Skip to main content Accessibility help
×
Home

Simple advecting structures and the edge of chaos in subcritical tokamak plasmas

  • Ben F. McMillan (a1), Chris C. T. Pringle (a2) and Bogdan Teaca (a2)

Abstract

In tokamak plasmas, sheared flows perpendicular to the driving temperature gradients can strongly stabilise linear modes. While the system is linearly stable, regimes with persistent nonlinear turbulence may develop, i.e. the system is subcritical. A perturbation with small but finite amplitude may be sufficient to push the plasma into a regime where nonlinear effects are dominant and thus allow sustained turbulence. The minimum threshold for nonlinear instability to be triggered provides a criterion for assessing whether a tokamak is likely to stay in the quiescent (laminar) regime. At the critical amplitude, instead of transitioning to the turbulent regime or decaying to a laminar state, the trajectory will map out the edge of chaos. Surprisingly, a quasi-travelling-wave solution is found as an attractor on this edge manifold. This simple advecting solution is qualitatively similar to, but simpler than, the avalanche-like bursts seen in earlier turbulent simulations and provides an insight into how turbulence is sustained in subcritical plasma systems. For large flow shearing rate, the system is only convectively unstable, and given a localised initial perturbation, will eventually return to a laminar state at a fixed spatial location.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org 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 @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ 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.

      Simple advecting structures and the edge of chaos in subcritical tokamak plasmas
      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.

      Simple advecting structures and the edge of chaos in subcritical tokamak plasmas
      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.

      Simple advecting structures and the edge of chaos in subcritical tokamak plasmas
      Available formats
      ×

