Hostname: page-component-7bb8b95d7b-s9k8s Total loading time: 0 Render date: 2024-09-18T20:13:37.216Z Has data issue: false hasContentIssue false
  • English
  • Français

Investigation of tokamak turbulent avalanches using wave-kinetic formulation in toroidal geometry

Part of: Focus on Fusion

Published online by Cambridge University Press:  22 April 2021

Camille Gillot Open the ORCID record for Camille Gillot [Opens in a new window] ,
Guilhem Dif-Pradalier Open the ORCID record for Guilhem Dif-Pradalier [Opens in a new window] ,
Xavier Garbet Open the ORCID record for Xavier Garbet [Opens in a new window] ,
Olivier Panico ,
Yanick Sarazin Open the ORCID record for Yanick Sarazin [Opens in a new window] ,
Robin Varennes  and
David Zarzoso Open the ORCID record for David Zarzoso [Opens in a new window]
Camille Gillot*
Affiliation:
IRFM, CEA, F-13108Saint-Paul-lez-Durance, France Ecole des Ponts ParisTech, Champs sur Marne, France
Guilhem Dif-Pradalier*
Affiliation:
IRFM, CEA, F-13108Saint-Paul-lez-Durance, France
Xavier Garbet
Affiliation:
IRFM, CEA, F-13108Saint-Paul-lez-Durance, France
Olivier Panico
Affiliation:
IRFM, CEA, F-13108Saint-Paul-lez-Durance, France
Yanick Sarazin
Affiliation:
IRFM, CEA, F-13108Saint-Paul-lez-Durance, France
Robin Varennes
Affiliation:
IRFM, CEA, F-13108Saint-Paul-lez-Durance, France
David Zarzoso
Affiliation:
PIIM, Aix Marseille Université, CNRS, UMR 7345, Marseille, France
*
†Email addresses for correspondence: camille.gillot@polytechnique.org, guilhem.dif-pradalier@cea.fr
†Email addresses for correspondence: camille.gillot@polytechnique.org, guilhem.dif-pradalier@cea.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The interplay between toroidal drift-wave turbulence and tokamak profiles is investigated using a wave-kinetic description. The coupled system is used to investigate the interplay between marginally stable toroidal drift-wave turbulence and geodesic acoustic modes (GAMs). The coupled system is found to be unstable. Notably, the most unstable mode corresponds to the resonance between the turbulent wave radial group velocity and the GAM phase velocity. For a low-field-side ballooned drift-wave growth, a background flow shear breaks the symmetry between inwards- and outwards-travelling instabilities. Although this turbulence–GAM coupling may not be the primary driver for avalanches in standard core ion temperature gradient simulations, this mechanism is generic and displays many of the expected features, and should be of interest in several other regimes, which include towards the edge or in the presence of energetic particles.

Keywords

fusion plasmaplasma confinementplasma instabilities
Type
Research Article
Information
Journal of Plasma Physics , Volume 87 , Issue 2 , April 2021 , 905870221
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adam, J. C., Laval, G. & Pesme, D. 1979 Reconsideration of quasilinear theory. Phys. Rev. Lett. 43 (22), 1671–1675.CrossRefGoogle Scholar
Anderson, J., Nordman, H., Singh, R. & Weiland, J. 2002 Zonal flow generation in ion temperature gradient mode turbulence. Phys. Plasmas 9 (11), 4500–4506.CrossRefGoogle Scholar
Anderson, J., Nordman, H., Singh, R. & Weiland, J. 2006 Zonal flow generation in collisionless trapped electron mode turbulence. Plasma Phys. Control. Fusion 48 (5), 651–661.CrossRefGoogle Scholar
