# Is the Earth’s magnetic field a constant ? a legacy of Poisson

J-L. Le Mouël  Université Paris Cité, Institut de Physique du globe de Paris,
CNRS UMR 7154, F-75005 Paris, France  F. Lopes<EMAIL_ADDRESS>Université Paris
Cité, Institut de Physique du globe de Paris, CNRS UMR 7154, F-75005 Paris,
France  V. Courtillot  Université Paris Cité, Institut de Physique du globe de
Paris, CNRS UMR 7154, F-75005 Paris, France  D. Giber  LGL-TPE - Laboratoire
de Géologie de Lyon - Terre, Planètes, Environnement, Lyon, France  J-B. Boulé
CNRS UMR7196, INSERM U1154, Museum National d’Histoire Naturelle, Paris,
F-75005, France

(August 28, 2024)

###### Abstract

In the report he submitted to the Académie des Sciences, Poisson imagined a
set of concentric spheres at the origin of the Earth’s magnetic field. It may
come as a surprise to many that Poisson as well as Gauss both considered the
magnetic field to be constant. We propose in this study to test this
surprising assertion for the first time evoked by Poisson (1826). First, we
will present a development of Maxwell’s equations in the framework of a static
electric field and a static magnetic field in order to draw the necessary
consequences for the Poisson hypothesis. In a second step, we will see if the
observations can be in agreement with Poisson (1826). To do so, we have chosen
to compare 1) the polar motion drift and the secular variation of the Earth’s
magnetic field, 2) the seasonal pseudo-cycles of day length together with
those of the sea level recorded by different tide gauges around the globe and
those of the Earth’s magnetic field recorded in different magnetic
observatories. We then propose a mechanism, in the spirit of Poisson, to
explain the presence of the 11-year in the magnetic field. We test this
mechanism with observations and finally we study closely the evolution of the
g10 coefficient of the IGFR over time.

###### Contents

  1. 1  Introduction
  2. 2 On the constancy of the magnetic field
    1. 2.1 Some consequences of Maxwell’s equations
    2. 2.2 The electrostatic field
    3. 2.3 The magnetostatic field
  3. 3 Some further remarks on section 2
  4. 4 Reconciling modern observations with Poisson’s theory
    1. 4.1 On the drift of the magnetic dipole
    2. 4.2 On the forced quasi-cycles of the magnetic field
    3. 4.3 On the 11-yr cycle and the magnetic field
    4. 4.4 On the international geomagnetic reference field
  5. 5 Conclusion

##  1  Introduction

The birth of geomagnetism as a science can be dated as August 8, 1269: on that
day Petrus Peregrinus wrote three letters ([1]; see f.i. Courtillot and Le
Mouël [2]) during the siege of the city of Lucera in the Italian region of
Puglie. The letters can be considered as the first scientific article on
geomagnetism, 331 years before the famous De Magnete by Gilbert ([3]).
Expressed in modern terms, Peregrinus wrote that the Earth had a magnetic
field with a dipolar structure and that some rocks or minerals were
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 ([4, 5, 6]), irregular variations such as events linked to
solar activity, the secular variation and its sudden jerks ([7, 8, 9]). In
order to handle an ever increasing data base, magnetic indices (eg aa, Dst,
Kp, etc …) were introduced (eg [10, 11, 12, 13, 14, 4, 15, 16, 17, 6]). It was
found that quasi periodical variations of the magnetic field ([4, 5, 6]), and
also sun spots ([18]) followed a [19] power law with exponent -5/3.

In a series of papers that started with Le Mouël in 1984 ([20]) and continued
with Jault et al. ([21]), and Jault and Le Mouël ([22, 23, 24]), these authors
found that the trends of (1) magnetic secular variation, (2) polar motion and
(3) length of day were 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 (see [24], system 6) is the same as that
generated by an internal gravitational wave in a rotating fluid (eg [25]) also
known as Proudman ([26]) flow. But this mechanism encountered serious
difficulties with the orders of magnitude of physical parameters such as CMB
topography. Also, the torque exerted by the fluid pressure and the
electromagnetic torque were too weak to validate the model, as concluded by
Jault and Le Mouël ([24]).

Actually, the model of a cylinder tangent to the core is very close in spirit
to that envisioned by Poisson ([27]). 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 ([27]) was the first
scientist to describe the magnetic field as a series of spherical harmonics, a
decade before Gauss did ([28]), 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 ([29, 30]), seven years
before Gauss ([31]) did.

It may come as a surprise to many that Poisson ([27]) as well as Gauss ([28]
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 Poisson ([27]) 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 ([28]), part 5, chapter 1,
paragraph 2, page 6, Gauss writes (our translation from the German): ”$\ldots$
magnetism consists only in galvanic currents (that is constant currents) that
persist in the smallest parts of the bodies $\ldots$”. Gauss develops his
theory very quickly (pages 18 to 23, paragraphs 14 to 27), without any
physical proof. And his mathematical proof is exactly that found in Legendre
([32]) or Laplace([33]) for the gravitational field. In contrast, the 130
pages of Poisson ([27]) memoir are devoted both to the full physical and
mathematical proofs of the magnetic field description.

Given that Poisson ([27]) work precedes and is more complete than that of
Gauss ([28]), 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.

We do not propose to follow, as Poisson ([27]) did, the description of a
magnetic field based on Maupertuis ([34]) principle of least action, but
rather to pursue our previous presentation of the laws of gravitation Lopes et
al. ([35]) following Lagrange ([36]) and extend it to the case (and
consequences) of the constancy of the electric and magnetic fields, giving
their full physical meaning to the moments. This is the aim of section 2. In
section 3 we try to explain the variations that one can measure within the
frame of Poisson ([27]) paradigm. We confront the previous theoretical
developments with modern observations in section 4 and conclude in section 5.

##  2 On the constancy of the magnetic field

###  2.1 Some consequences of Maxwell’s equations

One can learn a lot from a Lagrangian approach to the derivation of Maxwell’s
equations. Following Maupertuis ([34]), one only needs to know the action of a
moving charged particle in an electromagnetic (EM) field, associated with the
action of its interaction with that field, to derive the first pair of
equations :

$\displaystyle\textrm{rot}\textbf{E}=-\dfrac{1}{c}\dfrac{\partial\textbf{H}}{\partial
t}$ (1a) $\displaystyle\textrm{div}\textbf{H}=0$ (1b) Adding the action of
field EM, one obtains the second pair:

$\displaystyle\textrm{rot}\textbf{H}=\dfrac{1}{c}\dfrac{\partial\textbf{E}}{\partial
t}+\dfrac{4\pi}{c}\textbf{j}$ (1c)
$\displaystyle\textrm{div}\textbf{E}=4\pi\rho$ (1d)

Equations (1a) to (1d) link in a symmetrical way the magnetic field (H) to the
electric field (E). $c$ is the velocity of the charged particle $\rho$
associated to current density 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 (continuity111The 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)
also known as 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 $A_{i}(=\varphi,\textbf{A})$, where $\varphi$ is the
time component (called the scalar potential, linked to E) and A the space
component (called the vector potential and linked to H). Charges that move in
the field must obey the same decomposition ; one therefore introduces a
4-vector current density $j^{i}(=c\rho,\textbf{j})$, with a scalar charge
density ($\rho$) found in (1d) and a vector current density (j).

The EM field described by the Maxwell equations must be in one of the three
following forms: electrostatic, magnetostatic or a propagating field (wave
propagation) that will not be discussed further in this paper.

