∎

11institutetext: V. Lesur 22institutetext: Université de Paris, Institut de
physique du globe de Paris, CNRS, Paris, France
22email<EMAIL_ADDRESS>33institutetext: N. Gillet 44institutetext: Univ.
Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, UGE, ISTerre, 38000
Grenoble, France 55institutetext: M.D. Hammer 66institutetext: Division of
Geomagnetism and Geospace, DTU Space, Technical University of Denmark
77institutetext: M. Mandea 88institutetext: CNES – Centre National d’Etudes
Spatiales, 2 place Maurice Quentin 75039 Paris Cedex 01

# Rapid variations of Earth’s core magnetic field

Version from the: August 29, 2024

V. Lesur  N. Gillet  M.D. Hammer  M. Mandea

(Received: date / Accepted: date)

###### Abstract

Evidences of fast variations of the Earth’s core field are seen both in
magnetic observatory and satellite records. We present here how they have been
identified at the Earth’ surface from ground-based observatory records and how
their spatio-temporal structure is now characterised by satellite data. It is
shown how their properties at the core mantle boundary are extracted through
localised and global modelling processes, paying particular attention to their
time scales. Finally are listed possible types of waves in the liquid outer
core, together with their main properties, that may give rise to these
observed fast variations.

###### Keywords:

Geomagnetism Magnetic Field Modelling Core Time-scales

##  1 Introduction

The observed magnetic field above the Earth’s surface results from the
contribution of numerous sources situated inside the Earth, such as the core
and the lithosphere, or outside the Earth, such as the ionosphere and
magnetosphere. The dominant contribution is by far the field generated in the
core. Its longer time-changes, including millennial and longer periods, are
reconstructed from paleomagnetism (Constable and Johnson, 2005). The shortest
fluctuations likely reach sub-daily periods, in link with the rotating fluid
dynamics of the fluid core. However, the variations directly observable at the
Earth’ surface are on a limited range of time-scales. The upper limit is
slightly longer than a century, corresponding to the installation of the first
magnetic observatories (e.g., Matzka et al., 2010). The lower limit is poorly
known for two reasons. Firstly, the amplitude of the core signal gets weaker
towards high frequencies, so that external signals dominate the magnetic
records on periods shorter than $O(1)$ year. Secondly, the screening effect of
the conductive mantle, which filters out rapid signals originating from the
core, is not well estimated. The aim of this paper is to review our current
knowledge on the rapid variations of the core field, and the possible core
flow perturbations that generate them.

Very early in the study of the Earth’s magnetism it had been realised that the
magnetic field ${\bf B}$ is changing with time (e.g., Kono, 2007). The proper
description of this evolution came with the setting of magnetic observatories
from the middle of the 19th century onward. The general assumption was that
most of observed fast variations in ${\bf B}(t)$ were due to external fields
perturbations, while the dominant core field varied slowly over time. The
accumulation of long time series of magnetic observations led to an evolution
of this paradigm since, at observatory sites, the main field secular variation
(SV, or rate of change of the field, $\partial{\bf B}/\partial t$) often
appeared as linear trends with abrupt changes of slopes. These singular events
are clearly generated in the core and have been called "geomagnetic jerks"
(Courtillot et al., 1978). Early discussions on the origin of these events can
be found in Malin and Hodder (1982); Alldredge (1984). The evolution of this
secular variation has also been studied in terms of secular acceleration (SA,
or second time derivative, $\partial^{2}{\bf B}/\partial t^{2}$). Early models
of the SA can be found in Cain et al. (1967), who used OGO satellite data,
assuming the acceleration constant over time. The acceleration was also
assumed constant by Malin (1969) and Barraclough and Malin (1979), although in
these latter publications it is already noticed that the difference of SA
obtained by different authors likely reveal a temporal change. After the
magsat satellite mission in 1980, came the first robust representations of
magnetospheric fields (Langel and Estes, 1985a, b), as well as temporal
parameterisations of the core field changes using b-splines (Langel et al.,
1986; Bloxham and Jackson, 1992; Sabaka et al., 1997). In principle such
models are able to describe the SA and its evolution, but these are not
discussed in the aforementioned publications. After the launch of the Oersted
and champ satellite missions, it became possible to derive more accurate
magnetic field models. While preparing candidate models for the 10th igrf
generation (International Geomagnetic Reference Field, Maus et al., 2005a), it
became obvious that assuming a linear evolution of the magnetic field was not
adequate to fit the few years of available satellite data: including an
acceleration was necessary (Lesur et al., 2005; Olsen et al., 2005b; Maus et
al., 2005c). Soon after came models fitting these data and with a temporal
description of the core field using b-splines, initially of order four as in
the first versions of the chaos model (Olsen et al., 2006; Olsen and Mandea,
2008), and then five in the grimm model (Lesur et al., 2008). With this latter
model it was possible to follow the SA variations over five years, and to
identify a maximum of acceleration in 2006. From there on, numerous
publications (e.g., Chulliat and Maus, 2014; Finlay et al., 2016) have focused
on the acceleration patterns, highlighting pulses in the SA norm at inter-
annual periods. These are discussed in the following of this paper.

The crucial issue regarding field models derived from magnetic data is their
temporal resolution. Observatory time series show a temporal spectrum
$S(f)\propto f^{\alpha}$ with a slope $\alpha\simeq-4$ in the range of periods
from $\approx 70$ yrs down to a couple of years (De Santis et al., 2003). This
property, which is coherent with the existence of geomagnetic jerks, is also
recovered in time series of the geomagnetic Gauss coefficients (Lesur et al.,
2018). It is shared by the family of stochastic processes (of order 2) that is
differentiable only once in time (Gillet et al., 2013). For those, the
estimate of the SA is subject to the choice of sampling rate. With most field
models, an instantaneous measure of the SA is not available, but instead only
a filtered vision. This time resolution strongly depends on the spherical
harmonic degree: the larger the length-scale, the better the resolution in
frequency (see section §3.2). A magnetic signal probably exist at inter-annual
periods, which carries some information about the core physics. However, the
image of the shorter wavelengths is limited to long periods, which alters our
reconstruction of inter-annual SA patterns. This limitation precludes the
identification of the cutting frequency over which the slope $S(f)\propto
f^{-4}$ stops. This depends on the one hand on the conductance of the mantle
(Jault, 2015), and on the other hand on the relative importance of diffusion
and propagation of Alfvén waves in the fluid outer core (Aubert and Gillet,
2021).

There is a general agreement that the flow pattern in the liquid outer core
that generates most of the observed SV (by advection of the core field), is a
large gyre that flows westward near the core mantle boundary under Africa and
the Atlantic, and flows closer to the cylinder aligned with the rotation axis
and tangent to the inner core under the Pacific Ocean (e.g., Pais and Jault,
2008; Gillet et al., 2009, 2015; Aubert, 2014; Baerenzung et al., 2018). It
has been demonstrated however, that this flow has to vary in time to explain
the observed magnetic field variations at observatory sites (Bloxham, 1992;
Waddington et al., 1995). There are some indications that these flow
variations are mainly located under Eastern Asia, and Central America (e.g.,
Finlay et al., 2016), but they have also been identified close to the inner
core tangent cylinder (Livermore et al., 2017; Gillet et al., 2019). Torsional
waves with a periodicity around 6 years have also been identified in flow
models derived from ground based observations (Gillet et al., 2010). They
explain an independent 6-yr signal recorded in the length-of-day (Abarca del
Rio et al., 2000; Holme and De Viron, 2013; Chao et al., 2014), and their
physics determine the intensity of the magnetic field within the core (where
the potential field approximation drops due to electrical currents). The
magnetic signature of these waves at Earth’s surface is tiny (a couple of
nT.yr-1 – e.g., Cox et al. (2016)), and must be coherent within the set of
observation sites in order to imprint core flow models. This magnitude is
nevertheless too weak to entirely explain inter-annual SV changes (of the
order of $8$ nT.yr-1), which thus require the existence of other, non-zonal,
transient motions (Gillet et al., 2015; Finlay et al., 2016; Kloss and Finlay,
2019).

The remaining of this paper is organised as follow: in section §2 we first
come back on long series of observatory data and subsequently satellite (GVO)
data, from which we discuss the observed SV evolutions over time. We present
next in section §3 different modelling techniques and evaluate how these
affect the derived acceleration time scales. Then, possible flow structures
and waves that may generate observable signals at Earth surface are reviewed
in section §4. We finally conclude in section §5.

