# Relating the South Atlantic Anomaly and geomagnetic flux patches

Filipe Terra-Nova, Hagay Amit, Gelvam Hartmann, Ricardo I.F. Trindade, Katia J Pinheiro

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Filipe Terra-Nova, Hagay Amit, Gelvam Hartmann, Ricardo I.F. Trindade, Katia J Pinheiro. Relating the South Atlantic Anomaly and geomagnetic flux patches. Physics of the Earth and Planetary Interiors, 2017, 266, pp.39-53. 10.1016/j.pepi.2017.03.002. hal-02348660

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# Relating the South Atlantic Anomaly and geomagnetic flux patches

Filipe Terra-Nova¹,*, Hagay Amit¹, Gelvam A. Hartmann², Ricardo I. F. Trindade³, Katia J. Pinheiro⁴

¹ CNRS, Université de Nantes, Nantes Atlantiques Universités, UMR CNRS 6112, Laboratoire de Planétologie et de Géodynamique, 2 rue de la Houssinière, F-44000 Nantes, France.

² Instituto de Geociências, Universidade Estadual de Campinas, Rua João Pandiá Calógeras, 51, 13083-870, Campinas, Brasil.

³ Departamento de Geofísica, Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, Rua do Matão, 1226, Cidade Universitária, 05508-090, São Paulo, Brasil.

⁴ Observatório Nacional, Rua General José Cristino, 77, 20921-400, Rio de Janeiro, Brasil.

*Corresponding author: filipe.terranova@univ-nantes.fr

## Abstract

The South Atlantic Anomaly (SAA) is a region of weak geomagnetic field intensity at the Earth's surface, which is commonly attributed to reversed flux patches (RFPs) on the core-mantle boundary (CMB). While the SAA is clearly affected by the reversed flux region below the South Atlantic, we show that the relation between the intensity minimum at Earth's surface and RFPs is not straightforward. We map a field-dependent intensity kernel (Constable, 2007a) to study the relation between the radial geomagnetic field at the CMB and the field intensity at Earth's surface. Synthetic tests highlight the role of specific patches in determining the surface intensity minimum and demonstrate that the SAA
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can indeed be explained by a few intense patches. Supported by these synthetic tests, our analysis highlights the role of specific flux patches (reversed and normal) in determining the location of the SAA minimum. We show that the level of axial dipolarity of the field determines the stability of the relation between the SAA minimum and RFPs. The present position of the SAA minimum is determined by the interplay among several robust geomagnetic flux patches at the CMB. The longitude of the SAA minimum appears near the longitude of the Patagonia RFP due to the low-latitude normal flux patches (NFPs) near Africa and mid-Atlantic which diminish the effect of the Africa RFPs. The latitude of the SAA minimum is lower than the Patagonia RFP latitude due to the South Pacific high-latitude NFP and the axial dipole effect. The motion of the SAA minimum is explained by the motions and changes in intensity of these robust geomagnetic flux patches. Simple secular variation (SV) scenarios suggest that while the SAA path can be explained by advection, its intensity decrease requires magnetic diffusion. In addition these SV scenarios provide some speculative predictions for the SAA.

Keywords: South Atlantic Anomaly, geomagnetic flux patches, axial dipole, core-mantle boundary, kernel functions, secular variation.

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# 1 Introduction

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The South Atlantic Anomaly (SAA) is a region where the lowest geomagnetic field intensity values at the Earth's surface are observed. It is one of the most important features of the Earth's surface field where a strong non-dipolar field is present (Hartmann and Pacca, 2009). The SAA minimum intensity is currently located at inland South Brazil (Thébault et al., 2015). It has been decreasing with time (Hartmann and Pacca, 2009; Finlay et al., 2010), in conjunction with the rapidly decreasing geomagnetic dipole intensity (e.g. Finlay, 2008). The decreasing dipole is in part associated to the proliferation of reversed flux patches (RFPs) below the South Atlantic and their poleward motion (e.g. Gubbins, 1987; Hulot et al., 2002; Olson and Amit,
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2006; Finlay et al., 2012, 2016; Terra-Nova et al., 2015). The SAA has been commonly at-
tributed to RFPs at the core-mantle boundary (CMB) (e.g. Tarduno et al., 2015). Changes in
the SAA position and intensity are intrinsically related to the flow patterns at the top of the
core especially near RFPs (Aubert, 2015).