Copyright

Corresponding author

Email address for correspondence: b.f.mcmillan@warwick.ac.uk

References

Hide All
Balescu, R. 1960 Irreversible processes in ionized gases. Phys. Fluids 3, 52.
Beer, M. A., Cowley, S. C. & Hammett, G. W. 1995 Field-aligned coordinates for nonlinear simulations of tokamak turbulence. Phys. Plasmas 2 (7), 26872700.
Candy, J. & Waltz, R. E. 2003 Anomalous transport scaling in the DIII-D tokamak matched by supercomputer simulation. Phys. Rev. Lett. 91, 045001.
Casson, F. J.2011 Background $E\times B$ shear nonlinear benchmark (included in GKW distribution).
Casson, F. J., Peeters, A. G., Camenen, Y., Hornsby, W. A., Snodin, A. P., Strintzi, D. & Szepesi, G. 2009 Anomolous parallel momentum transport due to $E\times B$ flow shear in a tokamak plasma. Phys. Plasmas 16, 092303.
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.
Eyink, G. L. 2018 Cascades and dissipative anomalies in nearly collisionless plasma turbulence. Phys. Rev. X 8, 041020.
Friedman, B. & Carter, T. 2015 A non-modal analytical method to predict turbulent properties applied to the Hasegawa-Wakatani model. Phys. Plasmas 22, 012307.
Hahm, T. 1988 Nonlinear gyrokinetic equations for tokamak microturbulence. Phys. Fluids 31, 26702673.
Hatch, D. R., Jenko, F., Bratanov, V. & Navarro, A. B. 2014 Phase space scales of free energy dissipation in gradient-driven gyrokinetic turbulence. J. Plasma Phys. 80, 531.
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, 115003.
Highcock, E. G., Barnes, M., Parra, F. I., Schekochihin, A. A., Roach, C. M. & Cowley, S. C. 2011 Transport bifurcation induced by sheared toroidal flow in tokamak plasmas. Phys. Plasmas 18, 102304.
Horton, W. & Hasegawa, A. 1994 Quasi-two-dimensional dynamics of plasmas and fluids. Chaos 4 (2), 227251.
Howes, G. G., Dorland, W., Cowley, S. C., Hammett, G. W., Quataert, E., Schekochihin, A. A. & Tatsuno, T. 2008 Kinetic simulations of magnetized turbulence in astrophysical plasmas. Phys. Rev. Lett. 100, 65004.
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703716.
Landau, L. D. 1981 Phys. Z. Sowjet 10, 154 translated in The transport equation in the case of the Coulomb interaction, Collected Papers of L. D. Landau. Pergamon Press.
Lenard, A. 1960 Ann. Phys. NY 390 (10).
McMillan, B., Jolliet, S., Tran, T., Bottino, A., P., A. & Villard, L. 2009 Avalanchelike bursts in global gyrokinetic simulations. Phys. Plasmas 16, 022310.
Navarro, A. B., Morel, P., Albrecht-Marc, M., Carati, D., Merz, F., Görler, T. & Jenko, F. 2011 Free energy cascade in gyrokinetic turbulence. Phys. Rev. Lett. 106, 55001.
Navarro, A. B., Teaca, B., Jenko, F., Hammett, G. W. & Happel, T. 2014 Applications of large eddy simulation methods to gyrokinetic turbulence. Phys. Plasmas 21, 032304.
Parra, F. I., Barnes, M. & Peeters, A. G. 2011 Up-down symmetry of the turbulent transport of toroidal angular momentum in tokamaks. Phys. Plasmas 18 (6), 062501.
Peeters, A. G., Camenen, Y., Casson, F. J., Hornsby, W. A., Snodin, A. P., Strintzi, D. & Szepesi, G. 2009 The nonlinear gyro-kinetic flux tube code GKW. Comput. Phys. Commun. 180, 26502672.
Plunk, G. G., Navarro, A. B. & Jenko, F. 2015 Understanding nonlinear saturation in zonal-flow-dominated ion temperature gradient turbulence. Plasma Phys. Control. Fusion 57 (4), 045005.
Pringle, C., McMillan, B. F. & Teaca, B. 2017 A nonlinear approach to transition in subcritical plasmas with sheared flow. Phys. Plasmas 24, 122307.
Pringle, C. C., Willis, A. P. & Kerswell, R. R. 2012 Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos. J. Fluid Mech. 702, 415443.
Rincon, F., Ogilvie, G. I. & Proctor, M. R. E. 2007 Self-sustaining nonlinear dynamo process in Keplerian shear flows. Phys. Rev. Lett. 98, 254502.
Roach, C. M., Abel, I. G., Akers, R. J., Arter, W., Barnes, M., Camenen, Y., Casson, F. J., Colyer, G., Connor, J. W., Cowley, S. C. et al. 2009 Gyrokinetic simulations of spherical tokamaks. Plasma Phys. Control. Fusion 51 (12), 124020.
Rogers, B. N., Dorland, W. & Kotschenreuther, M. 2000 Generation and stability of zonal flows in ion-temperature-gradient mode turbulence. Phys. Rev. Lett. 85 (25), 53365339.
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, 4024.
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.
Schekochihin, A. A., Parker, J. T., Highcock, E. G., Dellar, P. J., Dorland, W. & Hammett, G. W. 2016 Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence. J. Plasma Phys. 82, 905820212.
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.
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, 15003.
Teaca, B., Navarro, A. B. & Jenko, F. 2014 The energetic coupling of scales in gyrokinetic plasma turbulence. Phys. Plasmas 21, 072308.
Waltz, R., Dewar, R. & Garbet, X. 1998 Theory and simulation of rotational shear stabilization of turbulence. Phys. Plasmas 5, 1784.
Watanabe, T.-H. & Sugama, H. 2006 Velocityspace structures of distribution function in toroidal ion temperature gradient turbulence. Nucl. Fusion 46 (1), 24.
Watanabe, T. H., Sugama, H., Nunami, M., Tanaka, K. & Nakata, M. 2012 Gyrokinetic simulations of entropy transfer in high ion temperature LHD plasmas. Plasma Phys. Control. Fusion 55 (1), 014017.
Wygnanski, I. J. & Champagne, F. H. 1973 On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech. 59 (2), 281335.
van Wyk, F., Highcock, E. G., Field, A. R., Roach, C. M., Schekochihin, A. A., Parra, F. I. & Dorland, W. 2017 Ion-scale turbulence in MAST: anomalous transport, subcritical transitions, and comparison to BES measurements. Plasma Phys. Control. Fusion 59 (11), 114003.
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), 905820609.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Simple advecting structures and the edge of chaos in subcritical tokamak plasmas

  • Ben F. McMillan (a1), Chris C. T. Pringle (a2) and Bogdan Teaca (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed