Plasma Physics Reports

, Volume 45, Issue 3, pp 179–194 | Cite as

Low-Frequency Continuous MHD Spectrum of Toroidally Rotating Tokamak Plasmas with Anisotropic Pressure

  • V. P. Lakhin
  • E. A. SorokinaEmail author


Continuous-spectrum equations for low-frequency ideal magnetohydrodynamic perturbations in toroidally rotating plasmas with anisotropic pressure in axisymmetric tokamaks are derived in the framework of the Chew–Goldberger–Low model. In the rotating coordinate system, these equations describe the toroidal coupling of Alfvén and slow magnetosonic modes due to the curvature of magnetic field lines, the pressure anisotropy, and the centrifugal and Coriolis effects. The derived general equations are applied to study the spectra of both the zonal flows and the general electromagnetic modes in low-pressure large-aspect-ratio tokamaks. The condition for the instability of zonal flows due to plasma stratification over the poloidal angle on a magnetic surface is obtained. It is shown that a similar instability takes place for the general modes localized in the vicinities of rational magnetic surfaces. Stabilization of this instability by the Alfvén effect for the modes localized far from rational surfaces is shown.



This work was supported in part by the Russian Science Foundation (project no. 14-22-00193) (the results presented in Section 4.2) and by the Russian Foundation for Basic Research (project no. 18-29-21041). The publication was prepared with support from “RUDN University Program 5-100.”


