Skip to main content

Gyrokinetic continuum simulation of turbulence in a straight open-field-line plasma

  • E. L. Shi (a1), G. W. Hammett (a2), T. Stoltzfus-Dueck (a1) (a3) and A. Hakim (a2)

Five-dimensional gyrokinetic continuum simulations of electrostatic plasma turbulence in a straight, open-field-line geometry have been performed using a full- $f$ discontinuous-Galerkin approach implemented in the Gkeyll code. While various simplifications have been used for now, such as long-wavelength approximations in the gyrokinetic Poisson equation and the Hamiltonian, these simulations include the basic elements of a fusion-device scrape-off layer: localised sources to model plasma outflow from the core, cross-field turbulent transport, parallel flow along magnetic field lines, and parallel losses at the limiter or divertor with sheath-model boundary conditions. The set of sheath-model boundary conditions used in the model allows currents to flow through the walls. In addition to details of the numerical approach, results from numerical simulations of turbulence in the Large Plasma Device, a linear device featuring straight magnetic field lines, are presented.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Gyrokinetic continuum simulation of turbulence in a straight open-field-line plasma
      Available formats
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Gyrokinetic continuum simulation of turbulence in a straight open-field-line plasma
      Available formats
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Gyrokinetic continuum simulation of turbulence in a straight open-field-line plasma
      Available formats
Corresponding author
Email address for correspondence:
Hide All
Angus, J. R. & Umansky, M. V. 2014 Modeling of large amplitude plasma blobs in three-dimensions. Phys. Plasmas 21 (1), 012514.
Belli, E. A. & Hammett, G. W. 2005 A numerical instability in an ADI algorithm for gyrokinetics. Comput. Phys. Commun. 172 (2), 119132.
Brizard, A. J. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421468.
Cagas, P., Hakim, A., Juno, J. & Srinivasan, B. 2017 Continuum kinetic and multi-fluid simulations of classical sheaths. Phys. Plasmas 24 (2), 022118.
Carter, T. A. 2006 Intermittent turbulence and turbulent structures in a linear magnetized plasma. Phys. Plasmas 13 (1), 010701.
Carter, T. A. & Maggs, J. E. 2009 Modifications of turbulence and turbulent transport associated with a bias-induced confinement transition in the Large Plasma Device. Phys. Plasmas 16 (1), 012304.
Chang, C. S., Ku, S., Diamond, P. H., Lin, Z., Parker, S., Hahm, T. S. & Samatova, N. 2009 Compressed ion temperature gradient turbulence in diverted tokamak edge. Phys. Plasmas 16 (5), 056108.
Chodura, R. 1982 Plasma–wall transition in an oblique magnetic field. Phys. Fluids 25 (9), 16281633.
Churchill, R., Canik, J., Chang, C., Hager, R., Leonard, A., Maingi, R., Nazikian, R. & Stotler, D. 2016 Kinetic simulations of scrape-off layer physics in the DIII-D tokamak. Nucl. Mater. Energy.
Cohen, R. H. & Xu, X. Q. 2008 Progress in kinetic simulation of edge plasmas. Contrib. Plasma Phys. 48 (1–3), 212223.
Dimits, A. M. 2012 Gyrokinetic equations for strong-gradient regions. Phys. Plasmas 19 (2), 022504.
Dimits, A. M., Bateman, G., Beer, M. A., Cohen, B. I., Dorland, W., Hammett, G. W., Kim, C., Kinsey, J. E., Kotschenreuther, M., Kritz, A. H. et al. 2000 Comparisons and physics basis of tokamak transport models and turbulence simulations. Phys. Plasmas 7 (3), 969983.
Dorf, M. A., Dorr, M. R., Hittinger, J. A., Cohen, R. H. & Rognlien, T. D. 2016 Continuum kinetic modeling of the tokamak plasma edge. Phys. Plasmas 23 (5), 056102.
Dorland, W. & Hammett, G. W. 1993 Gyrofluid turbulence models with kinetic effects. Phys. Fluids B 5 (3), 812835.
Dudson, B. D., Allen, A., Breyiannis, G., Brugger, E., Buchanan, J., Easy, L., Farley, S., Joseph, I., Kim, M., McGann, A. D. et al. 2015 BOUT++: recent and current developments. J. Plasma Phys. 81, 365810104.
Dudson, B. D. & Leddy, J. 2017 Hermes: global plasma edge fluid turbulence simulations. Plasma Phys. Control. Fusion 59 (5), 054010.