###  2.2 The electrostatic field

Equations (1a) and (1d) reduce to:

$\displaystyle\textrm{div}\textbf{E}=4\pi\rho$ (2a)
$\displaystyle\textrm{rot}\textbf{E}=0$ (2b)

E derives from scalar potential ($\varphi$):

$\textbf{E}=-\textrm{grad}\ \varphi$

leading to the Poisson equation:

$\textrm{div}(\textrm{grad})\ \varphi=\Delta\varphi=-4\pi\rho$ (2c)

In a vacuum ($\rho=0$), the scalar potential verifies the Laplace equation:

$\Delta\ \varphi=0$

The field produced by a point charge ($e$) will be directed along the vector
having the charge as one of its extremities. E is a radial field. The absolute
value of E depends only on the distance $R$ to $e$. Applying the divergence
theorem to (2a):

$\textrm{div}\textbf{E}\longmapsto\iiint_{V}\textrm{div}\textbf{E}\
dV=\iint_{S}\textbf{E}.d\textbf{S}$

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

$\oint\textbf{E}d\textbf{f}=4\pi\int edV$.

Finally, in vector form:

$\textbf{E}=\dfrac{e\textbf{R}}{R^{3}}$ (2d)

Thus, the field produced by a point charge is inversely proportional to the
square of the distance to $e$ (Coulomb’s law). The potential associated to
this field is:

$\varphi=\dfrac{e}{\textbf{R}}$ (2e)

and for a system of charges:

$\varphi=\sum_{i}\dfrac{e_{i}}{\textbf{R}_{i}}$ (2f)

Let us observe this field (2f) at a distance that is large compared to the
charge system’s dimension, that is far enough so that their relative motions
can be considered as constant (in the sense of Lagrangian mechanics). This
allows one to use the concept of ”moment”.

Let us choose a coordinate system whose origin lies within the charge system
and let $r_{i}$ be their respective vector radii. The total observed potential
at point $R_{o}$ is:

$\varphi=\sum\dfrac{e_{a}}{|\textbf{R}_{o}-\textbf{r}_{i}|}$ (3a)

For $R_{o}\gg r_{i}$ and thanks to the generalized Maclaurin expansion given
by [32], we can develop (3a) in a series of powers of $\dfrac{r_{i}}{R_{o}}$,
using the first order formula:

$f(\textbf{R}_{o}-\textbf{r})\approx f(\textbf{R}_{o})-\textbf{r}\
\textrm{grad}f(\textbf{R}_{o})$

thus (3a) becomes:

$\varphi=\dfrac{\sum e_{i}}{\textbf{R}_{o}}-\sum
e_{i}\textbf{r}_{i}\textrm{grad}\dfrac{1}{\textbf{R}_{o}}$ (3b)

The sum $\textbf{d}=\sum e_{i}\textbf{r}_{i}$ is the dipolar moment of the
charge system. By analogy with a system of masses, this dipolar moment is the
mathematical equivalent to the tensor of moments of inertia of order 2 in
geodesy. If the sum of charges is zero, the dipolar moment does not depend on
the choice of the origin ($\textbf{r}^{‘}=\textbf{r}+\textbf{k}$), $k$
constant, $\sum e_{i}=0$, then $\textbf{d}^{{}^{\prime}}=\sum
e_{i}\textbf{r}_{i}^{{}^{\prime}}=\sum e_{i}\textbf{r}_{i}+\textbf{k}\sum
e_{i}=\textbf{d}$) .

The potential at large distances can be written:

$\varphi=-\textbf{d}\nabla\dfrac{1}{\textbf{R}_{o}}=\dfrac{\textbf{d}\textbf{R}_{o}}{R_{o}^{3}}$
(3c)

and:

$\begin{split}\textbf{E}&=-\textrm{grad}\varphi\quad(=(\textbf{d}\nabla)\nabla\dfrac{1}{\textbf{R}_{o}})\\\
&=-\textrm{grad}\dfrac{\textbf{d}\textbf{R}_{o}}{R_{o}^{3}}=-\dfrac{1}{R_{o}^{3}}\textrm{grad}(\textbf{d}\textbf{R}_{o})-(\textbf{d}\textbf{R}_{o})\textrm{grad}\dfrac{1}{R_{o}^{3}}\\\
&=-\dfrac{\textbf{d}}{R_{o}^{3}}\nabla\textbf{R}_{o}-\dfrac{\textbf{R}_{o}}{R_{o}^{3}}\nabla\textbf{d}-\textbf{dR}_{o}\nabla\dfrac{1}{R_{o}^{3}}\\\
&=-\dfrac{\textbf{d}}{R_{o}^{3}}-0+\dfrac{3\textbf{dR}_{o}}{R_{o}^{3}}\\\
&=\dfrac{3(\textbf{nd})\textbf{n}-\textbf{d}}{R_{o}^{3}}\end{split}$

where 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:

$E_{z}=d.\ \dfrac{3\textrm{cos}^{2}\theta-1}{R_{o}^{3}},\quad E_{z}=d.\
\dfrac{3\textrm{sin}\theta\textrm{cos}\theta}{R_{o}^{3}}$ (3d)

and the radial and tangential components are:

$E_{r}=d.\ \dfrac{2\textrm{cos}\theta}{R_{o}^{3}},\quad E_{\theta}=-d.\
\dfrac{\textrm{sin}\theta}{R_{o}^{3}}$ (3e)

$\theta$ being the angle between $z$ and $\textbf{R}_{o}$. We note that
equations (3d) and (3e) are the same as those of the components of a dipolar
magnetic field with d being replaced by $\dfrac{\mu_{0}}{4\pi}$(eg Le Mouël in
[37], chapter 26, page 40, system 24).

As done by [33], one can always develop the scalar potential $\varphi$ as a
sum of contributions in ascending powers of $\dfrac{1}{\textbf{R}_{0}}$, the
term $\varphi^{(n)}$ being proportional to $\dfrac{1}{\textbf{R}_{o}^{n+1}}$,

$\varphi=\varphi^{(0)}+\varphi^{(1)}+\varphi^{(2)}+\ldots$ (4a)

The first term $\varphi^{(0)}$ is determined by the sum of all charges or
masses for Laplace ([33]), for whom this term can never be zero. As we have
just seen, when the sum of electric charges is zero, one is led to the
electrostatic components (3d) and (3e). The second term, $\varphi^{(1)}$ is
the dipolar one, determined by its dipolar moment d. One can continue the
development in a Legendre ([32]) series. The next term would be:

$\varphi^{(2)}=\dfrac{1}{2}\sum ex_{i}x_{j}\dfrac{\partial^{2}}{\partial
X_{i}\partial X_{j}}\dfrac{1}{R_{o}}$ (4b)

where the $x$ coordinates are the components of r and $X$ those of
$\textbf{R}_{o}$. We note that:

$\Delta\dfrac{1}{R_{o}}\equiv\delta_{ij}\dfrac{\partial^{2}}{\partial
X_{i}\partial X_{j}}\dfrac{1}{R_{o}}=0$

One can then write (4b) as:

$\varphi^{(2)}=\dfrac{1}{2}\sum
e(x_{i}x_{j}-\dfrac{1}{3}r^{2}\delta_{ij})\dfrac{\partial^{2}}{\partial
X_{i}\partial X_{j}}\dfrac{1}{R_{o}}$

The tensor $D_{ij}=\sum e(3x_{i}x_{j}–r^{2}\delta_{ij})$ is the quadrupolar
moment of the charge system. Thus:

$\varphi^{(2)}=\dfrac{D_{ij}}{6}\dfrac{\partial^{2}}{\partial X_{i}\partial
X_{j}}\dfrac{1}{R_{o}}$ (4c)

or, as done for the dipolar term in (3c):

