Skip to main content Accessibility help
×
Home

Threshold for the destabilisation of the ion-temperature-gradient mode in magnetically confined toroidal plasmas

  • A. Zocco (a1) (a2), P. Xanthopoulos (a1), H. Doerk (a2), J. W. Connor (a3) and P. Helander (a1)...

Abstract

The threshold for the resonant destabilisation of ion-temperature-gradient (ITG) driven instabilities that render the modes ubiquitous in both tokamaks and stellarators is investigated. We discover remarkably similar results for both confinement concepts if care is taken in the analysis of the effect of the global shear ${\hat{s}}$ . We revisit, analytically and by means of gyrokinetic simulations, accepted tokamak results and discover inadequacies of some aspects of their theoretical interpretation. In particular, for standard tokamak configurations, we find that global shear effects on the critical gradient cannot be attributed to the wave–particle resonance destabilising mechanism of Hahm & Tang (Phys. Plasmas, vol. 1, 1989, pp. 1185–1192), but are consistent with a stabilising contribution predicted by Biglari et al. (Phys. Plasmas, vol. 1, 1989, pp. 109–118). Extensive analytical and numerical investigations show that virtually no previous tokamak theoretical predictions capture the temperature dependence of the mode frequency at marginality, thus leading to incorrect instability thresholds. In the asymptotic limit ${\hat{s}}\unicode[STIX]{x1D704}\ll 1$ , where $\unicode[STIX]{x1D704}$ is the rotational transform, and such a threshold should be solely determined by the resonant toroidal branch of the ITG mode, we discover a family of unstable solutions below the previously known threshold of instability. This is true for a tokamak case described by a local ${\hat{s}}-\unicode[STIX]{x1D6FC}$ local equilibrium, and for the stellarator Wendelstein 7-X, where these unstable solutions are present even for configurations with a small trapped-particle population. We conjecture they are of the Floquet type and derive their properties from the Fourier analysis of toroidal drift modes of Connor & Taylor (Phys. Fluids, vol. 30, 1987, pp. 3180–3185), and to Hill’s theory of the motion of the lunar perigee (Acta Math., vol. 8, 1886, pp. 1–36). The temperature dependence of the newly determined threshold is given for both confinement concepts. In the first case, the new temperature-gradient threshold is found to be rather insensitive to the temperature ratio $T_{i}/T_{e}$ , at least for $T_{i}/T_{e}\lesssim 1$ , and to be a growing function of the density gradient scale for $T_{i}/T_{e}\gtrsim 1$ . For Wendelstein 7-X, the new critical temperature gradient is a growing function of the temperature ratio. The importance of these findings for the assessment of turbulence in stellarators and low-shear tokamak configurations is discussed.

  • 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.

      Threshold for the destabilisation of the ion-temperature-gradient mode in magnetically confined toroidal 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.

      Threshold for the destabilisation of the ion-temperature-gradient mode in magnetically confined toroidal 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.

      Threshold for the destabilisation of the ion-temperature-gradient mode in magnetically confined toroidal plasmas
      Available formats
      ×

