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Multipole analysis of the international geomagnetic reference field
1970, Phys. Earth Planet. Interiors 2, 163—165. North-Holland Publishing Company, Amsterdam
MULTIPOLE ANALYSIS OF THE INTERNATIONAL GEOMAGNETIC REFERENCE FIELD
D. E. WINCH and R. W. JAMES Department of Applied Mathematics, University of Sydney, Sydney, N.S. W., 2006, Australia Received 7 September 1969
The method of multipole analysis is applied to the International Geomagnetic Reference Field for 1965. Rates of change of the various multipole parameters are determined from the secular
variation coefficients, and the secular variation field is thereby separated into drifting and non-drifting components.
At the Symposium of the International Association of Geomagnetism and Aeronomy (IAGA) on the Description of the Earth’s Magnetic Field, held in Washington, D.C., October 1968, a composite of representations submitted by several authors was selected as the International Geomagnetic Reference Field (IGRF) appropriate for epoch 1965.0. At the same meeting, the World Magnetic Survey Board endorsed the choice of the IGRF, which was subsequently adopted by the IAGA at the General Scientific Assembly held in Madrid, September 1969. The IGRF is a series of spherical harmonic coefficients in geocentric spherical coordinates, having 80 coefficients for the main field and a further 80 coefficients for the secular variation (SV) field. The IGRF is intended to apply for the period 1955 to 1972. The nth order spherical harmonic component of the geomagnetic potential is usually represented by
Eq. (2) is a generalisation of the case n = 1, corresponding to the familiar geomagnetic dipole, and is thus called the multipole representation of V~.Know!edge of the axes ii,,, fixes the geometrical structure of the nth order field and it remains for M~to determine 2/n! in eq. (2) the intensity at any point. The factor a~ + is such that the multipole strength M~has units of magnetic induction and a magnitude comparable to the rms intensity of the nth order field on the Earth’s surface (JAMES, 1969). Conversion of the multipole representation (2) to the spherical harmonic representation (1) is simple if use is made of certain recurrence relations (JAMES, 1967); an example of such a conversion for n = 5 has been given by WINCH (1967). The inverse problem of convertingGaussiancoefficientstomultipoleparametersfor any order n is best carried out by means of a reduction of order technique based on the above-mentioned recurrence relations (JAMES, 1968). The multipole para-
=
a(a/r)”~1 ~
o
(g’ cos nu/+h~’sin m4))P~’(cos ~ (1)
where a is the mean radius of the Earth and 0, ~j,are geographic colatitude and east longitude respectively, According to Maxwell’s theory of poles of spherical harmonic functions an alternative formulation is available: V~=
(—
meters of the 1965 IGRF are listed in table 1. The secular trends tn the multtpole parameters M~, u~ reflect the changing patterns of the geomagnetic field on the Earth’s surface and the determination of the time derivatives J~f~ and à,~permits decomposition of the SY field at a particular epoch into components associated with longitudinal and meridional drifts of the multipole axes and a non-drifting component associated with the variation in multipole strength (WINcH, 1968). The IGRF SV coefficients have been
1)”a”~2(M~/n!)(U,, 1
.
V) (u,,2 V). .(u,,,,’ V) 1 .
