., Xiong, Z., Dorr, M., Hittinger, J., Bodi, K., Candy, J., Cohen, B., Cohen, R., Colella, P., Kerbel, G. et al. 2007 Edge gyrokinetic theory and continuum simulations. Nucl. Fusion 47, 809–816.Google Scholar
Zeiler, A., Drake, J. F. & Rogers, B. 1997 Nonlinear reduced Braginskii equations with ion thermal dynamics in toroidal plasma. Phys. Plasmas 4 (1997), 2134.Google 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.Google 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.Google Scholar
Zweben, S., Boedo, J., Grulke, O., Hidalgo, C., Labombard, B., Maqueda, R., Scarin, P. & Terry, J. 2007 Edge turbulence measurements in toroidal fusion devices. Plasma Phys. Control. Fusion 49 (7), S1–S23.Google Scholar
Zweben, S., Maqueda, R., Stotler, D., Keesee, A., Boedo, J., Bush, C., Kaye, S., Leblanc, B., Lowrance, J., Mastrocola, V. et al. 2004 High-speed imaging of edge turbulence in NSTX. Nucl. Fusion 44 (1), 134–153.CrossRefGoogle Scholar