# Is the Earth’s Magnetic Field a Constant? A Legacy of Poisson

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. On the Constancy of the Magnetic Field

#### 2.1. Some Consequences of Maxwell’s Equations

**H**) to the electric field (

**E**). c is the light velocity, and $\rho $ is the charged particle associated with the current density $\mathbf{j}$. It is important to note that, without knowing the action of

**EM**, one has access to important properties: the space component, which is actually the field H, is conserved (1b). By analogy to Euler’s (the continuity equation from fluid mechanics) equation, “magnetic charges” do not exist. Equation (1a) implies that as soon as

**H**varies with time, a field

**E**that is perpendicular (rot) and in quadrature with

**H**is created (instantly, hence the term $1/c$). The second pair of equations is fully symmetrical. Equation (1d) implies that there exists an "electric charge" ($\rho $) that locally deforms the field

**E**. In a vacuum, (1d) has exactly the same physical meaning as (1b). But one must add a current density to the time variation of

**E**to propagate the field

**H**(1c). A side note on (1c), it is also known as the Maxwell–Ampere equation. The classical understanding is that magnetic fields can be generated in two different ways, either by electrical currents (Ampere’s theorem), or by time changes of field

**E**, or the sum of both. The Lagrangian approach clarifies the picture. An EM field is defined by its 4-vector potential ${A}_{i}(=\phi ,\mathbf{A})$, where $\phi $ is the time component (called the scalar potential, linked to

**E**) and $\mathbf{A}$ is the space component (called the vector potential and linked to

**H**). Charges that move in the magnetic field must obey the same decomposition; one, therefore, introduces a 4-vector current density ${j}^{i}(=c\rho ,\mathbf{j})$, with a scalar charge density ($\rho $) found in (1d) and a vector current density ($\mathbf{j}$).

**B**and

**H**, which are only truly distinct in magnetized material, are measured by the same number.

#### 2.2. The Electrostatic Field

**E**derives from scalar potential ($\phi $):

**E**is a radial field. The absolute value of

**E**depends only on the distance R to e. Applying the divergence theorem to (2a),

**E**across a spherical surface with radius R centered on e is $4\pi {R}^{2}\mathbf{E}$ and also equals $4\pi e$ from Gauss’s theorem,

**k**constant, $\sum {e}_{i}=0$, then ${\mathbf{d}}^{{}^{\prime}}=\sum {e}_{i}{\mathbf{r}}_{i}^{{}^{\prime}}=\sum {e}_{i}{\mathbf{r}}_{i}+\mathbf{k}\sum {e}_{i}=\mathbf{d}$).

**n**is the unit vector oriented towards ${R}_{o}$. At large distances, the potential is inversely proportional to the square of distance and

**E**to its cube.

**E**is axially symmetrical about

**d**. In a plane where the direction of

**d**is that of the z axis, the Cartesian components of

**E**are

**E**, Coulomb’s law (2e) imposes itself, and the field is radial. In the case of a system of charges, if one is too close to the system, the interactions of the charges forbid one to use the Lagrangian concept of moment. Thus, one must remain far from the system. But as found by Legendre ([32]) and Laplace ([33]), when attempting to define the shape of the attraction field of masses, it is seen that the

**E**field involves the same constraints, i.e., is electrostatic. It is only because we are with a static field that the notion of the inverse of a distance takes its full meaning and that we can develop it into spherical harmonics.

#### 2.3. The Magnetostatic Field

**H**, created by charges in finite motion, remaining in a finite region of space (1b), whose impulses always retain finite values, has a stationary character that we wish to analyze further.

**A**being defined in a non-unequivocal way, one can impose an arbitrary condition, such as $\mathrm{div}\phantom{\rule{4pt}{0ex}}\mathbf{A}=0$. The previous line then becomes the Poisson equation:

**E**, one only has the effect of fixed charges or motion as a rigid block, whereas for

**H**, what counts is the uniform velocity of charges imposed by (1b). This is the reason why Poisson [27]’s title is Du magnétisme en mouvement (of magnetism in motion). And this is the main reason why one cannot write a physical description of a magnetic field as a series of spherical harmonics.