##  2 Core field variability observed from magnetic data

###  2.1 Ground magnetic observatories

A strong motivation in running magnetic observatories is the monitoring of the
Earth’s main magnetic field evolution originating in the fluid outer core. The
importance of long-term observations in support of this endeavour is
illustrated in Figure (1) where (irregular) declination measurements at a
number of locations in the vicinity of Paris, and regular measurements at the
observatories in Saint Maur des Fossés (Val de Marne, France, 1883-1902), Val
Joyeux (Yvelines, France, 1903-1935) and Chambon-la-Forêt (1936-present) are
plotted together. The Paris declination series starts as early as in the 16th
century, with the first measurement performed in 1541, giving a value of
$7^{\circ}E$. Let us underline that all declination measurements made in the
Paris area (Issy-les-Moulineaux, Montsouris et Montmorency) and in previous
French magnetic observatories are reduced to the current location of Chambon-
la-Forêt. A detailed discussion of the applied corrections is given in
Alexandrescu et al. (1996). The plot shows that, over the last 479 years, the
magnetic declination has varied between extreme values reaching
$13^{\circ}30^{\prime}E$ in 1580 to $22^{\circ}30^{\prime}W$ in 1814. Long
data series, such as those in Figure (1), have been compiled for example for
London, Rome and Munich by Malin and Bullard (1981); Cafarella et al. (1992);
Korte et al. (2009), respectively. These series are important in
characterising the scales of temporal variations in the core-generated field.

These unique series allow to easily compute the SV in a single position and to
estimate occurrence date for geomagnetic jerks, mainly before the 20th
century. In Figure (1) the SV is computed as annual mean differences. To
enhance rapid events and reduce solar cycle effects, the first time derivative
is computed after applying an 11-yr smoothing low-pass filter window which
makes the changes in the trend more rounded. Nevertheless, the geomagnetic
jerks can still be clearly identified. The figure clearly shows that, prior to
the 20th century, one of the most prominent geomagnetic jerks appears around
1870. This event is also observed in four other European locations
(Alexandrescu et al., 1997) and has recently been detected in the Munich curve
(Korte et al., 2009), although a few years earlier. Going farther back in
time, there is evidence of changes in the secular variation trend, supported
by measurements around epochs 1600, 1665, 1700, 1730, 1750, 1760, 1770, 1810,
1870, 1890, and 1900. These dates are close to those detected by Qamili et al.
(2013): 1603, 1663, 1703, 1733, 1751, 1763, 1770, 1810, 1868, 1870, 1888, and
1900, when analysing the temporal behaviour of the difference between
predicted and actual geomagnetic field model values for successive intervals
from 1600 to 1980, based on the gufm1 geomagnetic model (Jackson et al.,
2000).

Figure 1: Compiled records of the observed declination ($D$, in red) and its
estimated secular variation (SV, in blue), in the vicinity of Paris, France.
All records have been reduced to a single position at the current site of the
French national magnetic observatory in Chambon-la-Forêt (clf – Loiret,
France).

The typical SV time scale spans years to centuries, and its typical magnitude
is $10-100$ nT.yr-1. These values help to define the observational
requirements for magnetic observatories: they should run for many years,
achieve measurement accuracy in all components of the field vector of around 1
nT, and maintain the long-term stability needed to resolve the typical secular
change signal. With the above recommendation the geomagnetic observatories
represent a unique dataset to understand the secular variation over long-term
periods. Indeed, the first geomagnetic jerk has been detected was identified
in geomagnetic observatories series (Courtillot et al., 1978). In this
pioneering paper, it was defined as an abrupt turning point separating the
otherwise linear trends of the East component of SV prior to and after 1970 at
several northern hemisphere observatories.

The geomagnetic jerks can be observed in the various field derivatives (core
field, secular variation and secular acceleration) with a wide variety of
detection methods. A detection method needs to consider several factors linked
to the available data: noise content in the data and its origin, the
asynchronous form of a jerk in each field component, temporal and spatial
scales at which an event is significant enough to be seen as a jerk. These
factors are briefly discussed here:

  * -

the quality of data varies back in time. Systematic geomagnetic field
observations at multiple locations, for the three field components, began with
the establishment, by Gauss, of the Göttingen Magnetic Union in 1834. Before
that date only directional observations were available, so the
characterisation of the secular variation via the full vector field was not
possible. Only over the 20th century the three field components are available
for a set of observatories allowing comparison between different locations and
different field components.

  * -

the first data analyses were focused on the East component, mainly for two
reasons. The first analyses considered mainly European observatories and the
geomagnetic jerks are generally clearly noted in this component time series
(see Figure 2). It should be noted that the behaviour of the East component
secular variation is different for clf (48.025∘N, 2.260∘E) and ngk (52.07∘N,
12.68∘E) observatories, even if these two observatories are relatively close.

  * -

the magnetic field Eastern component is less affected by the external sources,
i.e. electrical currents flowing in the ionosphere and magnetosphere of the
Earth. Different methods have been applied to minimise the external field
contributions in geomagnetic series (monthly or annual means), via geomagnetic
indices or a combination of field models and magnetic indices, or empirical
techniques (e.g., De Michelis et al., 2000; Verbanac et al., 2007; Wardinski
and Holme, 2011). Different attempts have been also applied to parameterise
the external field sources during the modelling processes (see section §3.2
and references there in). These models are also suitable to obtain series of
data and to investigate geomagnetic jerks, mainly when considering three field
components.

  * -

it is well-known that the geomagnetic observatories are located mainly on
continental areas and that the northern hemisphere is better covered than the
southern hemisphere. It is then important to recall the effect of the uneven
distribution of observatories in the definition of geomagnetic jerk
characteristics. Identifying regional jerks have been possible not only
because more and more sophisticated methods have been applied (see Mandea et
al., 2010; Brown et al., 2013), but also due to the usage of full field
geometry.

Figure 2: In red is shown the secular variation for the East component at
Chambon-la-Forêt (clf – 48.025∘N, 2.260∘E), France and, in blue, at Niemegk
(ngk – 52.07∘N, 12.68∘E), Germany.

Although geomagnetic jerks are clearly of internal origin and are well
identified events in magnetic observatories time series, they are less and
less seen as singular events in term of flow dynamics. Study of recent
magnetic field models and progresses in numerical geodynamo models suggest
that they are natural features of the magnetic field generation in the Earth’s
core (see Section §4).

In order to get a better understanding of the recorded signal variability at
magnetic observatories, in Figure (3) are presented the time series of the
three components of the magnetic field in an Earth Centred Earth Fixed system
of coordinates (ECEF –i.e. Z is the Earth’s rotation axis, X points towards
Greenwich meridian and Y complete this system), together with their Fourier
spectra. Data are daily means. The time series have been corrected for
recorded jumps, interpolated for missing values using smoothed cubic-splines
and "detrended" by subtracting a linear polynomial going through the first and
last point of the series. Figure (3) thus display deviations from a linear
evolution of the recorded data. All plots have the same scales, and deviations
are much larger in Hermanus observatory (her, -34.425∘N, 19.225∘E, South-
Africa) than on M’bour and Chambon-la-Forêt observatories, (mbo, 14.392∘N,
-16.958∘E, Senegal) and (clf, 48.025∘N, 2.260∘E, France), respectively.
Because of the choice of coordinate system a large part of the signals
generated in the magnetosphere is concentrated in the Z component. These
signals are associated in the Fourier spectra with peaks at one year and half-
year periods. They are also related to excess of energy around 27-day periods
and other sub-harmonics. The 11-yr solar cycle generates small peaks in Z
components at all observatories. However, it is extremely difficult to
identify specific periodicities for signals generated in the core, as the
external field contributions dominate the spectra up to periods as large as 2
to 5 years, depending on locations and components. There is a clear excess in
energy at M’bour in the Y component for periods centred on 6.5 years, and also
for periods from 13 to 15 years. Less distinctive are these same periods for
Hermanus observatory. However, at Chambon-la-Forêt, in spite of a very long
series starting the 1st January 1936, these periods do not dominate the
spectra. The only unexpected maximum is at 1.7 years periods in the Y
component. All observatories at the longest periods present an increasing
trend that, as expected, roughly correspond to a $p^{4}$ behaviour, with $p$
the period. However, these observations indicate clearly the difference in the
content of the observatory records depending on their regions.

