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Forward and inverse problems in fundamental and applied magnetohydrodynamics

  • Andre Giesecke
  • Frank Stefani
  • Thomas Wondrak
  • Mingtian Xu
Review

Abstract

This minireview summarizes the recent efforts to solve forward and inverse problems as they occur in different branches of fundamental and applied magnetohydrodynamics. For the forward problem, the main focus is on the numerical treatment of induction processes, including self-excitation of magnetic fields in non-spherical domains and/or under the influence of non-homogeneous material parameters. As an important application of the developed numerical schemes, the functioning of the von-Kármán-sodium (VKS) dynamo experiment is shown to depend crucially on the presence of soft-iron impellers. As for the inverse problem, the main focus is on the mathematical background and some initial practical applications of contactless inductive flow tomography (CIFT), in which flow induced magnetic field perturbations are utilized to reconstruct the velocity field. The promises of CIFT for flow field monitoring in the continuous casting of steel are substantiated by results obtained at a test rig with a low-melting liquid metal. While CIFT is presently restricted to flows with low magnetic Reynolds numbers, some selected problems from non-linear inverse dynamo theory, with possible applications to geo- and astrophysics, are also discussed.

Keywords

Inverse Problem European Physical Journal Special Topic Forward Problem Submerged Entry Nozzle Magnetic Reynolds Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    J. Wicht, A. Tilgner, Space Sci. Rev. 152, 501 (2010)ADSCrossRefGoogle Scholar
  2. 2.
    C.A. Jones, M.J. Thompson, S.M. Tobias, Space Sci. Rev. 152, 591 (2010)ADSCrossRefGoogle Scholar
  3. 3.
    A. Gailitis, et al., Phys. Rev. Lett. 84, 4365 (2000)ADSCrossRefGoogle Scholar
  4. 4.
    R. Stieglitz, U. Müller, Phys. Fluids 13, 561 (2001)ADSCrossRefGoogle Scholar
  5. 5.
    R. Monchaux, et al., Phys. Rev. Lett. 98, 044502 (2007)ADSCrossRefGoogle Scholar
  6. 6.
    F. Stefani, A. Gailitis, G. Gerbeth, Zeitschr. Angew. Math. Mech. 88, 930 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    F. Stefani, Th. Gundrum, G. Gerbeth, Phys. Rev. E 70, 056306 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    A. Giesecke, C. Nore, F. Stefani, G. Gerbeth, J. Léorat, W. Herreman, F. Luddens, J.-L. Guermond, New J. Phys. 14, 053005 (2012)ADSCrossRefGoogle Scholar
  9. 9.
    A. Giesecke, F. Stefani, G. Gerbeth, Phys. Rev. Lett. 104, 044503 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    T.G. Cowling, Mon. Not. R. Astr. Soc. 140, 39 (1934)Google Scholar
  11. 11.
    E. Bullard, H. Gellman, Phil. Trans. R. Soc. London A 247, 213 (1954)MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    F. Stefani, A. Giesecke, G. Gerbeth, Theor. Comp. Fluid Dyn. 23, 405 (2009)zbMATHCrossRefGoogle Scholar
  13. 13.
    C.B. Forest, et al., Magnetohydrodynamics 38, 107 (2002)ADSGoogle Scholar
  14. 14.
    K.-H. Rädler, M. Rheinhardt, E. Apstein, H. Fuchs, Nonl. Proc. Geophys. 9, 171 (2002)ADSCrossRefGoogle Scholar
  15. 15.
    F. Stefani, G. Gerbeth, A. Gailitis, Velocity profile optimization for the Riga dynamo experiment, edited by A. Alemany, Ph. Marty, J.-P. Thibault, Transfer Phenomena in Magnetohydrodynamic and Electroconducting Flows (Kluwer, Dordrecht, 1999), p. 31Google Scholar