**n**is the unit vector along direction ${\mathbf{R}}_{o}$. If the ratio of mass to charge is the same for all charges in the system, then

**p**of the charge and

**H**. The (time-averaged) force exerted on the system (by the field

**H**) over time is 0. Indeed, according to the Lorentz force ([39]) $\mathcal{F}=\sum {\displaystyle \frac{e}{c}}\mathbf{v}\times \mathbf{H}={\displaystyle \frac{d}{dt}}\sum {\displaystyle \frac{e}{c}}\mathbf{r}\times \mathbf{H}$, $\mathcal{F}$ is the temporal derivative (taken between two finite times) of a quantity involving

**H**. It is known as Maxwell’s 5th equation. On the other hand, the time average of the moment of forces is $\mathcal{K}=\sum {\displaystyle \frac{e}{c}}\mathbf{r}\times (\mathbf{v}\times \mathbf{H})$which is not 0. Writing explicitly the double vector product,

**E**with central symmetry, due to a motionless particle. Let us shift from a motionless system of coordinates to a system undergoing a uniform rotation about an axis passing through the motionless particle. The velocity

**v**of the particle in the new system is linked to its velocity

**v**in the old system by ${\mathbf{v}}^{{}^{\prime}}=\mathbf{v}+\Omega \times \mathbf{r}$, where $\mathbf{r}$ is the particle’s vector radius and $\Omega $ the angular velocity of the rotating coordinate system. In the fixed system, the Lagrangian of charges is

**H**such that one can neglect terms in ${\mathbf{H}}^{2}$, the Lagrangian takes the form

**E**with central symmetry and a weak and uniform field

**H**is equivalent to the behavior of the same charge system in field

**E**with respect to a uniformly rotating coordinate system with angular velocity $\Omega $. This is Larmor’s theorem (cf. [40], chapter VI, [41]).

## 3. Some Further Remarks on Section 2

**E**or

**H**field implies wave propagation, hence physics described by Helmholtz equations, not Legendre. Let us describe what happens with a variable field in the equations of Section 2.

- The position term ${r}_{i}^{l}$ of each moving particle fluctuates with time so that the Legendre–Laplace condition (5a) is not satisfied any more, that is, the inverse distance $\frac{1}{|{\mathbf{R}}_{o}-\mathbf{r}|}$ is no more a natural solution of the Laplacian. One would need to introduce time but then the Laplacian would have to be replaced by a Dalembertian, i.e., a different problem.
- It is the number and/or quality of the charges that would change with time. But then the nature of the core would change with time, and one would need to find a physical mechanism that would explain how the field intensity could decrease (as is the case at present), yet could have increased and even reversed in the past.

**H**= 0. What enters a volume element and what leaves it is constant: the field is stationary.

## 4. Reconciling Modern Observations with Poisson’s Theory

#### 4.1. On the Drift of the Magnetic Dipole

#### 4.2. On the Forced Quasi-Cycles of the Magnetic Field

**SSA**in order to extract the annual and semi-annual components from the same three time series (sea level at Brest, magnetic field at CLF and length of day; this sub-section and Figure 4a–c).

**SSA**components of X, Y and Z at the five magnetic observatories. These figures illustrate the different modulation (“wave”) patterns associated with annual and semi-annual forcings. There is an ongoing debate as to their origins (e.g., [70,71,72,73,74]).

**CLF**(Figure 9a), Simons Bay-HER (Figure 9b) and Newlyn-HAD (Figure 9c) couples are in phase opposition with the sea level; the two other field components Y and Z are in phase with the sea level (with a small phase drift over the 40 years of the record). These three couples happen to be located on the same magnetic meridian. The same holds for the Newcastle-

**CNB**couple (Figure 9e), with a slightly larger phase drift.

**CNB**(Figure 9e), X is in phase opposition, Y in phase and Z in quadrature in 1980 and the three drift respectively to opposition, quadrature and opposition in 2020. We note that in Hartland, Z does not have a semi-annual component (Figure 9c), and it is quite small in Hermanus (Figure 9b). We tested tens of potential tide gauge/magnetic observatory couples, whose results are not good enough to be reported; we just note that some 80% of them have no semi-annual Z.

**SSA**determined) trends. We are in a position to strengthen our physical understanding of this link.

**SSA**annual components of the sea-level and magnetic components and their ratios for the five couples of stations of Figure 7. In the 1980–2022 period, these ratios are the same at 7 mm/nT for CLF, HAD and HER, almost the same at 8 mm/nT forCNB and not so different at 10 mm/nT for KNZ. Given the complexity of sea-level physics and geomagnetism, there is no a priori reason why the ratios should be constant, unless Equation (8a) holds, which seems to be the case. This result vindicates Poisson’s approach ([27]): fluid motions in the core are similar to those at the surface; because they are charged, the motifs of variations in the Earth’s magnetic field are those of the sea surface and the atmosphere.

