Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause
We present a new model of the magnetospheric magnetic field. Using the finite element method, ChapmanFerraro problem is solved numerically in the considered approach. The whole magnetic field is a sum of: the dipole field, the field, produced by the internal current systems (cross-tail, Birkeland, r...
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| Cite this: | Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause / P.S. Dobreva, M.D. Kartalev // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 73-76. — Бібліогр.: 22 назв. — англ. |
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| citation_txt | Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause / P.S. Dobreva, M.D. Kartalev // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 73-76. — Бібліогр.: 22 назв. — англ. |
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| description | We present a new model of the magnetospheric magnetic field. Using the finite element method, ChapmanFerraro problem is solved numerically in the considered approach. The whole magnetic field is a sum of: the dipole field, the field, produced by the internal current systems (cross-tail, Birkeland, ring currents) and the field induced by the magnetopause currents. In contrast to similar earlier models, the internal magnetospheric magnetic fields are taken from Tsyganenko data-based model. The magnetosphere boundary could be arbitrary (generally non-axisymmetric). Input model parameters are the solar wind parameters, the Dst index and the dipole tilt angle. We discuss some results, obtained in three dimensional solution of the Neumann-Dirichlet problem corresponding to a closed magnetosphere.
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Numerical modeling of the magnetosphere with data based internal
magnetic �eld and arbitrary magnetopause
P. S. Dobreva, M. D. Kartalev
Institute of Mechanics, Bulgarian Academy of Sciences, Acad G. Bonchev Street block 4, So�a 1113, Bulgaria
polya2006@yahoo.com
We present a new model of the magnetospheric magnetic �eld. Using the �nite element method, Chapman-
Ferraro problem is solved numerically in the considered approach. The whole magnetic �eld is a sum of:
the dipole �eld, the �eld, produced by the internal current systems (cross-tail, Birkeland, ring currents)
and the �eld induced by the magnetopause currents. In contrast to similar earlier models, the internal
magnetospheric magnetic �elds are taken from Tsyganenko data-based model. The magnetosphere boundary
could be arbitrary (generally non-axisymmetric). Input model parameters are the solar wind parameters,
the Dst index and the dipole tilt angle. We discuss some results, obtained in three dimensional solution of
the Neumann-Dirichlet problem corresponding to a closed magnetosphere.
Introduction
When the solar wind interacts with and is de�ected by the Earth's intrinsic magnetic �eld, the magne-
tosphere is formed. Since the discovery of the Earth's magnetosphere in 1958 by Explorer 1, considerable
e�orts of many researchers were devoted to its description. Many models, describing both the magneto-
spheric magnetic �eld and the magnetopause boundary itself, have been created during the last decades.
The early models have two principal limitations: the shape of the magnetopause is given by a surface (a
paraboloid, an ellipsoid, a semi-sphere and a cylinder), which is usually axisymmetric and does not depend
on the parameters of the solar wind. The second main shortcoming in the magnetosphere modeling is the
inability to include the e�ect of all main magnetospheric sources.
In one of the �rst magnetospheric models a vacuum magnetospheric magnetic �eld con�guration is con-
sidered, containing only the dipole source at the origin and its corresponding shielding �eld [8]. The model
is concentrated on a solution of the problem of calculating the shape of the magnetospheric boundary. An
iterative algorithm is used, and the criteria for reaching convergency is achievement of the pressure balance.
It is supposed that at any boundary point a local balance between the ram pressure of the solar wind and
the magnetic pressure inside the cavity is maintained.
The contributions of all main current systems inside the magnetosphere, taken from Tsyganenko 96
[19] model, are included in the model of Sotirelis [13], where the pressure outside is estimated within the
Newtonian approximation. Before that, in the Kartalev model [4] outside pressure was calculated solving
gas dynamics equations in the magnetosheath. Magnetic �eld inside is presented by the model of Stern [14].
