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SCIENCE CHINA Earth Sciences, Volume 64 , Issue 10 : 1798-1812(2021) https://doi.org/10.1007/s11430-020-9786-9

3D finite-element modeling of Earth induced electromagnetic field and its potential applications for geomagnetic satellites

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  • ReceivedSep 5, 2020
  • AcceptedMay 7, 2021
  • PublishedJul 6, 2021

Abstract


Funded by

the Theory and Application of Resource and Environment Management in the Digital Economy Era by National Natural Science Foundation of China(Grant,No.,72088101)

the National Natural Science Foundation of China(Grant,Nos.,41922027,41830107,41811530010)

Innovation-Driven Project of Central South University(Grant,No.,2020CX0012)

the National Natural Science Foundation of Hunan Province of China(Grant,No.,2019JJ20032)

Macau Foundation and the pre-research project on Civil Aerospace Technologies No. D020308 and D020303 funded by China’s National Space Administration.


Acknowledgment

Special thanks are given to Zdeněk MARTINEC from Dublin Institute for Advanced studies for kindly providing analytic solutions and Anna KELBERT from United States Geological Survey for kindly offering benchmark solutions for comparison. We thank Chaojian CHEN from ETH Zurich and Jianping LI from Guangzhou Marine Geological Survey of China Geological Survey for their helpful discussion. We would like to thank the responsible editor and two anonymous reviewers for their valuable comments, which have greatly improved the quality of this paper. This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 72088101, 41922027, 41830107, 41811530010), Innovation-Driven Project of Central South University (Grant No. 2020CX0012), the National Natural Science Foundation of Hunan Province of China (Grant No. 2019JJ20032), Macau Foundation and the pre-research project on Civil Aerospace Technologies funded by China’s National Space Administration (Grant Nos. D020308, D020303).


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  • Figure 1

    The spherical coordinate system (a) and the illustration of Earth electrical conductivity model (b). The notation O is the Earth’s center, the xy plane is the equatorial plane, the z-axis points from the Earth’s center to the north pole. The notations r, θ, ϕ are the radial distance, colatitude, and longitude respectively, while Br, Bθ, Bϕ are respectively the radial, colatitudinal, and longitudinal components of the magnetic induction vector. The Earth electrical conductivity model includes (1) the surface thin shell of variable conductance S(θ,ϕ), which is used to represent the nonuniform distribution of the oceans and continents; (2) the crustal and mantle; and (3) the conductive core.

  • Figure 2

    The distribution of the primary magnetic vector potential on the atmosphere boundary (at a distance of 5 radii from the Earth’s center).

  • Figure 3

    Illustration of a nested sphere model (Martinec, 1998).

  • Figure 4

    Tetrahedral meshes with (a) and without (b) the atmosphere of the nested sphere model. The notation 1 denotes the Earth with a conductivity of 1 S m−1, the notation 2 means the abnormal body with a conductivity of 10 S m−1 and the notation 3 denotes the atmosphere with a conductivity of 10−8 S m−1.

  • Figure 5

    The convergence of the relative residual norm as a function of the iterations for the nested sphere model at a period of 242 days.

  • Figure 6

    Comparison of our solution to the analytic solution (Martinec, 1998) for the nested sphere model at a period of 242 days.

  • Figure 7

    Illustration of a benchmark model (Kelbert et al., 2014).

  • Figure 8

    Tetrahedral meshes of the benchmark model. (a) The vertical slice; (b) illustration of the profile at zero longitude; (c) illustration of the profile at 45-degree colatitude. The notation 1 denotes the Earth with a conductivity of 0.01 S m−1, the notation 2 denotes the abnormal body with a conductivity of 1 S m−1.

  • Figure 9

    The convergence of the relative residual norm as a function of iterations for the benchmark model at a period of one day.

  • Figure 10

    Comparison of our solution to the integral equation solution (IE, Kuvshinov, 2008), the finite difference solution (FD, Kelbert et al., 2008) , and the previous finite element solution (FE, Ribaudo et al., 2012) for the benchmark model at zero longitude and 45-degree colatitude. The testing period is one day.

  • Figure 11

    Real and imaginary parts of the longitudinal component of the magnetic induction vector for the benchmark model at a period of one day. From top to bottom: the integral equation solution (IE, Kuvshinov, 2008), the finite difference solution (FD, Kelbert et al., 2008), the previous finite element solution (FE, Ribaudo et al., 2012) and our solution.

  • Figure 12

    Similar to Figure 11, but for the colatitudinal component.

  • Figure 13

    Similar to Figure 11, but for the radial component.

  • Figure 14

    Surface thin shell of variable conductance. Top: the linear-scale map; bottom: the log-scale map.

  • Figure 15

    The convergence of the relative residual norm as a function of iterations for a realistic 3D Earth model at a period of 6 h. The profile is at the Earth’s surface.

  • Figure 16

    Real and imaginary parts of the vertical component of the magnetic induction vector for the realistic 3D Earth model at a period of 6 h. Top: the integral equation solution (Kuvshinov, 2008); bottom: our solution. The profile is located at the Earth’s surface.

  • Figure 17

    Tetrahedral meshes for the 3D model. (a) and (b) are two slices of different directions, (c) is the locally enlarged mesh. The layered structures of (c) are respectively for the surface thin shell (Figure 14) and the underlying 1D layered model (Table 1).

  • Figure 18

    The convergence of the relative residual norm as a function of iterations for the 1D and 3D models at a period of 12 h. The computation is performed at 450 and 200 km altitudes.

  • Figure 19

    Real and imaginary parts of the vertical component of the magnetic induction vector at 450 and 200 km altitudes. From top to bottom: 1D model responses, 3D model responses and the ocean induction responses (3D-1D). The testing period is 2 h.

  • Figure 20

    Similar to Figure 19, but for a period of 12 h.

  • Figure 21

    Similar to Figure 19, but for a period of 2 days.

  • Table 1   1D layered conductivity model from Kelbert et al. (2014).

    Depths (km)

    Conductivities (S m−1)

    0–400

    0.01

    400–650

    0.1

    650–2871

    2.0

    2871–6371

    500,000

  • Table 2   Running parameters for 1D and 3D models at a period of 12 ha)

    Model

    Number of elements

    Number of nodes

    Number of unknowns

    Number of iterations

    Relative residual norm

    Computation time (minutes)

    1D

    9,657,944

    1,599,068

    6,396,272

    26

    8.39×10−7

    26.8

    3D

    10,167,842

    1,682,064

    6,728,256

    96

    9.95×10−7

    64.6

    The computation is performed at 450 and 200 km altitudes

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