#### 4.3. On the 11 yr Cycle and the Magnetic Field

**SSA**(Figure 10, black curve, bottom two rows). In the top row of Figure 10, we superimposed the 11 yr component from sunspots (pink) with that of aa. The two curves appear to be in quadrature: this is checked by offsetting the Schwabe cycle forward by exactly the 11/4 yr (second row). The explanation for this observation is the following: the torque exerted by Jupiter acts directly on sunspots, while the aa index is the difference between two antipodal observatories. Thus, the aa index is a derivative operator. This is likely why the 11 yr cycle is prominent in aa but minor in the X, Y and Z components. The same accounts for the phases of aa and lod: we integrate the 11 yr component of lod (black curve in 3rd row of Figure 10) and see it is in phase with aa. And finally, according to [36], Jupiter does act on the Earth’s rotation, as shown by Lopes et al. ([49]) and the Jupiter–Earth distance (blue curve bottom row in Figure 10). This last result provides a good illustration of Equation (8f):

#### 4.4. On the International Geomagnetic Reference Field

**IGRF**) is published every five years ([78]). Given the fact that there is no magnetic monopole, the first source term (i.e., “Gauss coefficient”) is the axial dipole ${g}_{1,0}$, an imaginary source at the center of the Earth. The other terms of the Fourier expansion on the base functions $\mathrm{cos}$ and $\mathrm{sin}$ are written as ${g}_{l,m}$ and ${h}_{l,m}$.

**IGRF**axial dipole ${g}_{1,0}$ since 1900. With Poisson ([27]) and Le Mouël’s ([20]) hypothesis in mind, and given some of the results in the previous sections of this paper (similar behavior of the annual

**SSA**components of sea level, rotation axis and magnetic field), it is natural to compare the behavior of the intensity of the

**IGRF**dipole with polar motion (${m}_{1}$,${m}_{2}$) or the equivalent parameter $\theta $ of [33]. This is done in Figure 11a.

**H**. Since $\Omega $ is connected to ${m}_{1}$ and ${m}_{2}$ through the Liouville–Euler equations,

**H**is also connected to ${m}_{1}$ and ${m}_{2}$. But the key coordinate is ${\dot{m}}_{3}$, which is the length of day. This is the reason why the second derivation of ${g}_{1,0}$ agrees better with the Markowitz–Stoyko drift, which is, as we saw, the second derivative of lod.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Polar Motion

**Figure A2.**Polar motion from IERS. The ${m}_{1}$ and ${m}_{2}$ components of polar motion from IERS (from 1846 to the present). From Lopes et al. [67].