Many models are based on an alternative approach, where the magnetopause is not a result of pressure
balance equation, but has a given predetermined shape. Numerical technics, solving the Chapman-Ferraro
problem, were applied in the models of Voigt [21, 22], Stern [14], To�oletto [6, 16]. The problem of �nding
a solution of the Laplace equation for the magnetic �eld potential is solved analytically in [14, 21, 22], and
numerically, using the �nite element method in [6]. The magnetic �eld potential is calculated mainly for
the dipole �eld [21], the ring current is included in [14]. Numerical solution, shielding the internal magnetic
�eld from Hilmer & Voigt [2] model is presented in [6]. A paraboloid magnetopause shape is used in [14], a
hemisphere elongated by a cylinder in the tail in [21], and a �xed arbitrary form in [6].
Satellite missions provided an opportunity to gather a great amount of magnetic �eld measurements.
During the last two decades, magnetospheric magnetic �eld models based on empirical measurements, such
as those of Tsyganenko & Usmanov 1982 [17], Tsyganenko 1995 [19], Tsyganenko 2001 [20], are widely used.
Latest versions of Tsyganenko models � Tsyganenko 1996 and Tsyganenko 2001, referred here as T96 and
T01, include separate contributions from all magnetospheric sources � the ring, cross-tail and Birkeland
73
Advances in Astronomy and Space Physics P. S. Dobreva, M. D. Kartalev
currents, and a magnetopause, based on real measurements � the form of Sibeck 1991 [11] is used in T96,
and that of Shue 1998 [10] � in T01.
We describe some numerical aspects of the presented magnetospheric magnetic �eld model. Next section
presents the technique for solving the Chapman-Ferraro problem using an application of the �nite element
method. The capabilities of the numerical method in simulation of some typical features of magnetic �eld
topology are presented in the last section.
Computational method
The section is devoted to a short description of a method of calculation the Chapman-Ferraro currents.
This magnetospheric model is based on the development and improvement of earlier created � 2D, Kartalev
1995 [3] and simpli�ed 3D, Koitchev 1999 [7] � models.
At each point inside the cavity of the magnetosphere, the net magnetic �eld B is presented as a sum:
B = Bd + Bt + Br + Bb + Bs, (1)
where Bd is the dipole �eld, and Bt, Br, Bb are the �elds of the cross-tail, ring and Birkeland currents
correspondingly, Bs is the magnetopause currents �eld, that has to be determined in the process of solution.
The sources of magnetic �eld, such as the ring, tail and Birkeland currents, are incorporated in our model
through the empirical model of Tsyganenko. We use two of the latest variants of Tsyganenko models � T96
and T01. The Tsyganenko model is modi�ed in order to satisfy the current physical problem formulation. A
prescribed magnetopause shape, based on the Sibeck 1991 [11] ellipsoid is used in T96, and the Shue 98 [10]
form in T01. Instead of using the above mentioned empirical forms, we �t the Tsyganenko model boundary
to an arbitrary surface. That surface can be obtained numerically, from the numerical magnetosheath-
magnetosphere model, developed in Geospace hydrodynamics laboratory at the Institute of Mechanics, part
of which is the magnetospheric model, described here. In that case the magnetopause surface is determined
by requiring the pressure balance relation to be ful�led. The magnetopause can also be initialized as having
a data-based shape (see examples in [9, 10, 11]).
As the magnetopause �eld is a function of its shape, it has to be calculated in correspondence to the
given magnetopause surface. Hence the �eld induced by the magnetopause currents in Tsyganenko model is
replaced by numerically calculated one.
It is supposed, that the unknown �eld Bs is divergence and curl free:
div Bs = 0, rotBs = 0, (2)
thus the �eld potential U , a harmonic scalar function, exists. The potential is found out from the Laplace
equation:
4U = 0, (3)
with Neumann boundary condition:
(Bs,n) = ∂U/∂n = − [(Bd,n) + (Bt,n) + (Br,n) + (Bb,n)] , (4)
applied at the boundary points, where n is the vector normal to the surface. The restriction of zero normal
magnetic �eld component at the boundary is implied in the present case, corresponding to a closed magne-
tosphere con�guration. When the �eld potential is already calculated from (3) and (4), the magnetic �eld
Bs is determined as its gradient:
Bs = ∇U. (5)
Beside physics, another aspect of prime importance is the computational one. Development of adequate
magnetosphere models requires not only understanding the physical processes in the magnetosphere, but
also �nding an appropriate method for solving such problems. The reliability of the method is indicated by
the accuracy of solution, the �exibility in the description of three-dimensional regions with a complicated
geometry, and last but not least, by the required calculation time.
