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Background Information Magnetic field models
Dipole approximations

Dipole approximations of the geomagnetic field

Introduction

It is customary to express the geomagnetic field as the gradient of a scalar potential V. The function V is usually expressed as an orthogonal expansion in spherical harmonics.

We introduce a system of orthogonal Cartesian coordinates (x, y, z) with origin at the centre of the Earth. The Z-axis coincides with the Earth's rotation axis, the X-axis lies in the equatorial plane and is directed towards the Greenwich meridian, and the Y-axis completes the right-handed coordinate system. In this frame of reference, we define a system of spherical coordinates (r, theta, phi) whose polar axis coincides with the Z-axis. The transformation formulae from Cartesian coordinates to spherical coordinates are:

In the spherical coordinate system, the expansion of V in spherical harmonics can be written as:

where RE is the mean radius of the Earth (6371.2 km), and Pnm(cos theta) are the quasi Schmidt- normalized associated Legendre functions (See section on Spherical functions and their normalisations). This expansion is an infinite series, but in practice it is usually limited to n = 10 or n = 12. The coefficients gnm and hnm are determined experimentally by combining Earth-based and satellite measurements of the geomagnetic field, and are time-dependent. DIVISION I WORKING GROUP I of the International Association for Geomagnetism and Aeronomy (IAGA) defined and periodically updates the International Geomagnetic Reference Field (IGRF). The IGRF consists of a series of sets for the coeffients in the expansion given above, corresponding to different epochs, in units of nT. IAGA has up to now published Definitive Geomagnetic Reference Field (DGRF) coefficients for epochs between 1945 and 1990, with intervals of five years, and IGRF for epoch 1995 and 2000. In all the IAGA models, the summation over n in the expansion goes to n = 10.

The centred dipole model

The main contribution in the spherical harmoic expansion comes from the terms with n = 1. It can be identified with the field produced by a dipole with centre coinciding with the centre of the Earth and dipole axis inclined with respect to the polar axis. The contributions of the other terms can be considered to be a perturbation of the main dipole field.

The direction defined by the Earth-centred dipole is called the geomagnetic axis. The sum of terms with n = 1 in the expansion reduces to one term in the spherical coordinate system with centre at the Earth's centre and polar axis coinciding with the geomagnetic axis. This new coordinate system (r1, theta1, phi1) is called the system of geomagnetic coordinates.

Let thetan and phin denote the colatitude and azimuth of the point of intersection of the geomagnetic axis with the Earth's surface in the northern hemisphere. The transformation from geographic Cartesian coordinates (x, y, z) to geomagnetic Cartesian coordinates (x1, y1, z1) is then given by:


In geomagnetic coordinates, the expansion takes the form:


The coefficients g'nm and h'nm can be related to the coefficients gnm and hnm by transforming the spherical harmonics in the expansion with Wigner's formula (Bernard et al., 1969). In particular, we find:

The angles thetan and phin are determined by the conditions:

whence
and
where
and

The expansion then reduces to

The geomagnetic field is often approximated by the first term in this expansion. This approximation is called the centred dipole model. Its parameters are completely specified by the coefficients g10, g11, and h11. The dipole moment M of the centred dipole is given by


Substituting the IGRF 2000 values for g10, g11, and h11, we find that

 B0 = 3.01153 104 nT 
 M = 7.788 1022 A m2
thetan =  10.46 ° 
phin =  -71.57 ° 

for epoch 2000. Table I lists the positions of the northern centred dipole pole calculated with DGRF 1945-1990 and IGRF 1995-2000 .