Figure 3: First row: Detrended three components of the magnetic field recorded
at observatory sites in ECEF system of coordinates. Second row: Fourier
spectra of the corresponding time series where $Y$ and $Z$ components have
been shifted up and down, respectively, for clarity. The three presented
observatories are from left to right:  clf, mbo, her (see text for details).
The slope $\propto p^{4}$, with $p$ the period, is indicated with a black
line.

###  2.2 Virtual observatories

With the advent of magnetic field measurements from low Earth orbiting
satellites, smaller spatial structures of the core field changes can be mapped
due to almost complete global coverage. The last twenty years of continuous
space records of the magnetic field, provide valuable insights into the field
and how it is changing in space and time. In particular, magnetic field
measurements from the Danish Oersted satellite (1999-2014), the German champ
satellite (2000-2010) and the European Swarm satellite trio mission (2013-)
have dramatically increased the constraint from geomagnetic data (Olsen and
Stolle, 2012; Domingos et al., 2019). In addition to these, calibrated
CryoSat-2 satellite measurements (2010-) can be used to generate further
constraints on the field variations, in particular during the temporal gap
2010–2014 between champ and Swarm (Olsen et al., 2020). Although the physical
mechanisms responsible for most of the field changes take place over time
scales longer than the satellite era (the past 20 years), shorter inter-annual
fluctuations may be characterised with satellite data alone.

The Geomagnetic Virtual Observatory (gvo) technique was first developed by
Mandea and Olsen (2006); Olsen and Mandea (2007), as a way of making satellite
field measurements easily accessible, through a regression of these into local
time series of the field at pre-specified locations. Such gvo time series
consist of vector field estimates at mean orbital altitude. Each gvo datum is
derived using satellite measurements in the near vicinity of a given gvo
location, and within a chosen time window. As for ground-based records, by
taking annual differences of these locally derived gvo field estimates, SV
time series can be obtained. To address issues of insufficient local time
sampling and contamination from external sources such as the magnetospheric
ring current and ionospheric current systems (Olsen and Mandea, 2007; Beggan
et al., 2009; Shore, 2013), additional data processing steps, changes to the
modelling scheme and de-noising procedures have been incorporated into the gvo
algorithm (Hammer et al., 2021a). In order to make robust gvo field estimates,
a pre-whitening step is applied wherein main field predictions ${\bf B}({\bf
r}_{GVO})$ for each data point of position ${\bf r}_{GVO}$ are subtracted off,
and at a later stage added back for the gvo epoch under consideration. The
IGRF model (which has linear SV) is typically used for determining ${\bf
B}({\bf r}_{GVO})$, however the choice of model in this pre-whitening step is
of little importance. The gvo time series are typically produced at 1-month or
4-month cadence and provided on a global grid of 300 equally spaced locations.
Comparisons of the gvo time series with independent ground observatory records
and spherical harmonic field models (see Figure 4) have demonstrated good
agreement on all three vector field components (Hammer et al., 2021a). The
figure shows SV time series, computed as annual differences, at the three
ground observatory stations Ascension Island (East component), Honolulu
(Radial component) and Hermanus (South component). Note here that the gvo
estimates have been mapped to ground in order to ease the visual comparison.

Figure 4: Comparison of SV times series at three example ground observatories
for the East ($\phi$), Radial ($r$), and South ($\theta$) components
respectively. Shown are SV from annual differences of revised monthly means
(black dots) (see Olsen et al., 2014), and 4-monthly gvos (mapped to ground)
derived from champ (red dots), CryoSat-2 (purple dots) and Swarm (blue dots)
measurements. Also shown are predictions of the CM6 (Sabaka et al., 2020)
(cyan line) and chaos-7.5 (Finlay et al., 2020) (green line) models.

Figure (5) presents a map of radial SV time series over 1999–2020, at 300
globally distributed gvos, with 4 month cadence. In order to produce such
composite time series, the gvos have been mapped to a common altitude of 700
km. This is performed by subtracting from each gvo series the difference
between the field at 700 km and at the mean orbital altitude (separately for
each satellite mission), as determined from the chaos-7.2 model (Finlay et
al., 2020). The map in Figure (5) allows the visual inspection of patterns of
sub-decadal SV trends and changes. Low latitude regions display particularly
strong variations, which do not occur simultaneously over all longitudes.
Distinctive "$\Lambda$" and "$V$" shaped behaviour occur regionally over time
spans ranging from 5 to 10 years. They can be observed for instance over the
South-Atlantic region around 2007 and again in 2014, over Indonesia around
2014, and in the Pacific region around 2017. gvos are thus suitable for
studying inter-annual and longer field changes at satellite altitude. These
likely originate from changes in the outer core fluid motions (see section
§4). The gvo dataset, together with their error estimates, are suitable for
core flow inversions (Whaler and Beggan, 2015; Kloss and Finlay, 2019; Rogers
et al., 2019) and data assimilation studies that combine information from
geodynamo numerical models with signals observed in the gvo time series
(Barrois et al., 2018; Huder et al., 2019).

Figure 5: Composite gvo time series of the radial SV ($\partial B_{r}/\partial
t$ – black dots) over the period 1999–2020, computed as annual differences of
4 monthly gvos mapped at an altitude of 700km. gvo locations are marked with a
red cross. The gvo time series have been derived using measurements from
Oersted data (1999-2004), champ data (2002-2010), chaos-6x9 calibrated
CryoSat-2 data (2010-2014, see Olsen et al., 2020) and data from the Swarm
trio mission (2014-2020). The scale of the time series is shown in the bottom
left corner, with the y-axis being 20 nT.yr-1 and the x-axis going from 1999
to 2020. Adapted from (Hammer et al., 2021b).

##  3 Geomagnetic field modelling

In order to study core field changes occurring over periods from years to
decades, the course of action is typically either to analyse magnetic changes
as recorded at a particular location, or to analyse spherical harmonic field
models (Mandea et al., 2010). Although the temporal resolution of ground
stations is remarkable, their uneven spatial distribution renders global
studies of core field changes challenging (Mandea and Olsen, 2006).
Alternatively, the construction of field models (e.g., Finlay et al., 2020;
Ropp et al., 2020; Sabaka et al., 2020) requires the addition of prior
information in order to reduce the non-uniqueness of the geomagnetic inverse
problem. Models using a spherical harmonic representation come with attractive
mathematical benefits – e.g., orthogonality relations and the addition theorem
(Olsen et al., 2010) and have simple means of specifying the model complexity
as a function of harmonic degree. However, since spherical harmonics are
global support functions, it is preferable to estimate such models with an
homogeneous data coverage over the Earth. With real data, specific sources
dominate the signals over some regions and it is therefore appealing to
consider local methods for studying the magnetic field. We detail in this
section local and global methods for modelling the core field.

###  3.1 Local modelling techniques

Modelling the geomagnetic field is typically done using spherical harmonic
expansions, which remains the preferred tool because of their convenient
mathematical properties. However, other modelling strategies focusing on local
techniques have been proposed, which have typically been developed for
modelling the field on regional scales. Local modelling strategies include (i)
considering a limited area with a flat-Earth approximation (Rectangular and
Cylindrical Harmonic Analysis, see Alldredge, 1981, 1982), or with basis
functions defined on a spherical cap (Spherical Cap Harmonic Analysis, see
Haines, 1985) or (Revised Spherical Cap Harmonic Analysis, see Thébault et
al., 2004), (ii) using basis functions such as wavelets (Chambodut et al.,
2005; Holschneider et al., 2003), "quasi-local" functions (Lesur, 2006),
Harmonic Splines (Shure et al., 1982; Geese et al., 2010) or Slepian functions
(Beggan et al., 2013; Plattner and Simons, 2015), (iii) using spherical
harmonic models with localised constraints (Constable et al., 1993), (iv)
using equivalent source representations such as dipoles (Mayhew and Estes,
1983) and monopoles (Hodder, 1982; O’Brien and Parker, 1994).