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RFPs may emerge by expulsion of toroidal field lines due to fluid upwelling at the top of the core (Bloxham, 1986), therefore RFPs may be paired with normal flux patches (NFPs) (Jackson, 2003; Terra-Nova et al., 2016). Recently, Finlay et al. (2016) combined observed field models and equatorially symmetric core flow models to show that an RFP located below Patagonia breaks the symmetry of advective sources of the axial dipole moment (ADM) changes thus causing its decrease. Although Olson and Amit (2006) used a distinctive helical core flow model, they also found a similar pattern of advective ADM sources with symmetry breaking below South America. We will show that the Patagonia RFP also determines the longitude of the SAA minimum because this patch does not pair with an NFP.

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The SAA persistence is under debate. Aubert (2015) obtained geomagnetic forecasting
based on a data assimilation technique that relies on statistical properties of a numerical dy-
namo model that reproduces some robust historical field and secular variation features. He
predicted that the SAA will drift westward to eastern Pacific Ocean until 2115. Based on local
field intensity measurements of African artefacts, Tarduno et al. (2015) argued for persistent
recurrence of low field intensity associated to the SAA. They suggested that the SAA location
is ancient due to prescribed positions of RFPs below Africa which are controlled by the het-
erogeneous lower mantle. Overall, it is under debate whether the current location of the SAA
is quasi-stationary (due to boundary control) or strongly transient. While the SAA position is
clearly affected by the reversed flux region below the South Atlantic, we will show that relating
the SAA minimum to RFPs is not trivial.

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Previous studies related the surface components of Earth's magnetic field and the radial
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field at the CMB via kernel functions (e.g. Gubbins and Roberts, 1983; Johnson and Constable, 1997; Gubbins, 2004; Constable, 2007a). Johnson and Constable (1997) derived field-dependent kernel functions for declination and inclination to study the effect of low geomagnetic data coverage on global modeling. These methodologies provide the tools to derive a field-dependent kernel function to relate the radial field at the CMB to the surface intensity field (Constable, 2007a). In particular, this intensity kernel may be applied as an analytical tool to study the relation between the radial field at the CMB and the SAA.

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Based on kernel functions, Chulliat et al. (2010) studied the relation between RFPs at the CMB to magnetic pole motion at Earth's surface. Such a formalism has not yet been applied for the relation between RFPs at the CMB and the minimum intensity of the surface field as observed in the SAA region. Tracking of the SAA minimum was performed at Earth's surface (Hartmann and Pacca, 2009; Finlay et al., 2010; Aubert, 2015), and tracking of RFPs was performed at the CMB (Terra-Nova et al., 2015, 2016), but the relation between the SAA and RFPs has not been demonstrated. In this study, we use historical and modern geomagnetic field models to unravel the relations between geomagnetic flux patches and the SAA minimum location, motion and intensity changes.

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# 2 Theory and Methods

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## 2.1 Kernels

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To generally assess how the radial magnetic field at a given point at the CMB affects the field at a given point above the CMB we consider appropriate kernel functions (Gubbins and Roberts, 1983; Constable et al., 1993; Johnson and Constable, 1997). The following equation

