## In Igrf Worked Exercise By Magnetic Scalar Potential In Geomagnetism

GEOCENTRIC DISTANCE (EARTH RADII)

GEOCENTRIC DISTANCE IEARTH RADII)

GEOCENTRIC DISTANCE (EARTH RADII)

Maximum Deviations of Approximate Models From the Earth's Magnetic Field (From Harris and Lyle [19690

plus secondary currents or whirlpools near the core-mantle boundary which produce local dipoles. These secondary dipoles are then superimposed to produce the observed multipole nature of the field, as well as local anomalies, which are large surface areas where the magnetic field deviates appreciably from the dipole field. The creation and decay of the whirlpools may cause the secular drift. Another theory of the secular drift is that the core is rotating more slowly than the mantle and crust.

Although the exact nature of the field generator is unknown, the fact that it is internal suggests that the field can be conveniently described as a solution to a boundary value problem. The lack of surface electric currents implies that outside the Earth, the magnetic field, B, has zero curl,

which means that the field can be expressed as the gradient of a scalar potential, V

The absence of magnetic monopoles implies

Substituting Eq. (5-2) into Eq. (5-3) yields Laplace's equation:

which, because of the spherical nature of the boundary at the Earth's surface, has a solution conveniently expressed in spherical harmonics as

where a is the equatorial radius of the Earth; g™ and h™ are called Gaussian coefficients; r, 9, and 4> are the geocentric distance, coelevation, and east longitude from Greenwich; and P"(9) are the associated Legrendre functions. (See Appendix G for a further discussion of spherical harmonics.) The n = 1 terms are called dipole; the n = 2, quadrupole; the n = 3, octupole. The actual calculation of B from Eqs. (5-5) and (5-2) is explained in detail in Appendix H.

To use Eq. (5-5) to calculate the field at any point, the Gaussian coefficients must be known. It is the object of theories, such as the core-dynamo theory, to calculate them; however, success has been severely limited. The alternative is to determine the Gaussian coefficients empirically by doing a least-squares fit to magnetic field data using the coefficients as fitting parameters. Data consisting of both magnitude and direction is obtained from a series of magnetic observatories. Unfortunately, these observatories are not distributed uniformly so that the data is sparse in some regions of the Earth. More uniformly distributed data is obtained from field magnitude measurements made by satellites. Although there are some ! theoretical arguments that obtaining coefficients by simply fitting field magnitudes 1 is an ambiguous process [Stern, 1975], it appears to work quite well in practice [Cain, 1970]. ;

One set of Gaussian coefficients to degree 8 (n in Eq. (5-5)) and order 8 (m in j Eq. (5-5)), comprises the International Geomagnetic Reference Field (1GRF (1975)) [Leaton, 1976] and is given in Appendix H. The field model includes the first-order time derivatives of the coefficients in an attempt to describe the secular variation. Because of the lack of adequate data over a long enough period of time, the accuracy of this (or any) field model will degrade with time. In fact, the IGRF (1975) is an update of IGRF (1965) [Cain and Cain, 1971]. The IGRF (1965) should be used for the period 1955-1975 and the IGRF (1975) should be used for the period 1975-1980. The maximum and root-mean-square (RMS) errors in the field magnitude based on IGRF (1965) are given in Table 5-1 for 1975.

The estimated growth of the errors presented in the table was a factor of two j from 1970 to 1975. The errors in direction (i.e., in components of the field) are more difficult to estimate but should not be more than a factor of two greater than the magnitude data shown in Table 5-1. One value in Table 5-1 was verified by GEOS-3 data taken in a polar orbit at an altitude of 840 km [Coriell, 1975], [

Substitution of the magnetic field potential in Eq. (5-5) into Eq. (5-2) will show , that the strength of the dipole field decreases with the inverse cube of the distance from the center of the dipole, and that the quadrupole decreases with the inverse fourth power. Higher degree multipoles decrease even more rapidly. Thus, at the t t-

Table 5-1. Errors in the Field Magnitude Derived From the IGRF (1965) for 1975 (From Trombka and Cain )

DISTANCE FROM EARTH CENTER (EARTH RAOIII

DIPOLE FIELD' MAGNITUDE tflT)

MAXIMUM ERROR (nT)

RMS ERROR InT)

1 (SURFACE)

0 0