Here, we revisit a local approach called Subtractive Optimally Localised
Averages (sola). The sola technique allows for localised estimates of the
radial field and SV field at the core-mantle boundary (CMB), to be computed as
local spatial averages centred on a location of interest (Hammer and Finlay,
2019). These local spatial field averages are computed from averaging kernels
which are influenced by the used data. The sola technique provides information
on the resolution of the field estimate, in the form of the averaging kernels
and variance estimates of the locally averaged field. This information is
important when trying to resolve smaller length and time scales of the core
field signal.

The approach adopted in geomagnetism builds on the method originally developed
in helioseismology by Pijpers and Thompson (1994), and is a modified Backus-
Gilbert inversion type (Backus and Gilbert, 1968, 1970; Pujol, 2013). At
satellite altitude, the magnetic vector field can be linked to the radial
field at the CMB using the Green’s functions of the Neumann boundary value
problem (Gubbins and Roberts, 1983). Focusing on the core field, a dark quiet
time selection scheme is typically implemented and field model estimates of
the lithospheric field, the magnetospheric and ionospheric fields along with
their associated Earth induced fields are subtracted (Hammer et al., 2021b).
Since the CMB radial magnetic field is linearly related to the satellite
vector data, a sola estimate at a specified target location and time,
$(\mathbf{r}_{0},t_{0})$, at the CMB, can be written as a weighted linear
combination of the satellite magnetic data (Backus and Gilbert, 1970):

$\widehat{B}_{r}(\mathbf{r}_{0},t_{0})=\sum_{n}^{N}q_{n}(\mathbf{r}_{0},t_{0})\,d_{n}(\mathbf{r}_{n},t_{n}),$
(1)

where $q_{n}$ are sola weight coefficients and $d_{n}$ are the satellite data
$(n=1,...,N)$. They are related to the radial magnetic field,
$B_{r}(\mathbf{r}^{\prime},t_{n})$, integrated over the CMB as (Gubbins and
Roberts, 1983)

$d_{n}(\mathbf{r}_{n},t_{n})=\oint_{S^{\prime}}G(\mathbf{r}_{n},\mathbf{r}^{\prime})B_{r}(\mathbf{r}^{\prime},t_{n})dS^{\prime},$
(2)

where the CMB surface element at radius $r^{\prime}$ is
$dS^{\prime}=\mathrm{sin}\theta^{\prime}d\theta^{\prime}d\phi^{\prime}$ and
the data kernel, $G(\mathbf{r}_{n},\mathbf{r}^{\prime})$, is the spatial
derivative of the Green’s functions for the exterior Neumann boundary value
problem (e.g., Gubbins and Roberts, 1983; Barton and Barton, 1989). Combining
equations (2) and (1), it is possible to write the sola estimate as

$\widehat{B}_{r}(\mathbf{r}_{0},t_{0})=\oint_{S^{\prime}}\mathcal{K}(\mathbf{r}_{0},t_{0},\mathbf{r}^{\prime})\,B_{r}(\mathbf{r}^{\prime},t_{n})dS^{\prime}\text{,}$
(3)

where $\mathcal{K}(\mathbf{r}_{0},t_{0},\mathbf{r}^{\prime})$ is a spatial
averaging kernel. Thus, the sola weight coefficients together with the data
kernel, specifies the averaging kernel centered on position $\mathbf{r}_{0}$
as

$\mathcal{K}(\mathbf{r}_{0},t_{0},\mathbf{r}^{\prime})=\sum_{n}^{N}q_{n}(\mathbf{r}_{0},t_{0})\,G(\mathbf{r}_{n},\mathbf{r}^{\prime})\text{.}$
(4)

To determine the sola weight coefficients, an objective function that
describes the departure of the spatial averaging kernel towards a pre-
specified target kernel is minimised and information from a data error
covariance matrix is included (Hammer and Finlay, 2019). By varying the target
kernel width, sola estimates of different spatial averaging width (i.e.
resolution) emerges. Typically, a Fisher distribution is used as a target
kernel (Fisher, 1953), however other options are possible (Masters and
Gubbins, 2003).

When calculating sola field estimates, data from within a month are commonly
used and time-dependence neglected. In order to compute sola SV estimates,
data from within a window (typically 2 years) are used, where the time-
dependence is handled using a first order Taylor expansion assumed valid close
to a reference time (Hammer et al., 2021b). By taking differences between
consecutive sola SV field estimates, estimates of the SA at the CMB can be
derived, which is of interest when studying core dynamics (e.g., Finlay et
al., 2016; Chi-Durán et al., 2020). In particular, using the high quality
measurements from the Swarm mission, it is possible to take 1-yr differences
of sola SV estimates derived from 1 year data windows (Hammer et al., 2021b).

Because the sola method allows for local field estimates to be determined,
collections of individual sola estimates having the same resolution, can be
combined into making regional or global maps. Of particular interest for more
detailed studies, are regions of strong fluctuations in the field changes
observed at low latitudes. In order to examine such changes, the sola method
can be used for making SA estimates along the CMB equator in consecutive time
steps, such that the evolution of the SA can be mapped. Figure (6) presents
time-longitude (TL) plots of the SA evolution on the geographic equator
(centred on the Pacific region) at the CMB between 2015.0 and 2020.0 derived
from the radial field measurements of the Swarm satellites. The left plot
shows sola SA estimates derived from 1-yr differences of sola SV determined
from 1 year data windows, sliding in 1 month steps, while the right plot shows
the SA predictions from the chaos-7.2 model for spherical harmonic degrees 1
to 10 – i.e. a maximum wavelength $\sim$2200km at CMB. Here the sola SA
estimates have associated errors estimates of about $0.6$ nT.yr-2 and
averaging kernel widths of $42^{\circ}$, corresponding to length scales of
2500km at the CMB. Comparing the two TL-plots, similar large scale features of
the SA evolution with amplitudes of $\pm 2.5$ nT.yr-2, can be observed in both
the sola and chaos plots. However, the sola plot demonstrates increased
temporal resolution, and reveals coherent patterns that have been smoothed out
by the temporal regularisation of the chaos-7.2 model.

In the sola TL plot, interesting structures includes intense SA patches under
Central America, at longitudes from $240^{\circ}$ to $320^{\circ}$, observed
going from 2015.5 to 2018. Also observed are strong side-by-side positive and
negative SA features emerging in the Pacific region around 2017, at longitudes
$150^{\circ}$ to $220^{\circ}$, which are seen to have drifted approximately
$20^{\circ}$ westwards until 2020. Such rapid variation at low latitudes may
be caused by time variations of the liquid outer core flow, and could provide
important constraints on the equatorial dynamics of the outer core (Kloss and
Finlay, 2019). Responsible mechanisms for such flows may involve equatorially-
trapped MAC waves in a stratified layer at the core surface (Buffett and
Matsui, 2019; Chi-Durán et al., 2020), Magneto-Coriolis modes (Gerick et al.,
2020) or equatorial focusing of hydrodynamic waves related to turbulent
convection deep within the core (Aubert and Finlay, 2019). Possible mechanisms
are described in more details in section §4.

(a) sola radial SA, 1yr time windows, 1 month steps.

(b) chaos-7.2 radial SA.

Figure 6: Time-longitude plots of the radial SA field along the CMB
geographical equator, showing: 1yr differences of sola SV estimates derived
from 1-yr data windows moving in 1 month steps (left), and predictions from
the chaos-7.2 model for spherical harmonic degrees 1 to 10 (right). Adapted
from (Hammer et al., 2021b).