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reconstructs the field at the position vector from Earth's center $\mathbf{r}$:

$$ \mathbf{B}(\mathbf{r}) = \int_S \mathbf{G}(\mathbf{r}|\hat{s}) B_r(c, \phi', \theta') \sin\theta' d\phi' d\theta' \quad (1) $$

where $\mathbf{G}(\mathbf{r}|\hat{s})$ contains the kernel functions relating the CMB radial field $B_r$ with the field vector $\mathbf{B}$ at $\mathbf{r}$, $c$ is the CMB radius, $S$ is the CMB surface and $\hat{s}$ is its normal unit vector. The pairs $(\phi, \theta)$ and $(\phi', \theta')$ represent the position (longitude, colatitude) of a given point at $\mathbf{r}$ and at the CMB, respectively. The kernel function $G(\mu)$ for the Neumann problem in Laplace's equation is (Mikhlin, 1970):

$$ G(\mu) = \frac{c}{4\pi} \left( \ln \left( \frac{f + x - \mu}{1 - \mu} \right) - \frac{2x}{f} \right) \quad (2) $$

where $x = c/r$, $r$ is the radial distance, $f = (1 - 2x\mu + x^2)^{1/2}$ and $\mu = \cos\alpha$ where $\alpha$ is the angle between the points $(\phi, \theta)$ and $(\phi', \theta')$.

The kernel function for the radial field at Earth's surface (Gubbins and Roberts, 1983) is

$$ G_r = \left( \frac{dG}{dr} \right)_{r=a} = \frac{b^2(1-b^2)}{4\pi f^3} \quad (3) $$

where $b = c/a$ and $a$ is Earth's radius. The kernel functions for the horizontal vector field components are

$$ G_{\theta} = \frac{1}{r} \frac{\partial G}{\partial \mu} \frac{\partial \mu}{\partial \theta} = \frac{1}{r} \frac{\partial G}{\partial \mu} (\cos\theta \sin\theta' \cos(\phi - \phi') - \sin\theta \cos\theta') \quad (4) $$

$$ G_{\phi} = \frac{1}{r} \frac{\partial G}{\partial \mu} \frac{1}{\sin\theta} \frac{\partial \mu}{\partial \phi} = -\frac{1}{r} \frac{\partial G}{\partial \mu} (\sin\theta' \sin(\phi - \phi')) \quad (5) $$
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$$ \frac{1}{r} \frac{\partial G}{\partial \mu} = - \frac{x}{4\pi} \left( \frac{1 - 2x\mu + 3x^2}{f^3} + \frac{\mu}{f(f + x - \mu)} - \frac{1}{(1 - \mu)} \right) \quad (6) $$

107 Note that (4) and (5) depend on several functions of $(\phi, \theta, \phi', \theta')$ and thus cannot be expressed exclusively as a function of $\mu$. However, a kernel function for the horizontal field can be expressed exclusively by $\mu$ (for details see e.g. Gubbins and Roberts, 1983):

$$ G_h = (1 - \mu^2)^{1/2} \left( \frac{1}{r} \frac{\partial G}{\partial \mu} \right)_{r=a} \quad (7) $$

110 Fig. 1 shows $G_r$ and $G_h$ as functions of $\mu$ as well as $G_\theta$ and $G_\phi$ for an arbitrary surface point at $(0^\circ, 90^\circ)$. $G_r$ and $G_h$ have maxima at $\mu = 1$ and $\mu = 0.92$, corresponding to $\alpha = 0^\circ$ (i.e exactly beneath $(\phi, \theta)$) and $\alpha = 23^\circ$, respectively (e.g. Gubbins, 2004). $G_\theta$ and $G_\phi$ have a more complex configuration with extrema $23^\circ$ away from the surface point north/south and east/west, respectively. $G_\theta$ and $G_\phi$ have negative values which modulate the signal to their respective influenced surface components, unlike $G_h$ and $G_r$ that are positive everywhere.