$\dfrac{\partial^{2}}{\partial X_{i}\partial
X_{j}}\dfrac{1}{R_{o}}=\dfrac{X_{i}X_{j}}{R_{o}^{5}}-\dfrac{\delta_{ij}}{R_{o}^{3}}$

and since $\delta_{ij}D_{ij}=D_{ii}=0$, we have:

$\phi^{(2)}=\dfrac{D_{ij}n_{i}n_{j}}{2R_{o}^{3}}$ (4d)

$n_{i}$ and $n_{j}$ are the unit vectors starting from $\textbf{R}_{o}$ and
oriented along the two axes of the quadrupolar moment. The eigenvalues of the
tensor are such that $D_{ii}=0$ and are therefore linked by:

$D_{xx}=D_{yy}=-\dfrac{1}{2}D_{zz}$

If we write $D$ for component $D_{zz}$, the quadrupolar potential becomes,

$\varphi^{(2)}=\dfrac{D}{4\pi
R_{o}^{3}}(3\textrm{cos}^{2}\theta-1)=\dfrac{D}{2R_{o}^{3}}\mathcal{P}_{2}(\textrm{cos}\
\theta)$ (4e)

in which one introduces the $\mathcal{P}_{2}$ Legendre ([32]) polynomial. One
can generalize this construction; the l-order term is determined by a tensor
of order l, the 2l-polar moment.

The mathematical development above is independent of the physical problem
(gravity, magnetism, electricity) one is interested in. It serves to
illustrate the fundamental result from Legendre ([32]) regarding the theory of
gravitational attraction of masses as generalized by Laplace ([33]): as far as
one observes from far enough the sources, the inverse of distance
($\dfrac{1}{\textbf{R}_{o}-\textbf{r}}$) is given by:

$\dfrac{1}{|\textbf{R}_{o}-\textbf{r}|}=\dfrac{1}{\sqrt{\textbf{R}_{o}^{2}+\textbf{r}^{2}-2\textbf{rR}_{o}\textrm{cos}\
\chi}}=\sum_{l=0}^{\infty}\dfrac{\textbf{r}^{l}}{\textbf{R}_{o}^{l+1}}\mathcal{P}_{l}(\textrm{cos}\
\chi)$ (5a)

Let us introduce the pairs of spherical angles $\Theta$, $\Phi$ and $\theta$ ,
$\varphi$ formed respectively by vectors and with the given coordinate axes,
and apply the addition theorem of spherical functions:

$\mathcal{P}_{l}(\textrm{cos}\
\chi)=\sum_{m=-l}^{l}\dfrac{(l-|m|)!}{(l+|m|)!}\mathcal{P}_{l}^{|m|}(\textrm{cos}\
\Theta)\mathcal{P}_{l}^{|m|}(\textrm{cos}\ \theta)e^{-im(\Phi-\varphi)}$ (5b)

were $\mathcal{P}_{l}^{m}$ are associated Legendre polynomials. Let us also
introduce the spherical functions:

$\mathcal{Y}_{lm}(\theta,\varphi)=(-1)^{m}i^{i}\sqrt{\dfrac{(2l+1)(l-m)!}{4\pi(l+m)!}}\mathcal{P}_{l}^{m}(\textrm{cos}\
\theta)e^{im\varphi},\quad m\geq 0.$ (5c)

Integrating (5b) and (5c) in (5a), one finally obtains the expression for the
inverse of a distance on the sphere:

$\dfrac{1}{|\textbf{R}_{o}-\textbf{r}|}=\sum_{l=0}^{\infty}\sum_{m=-1}^{l}\dfrac{r^{l}}{R_{o}^{l+1}}\dfrac{4\pi}{2l+1}\mathcal{Y}^{*}_{lm}(\Theta,\Phi)\mathcal{Y}_{lm}(\theta,\varphi)$
(5d)

Developing each term in equation (3a), one obtains the expression for the term
of order l of the potential:

$\varphi^{(l)}=\dfrac{1}{R_{o}^{l+1}}\sum_{m=-l}^{l}\sqrt{\dfrac{4\pi}{2l+1}}\mathcal{Q}_{m}^{(l)}\mathcal{Y}^{*}_{(lm)}(\Theta,\Phi)$
(5e)

where the $2l+1$ quantities $\mathcal{Q}_{m}^{l}$ constitute the 2l-polar
moment of the system of charges, defined by:

$\mathcal{Q}_{m}^{l}=\sum_{i}e_{i}r^{l}_{i}\sqrt{\dfrac{4\pi}{2l+1}}\mathcal{Y}_{lm}(\theta_{i},\varphi_{i})$
(5f)

At this point, let us underline why we have taken the trouble to recall this
(at least in large part) classical derivation, which is likely taught in all
graduate and even undergraduate physics programs. In this section, we have
seen that in the case of a static field 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, ie 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.

We can now undertake the same analysis in the case of the magnetic field.

###  2.3 The magnetostatic field

As we have seen above, Maxwell’s equations (1b) and (1c) imply that the
magnetic 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.

The two equations are now:

$\displaystyle\textrm{div}\ \textbf{H}=0$ (6a) $\displaystyle\textrm{rot}\
\textbf{H}=\dfrac{4\pi}{c}\textbf{j}$ (6b)

The vector potential A associated to H is defined by $\textrm{rot}\
\textbf{A}=\textbf{H}$. Carried into (6b) it becomes:

$\textrm{grad}\
\textrm{div}\textbf{A}-\Delta\textbf{A}=\dfrac{4\pi}{c}\textbf{j}$

A being defined in a non unequivocal way, one can impose an arbitrary
condition, such as $\textrm{div}\ \textbf{A}=0$. The previous line then
becomes the Poisson equation:

$\Delta\textbf{A}=-\dfrac{4\pi}{c}\textbf{j}$ (6c)

(6c) is analogous to (2c), charge density $\rho$ being replaced by current
density $\dfrac{\textbf{j}}{c}$. By analogy with the electrical potential, we
can write:

$\textbf{A}=\dfrac{1}{c}\int\dfrac{\textbf{j}}{R}dV$ (6d)

where $R$ is the distance from observation point to volume element $dV$. The
integral in (6d) can be replaced by a sum and the current density by
$\rho\textbf{v}$. By analogy with the scalar potential (3a), the vector
potential becomes:

$\textbf{A}=\dfrac{1}{c}\sum\dfrac{e_{i}\textbf{v}_{i}}{R_{i}}$ (6e)

This is how charges are introduced in a vector linked to the magnetic field,
without risking the mistake of writing that the magnetic field derives from a
scalar potential.

As done above for the electrostatic field, one can calculate the effect of the
moving charges in a reference system with its origin within the charge
distribution, with the same notations for vectors $\textbf{r}_{i}$ and
distance $\textbf{R}_{o}$. (6e) becomes:

$\textbf{A}=\dfrac{1}{c}\sum\dfrac{e_{i}\textbf{v}_{i}}{|\textbf{R}_{o}-\textbf{r}_{i}|}$
(7a)

The main difference between scalar and vector potentials is that for 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.

However, some remarkable consequences can be derived. For instance, one can
always write in a development as a Legendre ([32]) series analogous to (3b),
to first order:

$\textbf{A}=\dfrac{1}{cR_{o}}\sum e\textbf{v}-\dfrac{1}{c}\sum
e\textbf{v}(\textbf{r}\nabla\dfrac{1}{R_{o}})$

One can write $\sum e{\textbf{v}}=\dfrac{d}{dt}\sum e\textbf{r}$, but the mean
value of the derivative varying in a finite interval is 0.