###  3.2 Global field models

Although building localised representations of the core magnetic field
presents numerous advantages, it still relies on a rough separation of the
different sources often based on outputs of other’s modelling efforts. We
therefore present here global magnetic field modelling techniques that
simultaneously consider the core and other contributions. Although different
types of representation are possible, magnetic field components are most
generally described as the negative gradient of potentials of internal or
external origins, respectively $V_{i}$ and $V_{e}$,

${\mathbf{B}}(\theta,\phi,r,t)=-\nabla\\{V_{i}(\theta,\phi,r,t)+V_{e}(\theta,\phi,r,t)\\}\,,$
(5)

themselves being decomposed into spherical harmonic as

$\begin{array}[]{ll}\displaystyle
V_{i}(\theta,\phi,r,t)=a\sum_{\ell=1}^{L_{i}}\sum_{m=0}^{m=\ell}\;\\{g_{\ell}^{m}(t)\,\cos(m\phi)+h_{\ell}^{m}(t)\,\sin(m\phi)\\}\;\left(\frac{a}{r}\right)^{\ell+1}&\displaystyle
P_{\ell}^{m}(\cos\theta)\,,\\\ \displaystyle
V_{e}(\theta,\phi,r,t)=a\sum_{\ell=1}^{L_{e}}\sum_{m=0}^{m=\ell}\;\\{q_{\ell}^{m}(t)\,\cos(m\phi)+s_{\ell}^{m}(t)\,\sin(m\phi)\\}\;\left(\frac{r}{a}\right)^{\ell}&\displaystyle
P_{\ell}^{m}(\cos\theta)\text{.}\end{array}$ (6)

Here, $(\theta,\phi,r)$ are colatitude, longitude and radius in geocentric
system of coordinates. The $P_{\ell}^{m}(.)$ are Legendre functions. The
$g_{\ell}^{m}$, $h_{\ell}^{m}$, $q_{\ell}^{m}$ and $s_{\ell}^{m}$ are the
Gauss coefficients to be estimated independently for each source, where
$h_{\ell}^{0}$ and $s_{\ell}^{0}$ are null. All these Gauss coefficients
depend on time $t$. In such a decomposition it is assumed that there are no
sources of magnetic field at the measurement point, which usually is true for
ground measurements, but it is not necessarily valid at satellite altitude, in
particular in the auroral regions with field-aligned currents (see Finlay et
al., 2017). The internal and external description refers to a sphere of radius
$a=6371.2$km, that is an approximation of the Earth’s surface. Note that with
satellite data, ionospheric fields are generated in between the measurement
position and the Earth’s surface, introducing some confusion in this simple
description. The same difficulty in principle exists for the lithospheric
field at the equator, that is clearly outside the reference sphere, but this
latter effect is generally neglected.

Over the 20 last years, the nearly continuous series of magnetic satellite
missions allow to build particularly high resolution, time dependent models of
the magnetic field. Examples of these models are the Comprehensive models
(Sabaka et al., 2016, 2020) , chaos models (Olsen et al., 2006; Finlay et al.,
2020), GRIMM models (Lesur et al., 2008, 2015), for the most known. This list
is far from being exhaustive. Special issues associated with the publication
of the IGRF generally give a better picture of the diversity and numbers of
magnetic models available (see e.g., Maus et al., 2005b; Finlay et al., 2010;
Thébault et al., 2015; Alken et al., 2020). If these models are different,
they agree overall on the core magnetic field and its SV strength and
direction at the Earth’s surface – see Alken et al. (2020) for more
information on statistics of the differences. An example is displayed in
Figure 7, for the epoch 2015.0, of the vertical down components at the CMB
from the mcm model (Ropp et al., 2020). It should be noted that in these
images, the spherical harmonic degree is limited to $\ell=$13, and that
shorter wavelength signals, that can be dominent and therefore strongly modify
the images, are not included.

Figure 7: Vertical down core magnetic field as estimated by the mcm model
(Ropp et al., 2020) truncated at spherical harmonic 13 (left) and its secular
variation (right). The field has been mapped at the CMB for epoch 2015.0. In
both cases continents are drawn to help location.

The agreement between models for the SA remains generally good but they
present significant differences in terms of amplitudes and locations of the
acceleration patterns. Maps of the vertical down SA component for 2006 and
2018, at the CMB, are displayed in Figures (8) for two models chaos-7 (Finlay
et al., 2020) and mcm (Ropp et al., 2020). As for the SV, the spherical
harmonic degree is limited to $\ell=$13, and shorter wavelength signals can
significantly modify these maps. Differences between the two models are
obvious even if there is a general agreement that strong acceleration patterns
are mainly located at the CMB along the equatorial regions, or at all
latitudes in between 90 and 120 degree East. Strongest accelerations are often
seen under Indonesia or Central America. These acceleration patterns have a
footprint at the Earth’s surface that is very large, in agreement with those
directly observed in Figure (5).

Figure 8: Vertical down core magnetic field acceleration as estimated by the
mcm model (Ropp et al., 2020) (left) or chaos-7 model (Finlay et al., 2020)
(right), truncated at the spherical harmonic $\ell=13$. The field has been
mapped at the CMB for year 2006 (first row) and 2018 (second row). Continents
are drawn to help location.

The observed differences between models result from the way the separation of
sources contributing to the measured magnetic field is achieved. The coverage
in space and time of the combined observatory and satellite data sets is such
that for temporal scales of the order of a few days to a few weeks, the
separation of large scale internal fields from external fields is suitable for
core field modelling. There may be difficulties for short spatial wavelengths,
or short temporal scales but these are generally irrelevant for the core
field. One can refer to Lesur et al. (2013) for a description on how fast
field variations in time leaks in small wavelength of a nearly static internal
field. Satellite data are also contaminated by non-potential magnetic fields
(i.e. magnetic field that cannot be described as gradients of potentials as in
equations (6)). These signals are mainly due to field aligned electric
currents linking the magnetosphere to the ionosphere. These non-potential
fields can be co-estimated with other sources (e.g., Sabaka et al., 2016;
Lesur et al., 2008), although, even if not modelled, it seems that their
contributions do not significantly affect the core field models derived from
satellite data. Therefore, the main difficulties that remain for precise core
field models are the separation of sources that are seen as internal from the
satellite altitude.

From the satellite altitude down, there are firstly the magnetic fields
generated in the ionosphere during night-times (day-time data are not usually
used for core field modelling). There have been several attempts to model
these fields, particularly at high latitudes (e.g., Lesur et al., 2008), but
they remain difficult to handle because satellites generally fly at nearly
sun-synchronous orbits and therefore over a few days acquire data only on a
narrow local time window. It should be noted that because the Earth rotates
under the ionosphere that is locked in local time, the ionospheric signals may
preferentially affect zonal Gauss coefficients (i.e. $g_{l}^{m}$ and
$h_{l}^{m}$ with $m=0$) of the core contribution. Secondly there are the
fields generated in the lithosphere. So far, there has been no successful
attempt to separate the lithospheric field from the core field. It follows
that the contribution of the lithospheric field is ignored at long
wavelengths. At shorter wavelengths, from spherical harmonic degree higher
than $\ell=$16, where the lithospheric field is dominent at the Earth’s
surface, the static core field contribution is unknown. At the same altitude,
signals generated in oceans can be well estimated when linked to tides because
of their well established periodicities (Grayver and Olsen, 2019). However,
the signals associated with general oceanic circulation have not been isolated
yet. There are probably other sources in the crust that have not been
identified, such as possibly the water cycle that has a strong signature in
gravity data. However, the field induced in the conductive crust and mantle by
the time varying external fields remain the major difficulty that precludes
the derivation of accurate core field models as both contributions overlap
over a large range of spatial and temporal scales (see e.g., Olsen et al.,
2005a). It is presently the isolation of long periods external fields (and
their induced counter-parts) that sets the temporal resolution of core field
models. Ambiguities may remain at the decadal period of the solar cycle (and
its harmonics), and more importantly we lack accurate understanding of the
strong semi-annual and annual periodicities that arise naturally in
ionospheric and external fields.

Generally the strategy used to separate these various "internal" fields from
the core field is based on a careful data selection, a partial modelling of
the different contributions, and often, a constraint (i.e. a damping or
regularisation) set on the core field contribution such that it does not vary
too rapidly in time. This approach has been used for a long time, and examples
of application to satellite data can be found in e.g., Olsen et al. (2006);
Lesur et al. (2008) or more recently Sabaka et al. (2020); Finlay et al.
(2020). The characteristic of this approach is that the core magnetic field SV
and SA are averaged over time. Furthermore, their temporal resolutions vary
depending on the spatial scale. As shown in Figure (9), with these approaches
the acceleration is likely underestimated at short periods and small
wavelengths – i.e. from spherical harmonic degree $\ell=5$ up. An alternative
approach has emerged, pioneered by McLeod (1996); Gillet et al. (2013), where
the temporal constraints applied to construct the core field model is derived
from what is known of the spatial and temporal behaviour of each source
contributing to the observed magnetic field. The temporal constraints are
imposed either through a priori time cross-covariances (Huder et al., 2020),
or by time-stepping associated stochastic equations (Huder et al., 2019;
Baerenzung et al., 2020; Ropp et al., 2020). As a result, the modelled SV and
SA are not averaged over time by construction, to the cost of having models
with large posterior uncertainties (Gillet et al., 2013; Holschneider et al.,
2016). We point for instance to Ropp et al. (2020) where outputs from
numerical dynamos are used as prior spatial information on the core field
behaviour. Whatever modelling technique is used, some temporal averaging may
still exist in the models due to the imperfect data coverage. In particular
the nearly continuous satellite era extends only over the last 20 years
precluding robust estimation of temporal periodicities in the core exceeding
10 years.