  1. 1.
    A. Fasoli, C. Gormenzano, H. L. Berk, B. Breizman, S. Briguglio, D. S. Darrow, N. Gorelenkov, W. W. Heidbrink, A. Jaun, S. V. Konovalov, R. Nazikian, J.‑M. Noterdaeme, S. Sharapov, K. Shinohara, D. Testa, et al., Nucl. Fusion 47, S264 (2007).CrossRefGoogle Scholar
  2. 2.
    V. S. Vlasenkov, V. M. Kulygin, V. M. Leonov, V. G. Merezhkin, V. S. Mukhovatov, N. N. Semashko, L. D. Sinitsyna, A. A. Panasenkov, and G. N. Tilinin, in Proceedings of the 6th International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Berchtesgaden, 1976, Vol. 1, p. 85.Google Scholar
  3. 3.
    E. D. Andryukhina, I. S. Danilkin, K. S. Dyabilin, and O. Fedyanin, in Proceedings of the 12th European Conference on Controlled Fusion and Plasma Physics, Budapest, 1985, ECA 9F (1), 481 (1985).Google Scholar
  4. 4.
    C. A. Cotrell and D. F. H. Start, Nucl. Fusion 31, 61 (1991).CrossRefGoogle Scholar
  5. 5.
    H. Yamada, K. Ida, H. Iguchi, S. Morita, O. Kaneko, H. Arimoto, M. Hosokawa, H. Idei, S. Kubo, K. Matsuoka, K. Nishimura, S. Okamura, Y. Takeiri, Y. Takita, C. Takahashi, et al., Nucl. Fusion 32, 25 (1992).ADSCrossRefGoogle Scholar
  6. 6.
    W. Zwingmann, L. G. Ericsson, and P. Stubberfield, Plasma Phys. Controlled Fusion 43, 1441 (2001).ADSCrossRefGoogle Scholar
  7. 7.
    T. Yamaguchi, K. Y. Watanabe, S. Sakakibara, Y. Narushima, K. Narihara, T. Tokuzawa, K. Tanaka, I. Yamada, M. Osakabe, H. Yamada, K. Kawahata, K. Yamazaki, and LHD Experimental Group, Nucl. Fusion 45, L33 (2005).ADSCrossRefGoogle Scholar
  8. 8.
    K. Watanabe, Y. Suzuki, S. Sakakibara, T. Yamaguchi, Y. Narushima, Y. Nakamura, K. Ida, N. Nakajima, H. Yamada, and LHD Experiment Group, Fusion Sci. Technol. 58, 160 (2010).CrossRefGoogle Scholar
  9. 9.
    Y. Asahi, Y. Suzuki, K. Watanabe, and W. A. Cooper, Plasma Fusion Res. 6, 2403123 (2011).ADSCrossRefGoogle Scholar
  10. 10.
    M. J. Hole, G. von Nessi, M. Fitzgerald, K. G. McClements, J. Svensson, and the MAST team, Plasma Phys. Controlled Fusion 53, 074021 (2011).ADSCrossRefGoogle Scholar
  11. 11.
    J. S. deGrassie, Plasma Phys. Controlled Fusion 51, 124047 (2009).ADSCrossRefGoogle Scholar
  12. 12.
    K. Ida and J. E. Rice, Nucl. Fusion 54, 045001 (2014).ADSCrossRefGoogle Scholar
  13. 13.
    J. E. Rice, Plasma Phys. Controlled Fusion 58, 083001 (2016).ADSCrossRefGoogle Scholar
  14. 14.
    J. P. Goedbloed, Phys. Fluids 18, 1258 (1975).ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    B. van der Holst, A. J. C. Beliën, and J. P. Goedbloed, Phys. Rev. Lett. 84, 2865 (2000).ADSCrossRefGoogle Scholar
  16. 16.
    B. van der Holst, A. J. C. Beliën, and J. P. Goedbloed, Phys. Plasmas 7, 4208 (2000).ADSCrossRefGoogle Scholar
  17. 17.
    J. P. Goedbloed, A. J. C. Beliën, B. van der Holst, and R. Keppens, Phys. Plasmas 11, 28 (2004).ADSCrossRefGoogle Scholar
  18. 18.
    S. Wang, Phys. Rev. Lett. 97, 085002 (2006).ADSCrossRefGoogle Scholar
  19. 19.
    C. Wahlberg, Phys. Rev. Lett. 101, 115003 (2008).ADSCrossRefGoogle Scholar
  20. 20.
    C. Wahlberg, Plasma Phys. Controlled Fusion 51, 085006 (2009).ADSCrossRefGoogle Scholar
  21. 21.
    V. P. Lakhin, V. I. Ilgisonis, and A. I. Smolyakov, Phys. Lett. A 374, 4872 (2010).ADSCrossRefGoogle Scholar
  22. 22.
    V. I. Ilgisonis, V. P. Lakhin, A. I. Smolyakov, and E. A. Sorokina, Plasma Phys. Controlled Fusion 53, 065008 (2011).ADSCrossRefGoogle Scholar
  23. 23.
    V. P. Lakhin and V. I. Ilgisonis, Phys. Plasmas 18, 092103 (2011).ADSCrossRefGoogle Scholar
  24. 24.
    J. W. Haverkort, H. J. de Blank, and B. Koren, J. Comp. Phys. 231, 981 (2012).ADSCrossRefGoogle Scholar
  25. 25.
    V. P. Lakhin and E. A. Sorokina, Phys. Plasmas 25, 072111 (2018).ADSCrossRefGoogle Scholar
  26. 26.
    N. N. Gorelenkov, H. L. Berk, E. Fredrickson, S. E. Sharapov, and JET EFDA Contributors, Phys. Lett. A 370, 70 (2007).ADSCrossRefGoogle Scholar
  27. 27.
    G. F. Chew, M. L. Goldberger, and F. E. Low, Proc. R. Soc. London A 236, 112 (1956).ADSCrossRefGoogle Scholar
  28. 28.
    L. E. Zakharov and V. D. Shafranov, in Reviews of Plasma Physics, Ed. by M. A. Leontovich and B. B. Kadomtsev (Consultants Bureau, New York, 1986), Vol. 11, p. 153.Google Scholar
  29. 29.
    V. I. Ilgisonis, Phys. Plasmas 3, 4577 (1996).ADSCrossRefGoogle Scholar
  30. 30.
    R. Iacono, A. Bondeson, F. Troyon, and A. Gruber, Phys. Fluids B 2, 1794 (1990).ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    A. Fujisawa, T. Ido, A. Shimizu, S. Okamura, K. Matsuoka, H. Iguchi, Y. Hamada, H. Nakano, S. Ohshima, K. Itoh, K. Hoshino, K. Shinohara, Y. Miura, Y. Nagashima, S.-I. Itoh, et al., Nucl. Fusion 47, S718 (2007).CrossRefGoogle Scholar
  32. 32.
    A. Fujisawa, Nucl. Fusion 49, 013001 (2009).ADSCrossRefGoogle Scholar
  33. 33.
    A. Fujisawa, K. Itoh, H. Iguchi, K. Matsuoka, S. Okamura, A. Shimizu, T. Minami, Y. Yoshimura, K. Nagaoka, C. Takahashi, M. Kojima, H. Nakano, S. Ohsima, S. Nishimura, M. Isobe, et al., Phys. Rev. Lett. 93, 165002 (2004).ADSCrossRefGoogle Scholar
  34. 34.
    G. S. Xu, B. N. Wan, M. Song, and J. Li, Phys. Rev. Lett. 91, 125001 (2003).ADSCrossRefGoogle Scholar
  35. 35.
    D. K. Gupta, R. J. Fonck, G. R. McKee, D. J. Schlossberg, and M. W. Shafer, Phys. Rev. Lett. 97, 125002 (2006).ADSCrossRefGoogle Scholar
  36. 36.
    G. R. McKee, P. Gohil, D. J. Schlossberg, J. A. Boedo, K. H. Burrell, J. S. deGrassie, R. J. Groebner, R. A. Moyer, C. C. Petty, T. L. Rhodes, L. Schmitz, M. W. Shafer, W. M. Solomon, M. Umansky, G. Wang, et al., Nucl. Fusion 49, 115016 (2007).ADSCrossRefGoogle Scholar
  37. 37.
    H. Ren, Phys. Plasmas 22, 072502 (2015).ADSCrossRefGoogle Scholar
  38. 38.
    Z. S. Qu, M. Fitzgerald, and M. J. Hole, Plasma Phys. Controlled Fusion 56, 075007 (2014).ADSCrossRefGoogle Scholar
  39. 39.
    A. V. Melnikov, personal communication, 2018.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.National Research Center “Kurchatov Institute”MoscowRussia
  2. 2.Peoples’ Friendship University of Russia (RUDN University)MoscowRussia

Personalised recommendations