Bak, P., Tang, C. & Wiesenfeld, K. 1987 Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59 (4), 381–384.CrossRefGoogle ScholarPubMed
Beyer, P., Benkadda, S., Garbet, X. & Diamond, P. H. 2000 Nondiffusive transport in tokamaks: three-dimensional structure of bursts and the role of zonal flows. Phys. Rev. Lett. 85 (23), 4892–4895.CrossRefGoogle ScholarPubMed
Brillouin, L. 1960 Wave PRopagation and Group Velocity. Academic Press.Google Scholar
Candy, J. & Waltz, R. E. 2003 Anomalous transport scaling in the DIII-D tokamak matched by supercomputer simulation. Phys. Rev. Lett. 91 (4).CrossRefGoogle ScholarPubMed
Carreras, B. A., Hidalgo, C., Sánchez, E., Pedrosa, M. A., Balbın, R., Garcıa-Cortés, I., van Milligen, B., Newman, D. E. & Lynch, V. E. 1996 Fluctuation-induced flux at the plasma edge in toroidal devices. Phys. Plasmas 3 (7), 2664–2672.CrossRefGoogle Scholar
Champeaux, S. & Diamond, P. H. 2001 Streamer and zonal flow generation from envelope modulations in drift wave turbulence. Phys. Lett. A 288 (3–4), 214–219.CrossRefGoogle Scholar
Charney, J. G. & Drazin, P. G. 1961 Propagation of planetary-scale disturbances from the lower into the upper atmosphere. J. Geophys. Res. 66 (1), 83–109.CrossRefGoogle Scholar
Citrin, J., Bourdelle, C., Casson, F. J., Angioni, C., Bonanomi, N., Camenen, Y., Garbet, X., Garzotti, L., Görler, T., Gürcan, O., et al. 2017 Tractable flux-driven temperature, density, and rotation profile evolution with the quasilinear gyrokinetic transport model QuaLiKiz. Plasma Phys. Control. Fusion 59 (12), 124005.CrossRefGoogle Scholar
Diamond, P. H. & Hahm, T. S. 1995 On the dynamics of turbulent transport near marginal stability. Phys. Plasmas 2 (10), 3640–3649.CrossRefGoogle Scholar
Diamond, P. H., Itoh, S.-I., Itoh, K. & Hahm, T. S. 2005 Zonal flows in plasma—a review. Plasma Phys. Control. Fusion 47 (5), 35–161.CrossRefGoogle Scholar
Dif-Pradalier, G., Hornung, G., Garbet, X., Ghendrih, P., Grandgirard, V., Latu, G. & Sarazin, Y. 2017 The $\boldsymbol {E}\times \boldsymbol {B}$ staircase of magnetised plasmas. Nucl. Fusion 57 (6), 066026.CrossRefGoogle Scholar
Dif-Pradalier, G., Hornung, G., Ghendrih, P., Sarazin, Y., Clairet, F., Vermare, L., Diamond, P. H., Abiteboul, J., Cartier-Michaud, T., Ehrlacher, C., et al. 2015 Finding the elusive $\boldsymbol {E}\times \boldsymbol {B}$ staircase in magnetized plasmas. Phys. Rev. Lett. 114 (8).CrossRefGoogle Scholar
Dodin, I. Y. & Fisch, N. J. 2012 Adiabatic nonlinear waves with trapped particles. I. General formalism. Phys. Plasmas 19 (1), 012102.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64 (10), 3652–3665.CrossRefGoogle Scholar
Garbet, X., Sarazin, Y., Beyer, P., Ghendrih, P., Waltz, R., Ottaviani, M. & Benkadda, S. 1999 Flux driven turbulence in tokamaks. Nucl. Fusion 39, 2063–2068.CrossRefGoogle Scholar
Garbet, X., Sarazin, Y., Ghendrih, P., Benkadda, S., Beyer, P., Figarella, C. & Voitsekhovitch, I. 2002 Turbulence simulations of transport barriers with toroidal velocity. Phys. Plasmas 9 (9), 3893–3905.CrossRefGoogle Scholar
Garbet, X., Varennes, R., Gillot, C., Dif-Pradalier, G., Sarazin, Y., Bourne, E., Grandgirard, V., Ghendrih, P., Zarzoso, D. & Vermare, L. 2020 Zonal instability and wave trapping (in preparation).CrossRefGoogle Scholar