## References

- Peregrinus, P. De Magnete, seu Rota Perpetui Motus, Libellus nunc Primum Promulgatus; Translated in English by Brother Arnold, M. Sc “The letters of Petrus Peregrinus, on the Magnet, A.D. 1269”; McGraw-Hill: New York, NY, USA, 1917. [Google Scholar]
- Courtillot, V.; Le Mouël, J.L. The study of Earth’s magnetism (1269–1950): A foundation by Peregrinus and subsequent development of geomagnetism and paleomagnetism. Rev. Geophys.
**2007**, 45. [Google Scholar] [CrossRef] [Green Version] - Gilbert, W. De Magnete; Courier Corporation: Washington, DC, USA, 1600. [Google Scholar]
- Mayaud, P.N. The aa indices: A 100-year series characterizing the magnetic activity. J. Geophys. Res.
**1972**, 77, 6870–6874. [Google Scholar] [CrossRef] - Courtillot, V.; Le Mouel, J.; Mayaud, P. Maximum entropy spectral analysis of the geomagnetic activity index aa over a 107-year interval. J. Geophys. Res.
**1977**, 82, 2641–2649. [Google Scholar] [CrossRef] - Le Mouël, J.; Lopes, F.; Courtillot, V. Singular spectral analysis of the aa and Dst geomagnetic indices. J. Geophys. Res. Space Phys.
**2019**, 124, 6403–6417. [Google Scholar] [CrossRef] - Ducruix, J.; Courtillot, V.; Le Mouël, J.L. The late 1960s secular variation impulse, the eleven year magnetic variation and the electrical conductivity of the deep mantle. Geophys. J. Int.
**1980**, 61, 73–94. [Google Scholar] [CrossRef] [Green Version] - Le Mouel, J.L.; Courtillot, V. Core motions, electromagnetic core-mantle coupling and variations in the Earth’s rotation: New constraints from geomagnetic secular variation impulses. Phys. Earth Planet. Inter.
**1981**, 24, 236–241. [Google Scholar] [CrossRef] - Courtillot, V.; Le Mouël, J. Geomagnetic secular variation impulses. Nature
**1984**, 311, 709–716. [Google Scholar] [CrossRef] - Bartels, J. Terrestrial-magnetic activity and its relations to solar phenomena. Terr. Magn. Atmos. Electr.
**1932**, 37, 1–52. [Google Scholar] [CrossRef] [Green Version] - Bartels, J.; Heck, N.; Johnston, H. The three-hour-range index measuring geomagnetic activity. Terr. Magn. Atmos. Electr.
**1939**, 44, 411–454. [Google Scholar] [CrossRef] - Chapman, S.; Bartels, J. Geomagnetism, vol. II: Analysis of the Data, and Physical Theories; Oxford Univ. Press: London, UK, 1940. [Google Scholar]
- Mayaud, P.N. Indices Kn, Ks et Km: 1964–1967; Editions du Centre National de la Recherche Scientifique: Paris, France, 1968. [Google Scholar]
- Mayaud, P. Une mesure planétaire d’activité magnétique basée sur deux observatoires antipodaux. Ann. Geophys
**1971**, 27, 67–70. [Google Scholar] - Mayaud, P. The annual and daily variations of the Dst index. Geophys. J. Int.
**1978**, 55, 193–201. [Google Scholar] [CrossRef] [Green Version] - Mayaud, P. Derivation, Meaning and Use of Geomagnetic Indices; American Geophysical Union: Washington, DC, USA, 1980; Volume 22. [Google Scholar]
- Le Mouël, J.L.; Blanter, E.; Chulliat, A.; Shnirman, M. On the semiannual and annual variations of geomagnetic activity and components. In Proceedings of the Ann Geophys; Copernicus Publications: Göttingen, Germany, 2004; Volume 22, pp. 3583–3588. [Google Scholar]
- Le Mouël, J.; Lopes, F.; Courtillot, V. Solar turbulence from sunspot records. Mon. Not. R. Astron. Soc.
**2020**, 492, 1416–1420. [Google Scholar] [CrossRef] - Kolmogorov, A.N. The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Ann. Sov. Acad.
**1941**, 30, 301–303. [Google Scholar] - Le Mouël, J. Outer-core geostrophic flow and secular variation of Earth’s geomagnetic field. Nature
**1984**, 311, 734–735. [Google Scholar] [CrossRef] - Jault, D.; Gire, C.; Le Mouël, J. Westward drift, core motions and exchanges of angular momentum between core and mantle. Nature
**1988**, 333, 353–356. [Google Scholar] [CrossRef] - Jault, D.; Le Mouël, J.L. The topographic torque associated with a tangentially geostrophic motion at the core surface and inferences on the flow inside the core. Geophys. Astrophys. Fluid Dyn.
**1989**, 48, 273–295. [Google Scholar] [CrossRef] - Jault, D.; Le Mouël, J.L. Core-mantle boundary shape: Constraints inferred from the pressure torque acting between the core and the mantle. Geophys. J. Int.
**1990**, 101, 233–241. [Google Scholar] [CrossRef] [Green Version] - Jault, D.; Le Mouël, J. Exchange of angular momentum between the core and the mantle. J. Geomag. Geoelec.
**1991**, 43, 111–129. [Google Scholar] [CrossRef] - Landau, L.; Lifshitz, E. Fluid Mechanics, V. 6 of Course of Theoretical Physics, 2nd ed.; Pergamon Press: Oxford, UK; New York, NY, USA; Beijing, China; Frankfurt, Germany; San Paulo, Brazil; Sydney, Australia; Tokyo, Japan; Toronto, ON, Canada, 1987. [Google Scholar]
- Proudman, J. On the dynamical theory of tides. Part II. Flat seas. Proc. London Math. 2nd Ser.