Numerical procedure, based on the �nite element method (see [1] and [15]) lies in solving equations (3)
and (4). The modeled region of the magnetosphere is divided into 3D elements, which are accepted to be
twenty-node isoparametric serendipity hexahedrons. The elements satisfy the consistency conditions [15].
This type of elements is accepted for at least two reasons: the type is desirable for achieving the necessary
smoothness of the approximated functions and their derivatives. It also allows a precise approximation of
3D curvilinear boundaries, such as the magnetosphere shape.
74
Advances in Astronomy and Space Physics P. S. Dobreva, M. D. Kartalev
We replace the continuum formulation by a discrete representation, thus converting the equation (3) and
the boundary condition (4) into a linear algebraic system with a semi-de�nite matrix. Alternating subspace
iteration method [5] is used for the numerical solution of the system. More details about the method and the
discretization algorithm can be found in the earlier papers (see [3, 7]). The method of solving the algebraic
system [5] allows the computer code to be run on a personal computer with mean capabilities in a reasonable
time frame.
Some results
Some features of the model described in the previous section are considered. Input parameters for the
model are: dynamic pressure of the solar wind (Dp), two components, By and Bz, of the interplanetary
magnetic �eld, Dst index and the angle of the dipole inclination (tilt angle).
The distribution of the total magnetic �eld in the main meridional (noon-midnight) plane is presented
in Fig. 1. The magnetic �eld inside is presented by all main sources: the Earth' internal �eld, the �eld of
cross-tail, ring, Birkeland currents, and the �eld of the magnetopause currents. The magnetopause �eld is
calculated numerically in our model, and the �eld from the other magnetospheric sources (the cross-tail,
ring and Birkeland currents) is given by the model of Tsyganenko � T01. The magnetospheric boundary is
described by the empirical shape of Shue 98 [10] (for Dp = 2 nPa and Bz = 0 nT). The input parameters
are mentioned in the �gure.
The magnetic �eld distribution calculated in our model can be compared with the results, obtained in
similar way and published earlier in the literature. The distribution of total magnetic �eld, calculated in
the main meridional plane is presented in Fig. 5 in [12]. Figure 5 in [12] illustrates similar behavior of the
magnetic �eld, although it is obtained under di�erent conditions � the magnetic �eld inside is given by a
version of Tsyganenko 87 [18] model.
In this study we demonstrated, that the method, applied in calculation the Chapman-Ferraro currents for
a given magnetopause form and a data-based related internal �eld, gives reliable magnetic �eld distribution
in the magnetosphere.
Figure 1: Magnitude of total magnetic �led |B| (in nT) in the main meridional XZ (GSM) plane. The �eld of the
magnetopause currents is numerically calculated by our model. The empirical form of Shue 98 is used as a boundary
of the magnetosphere. Input parameters: Dp = 2 nPa, By = Bz = 2 nT, Dst = −10, tilt= 0◦.
.
75
Advances in Astronomy and Space Physics P. S. Dobreva, M. D. Kartalev
Acknowledgement
We would like to thank the Bulgarian �Evrika� Foundation for �nancial support. This work is also
supported by the European Social Fund and Bulgarian Ministry of Education, Youth and Science under
Operative Program �Human Resources Development�, Grant BG051PO001-3.3.04/40.