Figure 9: Intensity of the SA at the core surface ($\log_{10}$ of nT.yr-2) as
a function of the period and the spherical harmonic degree, for the chaos-7
field model (Finlay et al., 2020). Adapted from Gillet (2019).

All these modelling techniques rely strongly on what is assumed by the
modeller regarding the magnetic field generated by different sources, and in
particular by the Earth’s core. To better understand the dynamics of the
liquid outer core, it is therefore important to define how fast the core field
varies with time at the Earth’s surface. The fastest variations in the core
field are necessarily smoothed out by (i) magnetic diffusion within the fluid
core and (ii) induction in the conductive mantle. However, the associated cut-
off periods are not well known. On the core side, the most extreme numerical
simulations of the geodynamo suggest that it could be of the order of 1 year.
This is hard to confront immediately to magnetic data because of the dominant
external signals towards high frequencies (see Figure (3)). On the mantle
side, the cut-off period remains difficult to estimate because of the poorly
constrained lowermost mantle conductivity (e.g., Kuvshinov et al., 2021). This
cut-off value is likely less than 1 year (see Jault, 2015). It is interesting
to see how much variability has been accepted for the core field or its SV in
the currently available models. Both quantities are characterised by the SV
and SA time scales, as defined by respectively

$\tau_{sv}(\ell)=\sqrt{\frac{\sum_{m}(g_{\ell}^{m})^{2}+(h_{\ell}^{m})^{2}}{\sum_{m}(\dot{g}_{\ell}^{m})^{2}+(\dot{h}_{\ell}^{m})^{2}}}$
(7)

and

$\tau_{sa}(\ell)=\sqrt{\frac{\sum_{m}(\dot{g}_{\ell}^{m})^{2}+(\dot{h}_{\ell}^{m})^{2}}{\sum_{m}(\ddot{g}_{\ell}^{m})^{2}+(\ddot{h}_{\ell}^{m})^{2}}}\,.$
(8)

In these equations we use the notations $\dot{x}=\partial x/\partial t$ and
$\ddot{x}=\partial^{2}x/\partial t^{2}$. From the combined analysis of
magnetic field models and numerical simulations, Christensen et al. (2012)
propose, for $1<\ell\leq 13$, $\tau_{sv}(\ell)\simeq 480/\ell$ (in fair
agreement with Lhuillier et al., 2011) and $\tau_{sa}(\ell)\simeq 11$ years
for $\ell\leq 10$. With the new generation of numerical geodynamo simulations
(Aubert, 2018), the $\tau_{sa}$ value decreases down to sub-decadal periods
(for the largest length-scales) as conditions closer to Earth-like are reached
(Aubert and Gillet, 2021).

Because the SV energy is relatively well constrained by observations,
$\tau_{sa}$ depends directly on the SA energy. The latter can be decomposed,
from the induction equation in the fluid outer core

$\dot{\mathbf{B}}=\nabla\times({\mathbf{u}}\times{\mathbf{B}})+\eta\Delta{\mathbf{B}}\,,$
(9)

into three terms as

$\ddot{\mathbf{B}}=\nabla\times(\dot{\mathbf{u}}\times{\mathbf{B}})+\nabla\times({\mathbf{u}}\times\dot{\mathbf{B}})+\eta\Delta\dot{\mathbf{B}}\,.$
(10)

Here ${\mathbf{u}}$ is the flow in the Earth’s liquid outer core, and $\eta$
the magnetic diffusivity. The first term in the righthand side of equation (9)
is the SV generated by the advection of the magnetic field by the flow,
whereas the second term is linked to magnetic diffusion. In equation (10), the
first term in the righthand side is the SA generated by the flow acceleration,
the second is the SA generated by the flow advecting the SV, and the last term
is the contribution from diffusion. Results from numerical dynamo experiments
suggest that the two first terms in equation (10) strongly dominate the last
one, and that the first term most often dominates over the second (Aubert,
2018; Aubert and Gillet, 2021). The second term of the same decomposition,
neglecting diffusion, was used by Lesur et al. (2010) to estimate a lower
bound for the acceleration energy. Presented in Figure (10), it can be seen as
an upper limit for an acceptable acceleration time scale $\tau_{sa}(\ell)$.
All recent magnetic field models have acceleration time scales for spherical
harmonic degree up to $\ell=13$ well below this limit.

In the same figure is shown $\tau_{sa}(\ell)$ derived from the Coupled-Earth
numerical dynamo model (Aubert et al., 2013). For this dynamo simulation (as
for the ones used by Christensen et al., 2012) transient phenomena such as SA
pulses are excluded because of a too weak magnetic to kinetic energy ratio.
This tends to produce larger estimates of $\tau_{sa}$ (see also Section §4 for
possible physical interpretations of SA pulses). Therefore, except for
spherical harmonic degree $\ell=$1 (where the Coupled-Earth dynamo model is
known to present too weak SV), it is expected that the estimates for
$\tau_{sa}$ from field models remain below the one derived from the Coupled-
Earth model. As seen in Figure (10), this weak constraint is respected by
recent magnetic field models (as shown for the chaos-7 and mcm models). For
the chaos series of models, the progress in modelling and the precise
adjustment of the smoothing constraints led to a significant decrease of the
time scales towards large spherical harmonic degrees. The mcm model that uses
alternative temporal representation (see above), presents even lower time
scale for degrees $\ell$ from 6 to 13, but slightly larger values at the
lowermost degrees. It is reassuring that all models seem to agree for
intermediate length-scales ($\ell$ around 5). Clearly, above degree $\ell=$13
estimates for $\tau_{sa}$ are poor for all models (the SA power drops as it is
badly constrained). However, there is no insurance yet that at lower spherical
harmonic degrees the time scales estimates are exact. It is possible that some
fast variations of the core field are accounted for within the induction part
of the models, or that rapid induced and ionospheric signals are modelled as
generated in the core. We note however, that the typical 11-yrs value for the
acceleration time scale obtained in Christensen et al. (2012) seems
overestimated, in agreement with the latest simulations of the geodynamo.

Figure 10: Acceleration time scales $\tau_{sa}(\ell)$ as a function of
spherical harmonic degree $\ell$, for year 2006, and as estimated with
successive versions of chaos field model (Finlay et al., 2020) and the mcm
model (Ropp et al., 2020). Are also shown a characteristic time scale derived
from the Coupled Earth numerical dynamo model (red line – Aubert et al.,
2013), and a time scale obtained neglecting the flow acceleration in the core
and the diffusion (black line – see the main text).

Figure 11: Time scales from 2000 to 2020, for a spherical harmonic degree
range $\ell\in[1;13]$ (units in $\log_{10}$ of years). Time scales are shown
for the mcm model (left) and chaos-7 model (right).

These time scales are varying with time over the satellite era. Figure (11)
presents temporal evolution of $\tau_{sa}(\ell)$ derived from the mcm model
(Ropp et al., 2020) and the chaos-7 model (Finlay et al., 2020) from 2000 to
2020. Edge effects are visible for the mcm model in 2000 and 2020. Both models
show a significant variability up to spherical harmonic degree $\ell=6$, above
which the chaos-7 SA appears smoother. However, values obtained with mcm are
associated with large uncertainties. A good agreement in terms of variability
is found for degrees 3, 4 and 5. Discrepancies are important for the spherical
harmonic degree $\ell=1$ where part of the variability coincides with
satellite data availability. Much more variability is obtained with the
chaos-7 model, and this difference with the mcm model is probably due to
different way of handling high latitude night-time ionospheric signal, and/or
induced fields.

As described above, the variability of the magnetic field is a key information
to better understand the outer core dynamics. Progresses have been tremendous
in terms of spatial and temporal description of this variability. In
particular this leads to the possibility to identify waves and other transient
phenomena in the core.