**1916**, 18, 21–50. [Google Scholar] - Poisson, S.D. Mémoire Sur la Théorie du Magnétisme en Mouvement; L’Académie des Sciences: Paris, France, 1826. [Google Scholar]
- Gauss, C.F.; Weber, W.E. Resultate aus den Beobachtungen des Magnetischen Vereins: im Jahre 1836; im Verlage der Dieterischen Buchhandlung: Göttingen, Germany, 1837; Volume 2. [Google Scholar]
- Poisson, S.D. Solution D’un Problème Relatif au Magnétisme Terrestre; L’Académie des Sciences: Paris, France, 1825. [Google Scholar]
- Leprêtre, B.; Le Mouël, J.L. Sur la mesure de l’intensité du champ magnétique terrestre et la distribution du magnétisme dans les aimants. CR Geosci.
**2005**, 337, 1384–1391. [Google Scholar] [CrossRef] - Gauss, C.F. Intensitas vis Magneticae Terrestris ad Mensuram Absolutam Revocata; Reprinted in: Werke, tome 5, Göttingen, Allemagne, 1877; Springer: Berlin/Heidelberg, Germany, 1832. [Google Scholar]
- Legendre, A.M. Recherches Sur L’attraction des Spheroides Homogenes; Bibliothèque Nationale de France: Paris, France, 1785. [Google Scholar]
- Laplace, P.S. Traité de Mécanique Céleste; de l’Imprimerie de Crapelet: Paris, France, 1799; Volume 1. [Google Scholar]
- Maupertuis, P.L.M. Essai de Cosmologie; de l’imprimerie d’Elie Luzac et fils: Paris, France, 1750. [Google Scholar]
- Lopes, F.; Courtillot, V.; Gibert, D.; Mouël, J.L.L.; Boulé, J.B. On power-law distributions of planetary rotations and revolutions as a function of aphelia, following Lagrange’s formulation. arXiv
**2023**, arXiv:2209.07213. [Google Scholar] [CrossRef] - Lagrange, J.L. Mécanique Analytique; Mallet-Bachelier: Paris, France, 1788. [Google Scholar]
- Landau, L.; Lifshitz, E. The Classical Theroy of Fields, V. 2 of Course of Theoretical Physics, 4th ed.; Pergamon Press: Oxford, UK; New York, NY, USA; Beijing, China; Frankfurt, Germany; San Paulo, Brazil; Sydney, Australia; Tokyo, Japan; Toronto, ON, Canada, 1994. [Google Scholar]
- Coulomb, J.; Jobert, G. Traité de Géophysique Interne; Sismologie et pesanteur; Masson: Paris, France, 1973; Volume 1. [Google Scholar]
- Lorentz, H.A. Simplified theory of electrical and optical phenomena in moving systems. K. Ned. Akad. Van Wet. Proc. Ser. B Phys. Sci.
**1898**, 1, 427–442. [Google Scholar] - Larmor, J. Aether and Matter: A Development of the Dynamical Relations of the Aether to Material Systems on the Basis of the Atomic Constitution of Matter, Including a Discussion of the Influence of the Earth’s Motion on Optical Phenomena, Being an Adams Prize Essay in the University of Cambridge; University Press: Cambridge, UK, 1900. [Google Scholar]
- Brillouin, L. A theorem of Larmor and its importance for electrons in magnetic fields. Phys. Rev.
**1945**, 67, 260. [Google Scholar] [CrossRef] - Heaviside, O. A gravitational and electromagnetic analogy. Electrician
**1893**, 31, 281–282. [Google Scholar] - Talwani, M.; Worzel, J.L.; Landisman, M. Rapid gravity computations for two-dimensional bodies with application to the Mendocino submarine fracture zone. J. Geophys. Res.
**1959**, 64, 49–59. [Google Scholar] [CrossRef] [Green Version] - Talwani, M. Computation with the help of a digital computer of magnetic anomalies caused by bodies of arbitrary shape. Geophysics
**1965**, 30, 797–817. [Google Scholar] [CrossRef] - Markowitz, W. Concurrent astronomical observations for studying continental drift, polar motion, and the rotation of the Earth. In Proceedings of the Symposium-International Astronomical Union; Cambridge University Press: Cambridge, UK, 1968; Volume 32, pp. 25–32. [Google Scholar]
- Stoyko, A. Mouvement seculaire du pole et la variation des latitudes des stations du SIL. In Proceedings of the Symposium-International Astronomical Union; Cambridge University Press: Cambridge, UK, 1968; Volume 32, pp. 52–56. [Google Scholar]
- Milanković, M. Théorie Mathématique des Phénomènes Thermiques Poduits Par la Radiation Solaire; Gauthier-Villars: Paris, France, 1920. [Google Scholar]
- Laskar, J.; Fienga, A.; Gastineau, M.; Manche, H. La2010: A new orbital solution for the long-term motion of the Earth. Astron. Astrophys.
**2011**, 532, A89. [Google Scholar] [CrossRef] [Green Version] - Lopes, F.; Le Mouël, J.; Courtillot, V.; Gibert, D. On the shoulders of Laplace. Phys. Earth Planet. Inter.
**2021**, 316, 106693. [Google Scholar] [CrossRef] - Bank, M.J.; Scafetta, N. Scaling, mirror symmetries and musical consonances among the distances of the planets of the solar system. Front. Astron. Space Sci.
**2022**, 8, 758184. [Google Scholar] [CrossRef] - Lopes, F.; Courtillot, V.; Gibert, D.; Le Mouël, J.L. Extending the Range of Milankovic Cycles and Resulting Global Temperature Variations to Shorter Periods (1–100 Year Range). Geosciences
**2022**, 12, 448. [Google Scholar] [CrossRef] - Mörth, H.; Schlamminger, L. Planetary motion, sunspots and climate. In Solar-Terrestrial Influences on Weather and Climate: Proceedings of a Symposium/Workshop Held at the Fawcett Center for Tomorrow, The Ohio State University, Columbus, OH, USA, 24–28 August 1978; Springer: Dordrecht, The Netherlands, 1979; pp. 193–207. [Google Scholar]
- Scafetta, N. Does the Sun work as a nuclear fusion amplifier of planetary tidal forcing? A proposal for a physical mechanism based on the mass-luminosity relation. J. Atmos. Sol. Terr. Phys.
