Earth's magnetic field
Spherical harmonics

The most common way of analyzing the global variations in the Earth's magnetic field is to fit the measurements to a set of spherical harmonics. This was first done by Carl Friedrich Gauss. Spherical harmonics are functions that oscillate over the surface of a sphere. They are the product of two functions, one that depends on latitude and one on longitude. The function of longitude is zero along zero or more great circles passing through the North and South Poles; the number of such nodal lines is the absolute value of the order m. The function of latitude is zero along zero or more latitude circles; this plus the order is equal to the degree ℓ. Each harmonic is equivalent to a particular arrangement of magnetic charges at the center of the Earth. A monopole is an isolated magnetic charge, which has never been observed. A dipole is equivalent to two opposing charges brought close together and a quadrupole to two dipoles brought together. A quadrupole field is shown in the lower figure on the right.

Spherical harmonics can represent any scalar field (function of position) that satisfies certain properties. A magnetic field is a vector field, but if it is expressed in Cartesian components X, Y, Z, each component is the derivative of the same scalar function called the magnetic potential. Analyses of the Earth's magnetic field use a modified version of the usual spherical harmonics that differ by a multiplicative factor. A least-squares fit to the magnetic field measurements gives the Earth's field as the sum of spherical harmonics, each multiplied by the best-fitting Gauss coefficient g or h.

The lowest-degree Gauss coefficient, g, gives the contribution of an isolated magnetic charge, so it is zero. The next three coefficients – g, g, and h – determine the direction and magnitude of the dipole contribution. The best fitting dipole is tilted at an angle of about 10° with respect to the rotational axis, as described earlier.
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