$\textbf{A}=-\dfrac{1}{c}\sum
e\textbf{v}(\textbf{r}\nabla\dfrac{1}{R_{o}})=\dfrac{1}{cR_{o}^{3}}\sum
e\textbf{v}(\textbf{rR}_{o})$ (7b)

Note that $\textbf{v}=\dot{\textbf{r}}$, and since $\textbf{R}_{o}$ is a
constant vector,

$\sum e(\textbf{R}_{o}\textbf{r})\textbf{v}=\dfrac{1}{2}\dfrac{d}{dt}\sum
e\textbf{r}(\textbf{R}_{o}\textbf{r})+\dfrac{1}{2}\sum
e[\textbf{v}(\textbf{r}\textbf{R}_{o})-\textbf{r}(\textbf{v}\textbf{R}_{o})]$

Carrying this expression in (7b), the mean value of the first term is again 0,
thus:

$\textbf{A}=\dfrac{1}{2cR_{o}^{3}}\sum
e[\textbf{v}(\textbf{rR}_{o})-\textbf{r}(\textbf{v}\textbf{R}_{o})]$ (7c)

One recognizes a vector product in (7c). Let us introduce the magnetic moment
$\mathfrak{m}=\dfrac{1}{2c}\sum e\textbf{r}\times\textbf{v}$. Equation (7c)
becomes:

$\textbf{A}=\nabla\dfrac{1}{R_{o}}\times\mathfrak{m}=\dfrac{\mathfrak{m}\times\textbf{R}_{o}}{R_{o}^{3}}$
(7d)

Whereas $\dfrac{1}{R_{o}}$ verifies Laplace’s equation, its modification by
the rotational of $\mathfrak{m}=\dfrac{1}{2c}\sum e\textbf{r}\times\textbf{v}$
does not. With the expression for the vector potential (7d), one can derive
the magnetic field. Given the formula $\textrm{rot}\ a\times b=(b\nabla)\
a-(a\nabla)\ b+a\ \textrm{div}\ b-b\ \textrm{div}\ a$, one finds:

$\textbf{H}=\textrm{rot}\
\mathfrak{m}\times\dfrac{\textbf{R}_{o}}{R_{o}^{3}}=\mathfrak{m}\
\textrm{div}\dfrac{\textbf{R}_{o}}{R_{o}^{3}}-(\mathfrak{m}\nabla)\dfrac{\textbf{R}_{o}}{R_{o}^{3}}$

Since with $\textbf{R}_{o}\neq 0$:

$\textrm{div}\dfrac{\textbf{R}_{o}}{R_{o}^{3}}=\textbf{R}_{o}\textrm{grad}\dfrac{1}{R_{o}^{3}}+\dfrac{1}{R_{o}^{3}}\textrm{div}\textbf{R}_{o}=0,$

and:

$(\mathfrak{m}\nabla)\dfrac{\textbf{R}_{o}}{R_{o}^{3}}=\dfrac{1}{R_{o}^{3}}(\mathfrak{m}\nabla)\textbf{R}_{o}+\textbf{R}_{o}(\mathfrak{m}\nabla\dfrac{1}{R_{o}^{3}})=\dfrac{\mathfrak{m}}{R_{o}^{3}}-\dfrac{3\textbf{R}_{o}(\mathfrak{m}\textbf{R}_{o})}{R_{o}^{5}},$

then:

$\textbf{H}=\dfrac{3\textbf{n}(\mathfrak{m}\textbf{n})-\mathfrak{m}}{R_{o}^{3}},$
(7e)

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

$\mathfrak{m}=\dfrac{1}{2c}\sum e\textbf{r}\times\textbf{v}=\dfrac{e}{2mc}\sum
m\textbf{r}\times\textbf{v}$

Finally, if all velocities of all charges are such that $v\ll c$, then
$m\textbf{v}$ is the impulsion p of the charge and:

$\mathfrak{m}=\dfrac{e}{2mc}\sum\textbf{r}\times\textbf{p}=\dfrac{e}{2mc}\mathcal{M}$
(8a)

where $\mathcal{M}=\sum\textbf{r}\times\textbf{p}$ is the kinetic moment of
the system. Here the ratio of magnetic to mechanical moment is a constant.

Let us now consider a system of charges placed in a constant, external
magnetic field H. The (time averaged) force exerted on the system is the
Lorentz force. Following Lorentz ([38]), it is known as Maxwell’s 5ft equation
and given by
$\mathcal{F}=\sum\dfrac{e}{c}\textbf{v}\times\textbf{H}=\dfrac{d}{dt}\sum\dfrac{e}{c}\textbf{r}\times\textbf{H}$,
that is zero. On the other hand, the time average of the moment of forces is
$\mathcal{K}=\sum\dfrac{e}{c}\textbf{r}\times(\textbf{v}\times\textbf{H})$
which is not 0. Writing explicitly the double vector product,

$\mathcal{K}=\sum\dfrac{e}{c}\\{\textbf{v}(\textbf{rH})-\textbf{H}(\textbf{vr})\\}=\sum\dfrac{e}{c}\\{\textbf{v}(\textbf{rH})-\dfrac{1}{2}\textbf{H}\dfrac{d}{dt}\textbf{r}^{2}\\}$

one obtains simply:

$\mathcal{K}=\mathfrak{m}\times\textbf{H}$

The Lagrangian of a system of charges placed in an external, constant and
uniform magnetic field is:

$\mathcal{L}_{H}=\sum\dfrac{e}{c}\textbf{Av}=\sum\dfrac{e}{2c}(\textbf{H}\times\textbf{r})\textbf{v}=\sum\dfrac{e}{2c}(\textbf{r}\times\textbf{v})\textbf{H}$
(8b)

Introducing the magnetic moment (8b) becomes simply,

$\mathcal{L}_{H}=\mathfrak{m}\textbf{H}$ (8c)

By analogy, in a uniform electric field, the Lagrangian of a system with 0
total charge and a dipolar moment includes the term:

$\mathcal{L}_{E}=\textbf{dE}$ (8d)

Let us consider a system of charges undergoing a finite motion in a field 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
$\textbf{v}^{{}^{\prime}}=\textbf{v}+\Omega\times\textbf{r}$ where 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:

$\mathcal{L}=\sum\dfrac{mv^{{}^{\prime}2}}{2}-\mathcal{U},$

where $\mathcal{U}$ is the potential energy of charges in the external field
E. In the new system the Lagrangian becomes:

$\mathcal{L}=\sum\dfrac{m}{2}(\textbf{v}+\Omega\times\textbf{r})^{2}-\mathcal{U}.$

If the ratio $\dfrac{e}{m}$ of charge to mass is the same for all particles,
and if one writes:

$\Omega=\dfrac{e}{2mc}\textbf{H}$

then, for sufficiently small values of H so that one can neglect terms in
$\textbf{H}^{2}$, the Lagrangian takes the form:

$\mathcal{L}=\sum\dfrac{mv^{2}}{2}+\dfrac{1}{2c}\sum
e(\textbf{H}\times\textbf{r})\textbf{v}-\mathcal{U}$ (8e)

It is remarkable that the two Lagrangians in (8b) and (8c) are the same. In
summary, the Lagrangian of charges in finite motion in an electric field
produced by a motionless rotating particle (8c) is the same as the Lagrangian
of a system of charges placed in a constant and uniform magnetic field (8b).
In slightly different, more readable terms, the behavior of a system of
charges with the same $\dfrac{e}{m}$ ratio executing a finite motion in a
field 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 [39], chapter VI, [40]).

