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Magnetic configuration effects on the Wendelstein 7-X stellarator

An Author Correction to this article was published on 11 September 2018

A Publisher Correction to this article was published on 03 July 2018

This article has been updated


The two leading concepts for confining high-temperature fusion plasmas are the tokamak and the stellarator. Tokamaks are rotationally symmetric and use a large plasma current to achieve confinement, whereas stellarators are non-axisymmetric and employ three-dimensionally shaped magnetic field coils to twist the field and confine the plasma. As a result, the magnetic field of a stellarator needs to be carefully designed to minimize the collisional transport arising from poorly confined particle orbits, which would otherwise cause excessive power losses at high plasma temperatures. In addition, this type of transport leads to the appearance of a net toroidal plasma current, the so-called bootstrap current. Here, we analyse results from the first experimental campaign of the Wendelstein 7-X stellarator, showing that its magnetic-field design allows good control of bootstrap currents and collisional transport. The energy confinement time is among the best ever achieved in stellarators, both in absolute figures (τE > 100 ms) and relative to the stellarator confinement scaling. The bootstrap current responds as predicted to changes in the magnetic mirror ratio. These initial experiments confirm several theoretically predicted properties of Wendelstein 7-X plasmas, and already indicate consistency with optimization measures.

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Fig. 1: Outline of Wendelstein 7-X and its magnet system.
Fig. 2: Radial profiles of important magnetic field properties.
Fig. 3: Waveforms of plasma discharges.
Fig. 4: Energy confinement times of Wendelstein 7-X.
Fig. 5: Temperature and density profiles for different magnetic configurations.
Fig. 6: Neoclassical modelling: radial electric field, energy fluxes and bootstrap current profiles.

Change history

  • 11 September 2018

    In the version of this Article originally published, and in the associated Publisher Correction, the members of the W7-X Team were not included. All versions of the Article, and the Publisher Correction, have now been amended to include these team members.

  • 03 July 2018

    In the version of this Article originally published, A. Mollén’s affiliation was incorrectly denoted as number 10; it should have been 1. Throughout the Article, some technical problems in typesetting meant that the tilde symbol above b and one instance of a superscript 2 were too high to be visible; see the correction notice for details. Finally, the citation to ref. 35 on page one of the Supplementary Information was incorrect; it should have been to ref. 36. These issues have now been corrected.


  1. 1.

    Spitzer, L. The stellarator concept. Phys. Fluids 1, 253–264 (1958).

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Helander, P. et al. Stellarator and tokamak plasmas: a comparison. Plasma Phys. Contr. Fusion 54, 124009 (2012).

    ADS  Article  Google Scholar 

  3. 3.

    Galeev, A. A., Sagdeev, R. Z., Furth, H. P. & Rosenbluth, M. N. Plasma diffusion in a toroidal stellarator. Phys. Rev. Lett. 22, 511–514 (1969).

    ADS  Article  Google Scholar 

  4. 4.

    Beidler, C. D. & Hitchon, W. N. G. Ripple transport in helical-axis advanced stellarators: a comparison with classical stellarators/torsatrons. Plasma Phys. Contr. Fusion 36, 317–353 (1995).

    ADS  Article  Google Scholar 

  5. 5.

    Yokoyama, M. et al. Core electron-root confinement (CERC) in helical plasmas. Nucl. Fusion 47, 1213–1219 (2007).

    ADS  Article  Google Scholar 

  6. 6.

    Galeev, A. A. & Sagdeev, R. Z. in Reviews of Plasma Physics Vol. 7 (ed. Leontovich, M. A.) 257–343 (Consultants Bureau, New York, NY, 1979)

  7. 7.

    Palumbo, D. Some considerations on closed configurations of magnetohydrostatic equilibrium. Nuovo Cim. B X53, 507–511 (1968).

    ADS  Article  Google Scholar 

  8. 8.

    Nührenberg, J. Development of quasi-isodynamic stellarators. Plasma Phys. Contr. Fusion 52, 124003 (2010).

    ADS  Article  Google Scholar 

  9. 9.

    Nührenberg, J. & Zille, R. Stable stellarators with medium β and aspect ratio. Phys. Lett. A 114, 129–132 (1986).

    ADS  Article  Google Scholar 

  10. 10.

    Galeev, A. A. Diffusion-electrical phenomena in a plasma confined in a tokamak machine. Sov. Phys. JETP 32, 752–757 (1971).

    ADS  Google Scholar 

  11. 11.

    Bickerton, R. J., Connor, J. W. & Taylor, J. B. Diffusion driven plasma currents and bootstrap tokamak. Nature 229, 110–112 (1971).

    ADS  Article  Google Scholar 

  12. 12.

    Zarnstorff, M. C. et al. Bootstrap current in TFTR. Phys. Rev. Lett. 60, 1306–1309 (1988).

    ADS  Article  Google Scholar 

  13. 13.