Gillot, C. 2016 Study of the interaction between drift wave turbulence and zonal flows in a magnetic confinement device. Master's thesis, Université Paris–Saclay.Google Scholar
Gillot, C. 2020 Model reduction for tokamak plasma turbulence: beyond fluid and quasi-linear descriptions. PhD thesis, Aix-Marseille Université.Google Scholar
Girardo, J.-B. 2015 Control of instabilities and turbulence by fast particles in fusion plasmas. Theses, Ecode doctorale de l'Ecole Polytechnique (EDX).Google Scholar
Girardo, J.-B., Zarzoso, D., Dumont, R., Garbet, X., Sarazin, Y. & Sharapov, S. 2014 Relation between energetic and standard geodesic acoustic modes. Phys. Plasmas 21 (9), 092507.CrossRefGoogle Scholar
Görler, T., Lapillonne, X., Brunner, S., Dannert, T., Jenko, F., Aghdam, S. K., Marcus, P., McMillan, B. F., Merz, F., Sauter, O., et al. 2011 Flux- and gradient-driven global gyrokinetic simulation of tokamak turbulence. Phys. Plasmas 18 (5), 056103.CrossRefGoogle Scholar
Gürcan, O. D., Garbet, X., Hennequin, P., Diamond, P. H., Casati, A. & Falchetto, G. 2009 Wave-number spectrum of drift-wave turbulence. Phys. Rev. Lett. 102 (25).CrossRefGoogle ScholarPubMed
Hager, R. & Hallatschek, K. 2012 The nonlinear dispersion relation of geodesic acoustic modes. Phys. Plasmas 19 (8), 082315.CrossRefGoogle Scholar
Hallatschek, K. & Biskamp, D. 2001 Transport control by coherent zonal flows in the core/edge transitional regime. Phys. Rev. Lett. 86 (7), 1223–1226.CrossRefGoogle ScholarPubMed
Hasegawa, A. & Mima, K. 1978 Pseudo-three-dimensional turbulence in magnetized nonuniform plasma. Phys. Fluids 21 (1), 87.CrossRefGoogle Scholar
Hwa, T. & Kardar, M. 1992 Avalanches, hydrodynamics, and discharge events in models of sandpiles. Phys. Rev. A 45 (10), 7002–7023.CrossRefGoogle ScholarPubMed
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
Ivanov, P. G., Schekochihin, A. A., Dorland, W., Field, A. R. & Parra, F. I. 2020 Zonally dominated dynamics and dimits threshold in curvature-driven ITG turbulence. J. Plasma Phys. 86 (5).CrossRefGoogle Scholar
Jimenez, J. & Whitham, G. B. 1976 An averaged Lagrangian method for dissipative wavetrains. Proc. R. Soc. Lond. A 349 (1658), 277–287.Google Scholar
Kaufman, A. N., Ye, H. & Hui, Y. 1987 Variational formulation of covariant eikonal theory for vector waves. Phys. Lett. A 120 (7), 327–330.CrossRefGoogle Scholar
Kaw, P., Singh, R. & Diamond, P. H. 2001 Coherent nonlinear structures of drift wave turbulence modulated by zonal flows. Plasma Phys. Control. Fusion 44 (1), 51–59.CrossRefGoogle Scholar
Landau, L. D. & Lifschitz, E. M. 1984 Electrodynamics of Continuous Media. Pergamon.Google Scholar
Laval, G., Pesme, D. & Adam, J.-C. 2018 Wave-particle and wave-wave interactions in hot plasmas: a French historical point of view. Eur. Phys. J. H 43 (4–5), 421–458.CrossRefGoogle Scholar
Lin, Z., Ethier, S., Hahm, T. S. & Tang, W. M. 2002 Size scaling of turbulent transport in magnetically confined plasmas. Phys. Rev. Lett. 88 (19).CrossRefGoogle ScholarPubMed
McDonald, S. W. 1988 Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves. Phys. Rep. 158 (6), 337–416.CrossRefGoogle Scholar