Let us now evaluate the variation of the mean kinetic moment $\mathcal{M}$.
The variation of the mean kinetic moment $\mathcal{M}$ is equal to the moment
$\mathcal{K}$ of forces applied to the system. Thus
$\mathcal{K}=\mathfrak{m}\times\textbf{H}=\dfrac{d\mathcal{M}}{dt}$. If the
ratio $\dfrac{e}{m}$ is the same for all the system’s particles, then
$\mathcal{M}$ is proportional to $\mathfrak{m}$ (cf 8a), and:

$\dfrac{d\mathcal{M}}{dt}=-\Omega\times\mathcal{M}.$ (8f)

Vector $\mathcal{M}$, and thus vector $\mathfrak{m}$, both rotate with angular
velocity $-\Omega$ about the field direction; its absolute value and the angle
it makes with respect to the field direction are constant.

##  3 Some further remarks on section 2

Most if not all modern studies of the geomagnetic field have it varying in
time, keeping its first order dipolar configuration and ‘ready’ to be analyzed
with spherical harmonics. But a time variable 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 only link between the field geometry and the physics of the problem, that
is the e charges and their positions in the reference system, is (5f) that we
recall here:

$\mathcal{Q}_{m}^{l}=\sum_{i}e_{i}r^{l}_{i}\sqrt{\dfrac{4\pi}{2l+1}}\mathcal{Y}_{lm}(\theta_{i},\varphi_{i})$

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
$\mathcal{Y}_{4,0}$.

One needs to visualize the shapes of spherical harmonics, that are
“physically” represented by the coefficients $\mathcal{Y}_{lm}$ . Figure 1
displays the first four axial multipoles, that is the eigenvectors
$\mathcal{Y}_{1,0}$, $\mathcal{Y}_{2,0}$, $\mathcal{Y}_{3,0}$ and
$\mathcal{Y}_{4,0}$ (from left to right and top to bottom) corresponding to
the so-called Gauss coefficients $g_{1,0}$, $g_{2,0}$, $g_{3,0}$ and
$g_{4,0}$. The eigenvectors are a constant of the problem, as long as this
representation retains a physical meaning. For example, a dipolar magnetic
field must always have its principal axis aligned with the axis going from the
origin to the geographic North pole. In this paradigm, a variable field would
involve only modifications of the space-time physics of term $e^{i}r^{l}_{i}$
in the sum (5f). But this constraint cannot be satisfied because either:

  1. 1.

the position term $r_{i}^{l}$ of each moving particle fluctuates with time, so
that the Legendre-Laplace condition (05a) is not satisfied any more, that is
the inverse distance $\dfrac{1}{|\textbf{R}_{o}-\textbf{r}|}$ 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;

  2. 2.

or 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.

Poisson [27] reasoning was simpler. As early as page 7 of his memoir, above
his first equation, he linked gravitation and magnetism. So did Gauss ([28]),
and this was much developed by Heaviside ([41]). This was more than a simple
physical analogy.

Recall that modern geophysicists use Talwani’algorithms ([42, 43]) to
determine the mass (respectively dipolar) distributions based on
microgravimetric (resp. magnetic) measurements using the same equation.
Poisson ([27]) writes: ’The differential volume element, corresponding to
point M’ [The magnetic volume], will have $h^{3}d\chi d\xi d\eta$ as its
expression; we will write $\mu^{‘}h^{3}d\chi d\xi d\eta$ for the amount of
free fluid it contains, $\mu^{{}^{\prime}}$ being positive or negative
according to this fluid being boreal or austral. This coefficient will be a
function of $\chi,\xi,\eta$ depending on the distribution of the two fluids
inside the magnetic elements. If they are moving, $\mu^{{}^{\prime}}$ will
vary with time; but the total quantity of free fluid belonging to the same
element must always be zero, so we will always have:

$\int\mu^{‘}d\chi d\xi d\eta=0\qquad(I)$

The integral extending to the entire volume of the magnetic element’.

This short quotation is from the very beginning of Poisson ([27]) derivation;
in more modern terms div H = 0. What enters a volume element and what leaves
it is constant: the field is stationary.

From Lagrange ’s standpoint ([36]), Poisson ([27]) implies that at the origin
of the magnetic field is a rather undeformable object, a solid body consisting
of a set of electromagnetic point sources whose respective distances do not
vary, or that can be considered as such at the distance at which its effects
are observed. We note that the tensor of a quadrupolar electrical moment
$D_{ij}=\sum e(3x_{i}x_{j}–r^{2}\delta_{ij})$ is similar to the order 2
inertia moment tensor of a rotating solid body $I_{ik}=\sum
m(x_{i}^{2}\delta_{ik}–x_{i}x_{k})$.

One then concludes that the space-time variations of electric charges that
would not result in a constant field would make it impossible to construct
either an electric or a magnetic moment.

##  4 Reconciling modern observations with Poisson’s theory

###  4.1 On the drift of the magnetic dipole

In sections 2 and 3, we have shown that the only way a magnetic field can be
written as the sum of multipolar potentials is that (similar to masses
composing a solid rotating body) charges generating the electric and magnetic
fields should move uniformly in space and time. In other words the field must
be stationary. We also know that the field intensity fluctuates and that this
secular variation is morphologically similar to the drift of the rotation
pole, called the Markowitz (or Markowitz-Stoyko, [44, 45]) drift. The annual
oscillations of the field are morphologically similar to those of the length
of the day ([21, 22, 23, 24]).

One can reconcile Poisson ([27]) and observations by picturing a dipole that
oscillates about geographical North, this oscillation sharing the same
excitation (forcings) that act on polar motion. Let us first illustrate this
idea. In Figure 2 we show the eigenvector $\mathcal{Y}_{1,0}$ associated with
the Gauss coefficient $g_{1,0}$. CLF stands for the Chambon-La-Forêt magnetic
observatory. Point P is one of the elements of [27]’s magnetic volume
composing the dipole. As seen in section 2, the dipole action at CLF is
characterized by the distance P-CLF on the sphere. If the dipole is tilted
(Figure 2, right) this distance changes, and so do the coordinates of CLF in
the two dipole reference systems (Figure 2, left and right).

Following Lagrange ([36]), a number of authors (eg [46, 47, 48, 49, 50, 35])
have shown how and how much astronomical forces influence Earth’s rotation, in
the same way its own weight perturbs the rotation of a spinning top, through
an exchange of angular moments. The same astronomical forces perturb the
number of sunspots, ie solar activity (eg [51, 52, 53, 54]). Since [33], we
know that masses at the surfaces of planets can re-organize under the
influence of stresses acting on the rotation pole. The Liouville-Euler system
of linear differential equations of first order runs this re-organization (cf
[55], section 3 for more details). From sections 2 and 3, moving charges may
be considered as a moving fluid in rotation about the dipole’s symmetry
axis.Any large scale fluid motions must correspond to a block rotation about
the Earth’s rotation axis, as shown by Laplace and Poincaré. There are no
other possibilities of natural motion at first order. Inside as well as at its
surface the pattern of fluid motion must be the same (cf [33, 56]; see [57,
58], for some illustrations).

Figure 2: Eigen vector $\mathcal{Y}_{1,0}$ associated with Gauss coefficient
$g_{1,0}$: to the left the dipole is axial

Possibly the longest series of magnetic measurements is that of declination D
compiled for Paris (France) by Alexandrescu et al. ([59, 60]). The series
starts in 1541 but we select values from 1781 onward, when the sampling
becomes more regular. The Brest sea level data start in 1807. The polar motion
data are defined as the modulus of the horizontal displacement of the pole in
a plane tangential to the Earth ($m_{1}+i\ m_{2}$); it starts in 1846 and
therefore sets the length of our analyzes.