**2012**, 81, 27–40. [Google Scholar] [CrossRef] [Green Version] - Stefani, F.; Giesecke, A.; Weber, N.; Weier, T. Synchronized helicity oscillations: A link between planetary tides and the solar cycle? Sol. Phys.
**2016**, 291, 2197–2212. [Google Scholar] [CrossRef] [Green Version] - Courtillot, V.; Lopes, F.; Le Mouël, J. On the prediction of solar cycles. Sol. Phys.
**2021**, 296, 1–23. [Google Scholar] [CrossRef] - Lambeck, K. The Earth’s Variable Rotation: Geophysical Causes and Consequences; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Poincaré, H. Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Bull. Astron. Obs. Paris
**1885**, 2, 109–118. [Google Scholar] - Courtillot, V.; Le Mouël, J.L.; Lopes, F.; Gibert, D. On Sea-Level Change in Coastal Areas. J. Mar. Sci.
**2022**, 10, 1871. [Google Scholar] [CrossRef] - Lopes, F.; Courtillot, V.; Le Mouël, J.L. Triskeles and Symmetries of Mean Global Sea-Level Pressure. Atmosphere
**2022**, 13, 1354. [Google Scholar] [CrossRef] - Alexandrescu, M.; Courtillot, V.; Le Mouël, J.L. Geomagnetic field direction in Paris since the mid-sixteenth century. Phys. Earth Planet. Inter.
**1996**, 98, 321–360. [Google Scholar] [CrossRef] - Alexandrescu, M.; Courtillot, V.; Le Mouël, J.L. High-resolution secular variation of the geomagnetic field in western Europe over the last 4 centuries: Comparison and integration of historical data from Paris and London. J. Geophys. Res. Solid Earth
**1997**, 102, 20245–20258. [Google Scholar] [CrossRef] - Golyandina, N.; Zhigljavsky, A. Singular Spectrum Analysis with R; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Lemmerling, P.; Van Huffel, S. Analysis of the structured total least squares problem for Hankel/Toeplitz matrices. Num. Algorithms
**2001**, 27, 89–114. [Google Scholar] [CrossRef] - Golub, G.H.; Reinsch, C. Singular value decomposition and least squares solutions. Linear Algebra
**1971**, 2, 134–151. [Google Scholar] - Le Mouël, J.L.; Lopes, F.; Courtillot, V. Sea-Level Change at the Brest (France) Tide Gauge and the Markowitz Component of Earth’s Rotation. J. Coast. Res.
**2021**, 37, 683–690. [Google Scholar] [CrossRef] - Lopes, F.; Le Mouël, J.L.; Gibert, D. The mantle rotation pole position. A solar component. CR Geosci.
**2017**, 349, 159–164. [Google Scholar] [CrossRef] - Lopes, F.; Courtillot, V.; Gibert, D.; Mouël, J.L.L. On two formulations of polar motion and identification of its sources. Geosciences
**2022**, 12, 398. [Google Scholar] [CrossRef] - Stephenson, F.R.; Morrison, L.V. Long-term changes in the rotation of the Earth: 700 BC to AD 1980. Philos. Trans. R. Soc. A
**1984**, 313, 47–70. [Google Scholar] - Gross, R.S. A combined length-of-day series spanning 1832–1997: LUNAR97. Phys. Earth Planet. Inter.
**2001**, 123, 65–76. [Google Scholar] [CrossRef] - Stewart, B. Terrestrial Magnetism; Encyclopedia Brittanica: Chicago, IL, USA, 1882. [Google Scholar]
- Stewart, B. On the cause of the solar-diurnal variations of terrestrial magnetism. Philos. Trans. R. Soc. A
**1886**, 8, 38. [Google Scholar] - Russell, C.; McPherron, R. Semiannual variation of geomagnetic activity. J. Geophys. Res.
**1973**, 78, 92–108. [Google Scholar] [CrossRef] - Lockwood, M.; Owens, M.J.; Barnard, L.A.; Haines, C.; Scott, C.J.; McWilliams, K.A.; Coxon, J.C. Semi-annual, annual and Universal Time variations in the magnetosphere and in geomagnetic activity: 1. Geomagnetic data. J. Space Weather. Space Clim.
**2020**, 10, 23. [Google Scholar] [CrossRef] - Lockwood, M.; Haines, C.; Barnard, L.A.; Owens, M.J.; Scott, C.J.; Chambodut, A.; McWilliams, K.A. Semi-annual, annual and Universal Time variations in the magnetosphere and in geomagnetic activity: 4. Polar Cap motions and origins of the Universal Time effect. J. Space Weather. Space Clim.
**2021**, 11, 15. [Google Scholar] [CrossRef] - Schwabe, H. Sonnenbeobachtungen im jahre 1843. von herrn hofrath schwabe in dessau. Astron. Nachrichten
**1844**, 21, 233. [Google Scholar] - Courtillot, V.; Le Mouel, J.L. Time variations of the Earth’s magnetic field: From daily to secular. Annu. Rev. Earth Planet Sci.
**1988**, 16, 389–476. [Google Scholar] [CrossRef] - Le Mouël, J.; Lopes, F.; Courtillot, V.; Gibert, D. On forcings of length of day changes: From 9-day to 18.6-year oscillations. Phys. Earth Planet. Inter.
**2019**, 292, 1–11. [Google Scholar] [CrossRef] - Alken, P.; Thébault, E.; Beggan, C.D.; Amit, H.; Aubert, J.; Baerenzung, J.; Bondar, T.; Brown, W.; Califf, S.; Chambodut, A.; et al. International geomagnetic reference field: The thirteenth generation. Earth Planets Space
**2021**, 73, 1–25. [Google Scholar] [CrossRef]