  16. 16.
    A. Gailitis, O. Lielausis, E. Platacis, G. Gerbeth, F. Stefani, Phys. Plasmas 11, 2838 (2004)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    F. Stefani, A. Gailitis, G. Gerbeth, Astron. Nachr. 332, 4 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    J.-L. Guermond, J. Léorat, C. Nore, Eur. J. Mech. B/Fluids 22, 555 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    J.-L. Guermond, L. Laguerre, J. Léorat, C. Nore, J. Comp. Phys. 221, 349 (2007)ADSzbMATHCrossRefGoogle Scholar
  20. 20.
    J.-L. Guermond, L. Laguerre, J. Léorat, C. Nore, J. Comp. Phys. 228, 2739 (2009)ADSzbMATHCrossRefGoogle Scholar
  21. 21.
    A. Giesecke, C. Nore, F. Plunian, R. Laguerre, A. Ribeiro, F. Stefani, G. Gerbeth, J. Léorat, J.-L. Guermond, Geophys. Astrophys. Fluid Dyn. 104, 249 (2010)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    A. Giesecke, C. Nore, F. Stefani, G. Gerbeth, J. Léorat, F. Luddens, J.-L. Guermond, Geophys. Astrophys. Fluid Dyn. 104, 505 (2010)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    C. Nore, J. Léorat, J.-L. Guermond, F. Luddens, Phys. Rev. E 84, 016319 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    A.B. Iskakov, S. Descombes, E. Dormy, J. Comp. Phys. 197, 540 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. 25.
    A.B. Iskakov, E. Dormy, Geophys. Astrophys. Fluid Dyn. 99, 481 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    A. Giesecke, F. Stefani, G. Gerbeth, Magnetohydrodynamics 44, 237 (2008)ADSGoogle Scholar
  27. 27.
    C. Gissinger, A. Iskakov, S. Fauve, E. Dormy, EPL 82, 29001 (2008)ADSCrossRefGoogle Scholar
  28. 28.
    A.J. Meir, P.G. Schmidt, Appl. Math. Comp. 65, 95 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    A.J. Meir, P.G. Schmidt, SIAM J. Num. Anal. 36, 1304 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    A.J. Meir, P.G. Schmidt, S.I. Bakhtiyarov, R.A. Overfelt, J. Appl. Mech. 71, 786 (2004)ADSzbMATHCrossRefGoogle Scholar
  31. 31.
    M. Seilmayer, F. Stefani, T. Gundrum, T. Weier, G. Gerbeth, M. Gellert, G. Rüdiger, Phys. Rev. Lett. 108, 244501 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    M. Bourgoin, P. Odier, J.-F. Pinton, Y. Ricard, Phys. Fluids 16, 2529 (2004)ADSCrossRefGoogle Scholar
  33. 33.
    A. Gailitis, Magnetohydrodynamics 6, 14 (1970)Google Scholar
  34. 34.
    W. Dobler, K.-H. Rädler, Geophys. Astrophys. Fluid Dyn. 89, 45 (1998)ADSCrossRefGoogle Scholar
  35. 35.
    P.H. Roberts, An introduction to magnetohydrodynamics (Elsevier, New York, 1967)Google Scholar
  36. 36.
    F. Stefani, G. Gerbeth, K.-H. Rädler, Astron. Nachr. 321, 65 (2000)ADSzbMATHCrossRefGoogle Scholar
  37. 37.
    M. Xu, F. Stefani, G. Gerbeth, J. Comp. Phys. 196, 102 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  38. 38.
    M. Xu, F. Stefani, G. Gerbeth, Phys. Rev. E 70, 056305 (2004)MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    F. Stefani, M. Xu, G. Gerbeth, F. Ravelet, A. Chiffaudel, F. Daviaud, J. Léorat, Eur. J. Mech./B Fluids 25, 894 (2006)ADSzbMATHCrossRefGoogle Scholar
  40. 40.
    R. Avalos-Zuniga, M. Xu, F. Stefani, G. Gerbeth, F. Plunian, Geophys. Astrophys. Fluid Dyn. 101, 389 (2007)ADSCrossRefGoogle Scholar
  41. 41.