Since we wish to find the relation between CMB features and the surface intensity, a kernel function that relates the radial field at the CMB to the field intensity at the surface is needed. Following Johnson and Constable (1997) the intensity kernel is

$$ G_F = \frac{\partial F}{\partial B_r} = \frac{1}{F} (B_\theta(a) G_\theta + B_\phi(a) G_\phi + B_r(a) G_r) \quad (8) $$

where $B_\theta$, $B_\phi$, and $B_r$ are the surface field components pointing to South, East and outward from the Earth's surface respectively, and the surface field intensity is $F = \sqrt{B_\theta^2 + B_\phi^2 + B_r^2}$. It is worth noting that $G_F$ also depends on the surface field components, unlike $G_\theta$, $G_\phi$, $G_r$ and $G_h$ that depend only on the geometry between CMB and surface points. Because $F$ is non-linear to $B_r$, it cannot be reconstructed by an analogous equation to (1). The kernel (8)
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reflects the sensitivity of the surface intensity to changes in the radial field on the CMB about
a given background field.

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## 2.2 Identification

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Previous studies characterized the SAA by an area at Earth's surface bounded by a certain
low field intensity value (e.g. Pavón-Carrasco and De Santis, 2016). Such a measure is indeed
relevant for spacecraft failures. We characterize the SAA by the point of minimum intensity (as
in e.g. Hartmann and Pacca, 2009; Finlay et al., 2010; Aubert, 2015). Although our measure
is not directly related to spacecraft safety, it does not rely on an arbitrary threshold and it
allows for temporal tracking which provides direct insight to core dynamics. Overall, the
time-evolution of the minimum intensity and the size of the area below an intensity threshold
are well anti-correlated (Pavón-Carrasco and De Santis, 2016) so in practice the location of the
minimum intensity as well as its value provide a satisfactory description of the SAA.

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We perform two steps to identify the position of the SAA minimum. First, we locate the
minimum value of the field intensity on a spherical surface of a 1° × 1° regular grid. Second,
we interpolate this grid point and its near neighbors using second degree polynomial functions
in longitude and latitude.

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To identify RFPs at the CMB snapshots we used the same topological algorithms of Terra-
Nova et al. (2015), with one adjustment. Here we use a stronger intensity criterion, due to
the higher spatial resolution of the historical and modern field models compared to that of
the archeomagnetic field models for which the algorithms were originally designed, in order
to filter out the more abundant weak RFPs in the historical and modern field models. Only
RFPs with peak values larger than 0.6 of the most intense RFP of the same snapshot pass the
criterion. As in Terra-Nova et al. (2016) we adjust these algorithms to also identify NFPs using

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a stronger intensity threshold of 0.75.

## 3 Synthetic tests

Intense geomagnetic flux patches at the CMB (Christensen et al., 2010) which are observed in historical field models (Jackson, 2003) are also robust features of the field on millennial timescales (Amit et al., 2011; Terra-Nova et al., 2015, 2016) and possibly on much longer timescales as well (Kelly and Gubbins, 1997; Constable, 2007b). Their surface expression is not trivial due to the mixing of upward continued spherical harmonic contributions. To gain insight into the role of such patches in localizing the minimum intensity at the Earth's surface, we performed several synthetic tests of simple radial magnetic field configurations to guide us in the interpretation of field models based on observations. We built synthetic radial magnetic fields from a background axial dipole field $B_r^D$ superimposed by several localized flux patches. This background field is essential to define the polarity of each patch (normal or reversed). The actual dipole is a sum of the background dipole and the dipole associated with the patches. For each synthetic model we calculated the Earth's surface intensity and identified its minimum (or in some cases minima).

Following Amit (2014), the radial field of a synthetic intense magnetic flux patch was modeled as a 2D isotropic Gaussian:

$$B_r^P = A_0 \left( \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{d^2}{2\sigma^2}} \right), \qquad (9)$$

where $A_0$ is the amplitude corresponding to the radial field peak at the center of the patch, $d$ is the great circle distance from the center of the magnetic flux patch to a point at the CMB and $\sigma$ is the standard deviation which characterizes the width of the patch. The amplitude and the
width are chosen to roughly mimic the morphology of Earth-like patches (Amit, 2014). The synthetic field $B_r(\phi, \theta)$ is then given by:

$$B_r(\phi, \theta) = B_r^D(\theta) + \sum_i (B_r^P(\phi, \theta))_i, \quad (10)$$

where $i$ denotes summation over multiple patches. The widths are identical for all patches except for the cases that simulate the effect of magnetic field stretching. The amplitudes of the patches were set to maintain $\int_S B_r dS = 0$.