**Figure 1.**Spherical harmonics (from left to right and top to bottom) ${\mathcal{Y}}_{1,0}$, ${\mathcal{Y}}_{2,0}$, ${\mathcal{Y}}_{3,0}$ and ${\mathcal{Y}}_{4,0}$.

**Figure 2.**Eigenvector ${\mathcal{Y}}_{1,0}$ associated with Gauss coefficient ${g}_{1,0}$: to the left, the dipole is axial.

**Figure 3.**Various comparisons between the trends of magnetic declination in Paris, mean sea level in Brest, and polar motion and lod. (

**a**) Superposition of SSA trends of (1) the Markowitz–Stoyko drift since 1846 (gray curve) (2) of the magnetic declination D in Paris since 1781 (red curve) and (3) of the mean sea level from Brest tide gauge since 1807 (blue curve). (

**b**) Superposition of the first time derivatives of the three trends in Figure 3a. (

**c**) On the top: superposition of the smoothed first time derivative of magnetic declination D in Paris since 1835 (red curve) and the smoothed length of day (blue curve) from Stephenson and Morisson ([68]) since 1835. On the bottom: the latter curve is offset by 60 yr.

**Figure 4.**Sums of the annual and semi-annual components of (

**a**) lod (

**b**) Brest tide gauge and (

**c**) the X magnetic field at CLF since 1962.

**Figure 5.**Enlargement and superposition of the annual and semi-annual components of the X magnetic component at CLF (red), lod (gray curve) and sea level at the Brest tide gauge (blue curve). (

**a**) Length of day versus X component of the magnetic field in Chambon-La-Forêt (

**b**) Sea level recorded by the Brest tide gauge versus X component of the magnetic field in Chambon-La-Forêt.

**Figure 6.**Comparison of annual plus semi-annual components of all three geomagnetic components at CLF with those of lod. (

**a**) Comparison of annual plus semi-annual components of all three geomagnetic components X, Y and Z at CLF (X red, Y green, Z blue) with those of lod (gray) 1980–1990). (

**b**) Annual plus semi-annual components of geomagnetic components X, Y and Z at CLF (X red, Y green, Z blue) and those of lod (1962–2022).

**Figure 7.**Associated couples of a magnetic observatory (red diamond) and a tide gauge (blue circles). See Table 1.

**Figure 8.**Time evolution of annual plus semi-annual components extracted from the 5 observatories listed in Table 1. (

**a**) Forced quasi-cycles associated with the X magnetic component at (from left to right and top to bottom) Chambon-La-Forêt (

**CLF**), Hermanus (HER), Hartland (HAD), Kanozan (KNZ) and Canberra (

**CNB**). See Table 1 and Figure 7. (

**b**) Forced quasi-cycles associated with the Y magnetic component at (from left to right and top to bottom) Chambon-La-Forêt (

**CLF**), Hermanus (HER), Hartland (HAD), Kanozan (KNZ) and Canberra (

**CNB**). See Table 1 and Figure 7. (

**c**) Forced quasi-cycles associated with the Z magnetic component at (from left to right and top to bottom) Chambon-La-Forêt (