We have applied Singular Spectrum Analysis (SSA, eg [61]) in order to extract
the trend (this sub-section) and the annual and semi-annual components (next
subsection) from the three time series of sea-level at Brest, magnetic field
at CLF and length of day. SSA uses the mathematical properties of descending
order diagonal matrices (Hankel, Toeplitz matrices, eg [62]) and their
orthogonalization by singular value decomposition (SVD, eg [63]).

In Figure 3(a), we superimpose the SSA trends of declination in CLF in red,
sea level at Brest in blue and polar motion in black. We have dealt with the
sea-level series in Le Mouël et al. ([64]), in which we compared variations of
the sea-level trends with the variations of Markowitz-Stoyko polar drift. We
have studied many aspects of polar motion in [65, 48, 66]. The time
derivatives of the three trends are shown in Figure 3(b). We have also used
the long data set of Stepheson and Morrison ([67]) and Gross ([68]) for lod
and compare its trend with the first derivative of magnetic declination
(Figure 3(c)).

The trends of the three derivatives display very similar patterns. We have
shown before that this pattern is linked to the ephemerids of Uranus ([64,
48]). The similarity between the derivative of the trend of the rotation pole
and sea-level is expected: as the Earth shifts, its fluid envelope shifts as a
solid in the same way. The two are almost in phase. For the magnetic field,
there is a $\sim$20-year phase lag. It is in quadrature with polar motion
around 1930, and catches up in the 1960s. This vindicates the mechanism
advocated by Le Mouël et al. ([20, 21]) and Jault et al. ([22, 23, 24]), that
is a transfer of moment at the core-mantle boundary. The phase lag would be
due to the roughness of the CMB.

Laplace ([33]) has shown theoretically that lod and pole motion are linked by
a first order derivative operator. We have verified this with observations in
Lopes et al. ([66]). Figure 3(c) compares the (5 yr smoothed) mean value of
the derivative of declination in Paris to the corresponding (5 yr smoothed)
mean value of length of day. The two series of mean values of solid Earth
motions are in quadrature and show a 60 yr oscillation ([66], Figure 04). This
60 yr period is clearly present in the 2-humps pattern of lod as well as in
the derivative of declination (blue curve, Figure 3(c)). As shown in section
2, the mechanical and magnetic moments that are at the origin of the
geomagnetic field are linked linearly (equation 8c). On another hand, the
mechanical moment and the polar rotation axis are linked through a first order
time derivative (equation 8f). In the spirit of Larmor’s theorem, we assume
that the two phenomena (rotation and magnetism) are linked by the same
external forcing ; thus the former should be compared to the derivative of the
latter. In the lower part of Figure 3(c), the mean value of lod has been
offset by 60 years, leading to an almost perfect match of the patterns. We
have not (yet ?) been able to explain that offset.

(a) Superposition of SSA trends of 1) the Markowitz-Stoyko drift since 1846
(gray curve) 2) of the magnetic declination $\mathcal{D}$ 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
3(a)

(c) On the top: superposition of the smoothed first time derivative of
magnetic declination $\mathcal{D}$ in Paris since 1835 (red curve) and the
smoothed length of day (blue curve) from Stephenson and Garrison ([67]) since
1835. On the bottom: the latter curve has been offset by 60 yr.

Figure 3: Comparison between the trends variations of the magnetic declination
in Paris, of the mean sea level recorded in Brest, of the polar motion drift
and of the length of day.

One cannot envision that the magnetic field (declination) variations would be
due to intensity variations of components X, Y and Z since as seen in section
2 the field must be constant (following Poisson ). Figure 3(b) is but an
extension to sea-level of an observation made by Le Mouël [20].

###  4.2 On the forced quasi-cycles of the magnetic field

Next, we extend the observation by Jault et Le Mouël ([24]) of a link between
annual and semi-annual oscillations of lod and the magnetic field. We have
applied 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 Figures 4(a), 4(b), 4(c)).

Given their phase and amplitude modulation, the superimposition of the semi-
annual and annual components generates fringe patterns (Figure 4(a), 4(b),
4(c)). These are better visualized in the enlarged Figure 5(a), 5(b) showing
the 1980-1990 decade. In Figure 5(a) the two components generate a
characteristic pattern with two humps that correlate between magnetic field
and length of day, with constant phases. In Figure 5(b) one of the two humps
for sea-level is subdued and looks more like a shifting step. The double hump
pattern has already been recognized by [24], their Figure 1. Based on Figures
4 and 5, one can propose that the annual forcing of the double humps is driven
by polar motion/rotation. As explained by Poisson ([27]), a moving charged
fluid tends to replicate the pattern of the motion; the different processes
seem to be in phase and constant. If these originate from fluid motion
tangential to the core, there will be a transfer of moment according to the
Liouville-Euler equations. Thus, based on Figures 4 and 5, one can propose
that the annual forcing of the double humps is driven by polar
motion/rotation.

We next wish to check whether the same can be said of the other geomagnetic
field components at CLF. Figure 6(a), 6(b) compares the pattern of lod to
those of X, Y and Z: Y behaves as X but Z does not have the two humps, only
one strong maximum per year (ie no semi-annual component).

(a) Sum of the annual and semi-annual components of sea-level at the Brest
tide gauge since 1962, extracted by SSA.

(b) Sum of the annual and semi-annual components of the X magnetic component
at CLF since 1962, extracted by SSA.

(c) Sum of the annual and semi-annual components of lod since 1962, extracted
by SSA.

Figure 4: Trends (a) and (b) first time derivatives of trends of polar motion
drift (black curve), magnetic declination at CLF (red curve) and sea level at
Brest (blue curve).

(a) Enlargement and superposition of the annual and semi-annual components of
the X magnetic component at CLF (red) and length of day (gray).

(b) Enlargement and superposition of the annual and semi-annual components of
the X magnetic component at CLF (red) and sea-level at the Brest tide gauge
(blue).

Figure 5: Comparison on the top, between seasonal components of lod (gray
curve) and X component of the magnetic field recorded in CLF (red curve); on
the bottom same comparison with seasonal component extracted from the Brest
tide gauge (blue curve).

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

We have attempted to check whether the observations made with the couple
”magnetic observatory-tide gauge” at CLF and nearby Brest could be extended to
other couples. Unfortunately there are not many such couples, particularly
since all data sets should be of sufficient length and quality. We have found
five couples listed in Table 1 below and shown on a world map (Figure 7).

Magnetic observatory |  Tide gauge
---|---
Chambon-La-Forêt (CLF, 2.26∘E, 48.02∘N) |  Brest (4.49∘W, 48.38∘N)
Hartland (HAD, 4.48∘W, 51∘N) |  Newlyn (5.54∘W, 50.10∘N)
Canberra (CNB, 149.36∘E, 35.32∘S) |  Newcaslte V (151.78∘E, 32.92∘S)
Hermanus (HER, 19.23∘W, 34.43∘S) |  Simons Bay (18.44∘E, 34.18∘S)
Kanozan (KNZ, 139.95∘E, 35.25∘N) |  Mera (139.82°E, 34.91∘N)
Table 1: List of couple ”magnetic observatory-tide gauge” Figure 7: Associated
couples of a magnetic observatory (red diamond) and a tide gauge (blue
circles). See Table 1.

Figures 8(a) to 8(c) show the combined annual and semi-annual 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 (eg [69, 70, 71, 72,
73]).