    Murakami, M. et al. Bootstrap-current experiments in a toroidal plasma-confinement device. Phys. Rev. Lett. 66, 707–710 (1991).

    ADS  Article  Google Scholar 

  14. 14.

    Helander, P. & Nührenberg, J. Bootstrap current and neoclassical transport in quasi-isodynamic stellarators. Plasma Phys. Contr. Fusion 51, 055004 (2009).

    ADS  Article  Google Scholar 

  15. 15.

    Landreman, M. & Catto, P. J. Omnigenity as generalized quasisymmetry. Phys. Plasmas 19, 056103 (2012).

    ADS  Article  Google Scholar 

  16. 16.

    Hirsch, M. et al. Major results from the stellarator Wendelstein 7-AS. Plasma Phys. Contr. 50, 053001 (2008).

    ADS  Article  Google Scholar 

  17. 17.

    Sunn Pedersen, T. et al. Confirmation of the topology of the Wendelstein 7-X magnetic field to better than 1:100,000. Nat. Commun. 7, 13493 (2016).

    ADS  Article  Google Scholar 

  18. 18.

    Bozhenkov, S. A. et al. Effect of error field correction coils on W7-X limiter loads. Nucl. Fusion 57, 126030 (2017).

    ADS  Article  Google Scholar 

  19. 19.

    Sunn Pedersen, T. et al. Plans for the first plasma operation of Wendelstein 7-X. Nucl. Fusion 55, 126001 (2015).

    ADS  Article  Google Scholar 

  20. 20.

    Klinger, T. et al. Performance and properties of the first plasmas of Wendelstein 7-X. Plasma Phys. Contr. Fusion 59, 014018 (2017).

    ADS  Article  Google Scholar 

  21. 21.

    Wolf, R. C. et al. Major results from the first plasma campaign of the Wendelstein 7-X stellarator. Nucl. Fusion 57, 102020 (2017).

    ADS  Article  Google Scholar 

  22. 22.

    Sunn Pedersen, T. et al. Key results from the first plasma operation phase and outlook for future performance in Wendelstein 7-X. Phys. Plasmas 24, 055503 (2017).

    ADS  Article  Google Scholar 

  23. 23.

    Wolf, R. C. et al. Wendelstein 7-X program—Demonstration of a stellarator option for fusion energy. IEEE Trans. Act. Plasma Sci. 44, 1466–1471 (2016).

    ADS  Article  Google Scholar 

  24. 24.

    Wurden, G. A. et al. Limiter observations during W7-X first plasmas. Nucl. Fusion 57, 056036 (2017).

    ADS  Article  Google Scholar 

  25. 25.

    Yamada, H. et al. Characterization of energy confinement in net-current free plasmas using the extended International Stellarator Database. Nucl. Fusion 45, 1684–1693 (2005).

    ADS  Article  Google Scholar 

  26. 26.

    Stroth, U. et al. Energy confinement scaling from the International Stellarator Database. Nucl. Fusion 36, 1063–1077 (1996).

    ADS  Article  Google Scholar 

  27. 27.

    Dinklage, A. et al. Physics model assessment of the energy confinement time scaling in stellarators. Nucl. Fusion 47, 1265–1273 (2007).

    ADS  Article  Google Scholar 

  28. 28.

    Hirsch, M. et al. Confinement in Wendelstein 7-X limiter plasmas. Nucl. Fusion 57, 086010 (2017).

    ADS  Article  Google Scholar 

  29. 29.

    Doyle, E. J. et al. Chapter 2: Plasma confinement and transport. Nucl. Fusion 47, S18–S127 (2007).

    Article  Google Scholar 

  30. 30.

    Turkin, Yu. et al. Neoclassical transport simulations for stellarators. Phys. Plasmas 18, 022505 (2011).

    ADS  Article  Google Scholar 

  31. 31.

    Turkin, Yu, Maaßberg, H., Beidler, C. D., Geiger, J. & Marushchenko, N. Current control by ECCD for W7-X. Fusion Sci. Technol. 50, 387–394 (2006).

    Article  Google Scholar 

  32. 32.

    Bosch, H.-S. et al. Final integration, commissioning and start of the Wendelstein 7-X stellarator operation. Nucl. Fusion 57, 116015 (2017).

    ADS  Article  Google Scholar 

  33. 33.

    Marsen, S. et al. First results from protective ECRH diagnostics for Wendelstein 7-X. Nucl. Fusion 57, 086014 (2017).

    ADS  Article  Google Scholar 

  34. 34.

    Moseev, D. et al. Inference of the microwave absorption coefficient from stray radiation measurements in Wendelstein 7-X. Nucl. Fusion 57, 036013 (2017).

    ADS  Article  Google Scholar 

  35. 35.

    Wauters, T. et al. Wall conditioning by ECRH discharges and He-GDC in the limiter phase of Wendelstein 7-X. Nucl. Fusion 58, 066013 (2018).

    ADS  Article  Google Scholar 

  36. 36.

