Geomagnetic secular variation

Builds on Earth's magnetic field Gauss's spherical harmonic analysis of the geomagnetic field

Definition

Let $B_{\mathrm{int}}(x, t)$ be the internal part of Earth's magnetic field, averaged over times of about a year to remove external variations. The geomagnetic secular variation is its rate of change $\partial_t B_{\mathrm{int}}(x, t)$ , on time scales from about a year to several centuries. Let $g_n^m(t)$ and $h_n^m(t)$ be the internal Gauss coefficients of Gauss's spherical harmonic analysis. The secular variation coefficients are $\dot g_n^m = \frac{d g_n^m}{dt}$ and $\dot h_n^m = \frac{d h_n^m}{dt}$ , and $\partial_t B_{\mathrm{int}} = -\nabla \Bigl( a \sum_{n \ge 1} \sum_{m=0}^{n} \Bigl(\frac{a}{r}\Bigr)^{n+1} \bigl(\dot g_n^m \cos m\phi + \dot h_n^m \sin m\phi\bigr) P_n^m(\cos\theta) \Bigr).$

Remarks

Example: in the axial dipole part, $B_r$ at the north geographic pole equals $2 g_1^0$ . So a coefficient $\dot g_1^0 = 10$ nT per year changes it by $20$ nT per year. Since $g_1^0 < 0$ at present, a positive $\dot g_1^0$ means a weakening dipole. Non-example: the daily variation and magnetic storms are changes of the external field with periods of hours to days, so they are not secular variation. Reversals and excursions are changes over thousands of years and more, and are usually treated separately, although secular variation is part of the same core process. Abrupt changes in the second time derivative of the field, called geomagnetic jerks, interrupt otherwise smooth secular variation every few years to decades.

Reading path

To understand this page from the ground up, read these 63 pages in order. The foundations come first.

Associated Legendre polynomials concept
Cross product concept
Euclidean space object
Function concept
Infimum concept
Laplace's equation concept
Magnetic field concept
Maxwell's equations concept
Parametrised surface concept
Spherical harmonics concept
Supremum concept
Fréchet derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Jordan measure definition
Surface area concept
Tangent plane and normal vector of a surface concept
Differentiable function of several variables concept
Iterated integral definition
Limit of a function of several variables concept
Partial derivative concept
Parametrised curve concept
Vector field concept
Orientable surface concept
Surface integral concept
Gradient concept
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Divergence concept
Line integral of a vector field concept
Flux through a surface concept
Scalar potential concept
Double integral definition
Differentiable on an open set definition
Gradient definition
Jacobian matrix definition
Higher partial derivatives definition
Biot–Savart law statement
Fubini's theorem statement
Triple integral definition
Iterated derivative definition
C^k function characterised by continuous derivatives statement
Parametrised curve definition
Vector field definition
Closed curve definition
Rectifiable curve definition
Parametrised surface definition
Divergence definition
Line integral of a vector field definition
Tangent plane and normal vector of a surface definition
Green's theorem statement
Surface area definition
Divergence theorem on a rectangle in the plane statement
Surface integral definition
Orientable surface definition
Flux through a surface definition
Divergence theorem statement
Gauss's law for magnetism statement
Earth's magnetic field definition
Gauss's spherical harmonic analysis of the geomagnetic field statement
Geomagnetic secular variation definition

Tag	`TE7MC2EXEC`
URI	`m2://TE7MC2EXEC`
Kind	definition
Writing	W0 stub
MSC	86A25
Defines	Geomagnetic secular variation
Uses	Earth's magnetic field, Gauss's spherical harmonic analysis of the geomagnetic field
Changed	3 November 2026

Geomagnetic secular variation

Definition

Remarks

Reading path

Further reading

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