Toroidal-Poloidal Decompositions of Electromagnetic Green’s Functions in Geomagnetic Induction

Reference work entry


In this chapter, we make use of electromagnetic Green’s functions to analyze quasi-static electromagnetic fields outside and inside the domain of external current sources. Scalar potential representation of the magnetic field in a source-free domain is readily derived from the physical currents using Green’s functions. It may be shown that, although the electric field generated by external sources has a poloidal mode component, this component causes no induction inside a conductive ground and vanishes at the surface of a conductive ground in the transverse directions. An effective external current source represented as a toroidal source is thus justified for low-frequency natural source induction such as natural source magnetotellurics and geomagnetic global inductions.


Transverse Electric Toroidal Field Poloidal Field Spherical Harmonic Degree Helmholtz Decomposition 
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  1. Backus G (1986) Poloidal and toroidal fields in geomagnetic field modeling. Rev Geophys 24(1):75–109MathSciNetCrossRefGoogle Scholar
  2. Chew WC (1995) Waves and fields in inhomogeneous media. IEEE, New YorkGoogle Scholar
  3. Jackson JD (1998) Classical electrodynamics. Wiley, New YorkGoogle Scholar
  4. Merzbacher, E. (1998) Quantum Mechanics, 3rd edn., John Wiley & Sons, Inc.Google Scholar
  5. Schmitt BJ, Wahl W (1992) Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. In: Heywood JG, Masuda K, Rautmann R, Solonnikov VA (eds) The Navier-Stokes equations II – theory and numerical methods. Lecture notes in mathematics, vol 1530. Springer, Berlin/Heidelberg, pp 291–305CrossRefGoogle Scholar
  6. Sun J, Egbert GD (2012) Spherical decomposition of electromagnetic fields generated by quasi-static currents. Int J Geomath 3:279–295MathSciNetCrossRefzbMATHGoogle Scholar
  7. Wang Q, Ronneberger O, Burkhardt H (2008) Fourier analysis in polar and spherical coordinates, internal report.

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute of GeophysicsETH ZürichZürichSwitzerland

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