Copyright

Corresponding author

Email address for correspondence: alessandro.zocco@ipp.mpg.de

References

Hide All
Bhattacharjee, A., Sedlak, J. E., Similon, P. L., Rosenbluth, M. N. & Ross, D. W. 1983 Drift waves in a straight stellarator. Phys. Fluids 26 (4), 880882.
Biglari, H., Diamond, P. H. & Rosenbluth, M. N. 1989 Toroidal ion-pressure-gradient-driven drift instabilities and transport revisited. Phys. Fluids B 1 (1), 109118.
Candy, J., Waltz, R. E. & Rosenbluth, M. N. 2004 Smoothness of turbulent transport across a minimum-q surface. Phys. Plasmas 11 (5), 18791890.
Connor, J. W. & Hastie, R. J. 2004 Microstability in tokamaks with low magnetic shear. Plasma Phys. Control. Fusion 46 (10), 1501.
Connor, J. W., Hastie, R. J. & Taylor, J. B. 1978 Shear, periodicity, and plasma ballooning modes. Phys. Rev. Lett. 40, 396.
Connor, J. W. & Taylor, J. B. 1987 Ballooning modes or Fourier modes in a toroidal plasma? Phys. Fluids 30 (10), 31803185.
Coppi, B., Laval, G., Pellat, R. & Rosenbluth, M. N. 1968 Collisionless microinstabilities in configurations with periodic magnetic curvature. Plasma Phys. 10 (1), 1.
Coppi, B., Rosenbluth, M. N. & Sagdeev, R. Z. 1967 Instabilities due to temperature gradients in complex magnetic field configurations. Phys. Fluids 10 (3), 582.
Dewar, R. L. 1997 Spectrum of the ballooning Schrödinger equation. Plasma Phys. Control. Fusion 39 (3), 453.
Dickinson, D., Roach, C. M., Skipp, J. M. & Wilson, H. R. 2014 Structure of micro-instabilities in tokamak plasmas: stiff transport or plasma eruptions? Phys. Plasmas 21, 010702.
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.
Faber, B. J., Pueschel, M. J., Proll, J. H. E., Xanthopoulos, P., Terry, P. W., Hegna, C. C., Weir, G. M., Likin, K. M. & Talmadge, J. N. 2015 Gyrokinetic studies of trapped electron mode turbulence in the helically symmetric experiment stellarator. Phys. Plasmas 22 (7), 072305.
Fried, B. D. & Conte, S. D. 1961 The Plasma Dispersion Function. Academic.
Geiger, J., Beidler, C. D., Feng, Y., Maassberg, H., Marushchenko, N. B. & Turkin, Y. 2015 Physics in the magnetic configuration space of W7-X. Plasma Phys. Control. Fusion 57 (1), 014004.
Hahm, T. S. & Tang, W. M. 1989 Properties of ion temperature gradient drift instabilities in H-mode plasmas. Phys. Fluids B 1 (6), 11851192.
Hastie, R. J. & Hesketh, K. W. 1981 Kinetic modifications to the MHD ballooning mode. Nucl. Fusion 21 (6), 651.
Hill, G. W. 1886 On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon. Acta Math. 8 (1), 136.
Horton, W., Choi, D. & Tang, W. M. 1981 Toroidal drift modes driven by ion pressure gradients. Phys. Fluids 24 (6), 10771085.
Jenko, F., Dorland, W. & Hammett, G. W. 2001 Critical gradient formula for toroidal electron temperature gradient modes. Phys. Plasmas 8 (9), 40964104.
Jenko, F., Dorland, W., Kotschenreuther, M. & Rogers, B. N. 2000 Electron temperature gradient driven turbulence. Phys. Plasmas 7 (5), 19041910.
Kadomtsev, B. B. & Pogutse, O. P. 1970 Reviews of Plasma Physics, vol. 5. Taylor and Francis.
Mantica, P., Strintzi, D., Tala, T., Giroud, C., Johnson, T., Leggate, H., Lerche, E., Loarer, T., Peeters, A. G., Salmi, A. et al. 2009 Experimental study of the ion critical-gradient length and stiffness level and the impact of rotation in the JET tokamak. Phys. Rev. Lett. 102, 175002.
Owen, D. B. 1956 Tables for computing bivariate normal probabilities. Ann. Math. Statist. 27 (4), 10751090.
Plunk, G. G., Helander, P., Xanthopoulos, P. & Connor, J. W. 2014 Collisionless microinstabilities in stellarators. III. The ion-temperature-gradient mode. Phys. Plasmas 21 (3), 032112.
Proll, J. H. E., Xanthopoulos, P. & Helander, P. 2013 Collisionless microinstabilities in stellarators. II. Numerical simulations. Phys. Plasmas 20 (12), 122506.
Romanelli, F. 1989 Ion temperature-gradient-driven modes and anomalous ion transport in tokamaks. Phys. Fluids B 1 (5), 1018.
Romanelli, F. & Zonca, F. 1993 The radial structure of the ion-temperature-gradient-driven mode. Phys. Fluids B 5 (11), 40814089.
Rosenbluth, M. N. & Longmire, C. L. 1956 Stability of plasmas confined by magnetic fields. Ann. Phys. 1 (2), 120.
Rudakov, L. I. & Sagdeev, R. Z. 1961 O neustoychivosti neodnorodnoy razrezhennoy plazmyi v silnom magnitnom pole. Dokl. Akad. Nauk CCCP 138, 581.
Taylor, J. B. & Hastie, R. J. 1968 Stability of general plasma equilibria – I formal theory. Plasma Phys. 10 (5), 479.
Taylor, J. B. & Wilson, H. R. 1996 Plasma rotation and toroidal drift modes. Plasma Phys. Control. Fusion 38 (11), 1999.
Terry, P., Anderson, W. & Horton, W. 1982 Kinetic effects on the toroidal ion pressure gradient drift mode. Nucl. Fusion 22 (4), 487.
Tricomi, F. G. 1950a Asymptotische eigenschaften der unvollständigen gammafunktion. Math. Z. 53 (2), 136148.
Tricomi, F. G. 1950b Sulla funzione gamma incompleta. Annali di Matematica Pura ed Applicata 31 (1), 263279.
Xanthopoulos, P., Cooper, W. A., Jenko, F., Turkin, Y., Runov, A. & Geiger, J. 2009 A geometry interface for gyrokinetic microturbulence investigations in toroidal configurations. Phys. Plasmas 16 (8), 082303.
Zocco, A., Plunk, G. G., Xanthopoulos, P. & Helander, P. 2016 Geometric stabilization of the electrostatic ion-temperature-gradient driven instability. I. Nearly axisymmetric systems. Phys. Plasmas 23 (8).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Threshold for the destabilisation of the ion-temperature-gradient mode in magnetically confined toroidal plasmas

  • A. Zocco (a1) (a2), P. Xanthopoulos (a1), H. Doerk (a2), J. W. Connor (a3) and P. Helander (a1)...

Metrics

Altmetric attention score

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