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), 14671480.
Dumbser, M., Balsara, D. S., Toro, E. F. & Munz, C.-D. 2008 A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227 (18), 82098253.
Durran, D. R. 2010 Numerical Methods for Fluid Dynamics With Applications to Geophysics, Texts in Applied Mathematics, vol. 32. Springer.
Eich, T., Leonard, A., Pitts, R., Fundamenski, W., Goldston, R., Gray, T., Herrmann, A., Kirk, A., Kallenbach, A., Kardaun, O. et al. 2013 Scaling of the tokamak near the scrape-off layer H-mode power width and implications for ITER. Nucl. Fusion 53 (9), 093031.
Fisher, D. M. & Rogers, B. N. 2017 Two-fluid biasing simulations of the large plasma device. Phys. Plasmas 24 (2), 022303.
Fisher, D. M., Rogers, B. N., Rossi, G. D., Guice, D. S. & Carter, T. A. 2015 Three-dimensional two-fluid Braginskii simulations of the large plasma device. Phys. Plasmas 22 (9), 092121.
Friedman, B., Carter, T. A., Umansky, M. V., Schaffner, D. & Dudson, B. 2012 Energy dynamics in a simulation of LAPD turbulence. Phys. Plasmas 19 (10), 102307.
Friedman, B., Carter, T. A., Umansky, M. V., Schaffner, D. & Joseph, I. 2013 Nonlinear instability in simulations of Large Plasma Device turbulence. Phys. Plasmas 20 (5), 055704.
Gekelman, W., Pfister, H., Lucky, Z., Bamber, J., Leneman, D. & Maggs, J. 1991 Design, construction, and properties of the large plasma research device–The LAPD at UCLA. Rev. Sci. Instrum. 62 (12), 28752883.
Gekelman, W., Pribyl, P., Lucky, Z., Drandell, M., Leneman, D., Maggs, J., Vincena, S., Compernolle, B. V., Tripathi, S. K. P., Morales, G. et al. 2016 The upgraded Large Plasma Device, a machine for studying frontier basic plasma physics. Rev. Sci. Instrum. 87 (2), 025105.
Geraldini, A., Parra, F. I. & Militello, F. 2017 Gyrokinetic treatment of a grazing angle magnetic presheath. Plasma Phys. Control. Fusion 59 (2), 025015.
Goldston, R. 2012 Heuristic drift-based model of the power scrape-off width in low-gas-puff H-mode tokamaks. Nucl. Fusion 52 (1), 013009.
Gottlieb, S., Shu, C.-W. & Tadmor, E. 2001 Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (1), 89112.
Hahm, T. S., Wang, L. & Madsen, J. 2009 Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence. Phys. Plasmas 16 (2), 022305.
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, 388408.
Huba, J. D. 2013 NRL Plasma Formulary. Naval Research Laboratory.
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.
Kervalishvili, G. N., Kleiber, R., Schneider, R., Scott, B. D., Grulke, O. & Windisch, T. 2008 Intermittent turbulence in the linear VINETA device. Contrib. Plasma Phys. 48 (1–3), 3236.
Kinsey, J., Staebler, G., Candy, J., Waltz, R. & Budny, R. 2011 ITER predictions using the GYRO verified and experimentally validated trapped gyro-Landau fluid transport model. Nucl. Fusion 51 (8), 083001.
Korpilo, T., Gurchenko, A., Gusakov, E., Heikkinen, J., Janhunen, S., Kiviniemi, T., Leerink, S., Niskala, P. & Perevalov, A. 2016 Gyrokinetic full-torus simulations of ohmic tokamak plasmas in circular limiter configuration. Comput. Phys. Commun. 203, 128137.
Kotschenreuther, M., Dorland, W., Beer, M. A. & Hammett, G. W. 1995 Quantitative predictions of tokamak energy confinement from first principles simulations with kinetic effects. Phys. Plasmas 2 (6), 23812389.
Krommes, J. A. 2012 The gyrokinetic description of microturbulence in magnetized plasmas. Annu. Rev. Fluid Mech. 44, 175201.
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.
Ku, S., Hager, R., Chang, C., Kwon, J. & Parker, S. 2016 A new hybrid-Lagrangian numerical scheme for gyrokinetic simulation of tokamak edge plasma. J. Comput. Phys. 315, 467475.
Lee, W. W. 1987 Gyrokinetic particle simulation model. J. Comput. Phys. 72 (1), 243269.
van Leer, B. & Nomura, S. 2005 Discontinuous Galerkin for Diffusion. American Institute of Aeronautics and Astronautics.
Lenard, A. & Bernstein, I. B. 1958 Plasma oscillations with diffusion in velocity space. Phys. Rev. 112, 14561459.
LeVeque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press.
Liu, J.-G. & Shu, C.-W. 2000 A high-order discontinuous Galerkin method for 2D incompressible flows. J. Comput. Phys. 160 (2), 577596.
Loizu, J., Ricci, P., Halpern, F. D. & Jolliet, S. 2012 Boundary conditions for plasma fluid models at the magnetic presheath entrance. Phys. Plasmas 19 (12), 122307.
McMillan, B. F. & Sharma, A. 2016 A very general electromagnetic gyrokinetic formalism. Phys. Plasmas 23 (9), 092504.
Meyer, C. D., Balsara, D. S. & Aslam, T. D. 2014 A stabilized Runge–Kutta–Legendre method for explicit super-time-stepping of parabolic and mixed equations. J. Comput. Phys. 257, Part A, 594626.
Mosetto, A.2014 Turbulent regimes in the tokamak scrape-off layer. PhD thesis, École Polytechnique Fédérale de Lausanne.
Munz, C.-D. 1994 A tracking method for gas flow into vacuum based on the vacuum riemann problem. Math. Meth. Appl. Sci. 17 (8), 597612.
Naulin, V., Windisch, T. & Grulke, O. 2008 Three-dimensional global fluid simulations of cylindrical magnetized plasmas. Phys. Plasmas 15 (1), 012307.
Ng, J., Huang, Y.-M., Hakim, A., Bhattacharjee, A., Stanier, A., Daughton, W., Wang, L. & Germaschewski, K. 2015 The island coalescence problem: scaling of reconnection in extended fluid models including higher-order moments. Phys. Plasmas 22 (11), 112104.
Parker, S. E., Procassini, R. J., Birdsall, C. K. & Cohen, B. I. 1993 A suitable boundary condition for bounded plasma simulation without sheath resolution. J. Comput. Phys. 104 (1), 4149.
Parra, F. I. & Calvo, I. 2011 Phase-space Lagrangian derivation of electrostatic gyrokinetics in general geometry. Plasma Phys. Control. Fusion 53 (4), 045001.
Parra, F. I., Calvo, I., Burby, J. W., Squire, J. & Qin, H. 2014 Equivalence of two independent calculations of the higher order guiding center Lagrangian. Phys. Plasmas 21 (10), 104506.
Peterson, J. L. & Hammett, G. W. 2013 Positivity preservation and advection algorithms with applications to edge plasma turbulence. SIAM J. Sci. Comput. 35 (3), B576B605.
Popovich, P., Umansky, M. V., Carter, T. A. & Friedman, B. 2010a Analysis of plasma instabilities and verification of the BOUT code for the Large Plasma Device. Phys. Plasmas 17 (10), 102107.
Popovich, P., Umansky, M. V., Carter, T. A. & Friedman, B. 2010b Modeling of plasma turbulence and transport in the Large Plasma Device. Phys. Plasmas 17 (12), 122312.
Powers, E. 1974 Spectral techniques for experimental investigation of plasma diffusion due to polychromatic fluctuations. Nucl. Fusion 14 (5), 749.
Ribeiro, T. T. & Scott, B. 2005 Tokamak turbulence computations on closed and open magnetic flux surfaces. Plasma Phys. Control. Fusion 47 (10), 1657.
Ricci, P. & Rogers, B. N. 2010 Turbulence phase space in simple magnetized toroidal plasmas. Phys. Rev. Lett. 104, 145001.
Ricci, P., Rogers, B. N. & Brunner, S. 2008 High- and low-confinement modes in simple magnetized toroidal plasmas. Phys. Rev. Lett. 100, 225002.
Rogers, B. N. & Ricci, P. 2010 Low-frequency turbulence in a linear magnetized plasma. Phys. Rev. Lett. 104, 225002.
Rognlien, T. D., Brown, P. N., Campbell, R. B., Kaiser, T. B., Knoll, D. A., McHugh, P. R., Porter, G. D., Rensink, M. E. & Smith, G. R. 1994 2-D fluid transport simulations of gaseous/radiative divertors. Contrib. Plasma Phys. 34 (2–3), 362367.