## 3.1 Minimum surface intensity and core-mantle boundary patches

For fundamental understanding, as a first step we built eight different synthetic radial magnetic fields to study the relation between the positions of patches at the CMB and the position of the minimum field intensity point at the Earth's surface $F_{min}$. The synthetic fields setups are summarized in Table 1. The results of the synthetic tests, including the position and the intensity of $F_{min}$, the area of weak intensity and the axial dipole of the synthetic fields are also given in Table 1. Since the intensity is affected by the axial dipole, we calculated a relative value of minimum intensity normalized by that of the background axial dipole contribution $F_{min}/F_{min}^D$ ($F_{min}^D$ is identical for cases 1-8). The closer this ratio is to unity the lower the influence of the synthetic patches on the $F_{min}$ value. Likewise we monitored the relative axial dipole $m_z/m_z^D$. Lastly, we quantified the area of weak surface intensity $A$ which we defined as the portion of CMB surface with intensity values lower than $1.2F_{min}$ for each case. Smaller area corresponds to a sharper depression of surface intensity morphology, i.e. higher field roughness.

Fig. 2 shows the synthetic field models at the CMB and their intensity at the Earth's surface for cases 1, 2, 4 and 8. Case 1 has an RFP at each hemisphere both centered at latitudes 30° with
an offset of 120° longitude between them. It results in two $F_{min}$ with longitudes determined by the RFPs, but their latitudes are ~ 21° lower; without patches, both the magnetic equator and $F_{min}$ appear at the geographic equator (axial dipole effect). Case 2 has an RFP and an NFP in the Southern Hemisphere, again at latitude 30° with an offset of 120° longitude between them. The RFP determines the longitude of $F_{min}$ while the NFP determines the longitude of the maximum intensity along the equator (Fig. 2). The latitude of $F_{min}$ is once again lowered by the axial dipole effect. Case 4 also has an RFP and an NFP in the Southern Hemisphere, but at the same longitude, with the NFP at a higher latitude. The $F_{min}$ longitude is again determined by the RFP, but the proximity of the NFP reduces the $F_{min}$ latitude with respect to that of case 2. Case 8 has two RFPs and an NFP in the Southern Hemisphere, the twice stronger NFP amplitude balancing that of the RFPs. The RFPs are in the same locations as the patches in case 2, and the NFP appears in the same longitude of one RFP but at a higher latitude. The RFP which does not pair with the NFP dictates the longitude of the $F_{min}$. Again latitude reduction is associated to the axial dipole effect.

In almost all cases 1-8 the longitude of $F_{min}$ is determined by the longitude of RFPs. An exception is case 6 that does not contain RFPs. In this case the longitudinal position of the minimum field intensity is farthest from the NFPs (Table 1). Indeed in all cases $F_{min}$ tends to be far from NFPs.

In all cases $F_{min}$ is closer to the equator than to the RFPs. Cases 2-5 have two patches in the Southern Hemisphere, one RFP and one NFP. In cases 2 and 3 these patches have different longitudes, for case 2 the same latitude and for case 3 the NFP is at higher latitude. In both cases the $F_{min}$ latitude is the same, i.e. the latitude of the far NFP has little effect on the location of the $F_{min}$. Cases 4 and 5 have RFP and NFP in the same longitude, with the RFP closer to the equator than the NFP. The $F_{min}$ is closer to the equator than in cases 2 and 3, thus the proximity of the NFP appears to reduce the $F_{min}$ latitude. Comparing cases 4 and 5 we
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observe that the $F_{min}$ latitude is closer to the equator in the latter (where the RFP is at higher latitude) because the proximity of NFP to RFP reduces the RFP effect.