McDonald, S. W. 1991 Wave kinetic equation in a fluctuating medium. Phys. Rev. A 43 (8), 4484–4499.CrossRefGoogle Scholar
McDonald, S. W. & Kaufman, A. N. 1985 Weyl representation for electromagnetic waves: the wave kinetic equation. Phys. Rev. A 32 (3), 1708–1713.CrossRefGoogle ScholarPubMed
McMillan, B. F., Hill, P., Bottino, A., Jolliet, S., Vernay, T. & Villard, L. 2011 Interaction of large scale flow structures with gyrokinetic turbulence. Phys. Plasmas 18 (11), 112503.CrossRefGoogle Scholar
McMillan, B. F., Jolliet, S., Tran, T. M., Villard, L., Bottino, A. & Angelino, P. 2009 Avalanchelike bursts in global gyrokinetic simulations. Phys. Plasmas 16 (2), 022310.CrossRefGoogle Scholar
McMillan, B. F., Pringle, C. C. T. & Teaca, B. 2018 Simple advecting structures and the edge of chaos in subcritical tokamak plasmas. J. Plasma Phys. 84 (6).CrossRefGoogle Scholar
Moyal, J. E. & Bartlett, M. S. 1949 Quantum mechanics as a statistical theory. Math. Proc. Cambridge 45 (1), 99.CrossRefGoogle Scholar
Newman, D. E., Carreras, B. A., Diamond, P. H. & Hahm, T. S. 1996 The dynamics of marginality and self-organized criticality as a paradigm for turbulent transport. Phys. Plasmas 3 (5), 1858–1866.CrossRefGoogle Scholar
Parker, J. B. 2015 Zonal flows and turbulence in fluids and plasmas. PhD thesis, Princeton University.Google Scholar
Parker, J. B. 2016 Dynamics of zonal flows: failure of wave-kinetic theory, and new geometrical optics approximations. J. Plasma Phys. 82 (6).CrossRefGoogle Scholar
Pringle, C. C. T., McMillan, B. F. & Teaca, B. 2017 A nonlinear approach to transition in subcritical plasmas with sheared flow. Phys. Plasmas 24 (12), 122307.CrossRefGoogle Scholar
Qiu, Z., Chen, L. & Zonca, F. 2018 Kinetic theory of geodesic acoustic modes in toroidal plasmas: a brief review. Plasma Sci. Technol. 20 (9), 094004.CrossRefGoogle Scholar
Rath, F., Peeters, A. G., Buchholz, R., Grosshauser, S. R., Migliano, P., Weikl, A. & Strintzi, D. 2016 Comparison of gradient and flux driven gyro-kinetic turbulent transport. Phys. Plasmas 23 (5), 052309.CrossRefGoogle Scholar
Ruiz, D. E. 2017 A geometric theory of waves and its applications to plasma physics. PhD thesis, Princeton University.Google Scholar
Ruiz, D. E., Glinsky, M. E. & Dodin, I. Y. 2019 Wave kinetic equation for inhomogeneous drift-wave turbulence beyond the quasilinear approximation. J. Plasma Phys. 85 (1).CrossRefGoogle Scholar
Ruiz, D. E., Parker, J. B., Shi, E. L. & Dodin, I. Y. 2016 Zonal-flow dynamics from a phase-space perspective. Phys. Plasmas 23 (12), 122304.CrossRefGoogle Scholar
Sarazin, Y. & Ghendrih, P. 1998 Intermittent particle transport in two-dimensional edge turbulence. Phys. Plasmas 5 (12), 4214–4228.CrossRefGoogle Scholar
Sarazin, Y., Grandgirard, V., Abiteboul, J., Allfrey, S., Garbet, X., Ghendrih, P., Latu, G., Strugarek, A. & Dif-Pradalier, G. 2010 Large scale dynamics in flux driven gyrokinetic turbulence. Nucl. Fusion 50 (5), 054004.CrossRefGoogle Scholar
Sasaki, M. 2018 Trapping of turbulence clumps by geodesic acoustic modes. In AIP Conference Proceedings 1993, 020007.Google Scholar
Sasaki, M., Itoh, K., Kobayashi, T., Kasuya, N., Fujisawa, A. & Itoh, S.-I. 2018 a Propagation direction of geodesic acoustic modes driven by drift wave turbulence. Nucl. Fusion 58 (11), 112005.CrossRefGoogle Scholar
