Article Sidebar

Submitted
Apr 26, 2017
Published
Dec 1, 2012
Download
PDF
Statistic
Read Counter : 314 Download : 219

Main Article Content

Abstract

The objective is to introduce a semi-analytical method for solving axisymmetric problems of electromagnetic induction in thin spherical caps placed in a time-varying magnetic field due to an axial magnetic dipole or in a time-varying uniform axial magnetic field. This method provides approximate solutions to mathematically difficult mixed boundary-value problems governing the induction of electric currents in thin sheets, for arbitrary angles of the cap. Numerical results are given and discussed. The best approximations were obtained for sheets with integrated conductivity decreasing to zero towards the edge of the sheet. The case of uniform conductivity, characterized by weak singularities of the induced magnetic vector potential at the rim of the cap, yielded relatively large errors and is dealt with separately within an improved model. The method may be adequately extended to deal with other problems involving more complicated geometries, arbitrary electric conductivity distributions and inducing magnetic fields, in two or three dimensions, for various geophysical applications.

 

 

Keywords

Spherical cap Geomagnetic applications Axisymmetric case.

Article Details

References

  1. ABOU-DINA, M.S. and GHALEB, A.F. 2004. A variant of Trefftz's method by boundary Fourier expansion for solving regular and singular plane boundary-value problems. J. Comp. Appl. Math. (C.A.M.), 167: 363-387.
  2. ASHOUR, A.A. 1950. The induction of electric currents in a uniform circular disk. Quart. J. Mech. Appl. Math., 3(1): 119-128.
  3. ASHOUR, A.A. 1952.The induction of electric currents in a uniformly conducting circular disk by the sudden creation of magnetic poles. Quart. J. Mech. Appl. Math., 5(3): 379-384.
  4. ASHOUR, A.A. 1965. Electromagnetic induction in finite thin sheets. Quart. J. Mech. Appl. Math., 18(1): 73-86.
  5. ASHOUR, A.A. 1971. Electromagnetic induction in thin finite sheets having conductivity decreasing to zero at the edge, with geomagnetic applications. Geophys. J. R. Astr. Soc., 22: 489-498.
  6. ASHOUR, A.A. 1974a. Research Note: The condition at the surface of a thin conductor in which currents are induced. Geophys. J. R. Astr. Soc., 36: 235-237.
  7. ASHOUR, A.A. 1974b. Exact solutions for certain axisymmetric problems of induction in thin non-uniform sheets. J. Geophys. Res., 79(16): 2479-2486.
  8. CHAVE, A.D. and BOOKER, J.R. 1987. Electromagnetic induction studies. Reviews of Geophysics, 25(5): 989-1003.
  9. DOSS, S.S. and ASHOUR, A.A. 1971. Some results on the magnetic field of electric currents induced in a thin hemispherical shell of finite conductivity, with geomagnetic applications. Geophys. J. R. Astr. Soc., 22: 385-400.
  10. FAINBERG, E.B. 1980. Electromagnetic induction in the world ocean. Geophysial Surveys, 4: 157-171.
  11. GAINBERG, E.B., KUVSHINOV, A.V. and SINGER, B.Sh. 1990. Electromagnetic induction in a spherical earth with non-uniform oceans and continents in electric contact with the underlying medium. Geophys. J. Int., 102: 273-281.
  12. HEWSON-BROWNE, R.C. and KENDALL, P.C. 1976. Magnetotelluric modelling and inversion in three dimensions. Acta Geodet., Geophys. et Montanist. Acad. Sci. Hung., 11(3-4), 427-446.
  13. HUTSON, V.C.L., KENDALL, P.C. and MALIN, S.R.C. 1972. Computation of the solution of geomagnetic induction problems: A general method, with applications. Geophys. J. R. Astr. Soc., 28: 417-443.
  14. JACKSON, J.D. 1962. Classical Electrodynamics. John Wiley & Sons, Inc., New York, London.
  15. PRICE, A.T. 1949. The induction of electric currents in non-uniform thin sheets and shells. Quart. J. Mech. Appl. Math., 2(3): 283-310.
  16. RAVAL, U., WEAVER, J.T. and DAWSON, T.W. 1981. The ocean coast-effect re-examined. Geophys. J. R. Astr. Soc., 67: 115-123.
  17. SIEW, P.F. and HURLEY, D.G. 1988. Electromagnetic response of thin disks. J. Appl. Math. and Phys. (ZAMP), 39: 619-633.
  18. TARITS, P. 1994. Electromagnetic studies of global geodynamic processes. Surveys in Geophysics, 15: 20-238.
  19. THEBAULT, E., SCHOTT, J.J., MANDEA, M. and HOFFBECK, J.P. 2004. A new proposal for spherical cap harmonic modeling. Geophys. J. Int., 159: 83-103.
  20. THEBAULT, E., SCHOTT, J.J. and MANDEA, M. 2006a. Revised spherical cap harmonic analysis: Validation and properties. J. Geophys. Res.-Solid Earth, 111(5): B01102, 1-17.
  21. THEBAULT, E., MANDEA, M. and SCHOTT, J.J. 2006b. Modelling the lithospheric magnetic field over France by means of revised spherical cap harmonic analysis. J. Geophys. Res.-Solid Earth, 111: B05102, 1-13.
  22. VARENTSOV, IV. M. 1983. Modern trends in the solution of forward and inverse 3D electromagnetic induction problems. Geophysical Surveys, 6: 55-78.
  23. WEIDELT, P. 1971. The electromagnetic induction in two thin half-sheets. Zeitschrift für Geophysik, 37: 649-665.
  24. WOLF, D. 1983a. Inductive coupling between idealized conductors and its significance for the geomagnetic coast effect. J. Geophys., 52: 22-33.
  25. WOLF, D. 1983b. Singular solutions to Maxwell's equations and their significance for geomagnetic induction. Geophys. J. R. Astr. Soc., 75: 279-283.