**CLF**), Hermanus (HER), Hartland (HAD), Kanozan (KNZ) and Canberra (

**CNB**). See Table 1 and Figure 7.

**Figure 9.**Comparison between forced components extracted from each observatory/tide gauge couple. (

**a**) Comparison between forced components of sea level in Brest (grey curve) and respective oscillations of magnetic components X (red curve), Y (green curves) and Z (blue curve) in Chambon-La-Forêt. (

**b**) Same as Figure 9a for the tide gauge/magnetic observatory couple Simons Bay/Hermanus. (

**c**) Same as Figure 9a for the tide gauge/magnetic observatory couple Newlyn/Hartland. (

**d**) Same as Figure 9a for the tide gauge/magnetic observatory couple Mera/Kanozan. (

**e**) Same as Figure 9a for the tide gauge/magnetic observatory couple Newcastle/Canberra.

**Figure 10.**Eleven year quasi-cycles extracted by SSA from the geomagnetic index aa (red; top 3 rows), the sunspot series (pink; top 2 rows), the length of day (black; bottom 2 rows) and the ephemerids of Jupiter (blue; bottom row marked as distance of Earth from Jupiter).

**Figure 11.**Comparison between (

**a**) the evolution of ${g}_{1,0}$ from ([78]) and the Markowitz–Stoyko drift and (

**b**,

**c**) their corresponding time derivatives. (

**a**) Red dots =

**IGRF**${g}_{1}^{0}$ every 5 years since 1900 [78] and interpolation as the black curve. Blue curve =

**SSA**first component i.e., trend of polar motion, that is the Markowitz–Stoyko drift. (

**b**) First time derivatives of the two curves shown in Figure 11a. (

**c**) Second derivative in time of the curve corresponding to the coefficient ${g}_{1}^{0}$ versus the first derivative of the polar motion. The two curves are in phase possibly due to causality.

Magnetic Observatory | Tide Gauge |
---|---|

Chambon-La-Forêt (CLF, 2.26${}^{\circ}$ E, 48.02${}^{\circ}$ N) | Brest (4.49${}^{\circ}$ W, 48.38${}^{\circ}$ N) |

Hartland (HAD, 4.48${}^{\circ}$ W, 51${}^{\circ}$ N) | Newlyn (5.54${}^{\circ}$ W, 50.10${}^{\circ}$ N) |

Canberra (CNB, 149.36${}^{\circ}$ E, 35.32${}^{\circ}$ S) | Newcaslte V (151.78${}^{\circ}$ E, 32.92${}^{\circ}$ S) |

Hermanus (HER, 19.23${}^{\circ}$ W, 34.43${}^{\circ}$ S) | Simons Bay (18.44${}^{\circ}$ E, 34.18${}^{\circ}$ S) |

Kanozan (KNZ, 139.95${}^{\circ}$ E, 35.25${}^{\circ}$ N) | Mera (139.82° E, 34.91${}^{\circ}$ N) |

Couple Observatory–Tide Gauge | Ratio Sea Level/Magnetic Component | Order of Magnitude |
---|---|---|

CLF/Brest, between 1980 and 2000 | ∼100 mm/15 nT | ∼7 mm/nT |

HAD/Newlyn, between 1980 and 2005 | ∼100 mm/15 nT | ∼7 mm/nT |

CNB/Newcaste V, between 1980 and 2022 | ∼ 80 mm/10 nT | ∼8 mm/nT |

HER/Simons bay, between 1980 and 2000 | ∼100 mm/14 nT | ∼7 mm/nT |

KNZ/Mera, between 1980 and 1995 | ∼140 mm/14 nT | ∼10 mm/nT |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Le Mouël, J.-L.; Lopes, F.; Courtillot, V.; Gibert, D.; Boulé, J.-B.
Is the Earth’s Magnetic Field a Constant? A Legacy of Poisson. *Geosciences* **2023**, *13*, 202.
https://doi.org/10.3390/geosciences13070202

**AMA Style**

Le Mouël J-L, Lopes F, Courtillot V, Gibert D, Boulé J-B.
Is the Earth’s Magnetic Field a Constant? A Legacy of Poisson. *Geosciences*. 2023; 13(7):202.
https://doi.org/10.3390/geosciences13070202

**Chicago/Turabian Style**

Le Mouël, Jean-Louis, Fernando Lopes, Vincent Courtillot, Dominique Gibert, and Jean-Baptiste Boulé.
2023. "Is the Earth’s Magnetic Field a Constant? A Legacy of Poisson" *Geosciences* 13, no. 7: 202.
https://doi.org/10.3390/geosciences13070202