r
(2) 163
decomposed in such a manner and the results are given in table 2. Isoline maps and rms intensities derived from a table similar to table 2 show (JAMES, 1969) that
164
D. E. WINCH AND R. W. JAMES TABLE
—
n
I
Multipole parameters and rates of change
-
~
(y)
(y/yr)
2
30953 4922
—16.02 16.08
3
4220
5.15
4
2645
—0.36
5
1026
1.22
6
912
3.98
7
394
—0.37
8
213
0.19
i
O,,~ (°)
1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8
168.565 22.491 105.374 20.316 57.378 61.969 24.31 36.32 56.80 67.47 16.96 41.72 48.17 62.98 95.08 30.95 35.23 40.96 73.36 82.25 94.33 18.93 37.16 41.60 52.27 52.33 59.90 72.43 23.29 34.40 52.76 56.95 64.47 76.85 78.42 93.86
çl,,~ (°) 110.239 32.577 332.166 343.858 139.947 224.402 107.77 216.41 307.37 42.46 262.19 344.90 190.90 50.41 355.12 247.26 6.33 129.79 49.07 119.56 354.62 330.13 158.90 313.92 202.82 8.52 94.78 234.22 338.55 215.22 17.86 176.74 94.03 182.86 296.16 49.14
0,,, (°/yr)
(°/yr)
0.004 —0.10 0.20 —0.05 —0.02 —0.11 0.09 0.03 0.04 —0.05 —0.09 —0.24 —0.38 0.37 —0.61 0.02 0.04 0.22 —0.09 —0.06 0.13 —0.10 —0.38 0.14 —0.05 0.04 —0.20 0.15 0.26 0.47 —0.22 —1.39 —0.11 0.70 —0.10 0.19
—0.069 —0.49 —0.12 —0.38 0.03 —0.20 —0.24 —0.30 —0.04 —0.11 —0.40 0.00 —0.10 —0.21 0.15 0.01 0.21 —0.07 —0.27 —0.25 —0.09 —1.66 0.28 —0.19 —0.41 0.04 —0.19 0.03 —0.53 —0.11 —0.35 —2.13 —0.54 1.52 0.22 —0.01
Table I contains a list of multipole strengths M,,, and multipole axes in terms of 0,,,, q~,colatitude and east longitude respectively. Rates of change of the multipole parameters are also included. Note that the multipole axes may be reversed in direction in pairs, without changing the resulting geomagnetic potential.
SV is principally a drifting rather than a non-drifting phenomenon. The well-known tendency of westward drift of the Earth’s main geomagnetic field is reflected in the preponderance of minus signs in the rates of change of east longitude of the multipole axes, shown in the last column of table 1. However, some of the higher order fields have axes with large meridional drifts, and the amplification of such drifts on downward extrapolation to the core-mantle boundary suggests that meridional drifts may play a significant part in determining the structure of SY at the core-mantle boundary.
Table 1 shows that most of the non-dipole field strengths are increasing. However, the rate of decrease of the dipole strength is such that the magnetostatic energy of the IGRF exterior to the Earth is decreasing at about 0.09 % per year. References R. W. (1967) Pure AppI. Geophys. 68, 83. R. W. (1968) Austr. J. Phys. 21, 455. R. W. (1969) Austr. J. Phys. 22, 481. WINcif, D. E. (1967) Pure Appl. Geophys. 68, 90. JAMES,
JAMES, JAMES,
WINCH,
D. E. (1968) J. Geomag. Geoelect. 20, 205.
165
MULTIPOLE ANALYSIS OF THE I.G.R.F. TABLE
2
Analysis of the IGRF secular variation field n
m 0
2
4
5
6
8
ND
MD
LD
~
ND
MD
LD
15.3 8.7
15.7 1.1
—0.4 0.7
0.0 6.9
—2.3
—3.0
—1.9
2.5
—24.4 0.3 —1.6
—5.4 9.8 5.1
—13.9 —3.9 —8.1
—5.1 —5.6 —1.4
—11.8 —16.7
—6.6 0.4
—1.6 —0.7
—3.7 —16.5
2 3
0.2 —10.8 0.7 —3.8
1.6 —2.5 1.6 1.0
4.7 —2.1 —1.0 —3.1
—6.1 —6.2 0.2 —1.7
4.2 0.7 —7.7
—0.5 0.3 —0.2
3.2 0.8 0.7
1.4 —0.4 —8.1
0 1 2 3 4
—0.7 0.2 —3.0 —0.1 —2.1
—0.1 —0.1 —0.1 0.1 0.0
0.4 —1.3 —0.7 0.3 1.1
—1.0 1.6 —2.2 —0.5 —3.2
—0.1 1.6 2.9 —4.2
0.0 0.0 0.0 0.0
—0.9 1.5 0.4 —1.2
0.8 0.0 2.5 —3.1
0 2 3 4 5
1.9 1.1 2.9 0.6 0.0 1.3
—0.3 0.4 0.3 0.0 —0.2 —0.1
3.0 —0.1 1.9 1.6 0.4 0.6
—0.8 0.7 0.7 —1.0 —0.2 0.8
2.3 1.7 —2.4 0.8 —0.3
0.0 0.1 —0.1 —0.1 0.1
1.2 1.7 —1.8 0.3 —0.9
1.2 —0.2 —0.5 0.6 0.5
0 1 2 3 4 5 6
—0.1 —0.3 1.1 1.9 —0.4 —0.4 —0.2
—0.2 —0.3 0.0 1.0 0.0 0.0 0.5
0.5 0.1 0.4 0.4 0.1 —0.1 0.6
—0.4 —0.1 0.7 0.5 —0.5 —0.3 —0.1
—0.9 —0.4 2.0 —1.1 0.1 0.9
0.1 —0.5 —0.3 0.1 0.0 0.1
—0.2 0.3 —0.2 —0.7 —0.3 —0.1
—0.8 —0.3 2.5 —0.6 0.4 0.9
0 2 3 4 5 6 7
—0.5 —0.3 —0.7 —0.5 0.3 0.0 —0.2 —0.6
—0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.1 —0.6 0.0 0.2 —0.4 —0.2 0.0
—0.4 —0.4 —0.1 —0.5 0.1 0.4 0.0 —0.6
—1.1 0.3 0.4 0.2 0.4 0.2 0.3
0.1 0.0 0.0 0.0 0.0 0.0 0.0
—0.7 —0.3 0.0 —0.2 0.1 0.3 0.2
—0.4 0.5 0.4 0.4 0.4 —0.1 0.1
0 1 2 3 4 5 6 7 8
0.1 0.4 0.6 0.0 0.0 —0.1 0.3 —0.3 —0.5
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.1 0.1 —0.1 —0.1 —0.3 —0.1 —0.2 0.0
0.0 0.3 0.5 0.1 0.1 0.2 0.5 —0.1 —0.5
0.1 —0.2 —0.3 —0.2 —0.3 —0.4 —0.3 —0.3
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
—0.1 —0.2 —0.1 —0.1 —0.2 —0.3 0.1 —0.1
0.2 0.0 —0.2 —0.1 —0.1 —0.1 —0.4 —0.2
0 2
3
ii,,’”
0
-
ND: non-drifting component of SV associated with variations in the multipole strength M,,. MD: meridional drifting component of SV associated with variations in colatitude 0,,, of the multipole axes. LD: longitudinal drifting component of SV associated with variations in the east longitude ~,,, of the multipole axes.
MULTIPOLE ANALYSIS OF THE INTERNATIONAL GEOMAGNETIC REFERENCE FIELD
D. E. WINCH and R. W. JAMES Department of Applied Mathematics, University of Sydney, Sydney, N.S. W., 2006, Australia Received 7 September 1969
The method of multipole analysis is applied to the International Geomagnetic Reference Field for 1965. Rates of change of the various multipole parameters are determined from the secular
variation coefficients, and the secular variation field is thereby separated into drifting and non-drifting components.
At the Symposium of the International Association of Geomagnetism and Aeronomy (IAGA) on the Description of the Earth’s Magnetic Field, held in Washington, D.C., October 1968, a composite of representations submitted by several authors was selected as the International Geomagnetic Reference Field (IGRF) appropriate for epoch 1965.0. At the same meeting, the World Magnetic Survey Board endorsed the choice of the IGRF, which was subsequently adopted by the IAGA at the General Scientific Assembly held in Madrid, September 1969. The IGRF is a series of spherical harmonic coefficients in geocentric spherical coordinates, having 80 coefficients for the main field and a further 80 coefficients for the secular variation (SV) field. The IGRF is intended to apply for the period 1955 to 1972. The nth order spherical harmonic component of the geomagnetic potential is usually represented by
Eq. (2) is a generalisation of the case n = 1, corresponding to the familiar geomagnetic dipole, and is thus called the multipole representation of V~.Know!edge of the axes ii,,, fixes the geometrical structure of the nth order field and it remains for M~to determine 2/n! in eq. (2) the intensity at any point. The factor a~ + is such that the multipole strength M~has units of magnetic induction and a magnitude comparable to the rms intensity of the nth order field on the Earth’s surface (JAMES, 1969). Conversion of the multipole representation (2) to the spherical harmonic representation (1) is simple if use is made of certain recurrence relations (JAMES, 1967); an example of such a conversion for n = 5 has been given by WINCH (1967). The inverse problem of convertingGaussiancoefficientstomultipoleparametersfor any order n is best carried out by means of a reduction of order technique based on the above-mentioned recurrence relations (JAMES, 1968). The multipole para-
=
a(a/r)”~1 ~
o
(g’ cos nu/+h~’sin m4))P~’(cos ~ (1)
where a is the mean radius of the Earth and 0, ~j,are geographic colatitude and east longitude respectively, According to Maxwell’s theory of poles of spherical harmonic functions an alternative formulation is available: V~=
(—
meters of the 1965 IGRF are listed in table 1. The secular trends tn the multtpole parameters M~, u~ reflect the changing patterns of the geomagnetic field on the Earth’s surface and the determination of the time derivatives J~f~ and à,~permits decomposition of the SY field at a particular epoch into components associated with longitudinal and meridional drifts of the multipole axes and a non-drifting component associated with the variation in multipole strength (WINcH, 1968). The IGRF SV coefficients have been
1)”a”~2(M~/n!)(U,, 1
.
V) (u,,2 V). .(u,,,,’ V) 1 .
r
(2) 163
decomposed in such a manner and the results are given in table 2. Isoline maps and rms intensities derived from a table similar to table 2 show (JAMES, 1969) that
164
D. E. WINCH AND R. W. JAMES TABLE
—
n
I
Multipole parameters and rates of change
-
~
(y)
(y/yr)
2
30953 4922
—16.02 16.08
3
4220
5.15
4
2645
—0.36
5
1026
1.22
6
912
3.98
7
394
—0.37
8
213
0.19
i
O,,~ (°)
1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8
168.565 22.491 105.374 20.316 57.378 61.969 24.31 36.32 56.80 67.47 16.96 41.72 48.17 62.98 95.08 30.95 35.23 40.96 73.36 82.25 94.33 18.93 37.16 41.60 52.27 52.33 59.90 72.43 23.29 34.40 52.76 56.95 64.47 76.85 78.42 93.86
çl,,~ (°) 110.239 32.577 332.166 343.858 139.947 224.402 107.77 216.41 307.37 42.46 262.19 344.90 190.90 50.41 355.12 247.26 6.33 129.79 49.07 119.56 354.62 330.13 158.90 313.92 202.82 8.52 94.78 234.22 338.55 215.22 17.86 176.74 94.03 182.86 296.16 49.14
0,,, (°/yr)
(°/yr)
0.004 —0.10 0.20 —0.05 —0.02 —0.11 0.09 0.03 0.04 —0.05 —0.09 —0.24 —0.38 0.37 —0.61 0.02 0.04 0.22 —0.09 —0.06 0.13 —0.10 —0.38 0.14 —0.05 0.04 —0.20 0.15 0.26 0.47 —0.22 —1.39 —0.11 0.70 —0.10 0.19
—0.069 —0.49 —0.12 —0.38 0.03 —0.20 —0.24 —0.30 —0.04 —0.11 —0.40 0.00 —0.10 —0.21 0.15 0.01 0.21 —0.07 —0.27 —0.25 —0.09 —1.66 0.28 —0.19 —0.41 0.04 —0.19 0.03 —0.53 —0.11 —0.35 —2.13 —0.54 1.52 0.22 —0.01
Table I contains a list of multipole strengths M,,, and multipole axes in terms of 0,,,, q~,colatitude and east longitude respectively. Rates of change of the multipole parameters are also included. Note that the multipole axes may be reversed in direction in pairs, without changing the resulting geomagnetic potential.
SV is principally a drifting rather than a non-drifting phenomenon. The well-known tendency of westward drift of the Earth’s main geomagnetic field is reflected in the preponderance of minus signs in the rates of change of east longitude of the multipole axes, shown in the last column of table 1. However, some of the higher order fields have axes with large meridional drifts, and the amplification of such drifts on downward extrapolation to the core-mantle boundary suggests that meridional drifts may play a significant part in determining the structure of SY at the core-mantle boundary.
Table 1 shows that most of the non-dipole field strengths are increasing. However, the rate of decrease of the dipole strength is such that the magnetostatic energy of the IGRF exterior to the Earth is decreasing at about 0.09 % per year. References R. W. (1967) Pure AppI. Geophys. 68, 83. R. W. (1968) Austr. J. Phys. 21, 455. R. W. (1969) Austr. J. Phys. 22, 481. WINcif, D. E. (1967) Pure Appl. Geophys. 68, 90. JAMES,
JAMES, JAMES,
WINCH,
D. E. (1968) J. Geomag. Geoelect. 20, 205.
165
MULTIPOLE ANALYSIS OF THE I.G.R.F. TABLE
2
Analysis of the IGRF secular variation field n
m 0
2
4
5
6
8
ND
MD
LD
~
ND
MD
LD
15.3 8.7
15.7 1.1
—0.4 0.7
0.0 6.9
—2.3
—3.0
—1.9
2.5
—24.4 0.3 —1.6
—5.4 9.8 5.1
—13.9 —3.9 —8.1
—5.1 —5.6 —1.4
—11.8 —16.7
—6.6 0.4
—1.6 —0.7
—3.7 —16.5
2 3
0.2 —10.8 0.7 —3.8
1.6 —2.5 1.6 1.0
4.7 —2.1 —1.0 —3.1
—6.1 —6.2 0.2 —1.7
4.2 0.7 —7.7
—0.5 0.3 —0.2
3.2 0.8 0.7
1.4 —0.4 —8.1
0 1 2 3 4
—0.7 0.2 —3.0 —0.1 —2.1
—0.1 —0.1 —0.1 0.1 0.0
0.4 —1.3 —0.7 0.3 1.1
—1.0 1.6 —2.2 —0.5 —3.2
—0.1 1.6 2.9 —4.2
0.0 0.0 0.0 0.0
—0.9 1.5 0.4 —1.2
0.8 0.0 2.5 —3.1
0 2 3 4 5
1.9 1.1 2.9 0.6 0.0 1.3
—0.3 0.4 0.3 0.0 —0.2 —0.1
3.0 —0.1 1.9 1.6 0.4 0.6
—0.8 0.7 0.7 —1.0 —0.2 0.8
2.3 1.7 —2.4 0.8 —0.3
0.0 0.1 —0.1 —0.1 0.1
1.2 1.7 —1.8 0.3 —0.9
1.2 —0.2 —0.5 0.6 0.5
0 1 2 3 4 5 6
—0.1 —0.3 1.1 1.9 —0.4 —0.4 —0.2
—0.2 —0.3 0.0 1.0 0.0 0.0 0.5
0.5 0.1 0.4 0.4 0.1 —0.1 0.6
—0.4 —0.1 0.7 0.5 —0.5 —0.3 —0.1
—0.9 —0.4 2.0 —1.1 0.1 0.9
0.1 —0.5 —0.3 0.1 0.0 0.1
—0.2 0.3 —0.2 —0.7 —0.3 —0.1
—0.8 —0.3 2.5 —0.6 0.4 0.9
0 2 3 4 5 6 7
—0.5 —0.3 —0.7 —0.5 0.3 0.0 —0.2 —0.6
—0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.1 —0.6 0.0 0.2 —0.4 —0.2 0.0
—0.4 —0.4 —0.1 —0.5 0.1 0.4 0.0 —0.6
—1.1 0.3 0.4 0.2 0.4 0.2 0.3
0.1 0.0 0.0 0.0 0.0 0.0 0.0
—0.7 —0.3 0.0 —0.2 0.1 0.3 0.2
—0.4 0.5 0.4 0.4 0.4 —0.1 0.1
0 1 2 3 4 5 6 7 8
0.1 0.4 0.6 0.0 0.0 —0.1 0.3 —0.3 —0.5
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.1 0.1 —0.1 —0.1 —0.3 —0.1 —0.2 0.0
0.0 0.3 0.5 0.1 0.1 0.2 0.5 —0.1 —0.5
0.1 —0.2 —0.3 —0.2 —0.3 —0.4 —0.3 —0.3
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
—0.1 —0.2 —0.1 —0.1 —0.2 —0.3 0.1 —0.1
0.2 0.0 —0.2 —0.1 —0.1 —0.1 —0.4 —0.2
0 2
3
ii,,’”
0
-
ND: non-drifting component of SV associated with variations in the multipole strength M,,. MD: meridional drifting component of SV associated with variations in colatitude 0,,, of the multipole axes. LD: longitudinal drifting component of SV associated with variations in the east longitude ~,,, of the multipole axes.