Skip to main content
SpringerLink
Log in

Comparative Study of Boris and He-VPA for a Toroidally Rippled Tokamak

  • Original Research
  • Published:
Journal of Fusion Energy Aims and scope Submit manuscript

Abstract

Numerical schemes such as Boris solver and volume preserving algorithm (VPA) enable efficient calculations of plasma particle trajectories under the influence of electromagnetic and collisional forces. In this regard, trajectories of fusion-born alpha particles in an axisymmetric tokamak magnetic configuration have been calculated, using 4th order Runge–Kutta (RK4) technique, the Boris scheme and the VPA. It is observed that Boris scheme and VPA produce accurate trajectories for long (\(\ge 10^{3}\) bounce period) simulation time, while the RK4 scheme fails in this regard. Moreover, the total energy is well conserved in Boris and VPA, whereas in RK4, a spurious damping is introduced due to propagation of numerical errors. After demonstrating the superiority of the Boris algorithm and VPA, the computations are extended to magnetically perturbed configuration, namely the toroidal field ripples. It is observed that the resonance interaction, due to toroidal precession of banana orbits and ripple periodicity, causes unwanted radial spread. Whereas, the passing particles, which are not subjected to any resonance interactions, are not affected by ripple magnetic perturbation. In this regard, it is shown that both schemes successfully produce the super banana orbits. However, VPA based computations over-perform as far as energy conservation is concerned.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. D.J. Sigmar, G. Joyce, Plasma heating by energetic particles. Nucl. Fusion 11(5), 447 (1971)

    Article  Google Scholar 

  2. M. Khan, A. Zafar, M. Kamran, Fast ion trajectory calculations in tokamak magnetic configuration using symplectic integration algorithm. J. Fusion Energ. 34(2), 298 (2015)

    Article  Google Scholar 

  3. F.F. Chen, Introduction to Plasma Physics and Controlled Fusion, 3rd edn. (Springer, Berlin, 2015)

    Google Scholar 

  4. D.F.H. Start et al., DT fusion with ion cyclotron resonance heating in the JET tokamak. Phys. Rev. Lett. 80(21), 4681 (1998)

    Article  ADS  Google Scholar 

  5. Y.I. Kolesnichenko, The role of alpha particles in tokamak reactors. Nucl. Fusion 20(6), 727 (1980)

    Article  ADS  Google Scholar 

  6. R.J. Goldston, D.C. McCune, H.H. Towner, S.L. Davis, R.J. Hawryluk, G.L. Schmidt, New techniques for calculating heat and particle source rates due to neutral beam injection in axisymmetric tokamaks. J. Comput. Phys. 43(1), 61 (1981)

    Article  ADS  Google Scholar 

  7. M. Khan, K. Schoepf, V. Goloborodko, V. Yavorskij, Symplectic simulation of fast alpha particle radial transport in tokamaks in the presence of TF ripples and a neoclassical tearing mode. J. Fusion Energ. 31, 547 (2012)

    Article  ADS  Google Scholar 

  8. H. Mimata, K. Tani, H. Tsutsui, K. Tobita, S.T. Iio, R. Shimada, Numerical study of the ripple resonance diffusion of alpha particles in Tokamaks. Plasma Fusion Res. 4, 008 (2009)

    Article  ADS  Google Scholar 

  9. O. Buneman, Time-reversible difference procedures. J. Comput. Phys. 1(4), 517 (1967)

    Article  ADS  Google Scholar 

  10. O. Buneman, The advance from 2D electrostatic to 3D electromagnetic particle simulation. Comput. Phys. Commun. 12(1), 21 (1976)

    Article  ADS  Google Scholar 

  11. J. Dawson, One-dimensional plasma model. Phys. Fluids 5(4), 445 (1962)

    Article  ADS  Google Scholar 

  12. D. Tskhakaya, S. Kuhn, The magnetised plasma-wall transition: Theory and PIC simulation. Contrib. Plasma Phys. 44(5–6), 564 (2004)

    Article  ADS  Google Scholar 

  13. D. Tskhakaya, A. Soba, R. Schneider, M. Borchardt, E. Yurtesen, J. Westerholm, PIC/MC code BIT1 for plasma simulations on HPC. in Proceedings of 18-th Euromicro Conference on Parallel, Distributed and Network-based Processing, p. 476. IEEE Computer Society, 2010

  14. D.D. Tskhakaya Sr., L. Kos, D. Tskhakaya, Stability of the Tonks-Langmuir discharge pre-sheath. Phys. Plasmas 23(3), 032128 (2016)

    Article  ADS  Google Scholar 

  15. F.X. Bronold, K. Matyash, D. Tskhakaya, R. Schneider, H. Fehske, Radio-frequency discharges in oxygen. J. Phys. D Appl. Phys. 40(21), 6583 (2007)

    Article  ADS  Google Scholar 

  16. J. Gruenwald, D. Tskhakaya, J. Kovacic, M. Cercek, T. Gyergyek, C. Ionita, R. Schrittwieser, Comparison of measured and simulated electron energy distribution functions in low-pressure helium plasmas. Plasma Sources Sci. Technol. 22(1), 015203 (2013)

    Article  Google Scholar 

  17. D. Tskhakaya, One-dimensional plasma sheath model in front of the divertor plates. Plasma Phys. Control. Fusion 59(11), 114001 (2017)