Sasaki, M., Kasuya, N., Itoh, K., Hallatschek, K., Lesur, M., Kosuga, Y. & Itoh, S.-I. 2016 A branch of energetic-particle driven geodesic acoustic modes due to magnetic drift resonance. Phys. Plasmas 23 (10), 102501.CrossRefGoogle Scholar
Sasaki, M., Kobayashi, T., Itoh, K., Kasuya, N., Kosuga, Y., Fujisawa, A. & Itoh, S.-I. 2018 b Spatio-temporal dynamics of turbulence trapped in geodesic acoustic modes. Phys. Plasmas 25 (1), 012316.CrossRefGoogle Scholar
Schekochihin, A. A., Highcock, E. G. & Cowley, S. C. 2012 Subcritical fluctuations and suppression of turbulence in differentially rotating gyrokinetic plasmas. Plasma Phys. Control. Fusion 54 (5), 055011.CrossRefGoogle Scholar
Scott, B. & Smirnov, J. 2010 Energetic consistency and momentum conservation in the gyrokinetic description of tokamak plasmas. Phys. Plasmas 17 (11), 112302.CrossRefGoogle Scholar
Srinivasan, K. & Young, W. R. 2012 Zonostrophic instability. J. Atmos. Sci. 69 (5), 1633–1656.CrossRefGoogle Scholar
Staebler, G. M., Waltz, R. E., Candy, J. & Kinsey, J. E. 2013 New paradigm for suppression of gyrokinetic turbulence by velocity shear. Phys. Rev. Lett. 110, 055003.CrossRefGoogle ScholarPubMed
Suchy, K. 1981 Real Hamilton equations of geometric optics for media with moderate absorption. Radio Sci. 16 (6), 1179–1182.CrossRefGoogle Scholar
Taylor, G. I. 1915 I. Eddy motion in the atmosphere. Phil. Trans. R. Soc. Lond. A 215 (523–537), 1–26.Google Scholar
Villard, L., McMillan, B. F., Lanti, E., Ohana, N., Bottino, A., Biancalani, A., Novikau, I., Brunner, S., Sauter, O., Tronko, N., et al. 2019 Global turbulence features across marginality and non-local pedestal-core interactions. Plasma Phys. Control. Fusion 61 (3), 034003.CrossRefGoogle Scholar
Waltz, R. E. & Holland, C. 2008 Numerical experiments on the drift wave–zonal flow paradigm for nonlinear saturation. Phys. Plasmas 15 (12), 122503.CrossRefGoogle Scholar
Weinberg, S. 1962 Eikonal method in magnetohydrodynamics. Phys. Rev. 126 (6), 1899–1909.CrossRefGoogle Scholar
Whitham, G. B. 1965 A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22 (2), 273–283.CrossRefGoogle Scholar
van Wyk, F., Highcock, E. G., Schekochihin, A. A., Roach, C. M., Field, A. R. & Dorland, W. 2016 Transition to subcritical turbulence in a tokamak plasma. J. Plasma Phys. 82 (6).CrossRefGoogle Scholar
Zarzoso, D., Sarazin, Y., Garbet, X., Dumont, R., Strugarek, A., Abiteboul, J., Cartier-Michaud, T., Dif-Pradalier, G., Ghendrih, P., Grandgirard, V., et al. 2013 Impact of energetic-particle-driven geodesic acoustic modes on turbulence. Phys. Rev. Lett. 110 (12).CrossRefGoogle ScholarPubMed
Zhu, H., Zhou, Y. & Dodin, I. Y. 2018 a On the structure of the Drifton phase space and its relation to the Rayleigh–Kuo criterion of the zonal-flow stability. Phys. Plasmas 25 (7), 072121.CrossRefGoogle Scholar
Zhu, H., Zhou, Y. & Dodin, I. Y. 2020 Theory of the tertiary instability and the dimits shift from reduced drift-wave models. Phys. Rev. Lett. 124 (5).CrossRefGoogle ScholarPubMed
Zhu, H., Zhou, Y., Ruiz, D. E. & Dodin, I. Y. 2018 b Wave kinetics of drift-wave turbulence and zonal flows beyond the ray approximation. Phys. Rev. E 97 (5).CrossRefGoogle ScholarPubMed