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Case 7 is set as case 1 but with twice stronger RFPs amplitudes. This case has the lowest value of the ratio $m_z/m_z^D$ (Table 1) and $F_{min}$ is farthest from the equator, demonstrating the competition between the patches and the axial dipole in determining the latitude of $F_{min}$. Case 7 also has the lowest $F_{min}/F_{min}^D$ value, further demonstrating how strong RFPs can efficiently diminish locally the surface field intensity. Several cases may suggest that $m_z/m_z^D$ and $F_{min}/F_{min}^D$ are related. However, cases 3 and 4 have the same $m_z/m_z^D$ but $F_{min}/F_{min}^D$ is lower in case 3 since the NFP is not in the same longitude of the RFP as in case 4. Case 6 has $F_{min}/F_{min}^D$ larger than 1. In this case, in the absence of RFPs, NFPs enhance the $F_{min}$ value beyond the background field, i.e. the $F_{min}$ is located where the influence of the NFPs is the lowest. In case 8 the strong NFP gives a large $m_z/m_z^D$ and the strong RFP gives a low $F_{min}/F_{min}^D$. Overall, the lowest values of $F_{min}/F_{min}^D$ were associated to large amplitude RFPs (cases 7 and 8).

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Case 8 with higher amplitude patches shows strong local effects, as evidenced in its smallest weak intensity area $\mathcal{A}$ (see also case 7). Larger $\mathcal{A}$ is associated to the proximity between RFPs and NFPs (cases 4 and 5) or absence of RFPs (case 6). Lastly, we also show in Table 1 results of $F_{min}$ based on the geomagnetic field model CHAOS5 (Finlay et al., 2015) at 2003 expanded until spherical harmonic degree 14. Its $F_{min}$ is at a relatively high latitude 26° indicating strong RFPs effect. In addition, the small $\mathcal{A}$ value is associated with a rather sharp area morphology as expected from the complex field.

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Next we incorporate in our synthetic models some main features that are present in the current geomagnetic field in order to improve our understanding of the relation between CMB patches and the $F_{min}$ at more realistic conditions. In particular, as shown above, while recovering the SAA longitude appears to be feasible, the latitude seems more elusive. We built a series of field models (cases 9-12 in Table 1) that progressively add main features of the geomagnetic field in order to approach the SAA position.

Starting from case 9, we present a first attempt to get somewhat closer to the distribution of the more prominent geomagnetic flux patches, but still within a synthetic framework that allows understanding of the role of each patch. This synthetic field comprises four high-latitude NFPs (two at each hemisphere) and one equatorial NFP, as well as two RFPs in the Southern Hemisphere. The RFPs are at latitudes 30° and longitudes $\pm 60°$, while the high-latitude NFPs (all at latitudes 60°) form two pairs of same longitude in opposite hemispheres at longitudes $\pm 120°$ (Fig. 3). The equatorial NFP is located at the same longitude as one of the RFPs. The resulting $F_{min}$ longitude is determined by the other RFP. The latitude of the $F_{min}$ is again close to the equator.

As mentioned above the level of axial dipolarity plays a role in the $F_{min}$ latitude. With a purely axial dipolar field the $F_{min}$ is exactly at the equator. In case 10 we therefore strongly reduced the background field, with otherwise the same patches configuration as in case 9. The background field reduction leads to an increase in the $F_{min}$ latitude by 3°, but its longitude is less dictated by the South America RFP. In addition the $F_{min}$ area $A$ is larger, thus the $F_{min}$ is shallower and hence less well defined (see Table 1).

In case 11 we added a low-latitude NFP centered at the equator at longitude 0°, and we reduced the amplitude of the high-latitude NFP below south Pacific which has been decreasing
in the historical era (Amit et al., 2011). The weaker high-latitude NFP below south Pacific increases the influence of the South America RFP resulting in $F_{min}$ 4° farther away from the equator. The longitude is more dictated by the South America RFP than in cases 9-10 and is practically identical to that of the present-day geomagnetic field (Table 1). The new equatorial NFP is located right in between the longitudes of the two RFPs, hence its influence on the $F_{min}$ longitude is minor.

In case 12 the RFP below South America was dislocated to the south closer to its position in the current geomagnetic field (Finlay et al., 2015). This resulted in a remarkable increase of 9° in the $F_{min}$ latitude. Overall, case 12 gives an $F_{min}$ location rather close to the SAA position in the CHAOS5 model at 2003 (see Table 1).

The $m_z/m_z^D$ value decreases progressively from case 10 to 12, and consequently the $F_{min}$ latitude also progressively increases. Likewise, the $F_{min}/F_{min}^D$ value decreases from case 10 to 12 as the relative role of the RFPs increases. The area $A$ is also progressively reduced, indicating increasing roughness of the field with more localized and better defined $F_{min}$.

## 3.3 Minimum surface intensity secular variation scenarios

Here we use synthetic models to investigate some possible future scenarios for the SAA. For this purpose we recall the radial component of the magnetic induction equation at the top of the core (where the radial velocity vanishes)

$$ \frac{\partial B_r}{\partial t} = -\vec{u}_h \cdot \nabla B_r - B_r \nabla_h \cdot \vec{u}_h + \eta \left( \frac{1}{r_o^2} \frac{\partial^2}{\partial^2 r} (r^2 B_r) + \nabla_h B_r \right) \quad (11) $$

where $B_r$ is the radial field, $t$ is time, $\vec{u}_h$ is the tangential velocity vector, $\nabla_h = \nabla - \frac{\partial}{\partial r}$, $\eta$ is the magnetic diffusivity, $r_o$ is the core radius and $r$ is the radial coordinate. The term on the left hand side of (11) is the secular variation (SV). The terms on the right hand side of (11)
represent magnetic advection, stretching and radial and tangential diffusion, respectively. We chose case 12 as an initial configuration, and then applied simple kinematic effects of advection (non-uniform westward drift), stretching (contraction/expansion with increased/decreased amplitude) and radial diffusion (intensification) of selected patches. These kinematics effects correspond to some simple solutions of (11).

First we considered the translation of low- and mid-latitude patches, while high-latitude NFPs are kept stationary. Westward drift is a prominent feature of the historical field (Bullard et al., 1950). However, the drift is not uniform. SV peaks appear at the equator and mid-latitudes of the Southern Hemisphere (Finlay and Jackson, 2003; Aubert et al., 2013). Core flow models also exhibit differential rotation with stronger zonal flows at low- and mid-latitudes of the Southern Hemisphere (Amit and Olson, 2006; Holme and Olsen, 2006). In contrast, it has been proposed that high-latitude NFPs are locked to lower mantle thermal anomalies (Gubbins et al., 2007; Willis et al., 2007). Although these NFPs do exhibit some mobility (Amit et al., 2011), the motion of lower latitude patches is faster and more monotonous.

We applied a classical westward drift rate of 0.2°/yr (Bullard et al., 1950) over a period of 100 yr to the low- and mid-latitude patches only. In longitude the $F_{min}$ moves westward in a rather steady rate of 0.13°/yr (Fig. 4a). The $F_{min}$ latitude remains practically unchanged. The slower westward motion of the $F_{min}$ indicates an effect of the stationary NFPs. In particular, the presence of the NFP below South Pacific slows the $F_{min}$ as it drifts towards this NFP.

RFPs emergence may be related to upwelling at the top of the core (Bloxham, 1986; Aubert et al., 2008), whereas NFPs may be concentrated by downwelling near the edge of the tangent cylinder (Olson et al., 1999; Peña et al., 2016). We produced such stretching effects by changing a patch width $\sigma$ while compensating with an inverse change in its amplitude $A_0$. To
conserve magnetic flux the width and amplitudes changes obey the relation (Roberts, 2007):

$$ \frac{\sigma(t_0)}{\sigma(t_1)} = \frac{A_0(t_1)}{A_0(t_0)} \qquad (12) $$

where $t_0$ and $t_1$ denote two snapshots of the synthetic field. We applied width changes to the RFP below South America which strongly affects the $F_{min}$ location in case 12.