References
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[2] Hilmer R. V., Voigt G.-H. J. Geophys. Res., V. 100(A4), pp. 5613-5626 (1995)
[3] Kartalev M. D., Kaschiev M. S., Koitchev D. K. J. Comp. Phys., V. 119, pp. 220-230 (1995)
[4] Kartalev M. D., Nikolova V. I., Kamenetsky V. E., Mastikov I. P. Planet. Space Sci., V. 44, pp. 1195-1208 (1996)
[5] Kaschiev M.S. `Computational Processes', Moskow, Science, V. B, pp. 161-170 (1988)
[6] Kloucek P., To�oletto F. R. J. Comp. Phys., V. 150, pp. 549-560 (1999)
[7] Koitchev D. K., Kaschiev M. S., Kartalev M. D. `Finite Di�erence Methods: Theory and Applications', Nova Science
Publishers, Commack, NY, U.S.A., pp. 127-138, (1999)
[8] Mead G. D., Beard D. B. J. Geophys. Res., V. 69, pp. 1169-1179 (1964)
[9] Petrinec S. M., Russell C. T. J. Geophys. Res., V. 101(A1), pp. 137-152 (1996)
[10] Shue J.-H., Song P., Russell C. T. et al. J. Geophys. Res., V. 103(A8), pp. 17691-17700 (1998)
[11] Sibeck D. G., Lopez R. E., Roelof E. C. J. Geophys. Res., V. 96(A4), pp. 5489-5495 (1991)
[12] Sotirelis T. J. Geophys. Res., V. 101, pp. 15255-15264 (1996)
[13] Sotirelis T., Meng C. I. J. Geophys. Res., V. 104, pp. 6889-6898 (1999)
[14] Stern D. P. J. Geophys. Res., V. 90, pp. 10851-10863 (1985)
[15] Strang G., Fix G. `An Analysis of the Finite Element Method', Prentice-Hall, Englewood Cli�s, NJ, (1977)
[16] To�oletto F. R. Geophys. Res. Lett., V. 90, pp. 621-624 (1994)
[17] Tsyganenko N. A., Usmanov A. V. Planet. Space Sci., V. 30, pp. 985-998 (1982)
[18] Tsyganenko N. A. Planet. Space Sci., V. 35, pp. 1347-1358 (1987)
[19] Tsyganenko N. A. J. Geophys. Res., V. 100(A4), pp. 5599-5612 (1995)
[20] Tsyganenko N. A. J. Geophys. Res., V. 107(A8), pp. 1179-1195 (2002)
[21] Voigt G.-H. Z. Geophys., V. 38, pp. 319-346 (1972)
[22] Voigt G.-H. Planet. Space Sci., V. 29, pp. 1-20 (1981)
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| id | nasplib_isofts_kiev_ua-123456789-118974 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| isbn | 987-966-439-367-3 |
| language | English |
| last_indexed | 2025-12-07T18:59:59Z |
| publishDate | 2011 |
| publisher | Advances in astronomy and space physics |
| record_format | dspace |
| spelling | Dobreva, P.S. Kartalev, M.D. 2017-06-02T18:12:25Z 2017-06-02T18:12:25Z 2011 Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause / P.S. Dobreva, M.D. Kartalev // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 73-76. — Бібліогр.: 22 назв. — англ. 987-966-439-367-3 https://nasplib.isofts.kiev.ua/handle/123456789/118974 We present a new model of the magnetospheric magnetic field. Using the finite element method, ChapmanFerraro problem is solved numerically in the considered approach. The whole magnetic field is a sum of: the dipole field, the field, produced by the internal current systems (cross-tail, Birkeland, ring currents) and the field induced by the magnetopause currents. In contrast to similar earlier models, the internal magnetospheric magnetic fields are taken from Tsyganenko data-based model. The magnetosphere boundary could be arbitrary (generally non-axisymmetric). Input model parameters are the solar wind parameters, the Dst index and the dipole tilt angle. We discuss some results, obtained in three dimensional solution of the Neumann-Dirichlet problem corresponding to a closed magnetosphere. We would like to thank the Bulgarian "Evrika" Foundation for financial support. This work is also supported by the European Social Fund and Bulgarian Ministry of Education, Youth and Science under Operative Program "Human Resources Development", Grant BG051PO001-3.3.04/40. en Advances in astronomy and space physics Advances in Astronomy and Space Physics Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause Article published earlier |
| spellingShingle | Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause Dobreva, P.S. Kartalev, M.D. |
| title | Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause |
| title_full | Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause |
| title_fullStr | Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause |
| title_full_unstemmed | Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause |
| title_short | Numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause |
| title_sort | numerical modeling of the magnetosphere with data based internal magnetic field and arbitrary magnetopause |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/118974 |
| work_keys_str_mv | AT dobrevaps numericalmodelingofthemagnetospherewithdatabasedinternalmagneticfieldandarbitrarymagnetopause AT kartalevmd numericalmodelingofthemagnetospherewithdatabasedinternalmagneticfieldandarbitrarymagnetopause |