    M. Xu, F. Stefani, G. Gerbeth, J. Comp. Phys. 227, 8130 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  42. 42.
    M.L. Dudley, R.W. James, Proc. R. Soc. Lond. A 425, 407 (1989)MathSciNetADSCrossRefGoogle Scholar
  43. 43.
    L. Marié, C. Normand, F. Daviaud, Phys. Fluids 18, 017102 (2006)MathSciNetADSCrossRefGoogle Scholar
  44. 44.
    R. Moncheaux, et al., Phys. Fluids 21, 035108 (2009)ADSCrossRefGoogle Scholar
  45. 45.
    F. Petrelis, S. Fauve, Phil. Trans. R. Soc. A 368, 1595 (2010)ADSCrossRefGoogle Scholar
  46. 46.
    W. Dobler, P. Frick, R. Stepanov, Phys. Rev. E. 67, 056309 (2003)MathSciNetADSCrossRefGoogle Scholar
  47. 47.
    F. Ravelet, B. Dubrulle, F. Daviaud, P.A. Ratié, Phys. Rev. Lett. 109, 024503 (2012)ADSCrossRefGoogle Scholar
  48. 48.
    F. Stefani, G. Gerbeth, Inverse Problems 15, 771 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  49. 49.
    F. Stefani, G. Gerbeth, Inverse Problems 16, 1 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  50. 50.
    F. Stefani, G. Gerbeth, Meas. Sci. Techn. 11, 758 (2000)ADSCrossRefGoogle Scholar
  51. 51.
    T. Wondrak, V. Galindo, G. Gerbeth, Th. Gundrum, F. Stefani, K. Timmel, Meas. Sci. Techn. 21, 045402 (2010)ADSCrossRefGoogle Scholar
  52. 52.
    T. Wondrak, S. Eckert, G. Gerbeth, K. Klotsche, F. Stefani, K. Timmel, A. Peyton, N. Terzija, W. Yin, Metal. Mater. Trans. B 42, 1201 (2011)CrossRefGoogle Scholar
  53. 53.
    T. Wondrak, V. Galindo, G. Gerbeth, T. Gundrum, F. Stefani, K. Timmel, A. Peyton, W. Yin, S. Riaz, Ironmaking Steelmaking 39, 1 (2012)CrossRefGoogle Scholar
  54. 54.
    K. Timmel, S. Eckert, G. Gerbeth, F. Stefani, ISIJ Int. 50, 1134 (2010)CrossRefGoogle Scholar
  55. 55.
    P.H. Roberts, S. Scott, J. Geomagn. Geoelec. 17, 137 (1965)CrossRefGoogle Scholar
  56. 56.
    J. Aubert, A. Fournier, Nonl. Proc. Geophys. 18, 657 (2011)ADSCrossRefGoogle Scholar
  57. 57.
    K. Li, P.W. Livermore, A. Jackson, Phys. Rev. E 84, 056321 (2011)ADSCrossRefGoogle Scholar
  58. 58.
    F. Stefani, G. Gerbeth, Phys. Rev. E 67, 027302 (2003)ADSCrossRefGoogle Scholar
  59. 59.
    F. Stefani, G. Gerbeth, Phys. Rev. Lett. 94, 184506 (2005)ADSCrossRefGoogle Scholar
  60. 60.
    F. Stefani, G. Gerbeth, U. Günther, M. Xu, Earth Planet. Sci. Lett. 143, 828 (2006)ADSCrossRefGoogle Scholar
  61. 61.
    F. Stefani, M. Xu, L. Sorriso-Valvo, G. Gerbeth, U. Günther, Geophys. Astrophys. Fluid Dyn. 101, 227 (2007)MathSciNetADSCrossRefGoogle Scholar
  62. 62.
    M. Fischer, G. Gerbeth, A. Giesecke, F. Stefani, Inverse Problems 25, 065011 (2009)MathSciNetADSCrossRefGoogle Scholar
  63. 63.
    A. Tilgner, Phys. Rev. Lett. 100, 128501 (2008)ADSCrossRefGoogle Scholar
  64. 64.
    A. Giesecke, F. Stefani, J. Burguete, Phys. Rev. E 86, 066303 (2012)ADSCrossRefGoogle Scholar
  65. 65.
    N. Terzija, W. Yin, G. Gerbeth, F. Stefani, K. Timmel, T. Wondrak, A. Peyton, Flow Meas. Instrum. 22, 10 (2011)CrossRefGoogle Scholar
  66. 66.
    J. Simkanin, A. Tilgner, Geophys. Astrophys Fluid Dyn. 102, 205 (2008)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  • Andre Giesecke
    • 1
  • Frank Stefani
    • 1
  • Thomas Wondrak
    • 1
  • Mingtian Xu
    • 2
  1. 1.Helmholtz-Zentrum Dresden-RossendorfDresdenGermany
  2. 2.Shandong UniversityJinan City, Shandong ProvincePR China

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