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

Figures 9(a) to 10(b) allow one to compare the magnetic oscillations for X, Y
and Z with the corresponding sea level oscillation, one for each observatory
couple. Magnetic components X extracted from the Brest-CLF (Figure 9(a)),
Simons Bay-HER (Figure 9(b)) and Newlyn-HAD (Figure 9(c)) couples are in phase
opposition with sea-level; the two other field components Y and Z are in phase
with 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 Newcastlte-CNB couple (Figure 10(b)), with a slightly
larger phase drift.

(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 9(a) for the tide gauge/magnetic observatory couple Simons
Bay/Hermanus.

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

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

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

Figure 10: Comparison between forced components extracted from each couples
observatory/tide gauge according listed on the Table (1)

For the Japanese couple Mera/KNZ (Figure 10(a)), X is in phase with sea-level
in 1980, when Y and Z are in quadrature. After 40 years of slow drift, Z is in
phase, X and Y in quadrature. Finally, for the Australian couple Newcastle/CNB
(Figure 10(b)), 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 9(c)), and it is quite small in Hermanus (Figure 9(b)). We have 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.

The comparisons made in Figures 10 suggest strongly to us (though, granted,
they do not prove) a link between annual variations of sea-level and the
magnetic field. The correlations are of better quality than we might have
expected, despite differences in topography and geography in the vicinity of
the gauges. The same can be said of the (SSA determined) trends. We are in a
position to strengthen our physical understanding of this link.

In section 2 we have seen that if a measured magnetic field has its origin in
the uniform motion of charges in rotation about our planet’s symmetry axis,
the quantity and quality of charges cannot vary with time (or the dipole would
vanish). Then there is a link between the magnetic and mechanical moments. It
is expressed by equation (8a):

$\mathfrak{m}=\dfrac{e}{2mc}\sum\textbf{r}\times\textbf{p}=\dfrac{e}{2mc}\mathcal{M}$

Sea-level and magnetic variations are linked through polar motion (eg [20],
and section 2), here the length of day. Polar motion is forced by the Earth’s
revolution about the Sun. If the field is constant and the link exists, the
ratio $\dfrac{e}{2mc}$ must be constant. The (geo-)physical consequences
generated by these moments should be to first order the same for all torques
around the globe, and the variations in amplitude of the geophysical phenomena
involved should be proportional.

In Table 2 below, we evaluate the amplitudes of the SSA annual components of
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
7mm/nT for CLF, HAD and HER, almost the same at 8 mm/nT for CNB and not so
different at 10 mm/nT for KNZ. Given the complexity of sea-level physics and
geomagnetism, there was no a priori reason why the ratios should be constant,
unless equation (08) 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 sea-surface, and atmosphere.

couple observatory-tide gauge |  ratio sea level / magnetic component |  order of magnitude
---|---|---
CLF/Brest, between 1980 and 2000 |  $\sim$100 mm / 15 nT |  $\sim$7 mm/nT
HAD/Newlyn, between 1980 and 2005 |  $\sim$100 mm / 15 nT |  $\sim$7 mm/nT
CNB/Newcaste V, between 1980 and 2022 |  $\sim$ 80 mm / 10 nT |  $\sim$8 mm/nT
HER/Simons bay, between 1980 and 2000 |  $\sim$100 mm / 14 nT |  $\sim$7 mm/nT
KNZ/Mera, between 1980 and 1995 |  $\sim$140 mm / 14 nT |  $\sim$10 mm/nT
Table 2: List of couple ”magnetic observatory-tide gauge”

###  4.3 On the 11-yr cycle and the magnetic field

The 11-yr cycle is one of the better known variations of the Earth’s magnetic
field. It is considered as the same Schwabe ([74]) cycle that is found in
sunspot numbers. In the minds and logic of Laplace ([33]), Lagrange ([36]) and
Poisson ([27]), planets are responsible, through exchanges in moment, for a
number of astrophysical and geophysical phenomena. The origin of this 11-yr
cycle has likely its source in the revolution and moment of Jupiter. This
moment is directly connected to variations of distances to Jupiter in the
solar system. Jupiter exerts the largest torque of the 8 planets (and also
larger than the Sun’s, that is motionless at the time scales we are interested
in Lopes et al. [66]). This torque acts on (or modulates, or forces) sunspots
(eg [54]) as well as on polar rotation (eg [65, 48]).

In what we will call the Poisson \- Le Mouël ’s paradigm, the fluid’s rotation
generates the field; this same rotation must then force solar activity in the
form of sunspots. In the case of fluid mechanics, and as shown in Courtillot
and Le Mouël ([75]) figure 45 and in Le Mouël et al. ([6, 76]), a law of
natural turbulence appears, a [19] power law with exponent -5/3. This law is
found in sunspots as well as in variations of geomagnetic intensity (the
latter since as far back as at least 1 Myr, see [75]). The angular coordinate
$\theta$ and $\dfrac{d\psi}{dt}$, that define the location of the pole of
rotation on the sphere are the solutions of a system of differential
equations, one for $\theta$, that describes pole motion, the other for
$\dfrac{d\psi}{dt}$ that describes length of day. One is the derivative of the
other (see [50]). As is the case for all derivative operators, it amplifies
high frequency components. Le Mouël et al. ([77]) have shown that the 11-yr
component is a major component of length of day, whereas Lopes et al. ([65])
have shown that it was a minor one of polar motion.

Figure 11: 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).

In order to retain the homogeneity of our data sets and their temporal
resolution, we have integrated the 11-yr quasi-cycle extracted from length of
day by SSA (Figure 11, black curve, bottom two rows). In the top row of Figure
11, we have 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 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 11) 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. ([48]) and the Jupiter-Earth distance (blue curve bottom
row in Figure 11). This last result provides a good illustration of equation
(8d):

$\dfrac{d\mathcal{M}}{dt}=-\Omega\times\mathcal{M}.$

Polar motion as well as length of day are linked to $\Omega$ (eg Lambeck [55],
section 03). We have also seen in theory (and checked in the example above)
that the variation in the magnetic field is linked to moment $\mathcal{M}$. We
could say that magnetic field components (X,Y,Z) are to polar motion
($m_{1}$,$m_{2}$) what aa indices are to length of day. The link between
sections 2 and 3 above is the length of day.

###  4.4 On the international geomagnetic reference field

Since the work of Gauss ([28]), the decomposition of the geomagnetic field
into spherical harmonics has become normal practice (“routine”). We recall
however as a caveat the fact that an electric field or a gravity field can be
decomposed in SH because these fields are constant and their elementary
sources all have the same sign. Such is not the case with a magnetic field,
unless it is constant also (sections 2 and 3). In principle, the SH
decomposition of a magnetic field does not have physical significance.
Nevertheless a spherical harmonic decomposition of 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 (aka ”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
cos and sin are written as $g_{l,m}$ and $h_{l,m}$.

Figure 12(a) shows the monotonous decay of the IGRF axial dipole $g_{1,0}$
since 1900. With Poisson ’s ([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 12(a).

The time variations of $g_{1}^{0}$ and $\theta$ are indeed very close. The
first derivative of polar motion is in quadrature with the first derivative of
$g_{1}^{0}$ (Figure 12(b)) and in phase (opposition) with the second
derivative of $g_{1}^{0}$ (Figure 12(c)).The variation of dipole intensity
follows the variation of the angle $\theta$ between the rotation axis of the
planet and that of the dipole. Another example is given by the similarities
between magnetic declination in Paris and rotation axis, and between their
derivatives (Figures 3(a) and 3(b)).