##  4 Possible sources to inter-annual geomagnetic changes

The apparent periodicity of SA pulses (Chulliat and Maus, 2014; Finlay et al.,
2016) has motivated the search for waves as an origin to rapid SV changes. In
the presence of global rotation the fluid response to transient perturbations
consists of inertial waves whereas, within an ambient magnetic field, the
responses to perturbations are made of Alfvén waves. For the former waves the
restoring force comes from the Coriolis force, while for the latter it is the
Lorentz force (for a review, see Finlay et al., 2010; Jault and Finlay, 2015).
Their typical periods are comparable with the rotation and Alfvén times,
respectively:

$\tau_{\Omega}=1/2\Omega\mbox{ and }\tau_{A}=c/V_{A}\,,$ (11)

with $\Omega\simeq 7.3\times 10^{-5}$ rad.s-1 the Earth’s rotation rate,
$c=3485$ km the core radius, and $V_{A}=B/\sqrt{\rho\mu}$ the Alfvén speed
($\rho\simeq 10^{4}$ kg.m-3 is the core density and $\mu=4\pi\times 10^{-7}$
H.m-1 the magnetic permeability). For a magnetic field intensity $\approx 5$
mT within the core (Gillet et al., 2010), one gets $\tau_{A}\approx 2$ years
that is much longer than $\tau_{\Omega}$, as measured by the Lehnert number
$Le=\tau_{\Omega}/\tau_{A}\approx 10^{-4}$. For planetary scale Alfvén waves
to propagate, their periods must be much less than the diffusion time
$\tau_{\eta}=c^{2}/\eta$ (with $\eta\simeq 1$ m2.s-1 the magnetic
diffusivity). It is the case in the Earth’s core, as measured by the Lundquist
number $Lu=\tau_{\eta}/\tau_{A}\approx 10^{5}$.

When both rotation and magnetic field are present, the response to a
planetary-scale perturbation by inertial waves is thus much faster than that
by Alfvén waves. The predominance of the Coriolis force in the momentum budget
has two important consequences. First, transient flows tend to be almost
invariant along the rotation axis (Jault, 2008), or quasi-geostrophic (QG).
Second, for $Le\ll 1$ there should exist no large length-scales QG Alfvén
modes in Earth’s core, but instead magneto-Coriolis (MC) modes (Hide, 1966).
These eigen-modes in the presence of global rotation and an imposed field
divide for $Le\ll 1$ into slow and fast modes, whose periods are commensurate
with respectively

$\tau_{MC}^{s}\approx\tau_{A}^{2}(L/c)^{2}/\tau_{\Omega}\mbox{ and
}\tau_{MC}^{f}\approx\tau_{\Omega}c/L\,,$ (12)

with $L$ a typical length-scale (for a review, see Finlay et al., 2010). Fast
QG-MC modes are only weakly sensitive to the magnetic field and correspond to
Rossby modes (i.e. axially invariant inertial modes, see Zhang et al., 2001).
At large length-scales their periods remain much smaller than $\tau_{A}$ (it
spans approximately the range between 0.1 and 2 months for azimuthal wave
numbers $m\leq 10$ and radial wave numbers $N\leq 5$). Because their ratio of
magnetic to kinetic energy is weak their detection in magnetic data is very
difficult (furthermore in the monthly period range the core signal is hindered
by external sources – see Figure (3)). Conversely, slow QG-MC (or ‘
magnetostrophic’) modes are dominated by the magnetic energy, and their period
reaches centuries to millennia. Thus, a priori, they cannot explain SA pulses
(but see below).

Nevertheless one particular family of Alfvén modes exists. These ‘torsional’
modes are organised as cylinders co-axial with the rotation axis (geostrophic
cylinders). For these modes, the projection of the Coriolis force vanishes, so
that the fluid response to a perturbation naturally comes from the Lorentz
force (Braginsky, 1970). Detected from geomagnetic field models at a period of
6-yrs (Gillet et al., 2010), their outward propagation across the fluid core
in about 4 years constrains the intensity of the field in the bulk of the
core. However, their flow perturbation magnitude (of r.m.s. $\approx 0.3-0.5$
km.yr-1) is not large enough to explain alone the observed SA pulses, and
therefore non-axisymmetric flow contributions about 3 to 5 times larger are
required (Gillet et al., 2015, 2019; Kloss and Finlay, 2019).

Several possibilities have been proposed to explain such intense non-zonal
motions at inter-annual periods, among which are waves involving a stratified
layer at the top of the core. The restoring force then comes from an interplay
between the magnetic, Archimedes and Coriolis forces. Such ‘MAC’ waves are
appealing in the context of a heat flux at the CMB weaker than the adiabatic
flux, a scenario made possible with a relatively high thermal conductivity of
the core (Pozzo et al., 2012; Ohta et al., 2017), and motivated by
seismological studies (Helffrich and Kaneshima, 2010). The role of buoyancy in
the stable layer is governed by the steepness of the density profile
$\rho^{\prime}(r)=\rho(r)-\rho_{0}(r)$ below the CMB (with $\rho_{0}$ the
reference density profile for stability, the adiabatic one in the Boussinesq
approximation). It is measured by the Brunt-Vaïsala frequency

$N=\sqrt{-\frac{g}{\rho}\frac{d\rho^{\prime}}{dr}}\,,$ (13)

where $g(r)$ is the gravity acceleration. For reduced models of a hidden ocean
at the top of the core, the control parameters are only the density jump
$\Delta\rho$ across the layer, and the height $H$ of the stratified layer
(Braginsky, 1993, 1999). By tuning $N$ (or $H$ and $\Delta\rho$), it is
possible to match the inter-annual time-scales of interest for SA pulses
(e.g., Chulliat et al., 2015; Chi-Durán et al., 2020). Note that the
configuration of the background field has a significant influence on the
spatial distribution of the waves (see Knezek and Buffett, 2018). In
particular under some conditions MAC waves may be trapped in the equatorial
belt (Buffett and Matsui, 2019), as is the majority of SA pulses witnessed
over the satellite era.

Meanwhile, the existence of a stratified layer is still a debated issue from
several aspects: seismology (Irving et al., 2018), geodynamo simulations
(Gastine et al., 2020) and high-pressure-temperature experiments (Konôpkovà et
al., 2016). Furthermore, rapid SV events have been found in geodynamo
simulations, in the absence of any stratification, as the regime of parameters
gets closer to Earth’s conditions (Aubert, 2018). There, they are interpreted
as the signature of QG Alfvén waves carried by intense magnetic field lines
(Aubert and Finlay, 2019). Such Alfvén waves can exist in spite of the
Coriolis force because of their reduced extension in the cylindrical radial
direction. If the Coriolis force determines their QG nature, it seems to only
marginally affect their period, slightly above $\tau_{A}$ (Aubert and Gillet,
2021). Still the complexity of the background field within the dynamo
simulations, with strong heterogeneities, renders more subtle to decipher the
roles played by the Coriolis, Lorentz and diffusion terms along the wave
propagation (Aubert, 2019). Even though SV field models are limited in spatial
resolution, QG Alfvén waves might be detectable as they reach the equatorial
area, because of the spherical geometry of the core that (i) focuses the wave
energy and (ii) projects sharp gradients in the cylindrical radial axis onto
smoother spatial variations on the spherical boundary. The detection of such
QG Alfvén waves in dynamo simulations is supported by the finding of Gerick et
al. (2020): the periods of the slow QG-MC waves decrease to inter-annual
values as the spatial complexity of its modes increases. These eigen-solutions
furthermore focus the SV signals in the equatorial belt, and may explain the
minimum of radial velocity at $\pm 10^{\circ}$ in latitude recovered from
magnetic data (Gillet et al., 2015). They may therefore be the elementary
components needed to understand the SA derived from satellite data. The
several families of waves discussed above, and their main characteristics
regarding SA events, are summarised in Table 1.