Table I. Position of the centred dipole model northern pole
Model Latitude Longitude
DGRF 1945
DGRF 1950
DGRF 1955
DGRF 1960
DGRF 1965
DGRF 1970
DGRF 1975
DGRF 1980
DGRF 1985
DGRF 1990
IGRF 1995
IGRF 2000
78.47
78.47
78.46
78.51
78.53
78.59
78.69
78.81
78.97
79.13
79.30
79.54
291.47
291.15
290.84
290.53
290.15
289.82
289.53
289.24
289.10
288.89
288.59
288.43

The eccentric dipole model

In order to approximate the geomagnetic field with a dipole which is not necessarily located at the centre of the Earth, the criterion to be used to judge the best fit to the observed field has to be defined. The criterion adopted by Schmidt (1934) and described by Bartels (1936) is to minimize the terms of second order in the potential used in the spherical harmonic representation of the field. The eccentric dipole so obtained has the same moment as the centred dipole and the same orientation of its axis, but in terms of the geographic Cartesian coordinate system (x, y, z), it is not located at the centre but at a position (eta, zeta, xi) RE, where the quantities eta, zeta and xi can be derived from the Gauss coefficients.

It may be noted that the rectangular coordinate system used by Schmidt (1934) and Bartels (1936) differs from those now conventionally used and this circumstance has led to an erroneous designation of the dipole position in Chapman and Bartels (1940).

When reference is made to the eccentric dipole model of the Earth's magnetic field, it is now generally understood that the Schmidt (1934) criterion and its resulting mathematical formulation are applicable, even though other eccentric dipole models are possible (e.g. Bochev, 1969), and it is the Schmidt eccentric dipole model that is described here.

The dimensionless coordinate quantities xi, eta and zeta are given by:



where B0 is the reference field defined as for the centred dipole model, and


The distance from the dipole position to the centre is given by:


It follows from the above equations that the eccentric dipole is completely specified by the first eight Gauss coefficients. Substituting the values of the IGRF 2000 Gauss coefficients, the folllowing values are obtained:

 L0 = 1.028 108 nT2 
 L1 = -1.706 108 nT2 
 L2 = 1.252 108 nT2 
 E = -5.791 102 nT
which give:
eta = -0.06308
zeta = 0.04713
xi = 0.03149

The shifts in the x, y and z GEO directions are therefore -401.86 km, 300.25 km and 200.61 km, respectively, and the total distance shifted by the dipole is delta = 540.27 km. The direction of the shift is given by thetad = arccos (200.61/540.27) = 68.20°, and phid = 90° + arctan (401.86/200.61) = 143.24°, that is, it is toward the point 21.80°, 143.24°. This point is in the northwest Pacific, at the northern end of the Mariana Islands. The projection of the dipole position on the Earth's surface is shown in Figure I. Table II gives the position of the eccentric dipole for epochs 1945-2000.

Position of the eccentric dipole
Figure 1. Projection of the eccentric dipole position on the Earth's surface.


Table II. Position of the eccentric dipole.
Model X0
(km)
Y0
(km)
Z0
(km)
delta
(km)
DGRF 1945
DGRF 1950
DGRF 1955
DGRF 1960
DGRF 1965
DGRF 1970
DGRF 1975
DGRF 1980
DGRF 1985
DGRF 1990
IGRF 1995
IGRF 2000
-355.24
-359.03
-362.59
-365.90
-368.77
-373.13
-378.57
-385.41
-391.78
-396.49
-400.51
-401.86
175.47
190.67
203.52
214.78
223.78
230.96
237.02
247.49
258.51
270.82
282.84
300.25
92.33
101.29
110.75
122.42
133.56
146.40
159.83
170.21
178.73
185.88
192.87
200.61
406.83
418.95
430.30
441.58
451.57
462.60
474.38
488.63
502.26
514.87
526.89
540.27

Eccentric dipole axial poles

The eccentric dipole model for the Earth's magnetic field produces two different varieties of poles. The first of these are referred to as the axial poles (i.e. the two points on the Earth's surface where the eccentric dipole axis intersects the surface). Because of the displacement of the eccentric pole away from the Earth's centre the eccentric dipole axis and associated magnetic field, in particular, are not perpendicular to the surface at the axial poles. There are, however, two points where the magnetic field is perpendicular to the surface, which are referred to as the eccentric dipole dip poles.

The axis of the eccentric dipole is parallel to the centred dipole axis. This fact and the knowledge that the eccentric dipole axis passes through the point (eta, zeta, xi) RE in GEO rectangular coordinates enable the derivation of an expression for the axis in our basic geographic spherical polar coordinate system. The equation for a line passing through a point (x0, y0, z0) is:



where l,m and n are the direction cosines of the line. Converting to spherical coordinates and substituting l = sin thetan cos phin, m = sin thetan sin phin and n = cos thetan, which follow from the known orientation of the dipole axis in geographic coordinates, the following equation is obtained for the eccentric dipole axis:


Substituting r = RE in the expression above and carrying out the appropriate algebraic manipulations, the following equations are obtained for the intersection points:



Note that these equations are not identical to those used by Fraser-Smith (1987), where the approximation was made that cos theta is +/- 1 for the northern, respectively southern pole. Using the IGRF data for epoch 2000, the axial poles are found to be:

Table III lists the position of the eccentric axial pole positions for epochs 1945-2000.

Table III. Position of the eccentric dipole axial poles.
  North pole South pole
Model latitude
(deg)
longitude
(deg)
latitude
(deg)
longitude
(deg)
DGRF 1945
DGRF 1950
DGRF 1955
DGRF 1960
DGRF 1965
DGRF 1970
DGRF 1975
DGRF 1980
DGRF 1985
DGRF 1990
IGRF 1995
IGRF 2000
80.90
81.04
81.15
81.30
81.40
81.53
81.68
81.88
82.15
82.39
82.65
83.03
-83.86
-84.39
-84.94
-85.57
-86.27
-87.06
-87.99
-89.05
-90.05
-91.05
-92.20
-93.30
-75.52
-75.38
-75.25
-75.19
-75.13
-75.10
-75.11
-75.11
-75.15
-75.19
-75.24
-75.34
121.09
120.68
120.29
119.98
119.62
119.40
119.29
119.17
119.18
119.05
118.87
118.66

Eccentric dipole dip poles

The dip poles are located on the great circle defined by the intersection of the plane containing both the centred and the eccentric axes with the earth's surface. They are separated from their corresponding eccentric dipole axial poles by small angular distances along the great circle, with the direction of the separition being away from the local centric dipole pole (All centred and eccentric poles lie on the same great circle). At these points there is enough curvature of the dipole field lines away from the eccentric dipole axis to compensate for the small angle made by the axis with the Earth's surface and thus to bring the field lines perpendicular to the surface (see figure 2).

Geometry of the poles
Figure 2. Geometry used in the derivation of the equation for the dip poles. P is a general point and not necessarily the dipole pole. [Taken from Fraser-Smith (1987)].

The procedure to calculate the dip pole positions consists of the following steps:

The only significant feature in the first step is a change of the coordinates of the eccentric dipole. The eccentric dipole has CD coordinates (X0, Y0, Z0) and (delta, THETA, PHI). The change of coordinates from (delta, theta, phi) to (delta, THETA, PHI) is easily carried out with the transformation matrix described in the section centred dipole model.

The second step in the derivation is to obtain the CD azimuthal coordinates of the dip poles. Figure 2 shows the geometry required in this step of the derivation and it foolows that the CD azimuthal coordinates for the dip poles are the same and equal to PHId; the azimuthal angle does not play a further role at this stage of the derivation. The other CD coordinates of the dip poles are now obtained by resolving the magnetic field of the eccentric dipole, located at E in figure 2, along the tangent to the circle (representing the Earth's surface) at the general point P, equating the resolved field to zero, and then rearranging the resulting equations to obtain an expression for THETA. Remembering that the dipole at E is oriented along the axis DEA in figure 2, the two components of the dipole field at P are:



where re and thetae are the eccentric dipole polar coordinates of P in figure 2. The dip pole condition is then:


which, after substitution for Bre and Bthetae, becomes:


This equation now has to be solved for THETA. Applying the sine rule to the triangle OEP:


Further, if F is the foot of the perpendicular from E onto the line OP, we have RE = OF + FP, which gives:


This can be written as:

giving

The point P in figure 2 has CD coordinates (RE, THETA) and eccentric dipole coordinates (re, thetae) which, from the geometry of figure 2, implies the following two results:

giving

Substituting the expressions for tan (THETA - thetae) and tan thetae into the dip condition, and carrying out the necessary algebraic manipulation, the following expression for THETA is obtained:

where

This equation must be solved numerically for THETA, and being of the second order in cos THETA, it gives two values THETA1 and THETA2, corresponding to the north and south eccentric dipole dip poles.

The third and final step in the derivation is to convert the CD coordinates (RE, THETA1, 2, PHId) of the dip poles into geographic coordinates using the transformation matrix described in the section centred dipole model.

Using this procedure with the IGRF coefficients for epoch 2000, the following dip pole positions are obtained:


Figures 3 and 4 show the positions of the CD poles, the ED axial poles and the ED dip poles for the northern and southern hemisphere respectively. The dip poles are also listed in table IV.

Position of poles in the northern hemisphere
Figure 3. Position of the centred dipole poles, the eccentric dipole axial poles and the eccentric dip poles in the northern hemisphere.

Position of poles in the southern
hemisphere
Figure 4. Position of the centred dipole poles, the eccentric dipole axial poles and the eccentric dip poles in the southern hemisphere.


Table IV. Position of the eccentric dipole dip poles.
  North pole South pole
Model latitude
(deg)
longitude
(deg)
latitude
(deg)
longitude
(deg)
DGRF 1945
DGRF 1950
DGRF 1955
DGRF 1960
DGRF 1965
DGRF 1970
DGRF 1975
DGRF 1980
DGRF 1985
DGRF 1990
IGRF 1995
IGRF 2000
82.20
82.39
82.52
82.64
82.69
82.70
82.67
82.65
82.65
82.65
82.61
82.66
-132.81
-135.69
-138.28
-141.07
-143.36
-145.75
-148.33
-151.89
-155.76
-159.58
-163.37
-168.60
-68.97
-68.56
-68.19
-67.89
-67.62
-67.37
-67.16
-66.88
-66.66
-66.44
-66.25
-66.06
131.79
131.09
130.49
130.03
129.59
129.38
129.32
129.18
129.12
128.86
128.57
128.04

References

Bartels, J., The eccentric dipole approximating the Earth's magnetic field, J. Geophys. Res., 41, pp. 225-250, 1936.

Bernard, J., J.-C. Kosik, G. Laval, R. Pellat, J.-P. Philippon, Représentation Optimale du Potentiel Géomagnétique dans le Repère d'un Dipole Décentré, Incliné, Ann. Géophys., 25, pp. 659-665, 1969.

Bochev, A., A dipole approximating to the highest possible degree the Earth's magnetic field, Pure Appl. Geophys., 74, pp. 25-28, 1969.

Chapman, S. and J. Bartels, Geomagnetism, vol. 2, Oxford University Press, New York, pp. 639-668, 1940.

Fraser-Smith, A. C., Centered and Eccentric Geomagnetic Dipoles and Their Poles, 1600-1985, Rev. Geophys., 25, 1, pp. 1-16, 1987.

Heynderickx, D. and J. Lemaire, Improvements to Trapped Radiation Software, Trapped Radiation Environment Model Development, Technical Note 1, ESTEC Contract No. 9828/92/NL/FM, 1993.

Schmidt, A., Der magnetische Mittelpunkt der Erde und seine Bedeutung, Gerlands Beitr. Geophys., 41, pp. 346-358, 1934.



The description of the centred dipole model is based on Heynderickx and Lemaire (1993).
The description of the eccentric dipole model is based on the article of Fraser-Smith (1987).


Last update: Wed, 23 Mar 2016