Nonlinear Dynamics

, Volume 93, Issue 4, pp 1989–2001 | Cite as

Modeling of the mode dynamics generated by Madison Symmetric Torus machine utilizing a modified sine-Gordon equation

  • Hameed K. Ebraheem
  • Nizar J. AlkhateebEmail author
  • Ebraheem K. Sultan
Original Paper


In this paper, a new dynamic model is presented for the experimental data generated by the Madison Symmetric Torus (MST) machine. The model is based on a modified sine-Gordon (SG) dynamic equation. The modified sine-Gordon equation model effectively captures the behavior of the slinky mode in reversed-field pinch experiments. In addition, this paper demonstrates how the derived model accurately describes the behavior of the localized magnetohydrodynamic mode (slinky mode) that appears in reversed-field pinch toroidal magnetic confinement systems. The modified SG equation model is solved analytically by using the perturbation method. The resulting model is fit to match a variety of experimental results in the MST reversed-field pinch experiment. The efficacy of the newly developed model in effectively representing the slinky mode is verified by comparing obtained analytical solution to experimentally measured data.


Madison Symmetric Torus (MST) Magnetohydrodyamic (MHD) Sine-Gordon toroidal Dynamic modeling Reversed-field pinch (RFP) 



The authors would like to thank S. C. Prager, and A. F. Almagri from the MST scientific research group at University of Wisconsin-Madison for providing the experimental data taken at MST.


Funding was provided by Public Authority of Applied Education and Training (Grant No. TS16-11).


  1. 1.
    Ebraheem, H.K., Shohet, J.L., Scott, A.C.: Mode locking in reversed-field pinch experiments. Phys. Rev. Lett. 88, 235003 (2002)CrossRefGoogle Scholar
  2. 2.
    Shohet, J.L., Barmish, B.R., Ebraheem, H.K., Scott, A.C.: The sine-Gordon equation in reversed-field pinch experiments. Phys. Plasmas 11, 3887 (2004)CrossRefGoogle Scholar
  3. 3.
    Shohet, J.L.: The sine-Gordon equation in toroidal magnetic-fusion experiments. Eur. Phys. J. Special Topics 147, 191–207 (2007)CrossRefGoogle Scholar
  4. 4.
    Yagi, Y., Koguchi, H., Nilsson, J.-A.B., Bolzonella, T., Zanca, P., Sekine, S., Osakabe, K., Sakakita, H.: Phase and wall-locked modes found in a large reversed-field pinch machine. Jpn. J. Appl. Phys. 38, L780 (19990)Google Scholar
  5. 5.
    Hansen, A.K.: Kinematics of nonlinearly interacting MHD instabilities in a plasma. Ph.D. Thesis, University of Wisconsin -Madison (2000)Google Scholar
  6. 6.
    Hansen, A.K., Almagri, A.F., Den Hartog, D.J., Prager, S.C., Sarff, J.S.: Locking multiple resonant mode structures in the reversed-field pinch. Phys. Plasmas 5, 2942 (1998)CrossRefGoogle Scholar
  7. 7.
    Hegna, C.C.: Nonlinear tearing mode interactions and mode locking in reversed-field pinches. Phys. Plasmas 3, 4646 (1996)CrossRefGoogle Scholar
  8. 8.
    Fitzpatrick, R.: Formation and locking of the slinky mode in reversed field pinches. Phys. Plasmas 6, 1168 (1999)CrossRefGoogle Scholar
  9. 9.
    Tamano, T., Bard, W.D., Cheng, C., Kondoh, Y., Haye, R.J.L., Lee, P.S., Saito, M., Schaffer, M.J., Taylor, P.L.: Observation of a new toroidally localized kink mode and its role in reverse-field-Pinch plasmas. Phys. Rev. Lett. 59, 1444 (1987)CrossRefGoogle Scholar
  10. 10.
    Sarff, J.S., Assadi, S., Almagri, A.F., Cekic, M., Den Hartog, D.J., Fiksel, G., Hokin, S.A., Ji, H., Prager, S.C., Shen, W., Sidikman, K.L., Stoneking, M.R.: Nonlinear coupling of tearing fluctuations in the Madison Symmetric Torus. Phys. Fluids B 5, 2540 (1993)CrossRefGoogle Scholar
  11. 11.
    Bodin, H.A.B.: Reversed Field Pinch plasma. Nucl. Fusion 30, 1717 (1990)CrossRefGoogle Scholar
  12. 12.
    Ho, Y.L., Prager, S.C., Schnack, D.D.: Nonlinear behavior of the reversed field pinch with nonideal boundary conditions. Phys. Rev. Lett. 62, 1504 (1989)CrossRefGoogle Scholar
  13. 13.
    Almagri, A. F.: The effects of magnetic field errors on reversed-field pinch plasmas, Ph.D. thesis, University of Wisconsin-Madison (1990)Google Scholar
  14. 14.
    White, R., Fitzpatrick, R.: Effect of rotation and velocity shear on tearing layer stability in tokamak plasmas. Phys. Plasmas 22, (2015)Google Scholar
  15. 15.
    Fitzpatrick, R.: Phase locking of multi-helicity neoclassical tearing modes in Tokomak plasmas. Phys. Plasmas 22 (2015)Google Scholar
  16. 16.
    Xu Tao, Hu, Xi-Wei, Qi-Ming, Hu, Qing-Quan, Yu.: Locking of tearing modes by the error field. Chin. Phys. Lett. 22, 9 (2011)Google Scholar
  17. 17.
    Ivanov, N.V., Kakurin, A.M.: Locking of Small Magnetic Islands by Error Field in T-10 Tokamak. In: 38th EPS Conference on Plasma Physics (2011)Google Scholar
  18. 18.
    Fitzpatrick, R.: Linear and nonlinear response of a rotating tokomak plasma to a resonant error-field. Phys. Plasmas 21, (2014)Google Scholar
  19. 19.
    Scott, A.C.: Nonlinear Science: Emergence and Dynamics of Coherent Structures. Oxford University Press, Oxford (2006)Google Scholar
  20. 20.
    Scott, A.C.: Encyclopedia of Nonlinear Science. Taylor and Francis Group, New York (2005)zbMATHGoogle Scholar
  21. 21.
    Remoissenet, M.: Waves Called Solitons, 3rd edn. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  22. 22.
    Almagri, A.F., Assadi, S., Prager, S.C., Sarff, J.S., Kerst, D.W.: Locked modes and magnetic field errors in the Madison Symmetric Torus. Phys. Fluids B 4, 4080 (1992)CrossRefGoogle Scholar
  23. 23.
    McLaughlin, D.W., Scott, A.C.: Perturbation analysis of fluxon dynamics. Phys. Rev. A 18, 1652 (1978)CrossRefGoogle Scholar
  24. 24.
    Keener, J.P., Mc Laughlin, D.W.: Solitons under perturbations. Phys. Rev. A 16, 777 (1977)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley Inter-Science, New York (1974)zbMATHGoogle Scholar
  26. 26.
    Nayfeh, A.H.: Introduction to Perturbation Techniques. John Wiley & Sons, Hoboken (1981)zbMATHGoogle Scholar
  27. 27.
    Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Springer, New York (1991)zbMATHGoogle Scholar
  28. 28.
    Ho, Y.L., Prager, S.C.: Stability of a reversed field pinch with resistive and distant boundaries. Phys. Fluids 31, 1673 (1988)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electronic Engineering Technology, College of Technological Studies (CTS)The Public Authority of Applied Education and Training (PAAET)ShuwiekhKuwait

Personalised recommendations