Table 1: Waves in the inter-annual period range of interest for explaining SA changes, with their main physical ingredients. a the period of long period inertial (or Rossby) waves increases towards $\tau_{A}$ with increasing spatial complexity (Zhang et al., 2001). b detected from magnetic data (Gillet et al., 2010). c from the eigen-mode study by (Gerick et al., 2020). d detected in numerical simulations (Aubert and Finlay, 2019; Aubert and Gillet, 2021). † although essentially of magnetic nature, these waves still are inprinted by the Coriolis force (see text for details). e for their focalization in the equatorial area, see Buffett and Matsui (2019).  name | Coriolis | magnetic | stratification | must involve short | SA enhanced in
---|---|---|---|---|---
| force | field |  | wave lengths ? | the equatorial belt
slow inertial wavesa | yes | no | no | in the cylindrical | yes but carry too weak
|  |  |  | radial direction, yes | magnetic energy
torsional Alfvén wavesb | no | yes | no | not required for | cannot alone explain
|  |  |  | the gravest modes | the entire SA signal
QG-MC modesc | yes | yes | no | in the cylindrical | yes
|  |  |  | radial direction, yes |
QG-Alfvén wavesd | yes† | yes | no | yes | most often
MAC wavese | yes | yes | yes | yes (at least in the | possibly (depends on
|  |  |  | radial direction) | the background field)

One should keep in mind some caveats when interpreting the strong SA
components in models of the core magnetic field. First, a significant part of
the magnetic signal is likely associated with unresolved patterns, consequence
of the advection process (equation 9 & 10) that nonlinearly mixes wavelengths
(e.g., Pais and Jault, 2008; Gillet et al., 2019). This reduces the resolution
of the reconstructed core dynamics. Furthermore, the quasi-periodic nature of
observed SA pulses (Soloviev et al., 2017) remains questionable, because of
the limited spatio-temporal resolution of geomagnetic field models (Gillet,
2019), inherent in the difficult separation of magnetic sources (see section
3.2 and also Finlay et al., 2017). Indeed, when looking at rapid SV changes
from magnetic field models, the most intense patterns necessarily show up at
intermediate length-scales, as towards short wave-lengths fast variations are
generally smoothed out (see Figure 9 and Section 3.2).

Finally, whatever the source of SA pulses, the core dynamics do generate a
magnetic field with a temporal spectrum behaving approximately as $S(f)\propto
f^{-4}$ at observatory sites, or equivalently (in the case of an insulating
mantle) through series of geomagnetic Gauss coefficients at the core surface.
This spectral property has been recovered in dynamo simulations, even when run
at parameters far from Earth-like values, for the dipole moment (Olson et al.,
2012; Buffett and Matsui, 2015) as well as for non-dipole coefficients
(Bouligand et al., 2016). However, the range of frequencies relevant to
interpret inter-annual changes must also be considered in comparison with the
Alfvén time $\tau_{A}$ on the one side, and with the turn-over time
$\tau_{U}=c/U$ on the other side. Here $U$ is the typical flow speed in the
core, and $c$ the core radius, leading for $U\approx 20$ km.yr-1 to
$\tau_{U}\approx 150$ yr. The spectral range in $f^{-4}$ for the Earth’s core
is found for frequencies in the range $\tau_{A}\leq 1/f<\tau_{U}$. It is
numerically very expensive to reach geophysical values of the Alfvén number
$Al=\tau_{A}/\tau_{U}\approx 10^{-2}$ in numerical geodynamo simulations
(Schaeffer et al., 2017). When $Al=O(1)$, all the processes at work in Earth’s
core at periods from a few years to a century are indistinguishable. It is
only recently, by using specific parameterisation of turbulent processes, that
values of $Al$ significantly lower than $O(1)$ have been achieved (Aubert et
al., 2017). Only such extreme numerical experiments present a dynamics usable
to replicate the physics of rapid SV changes. They show that torsional waves
propagate predominantly outward, as recovered from magnetic records, and are
constantly excited. Furthermore, as numerical geodynamo parameters get closer
to Earth’s conditions, jerk events become ubiquitous, and the range where the
temporal spectrum for the SA is approximately flat (or equivalently that of
the main field $\propto f^{-4}$) becomes wider (Aubert, 2018; Aubert and
Gillet, 2021).

##  5 Conclusion

In this short overview we have described first the observational evidences of
fast variations of the magnetic field generated in the liquid outer core of
the Earth. These evidences directly derive from processing steps applied on
magnetic observatory or satellite data. Both show episodes of fast variations
of the geomagnetic field secular variation, mainly at mid- and low-latitudes.
The application of modelling techniques either localised or global, shows that
these fast variations are mainly due to strong, localised, spots of
acceleration at the core mantle boundary. These are generally, but not always,
located close to the equator under Indonesia or Central America, or in the
northern hemisphere along the meridian 100∘E.

In order to understand these observations, one of the pressing questions that
must be addressed is the temporal variability acceptable for core field
models. An answer to this question will necessarily come from a combination of
theoretical approaches that provide the possible characteristics
(periodicities, strength, location) of a signal generated in the core, and an
observational approach that extracts core field signatures from the complex
mix of signals observed at the Earth’s surface or in its immediate
environment. Regarding this latter approach, the data coverage provided by
satellite missions is such that the main difficulty to answer this question is
the characterisation and separation of the core field from sources of internal
origins, as seen from satellite altitudes. The fields induced in the mantle by
rapidly varying external fields and ionospheric contributions are the sources,
internal to the satellite sampling region, most difficult to separate.

Current field models present time scales for the secular acceleration much
shorter than previously thought, of the order 10 to 15 years for the largest
wavelengths (spherical harmonic degrees 1 and 2), down to 6 to 7 years from
spherical harmonic degrees 5 or 6 up. They present also a temporal variability
that has not been yet studied, but certainly carries information on the core
dynamics. However, these time scale values may still be over-estimated and
hide part of the complexity of the underlying dynamics. In particular
$\tau_{sa}$ remains much larger than the expected cut-off period (one year or
less) where the magnetic core field variation would be smoothed out by the
magnetic diffusion within the core or by electrical currents in the conductive
mantle. Nevertheless, it is possible that the obtained $\tau_{sa}$ values are
geophysically relevant since extrapolation of the current dynamo runs the
closest to Earth’s conditions suggest that time scales of $\approx$7 years are
possible.

However, these dynamo simulations do not explore configurations where there is
a stratification at the top of the core and alternative processes for
generating inter-annual motions. We have tried here to give an overview of the
possible transient responses that have been documented so far. It is not
absolutely an exhaustive picture, as we ignore possible forcings from Earth
nutation and precession that may be of importance as the Earth’s core deviates
from a perfect sphere (Le Bars et al., 2015). In a turbulent regime, these
forcings may also lead to QG flows and trigger dynamics at inter-annual time-
scales.

Currently, the scientific community has a limited knowledge on the three-
dimensional distribution of the mantle conductivity. In addition we lack data
at Earth’s surface to better describe ionospheric fields, and we have
difficulties to separate lithospheric or oceanic contributions from the core
field. It follows that the most promising path to better describe the core
field dynamics is to identify through theoretical studies their expected main
characteristics (e.g., propagating waves, time scales) and try to identify
their signals in observatory and satellite data sets. This can be done through
a co-estimation with core surface flow models (e.g., Barrois et al., 2018;
Baerenzung et al., 2018), or more sophisticated assimilation approaches based
either on dynamo simulations (e.g., Sanchez et al., 2019) or on reduced QG
equations (see Jault and Finlay, 2015; Gerick et al., 2020; Jackson and
Maffei, 2020), although these have not led yet to any application to
geophysical data. Given the time scales involved in the core field processes
and the core complex spatio-temporal dynamics, it is crucial to continuously
measure the magnetic field with satellites carrying magnetometers and in
magnetic observatories. Some information content on the core dynamics might be
obtained from other types of data, such as Earth’s rotation, gravity or
geodesy. The core variations signature in these data is certainly very weak,
however their use in synergy with magnetic data will lastly improve our
understanding of the liquid core dynamics.

###### Acknowledgements.

We thank the anonymous reviewer for remarks that help us improving the
manuscript. This work was partially supported by the CNES. It is based on
observations with magnetometers embarked on Swarm mission. NG contribution has
been also funded by ESA in the framework of EO Science for Society, through
contract 4000127193/19/NL/IA (SWARM + 4D Deep Earth: Core). MM: the research
leading to these results has received funding from the European Research
Council (ERC) GRACEFUL Synergy Grant No 855677. MDH was supported by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 772561) and also partly
by Swarm DISC activities, funded by ESA contract no. 4000109587. This is the
IPGP contribution number XXXX

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