If the field is constant, we have seen that there is a link between the
magnetic moment $\mathfrak{m}$ and the kinetic moment $\mathcal{M}$

$\mathfrak{m}=\dfrac{e}{2mc}\sum\textbf{r}\times\textbf{p}=\dfrac{e}{2mc}\mathcal{M}$
$\dfrac{d\mathcal{M}}{dt}=-\Omega\times\mathcal{M}$

We have checked this relation, using observations (Table 2) We have also shown
that, in the case of a rotating system of charges about the axis $Z$ (with
velocity $\Omega$), this relation becomes:

$\bm{\Omega}=\dfrac{e}{2mc}\textbf{H}$

This does express the link between the variations of the Earth’s rotation and
the magnetic field H. Since $\Omega$ is connected to $m_{1}$ and $m_{2}$
through the Liouville-Euler equations, then H is also connected to $m_{1}$ and
$m_{2}$. But the key coordinate is $\dot{m}_{3}$ that is the length of day.
This is the reason why the second derivation of to $g_{1,0}$ agrees better
with the Markowitz-Stoyko drift, which is as we have seen the second
derivative of lod.

(a) Red dots = IGRF $g_{1}^{0}$ every 5 years since 1900 [78] and
interpolation as the black curve. Blue curve = SSA first component ie trend of
polar motion, that is the Markowitz-Stoyko drift.

(b) First time derivatives of the two curves shown in figure 12(a)

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

Figure 12: Comparison between (a) the evolution of $g_{1,0}$ from ([78]) and
the Markowitz-Stoyko drift and (b-c) their corresponding time derivatives.

##  5 Conclusion

Our aim in this paper was to assess Poisson ’s pioneering work ([27]) on the
nature and origin of the Earth’s magnetic field. These works started with the
scientist’s recognition that the field had to be constant, which may seem
awkward to the modern physicist. Gauss ([28]) also considered that the
magnetic field was constant.

Le Mouël [20] observed that there was a strong morphological link between the
secular variation of the field ($\dfrac{d\textbf{H}}{dt}$) and the drift of
the rotation pole (polar motion $m_{1}$ and $m_{2}$), and another link between
the forced oscillations of the magnetic field H and the length of day
$\dot{m}_{3}$ (ie rotation velocity). Based on these observational facts, [20]
proposed a model, a rotating cylinder, that was in many ways identical to
Poisson ’s sphere, though he did not state whether the field was constant or
not. Poisson’s proof that indeed the magnetic field had to be constant
involved theoretical physics and mathematics but only a limited set of
observations. Based on a large number of observations, Le Mouël ([20]) and
collaborators built quasi “experimentally” the same theory as Poisson in a
suite of papers. These papers are used significantly in the literature; it is
not clear whether they are compatible with the complex dynamo models that are
often preferred today.

We have attempted in this paper to return to the original sources and to
reconstruct the development of geomagnetism, using “modern” language. Starting
from Maxwell’s equations, we have derived the equations for the electrostatic
and magnetostatic fields (section 2). We come to the same equations as Poisson
with an important difference: Poisson chose the Lagrangian approach to gravity
([32, 36, 33]), to formally derive the equations for the magnetic field (cf
[35]). To scientists of this epoch, there was no difficulty in reasoning in
magnetism in the same “classical” way one reasoned in gravity (as a side note,
this led to a beautiful piece by Heaviside [41], on the deep analogy between
magnetism and gravity).

We have pursued a more classical approach, and in section 2 we propose a
synthesis with comments of the equations of electromagnetism that can be found
in most physics graduate textbooks. One of the points we wish to emphasize is
that the magnetic field need not 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
using the vector potential (equation 6c) to return to their expression for the
magnetic field.

We have not implemented this last step because one can perform a decomposition
in spherical harmonics and write about 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, we have drawn 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 mechanical 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 ([33, 27]) and [20]. A magnetic field
can be written on a basis of spherical harmonics only if this field is
constant. Section 3 has dealt with the intrinsic theoretical consequences of a
constant dipolar field.

From the ideas presented in sections 2 and 3, we have concluded that in order
to satisfy all hypotheses and all theoretical results, it is sufficient that
the axis of symmetry of the magnetic dipole and the rotation axis can move one
with respect to the other, which is never the case with a development in
spherical harmonics (schematically illustrated in Figure 2).

Section 4 has presented tests of our hypotheses on observations and other
data. In Figure 3 we have shown that sea-level variations in Brest, variations
of declination in Chambon-la-Foret and Markowitz-Stoyko polar drift were
essentially the same, except for the phase of declination, that we interpret
as resulting from the nature of core-mantle coupling. We know from other tide-
gauges [79, 64], that the Brest gauge is representative of most northern
hemisphere gauges. [24] discussed the correlation and the model that would
link geomagnetic secular variations and polar drift; we extend the correlation
and the model to sea-level.

Next, we have focused on annual and semi-annual oscillations, as did
previously Jault and Le Mouël ([24]), here again bringing in the information
carried by sea-level. All these cycles, regardless of whether they correspond
to geomagnetic field components (X,Y,Z), length of day or sea-level, are in
phase or in phase opposition. This observation falsifies the hypothesis that
the field would include seasonal variations (which invoked the Russel and
McPherron [71] effect), unless the effect could also explain the presence of
the same cycles in the sea-level and length of day variations.

We have calculated the ratios of mean amplitude of variations in these cycles
of magnetic field components to associated tide gauges. Independent of local
and regional geography and topography, we find these ratios to be quasi
constant at $\sim$8mm/nT. This also agrees with the concept of a constant
field (relation 8a).

We have next come to testing the famous Schwabe $\sim$11 yr cycle.
Paradoxically it is rather weak in geomagnetic field components, and quite
strong in magnetic indices, such as aa. In a parallel way, the cycle is weak
in polar motion yet it is one of the main components of length of day. This is
readily understood in Laplace ’s paradigm ([33]): polar motion and length of
day describe the same phenomenon but differ by one order of derivation. In the
same way, aa is also a derivative operator (difference between two antipodal
observatories). We have shown that the 11yr component of the magnetic field is
in phase with the corresponding component of polar motion, when the latter
(lod) has been integrated (Figure 7). In a reciprocal way, polar motion is
linked to the derivative of magnetic components.

Our last test has been with the secular variation of the IGRF. Despite the
caveat that a dynamic magnetic field derives from a vector potential, not a
scalar potential, hence can in principle not be developed in spherical
harmonics (Poisson, Gauss), regular analyses of magnetic field models have
been produced at five year intervals for years since 1900. The secular
variation of that field has been monotonous and decreasing, as far as the
leading axial dipole field component $g_{1,0}$ is concerned. This behavior is
parallel to that of polar motion ($m_{1},m_{2}$, see Figure 12(a)) or also
$\dot{m}_{3}$.

These two lines of observations complement our series of tests of the validity
(and foresight) of Poisson’s derivation of Maxwell’s equations, Larmor’s
equation, the Liouville-Euler system, and in general of the distinction that
must be made between electric and magnetic fields. The parallel behaviors of
magnetism and rotational mechanics, illustrated by the Larmor formula, have
been put to test with modern data (observations) with success, as we have
endeavored to show in this paper. These results can be complemented by the
analysis of the responses at a series of “commensurate periods” that have
their origins in the ballet of (primarily) Jovian planets: these are the
”forcing factors” not only of variations in sea-level, and a number of
geophysical, atmospheric and heliophysical processes (eg [52, 53, 54]) but
also of Earth’s magnetic field as we have shown in this paper.

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