    Beidler, C. D. et al. Benchmarking of the mono-energetic transport coefficients—results from the International Collaboration on Neoclassical Transport in Stellarators (ICNTS). Nucl. Fusion 51, 076001 (2011).

    ADS  Article  Google Scholar 

  37. 37.

    Beidler, C. et al. Physics and engineering design for Wendelstein VII-X. Fusion Technol. 17, 148–168 (1990).

    Article  Google Scholar 

  38. 38.

    Hirshman, S. P., van Rij, W. I. & Merkel, P. Three-dimensional free boundary calculations using a spectral Green's function method. Comp. Phys. Comm. 43, 143–155 (1986).

    ADS  MathSciNet  Article  Google Scholar 

  39. 39.

    Bozhenkov, S. A. et al. Service oriented architecture for scientific analysis at W7-X. An example of a field line tracer. Fusion Eng. Des. 88, 2997–3006 (2013).

    Article  Google Scholar 

  40. 40.

    Hirshman, S. P., Shaing, K. C., van Rij, W. I. & Crume, E. C. Jr. Plasma transport coefficients for nonsymmetric toroidal confinement systems. Phys. Fluids 29, 2951–2959 (1986).

    ADS  Article  Google Scholar 

  41. 41.

    van Rij, W. I. & Hirshman, S. P. Variational bounds for transport coefficients in three‐dimensional toroidal plasmas. Phys. Fluids B 1, 563–569 (1989).

    ADS  Article  Google Scholar 

  42. 42.

    Landreman, M., Smith, H. M., Mollén, A. & Helander, P. Comparison of particle trajectories and collision operators for collisional transport in nonaxisymmetric plasmas. Phys. Plasmas 21, 042503–042503 (2014).

    ADS  Article  Google Scholar 

  43. 43.

    Marushchenko, N., Turkin, Yu & Maaßberg, H. Ray-tracing code TRAVIS for ECR heating, EC current drive and ECE diagnostic. Comp. Phys. Comm. 185, 165–176 (2014).

    ADS  Article  Google Scholar 

  44. 44.

    Feng, Y., Sardei, F., Kisslinger, J., Grigull, P., Mc Cormick, K. & Reiter, D. 3D edge modeling and island divertor physics. Contrib. Plasma Phys. 44, 57–69 (2004).

    ADS  Article  Google Scholar 

  45. 45.

    Effenberg, F. et al. Numerical investigation of plasma edge transport and limiter heat fluxes in Wendelstein 7-X startup plasmas with EMC3-EIRENE. Nucl. Fusion 57, 036021 (2017).

    ADS  Article  Google Scholar 

  46. 46.

    Krychowiak, M. et al. Overview of diagnostic performance and results for the first operation phase in Wendelstein 7-X. Rev. Sci. Instrum. 87, 11D304 (2016).

    Article  Google Scholar 

  47. 47.

    Endler, M. et al. Engineering design for the magnetic diagnostics of Wendelstein 7-X. Fusion Eng. Des. 100, 468–494 (2015).

    Article  Google Scholar 

  48. 48.

    Pablant, N. A. et al. Core radial electric field and transport in Wendelstein 7-X plasmas. Phys. Plasmas 25, 022508 (2018).

    ADS  Article  Google Scholar 

  49. 49.

    Krämer-Flecken, A. et al. Investigation of turbulence rotation in limiter plasmas at W7-X with a new installed poloidal correlation reflectometry. Nucl. Fusion 57, 066023 (2017).

    ADS  Article  Google Scholar 

  50. 50.

    Romé, M., Erckmann, V., Gasparino, U. & Karulin, N. Electron cyclotron resonance heating and current drive in the W7-X stellarator. Plasma Phys. Contr. Fusion 40, 511–530 (1998).

    ADS  Article  Google Scholar 

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This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 under grant agreement 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. This work is partially supported by the US Department of Energy under a project agreement with the Max Planck Institute for Plasma Physics.

Author information





A.D., C.D.B., P.H., T.S.P., R.C.W. and T.K. wrote the paper. A.D., T.S.P., S.B., F.E. and J.G. prepared the configuration changes and the discharge program. A.D., H.M., Y.T., T.A., A.A., C.D.B., F.E., Y.F., J.G., A.M., N.M., H.M.S. and O.S. did modelling and data validation. K.R., B.B., B.B., A.C., G.F., M.H., U.H., M.J., J.K., G.K., A.K.F., M.K., A.L., H.L., U.N., H.N., E.P., N.P., L.R., T.S., T.S., G.W., T.W., G.W. and D.Z. did measurements and data analysis.

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Correspondence to A. Dinklage.

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Supplementary Information

Supplementary notes, supplementary figures 1–2, supplementary tables 12

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Dinklage, A., Beidler, C.D., Helander, P. et al. Magnetic configuration effects on the Wendelstein 7-X stellarator. Nature Phys 14, 855–860 (2018).

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