    Article  ADS  Google Scholar 

  18. A. Kirschner, D. Tskhakaya, S. Brezinsek, D. Borodin, J. Romazanov, R. Ding, A. Eksaeva, Ch. Linsmeier, Modelling of plasma-wall interaction and impurity transport in fusion devices and prompt deposition of tungsten as application. Plasma Phys. Control. Fusion 60(1), 014041 (2017)

    Article  ADS  Google Scholar 

  19. C.K. Birdsall, A.B. Langdon, Plasma Physics via Computer Simulation (McGraw-Hill, New York, 1985)

    Google Scholar 

  20. J.P. Boris, R.A. Shanny, Proceedings: Fourth Conference on Numerical Simulation of Plasmas (Naval Research Laboratory, 1972)

  21. H. Qin, S. Zhang, J. Xiao, J. Liu, Y. Sun, W.M. Tang, Why is boris algorithm so good? Phys. Plasmas 20(8), 084503 (2013)

    Article  ADS  Google Scholar 

  22. G. Penn, P.H. Stoltz, J.R. Cary, J. Wurtele, Boris push with spatial stepping. J. Phys. G 29(8), 1719 (2003)

    Article  ADS  Google Scholar 

  23. P.H. Stoltz, J.R. Cary, G. Penn, J. Wurtele, Efficiency of a Boris-like integration scheme with spatial stepping. Phys. Rev. Accel. Beams 5(9), 094001 (2002)

    Article  ADS  Google Scholar 

  24. S. Zenitani, T. Umeda, On the boris solver in particle-in-cell simulation. Phys. Plasmas 25(11), 112110 (2018)

    Article  ADS  Google Scholar 

  25. M. Hoppe, A. Iantchenko, I. Strandberg, Simulation of charged particle orbits in fusion plasmas. Bachelor thesis, Chalmers University of Technology, 2015

  26. Y. He, Z. Zhou, Y. Sun, J. Liu, H. Qin, Explicit K-symplectic algorithms for charged particle dynamics. Phys. Lett. A 381, 568 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  27. Y. He, H. Qin, Y. Sun, J. Xiao, R. Zhang, J. Liu, Hamiltonian time integrators for Vlasov-Maxwell equations. Phys. Plasmas 22, 124503 (2015)

    Article  ADS  Google Scholar 

  28. H. Fehske, R. Schneider, A. Wei\(\beta\)e (ed.), Computational Many-Particle Physics. (Springer-Verlag, Berlin, 2008)

  29. J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, vol. 17 (Springer Science & Business Media, Berlin, 2013)

    MATH  Google Scholar 

  30. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31 (Springer Science & Business Media, Berlin, 2006)

    MATH  Google Scholar 

  31. R.I. McLachlan, G.R.W. Quispel, Splitting methods. Acta Numer. 11, 341 (2002)

    Article  MathSciNet  Google Scholar 

  32. F. Kang, S.Z. Jiu, Volume-preserving algorithms for source-free dynamical systems. Numer. Math. 71(4), 451 (1995)

    Article  MathSciNet  Google Scholar 

  33. Y. He, Y. Sun, J. Liu, H. Qin, Volume-preserving algorithms for charged particle dynamics. J. Comput. Phys. 281, 135 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  34. S.A. Chin, Symplectic and energy-conserving algorithms for solving magnetic field trajectories. Phys. Rev. E 77(6), 066401 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  35. R. Zhang, J. Liu, H. Qin, Y. Wang, Y. He, Y. Sun, Volume-preserving algorithm for secular relativistic dynamics of charged particles. Phys. Plasmas 22(4), 044501 (2015)

    Article  ADS  Google Scholar 

  36. V.A. Yavorskij, J.W. Edenstrasser, V.. Ya.. Goloborodko, S.N. Reznik, S.J. Zweben, Fokker-Planck modelling of delayed loss of charged fusion products in TFTR. Nucl. Fusion 38(10), 1565 (1998)

    Article  ADS  Google Scholar 

  37. P.N. Yushmanov, Generalized ripple-banana transport in a tokamak. Nucl. Fusion 23(12), 1599 (1983)

    Article  Google Scholar 

  38. P.N. Yushmanov, Confinement of toroidally trapped high-energy particles in a rippled magnetic field. JETP Lett. 35(12), 619 (1982)

    ADS  Google Scholar 

Download references

Author information

Author notes

  1. Sofia Khalid and Abdullah Zafar have contributed equally to this work.

Authors and Affiliations

  1. Department of Physics, COMSATS University Islamabad, Park Road, Chak Shahzad, Islamabad, 45550, Pakistan

    M. Kamran

  2. Department of Physics, Quaid-i-Azam University, Islamabad, 45320, Pakistan

    Sofia Khalid & Majid Khan

  3. Department of Engineering and Applied Physics, University of Science and Technology, Hefei, 230026, China

    Abdullah Zafar

  4. Department of Physics, Hazara University, Mansehra, 21300, Pakistan

    M. Ikram

Authors
  1. Sofia Khalid
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdullah Zafar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Majid Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Kamran
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. M. Ikram
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Kamran.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Project supported by Higher Education Commission (HEC), Pakistan (Grant No. 7659/Balochistan/NRPU/R&D/HEC/2017).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khalid, S., Zafar, A., Khan, M. et al. Comparative Study of Boris and He-VPA for a Toroidally Rippled Tokamak. J Fusion Energ 40, 19 (2021). https://doi.org/10.1007/s10894-021-00309-1

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10894-021-00309-1

Keywords

Search

Navigation