Schaffner, D. A., Carter, T. A., Rossi, G. D., Guice, D. S., Maggs, J. E., Vincena, S. & Friedman, B. 2012 Modification of turbulent transport with continuous variation of flow shear in the Large Plasma Device. Phys. Rev. Lett. 109, 135002.
Schaffner, D. A., Carter, T. A., Rossi, G. D., Guice, D. S., Maggs, J. E., Vincena, S. & Friedman, B. 2013 Turbulence and transport suppression scaling with flow shear on the Large Plasma Device. Phys. Plasmas 20 (5), 055907.
Schneider, R., Bonnin, X., Borrass, K., Coster, D. P., Kastelewicz, H., Reiter, D., Rozhansky, V. A. & Braams, B. J. 2006 Plasma edge physics with B2-Eirene. Contrib. Plasma Phys. 46 (1–2), 3191.
Scott, B. 1997 Three-dimensional computation of drift Alfvén turbulence. Plasma Phys. Control. Fusion 39 (10), 1635.
Scott, B. & Smirnov, J. 2010 Energetic consistency and momentum conservation in the gyrokinetic description of tokamak plasmas. Phys. Plasmas 17 (11), 112302.
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.
Shimada, M., Campbell, D., Mukhovatov, V., Fujiwara, M., Kirneva, N., Lackner, K., Nagami, M., Pustovitov, V., Uckan, N., Wesley, J. et al. 2007 Chapter 1: overview and summary. Nucl. Fusion 47 (6), S1.
Snyder, P. B., Hammett, G. W. & Dorland, W. 1997 Landau fluid models of collisionless magnetohydrodynamics. Phys. Plasmas 4 (11), 39743985.
Stangeby, P. C. 2000 Plasma Physics Series. Institute of Physics Publishing.
Stoltzfus-Dueck, T.2009 Tokamak edge turbulence and the approach to adiabatic response. PhD thesis, Princeton University.
Sugama, H. 2000 Gyrokinetic field theory. Phys. Plasmas 7 (2), 466480.
Taitano, W., Chacón, L., Simakov, A. & Molvig, K. 2015 A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation. J. Comput. Phys. 297, 357380.
Thakur, S. C., Xu, M., Manz, P., Fedorczak, N., Holland, C. & Tynan, G. R. 2013 Suppression of drift wave turbulence and zonal flow formation by changing axial boundary conditions in a cylindrical magnetized plasma device. Phys. Plasmas 20 (1), 012304.
Togo, S., Takizuka, T., Nakamura, M., Hoshino, K., Ibano, K., Lang, T. L. & Ogawa, Y. 2016 Self-consistent treatment of the sheath boundary conditions by introducing anisotropic ion temperatures and virtual divertor model. J. Comput. Phys. 310, 109126.
Wang, L., Hakim, A. H., Bhattacharjee, A. & Germaschewski, K. 2015 Comparison of multi-fluid moment models with particle-in-cell simulations of collisionless magnetic reconnection. Phys. Plasmas 22 (1), 012108.
Wersal, C. & Ricci, P. 2015 A first-principles self-consistent model of plasma turbulence and kinetic neutral dynamics in the tokamak scrape-off layer. Nucl. Fusion 55 (12), 123014.
Xu, X. Q. & Cohen, R. H. 1998 Scrape-off layer turbulence theory and simulations. Contrib. Plasma Phys. 38 (1–2), 158170.
Xu, X. Q., Xi, P. W., Dimits, A., Joseph, I., Umansky, M. V., Xia, T. Y., Gui, B., Kim, S. S., Park, G. Y., Rhee, T. et al. 2013 Gyro-fluid and two-fluid theory and simulations of edge-localized-modes. Phys. Plasmas 20 (5), 056113.
Yuan, L. & Shu, C.-W. 2006 Discontinuous Galerkin method based on non-polynomial approximation spaces. J. Comput. Phys. 218 (1), 295323.
Zhang, X. & Shu, C.-W. 2010 On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229 (23), 89188934.
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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Plasma Physics
  • ISSN: 0022-3778
  • EISSN: 1469-7807
  • URL: /core/journals/journal-of-plasma-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 13
Total number of PDF views: 164 *
Loading metrics...

Abstract views

Total abstract views: 487 *
Loading metrics...

* Views captured on Cambridge Core between 29th May 2017 - 23rd March 2018. This data will be updated every 24 hours.