Is the Earth’s Magnetic Field a Constant?

The birth of geomagnetism as a science can be dated to 8 August 1269 on that day, Petrus Peregrinus wrote three letters .Courtillot and Le Mouël during the siege of the city of Lucera in the Italian region of Puglie. The letters can be considered the first scientific article on geomagnetism, 331 years before the famous De Magnete by Gilbert . Expressed in modern terms, Peregrinus wrote that Earth has a magnetic field with a dipolar structure and that some rocks or minerals are magnetized. One could carve a sphere out of magnetite, pierce a hole through its center, and it would oscillate around the northward direction, a thought experiment that announced the compass. In the three centuries that unfurled since then, a lot has been discovered: the variation of inclination with latitude, the daily, annual and other periodic variations ، irregular variations, such as events linked to solar activity, the secular variation and its sudden jerks . In order to handle an ever increasing data base, magnetic indices were introduced It was found that quasi periodical variations of the magnetic field , and also sun spots followed a power law with exponent −5/3. In a series of papers that started with Le Mouël in 1984 and continued with Jault et al. , and Jault and Le Mouël , these authors found that the trends of (1) magnetic secular variation, (2) polar motion and (3) length of day (lod) are strongly correlated. They proposed to explain these observations with a coupling mechanism, in which flow in a cylinder tangent to the core and the rotation axis exchange torques at the core–mantle boundary. The solution for flow on the cylinder is the same as that generated by an internal gravitational wave in a rotating fluid also known as Proudman flow. But this mechanism encounters serious difficulties with the orders of magnitude of physical parameters, such as the core–mantle boundary (CMB) topography. Also, the torque exerted by the fluid pressure and the electromagnetic torque are too weak to validate the model as concluded by Jault and Le Mouël .Actually, the model of a cylinder tangent to the core is very close in spirit to that envisioned by Poisson . In the report he submitted to the Académie des Sciences, prior to an oral communication, Poisson imagined a set of concentric spheres in place of a cylinder. Poisson was the first scientist to describe the magnetic field as a series of spherical harmonics, a decade before Gauss did Part 5, chapter 1, Allgemeine Theorie des Erdmagnetismus). He also invented a technique to measure the absolute value of the horizontal component of the magnetic field , seven years before Gauss did.

It may come as a surprise to many that Poisson as well as Gauss both considered the magnetic field to be constant. This was an axiomatic basis for the development into spherical harmonics. There are very clear statements to this effect in the writings of both scientists. In the work of Poisson , page 49, one reads (our translation from the French): “We will assume that the hollow sphere be magnetized under the influence of a force that be the same in magnitude and direction for all its points, such as the magnetic action of Earth, for instance”. And in page 54: “Since time does not enter these formulae, a consequence is that, after the first instants of rotation, that we did not mention, the action of the rotating sphere on a given point will be constant in magnitude and in direction”. Gauss , part 5, chapter 1, paragraph 2, page 6, writes (our translation from the German): “. . . magnetism consists only in galvanic currents (that is, constant currents) that persist in the smallest parts of the bodies . . .”. Gauss develops his theory very quickly without any physical proof. And his mathematical proof is exactly that found by Legendre or Laplace for the gravitational field. In contrast, the 130 pages of Poisson’s memoir are devoted both to the full physical and mathematical proofs of the magnetic field description.

Given that Poisson’s work precedes and is more complete than that of Gauss , it is only fair to recognize that the former was the first to develop the magnetic field in spherical harmonics and to state that this magnetic field was constant..

In his pioneering work on the nature and origin of the Earth’s magnetic field, Poisson recognized that the field had to be constant. Gauss came to the same conclusion. This will seem awkward to the modern physicist. Poisson’s proof involved theoretical physics and mathematics but only a limited set of observations.

More recently, one of us Le Mouël observed that there were strong observational connections between the Earth’s magnetic field and its rotation, more precisely, the secular variation of the field and the drift of the rotation pole on one hand, and the forced oscillations of the magnetic field and the length of day on the other. Based on these observational facts, Le Mouël proposed to model the core source as a rotating cylinder when Poisson envisioned a sphere. Thus, based on observations, Le Mouël (1984) (with collaborators, in a suite of often cited papers) built quasi “experimentally”, 150 years later, almost the same theory as Poisson had done “theoretically”.

Starting from Maxwell’s equations, we derived the equations for the electrostatic and magnetostatic fields . We came to the same equations as Poisson, with an important difference: Poisson chose the Lagrangian approach to gravity (Legendre, 1785; Lagrange 1788; Laplace, 1799) to formally derive the equations for the magnetic field . To scientists of this epoch, there was no difficulty in reasoning the magnetism in the same “classical” way as one reasons in gravity. In Section 2, we propose a short tutorial on the equations of electromagnetism. This can be found in most graduate textbooks. But we emphasize the fact that the magnetic field does not need to derive from a scalar potential to be developed into spherical harmonics. However, most geomagnetists do make this hypothesis, invoking Stokes’s theorem: since magnetic measurements are made at the surface where almost no charge circulates, one can assume that the field derives from a scalar, not a vector potential. It is more physical, hence logical, to obtain the spherical harmonics from the electrostatic field, then use the vector potential to return to their expression for the magnetic field. One can perform a decomposition in spherical harmonics and consider multi-poles if and only if the motions of charges are finite and uniform, two conditions that are not met in a dynamic field. From this, one can draw a number of consequences for the magnetic field, the main one being that the magnetic moment of the charges that generate the field and their angular moment (thus, the motion of the rotation pole) are linked by Larmor’s relation. This is in agreement with the theoretical works of Laplace Poisson and . A magnetic field can be written on the basis of spherical harmonics only if this field is constant.