Fig. 4b shows that a wider South America RFP caused by local upwelling reduces the $F_{min}$ latitude while the $F_{min}$ longitude drifts eastward. This $F_{min}$ motion indicates that as the South America RFP weakens it somewhat loses ground to the Africa RFP in terms of determining the $F_{min}$ position. Note that the RFP expansion affects less the $F_{min}$ longitude than its latitude. In addition a wider RFP reduces the field roughness leading to a less localized and less well defined $F_{min}$. When the RFP is contracted by downwelling the $F_{min}$ position remains practically unchanged. This suggests that our choice of patches width is already rather concentrated and an asymptotic behavior is reached when further concentration is applied.

The detectability of diffusion in the observed geomagnetic SV is under debate. It has been demonstrated that field models can be constructed assuming frozen-flux (O'Brien et al., 1997; Jackson et al., 2007). In contrast, proliferation of RFPs, in particular below the Atlantic, has been proposed as a significant cause for the decrease of the geomagnetic dipole intensity (Gubbins, 1987; Olson and Amit, 2006; Finlay et al., 2012). Non-zero radial field integrals over regions bounded by null-flux curves provide evidence for flux expulsion (Gubbins and Bloxham, 1986; Chulliat and Olsen, 2010), though such calculations might be biased by imprecise topology of null-flux contours (Gillet et al., 2013). Core flow inversions from geomagnetic SV commonly neglect diffusion based on large magnetic Reynolds number estimates (Roberts and Scott, 1965). However, numerical dynamos models show clear radial diffusion SV contributions at the top of the shell even for Earth-like magnetic Reynolds numbers (Amit
and Christensen, 2008; Finlay et al., 2016).

We applied amplitude changes to selected patches of case 12 while maintaining their width fixed, in order to mimic radial diffusion SV. The amplitude range considered here corresponds to the amplification range of the total reversed flux in the Southern Hemisphere over 100 yr (Olson and Amit, 2006). RFPs were intensified/dissipated while Northern Hemisphere high-latitude NFPs were dissipated/intensified, respectively, to compensate for the change in the total magnetic flux. The results in Fig. 4c show that the $F_{min}$ longitude is practically constant for any RFPs amplitude. RFPs dissipation clearly reduces the $F_{min}$ latitude due to an increasing axial dipole effect. RFPs intensification gives initially higher $F_{min}$ latitude, but further intensification surprisingly reduces the $F_{min}$ latitude.

Lastly, we explore the effects of the SV kinematic scenarios on the value of minimum intensity and its area $A$ (Fig. 5). The effects of westward drift, which were found to be prominent in changing the $F_{min}$ longitude (Fig. 4), are the least efficient in changing $F_{min}/F_{min}^D$ and $A$. The ratio $F_{min}/F_{min}^D$ and the area $A$ are reduced/increased by contraction/expansion due to stretching, respectively. Intensification/dissipation due to diffusion reduce/increase $F_{min}$ and the area $A$, respectively. Diffusion is by far the most efficient mechanism in terms of intensity change.

## 3.4 Summary

Our main findings from the synthetic tests are: (i) the longitude of the $F_{min}$ is clearly dictated by RFPs; (ii) the $F_{min}$ tends to be far away from the NFPs; (iii) the latitude of the $F_{min}$ is affected by the competition between the axial dipole and the RFPs. Relatively strong axial dipole will result in low-latitude $F_{min}$, whereas relatively strong RFPs will give $F_{min}$ closer to the latitude of the RFPs; (iv) NFPs may cause a reduction in the $F_{min}$ latitude if located next to
the RFP that determines the $F_{min}$ longitude; (v) strong RFPs give more pronounced intensity minimum confined to a smaller area.

The examined SV scenarios provide some speculative predictions for the SAA: (i) westward drift of low- and mid-latitude patches will lead to SAA westward drift, though at a lower speed; (ii) South America RFP dispersion by upwelling will move the SAA to lower latitudes and to a lesser extent eastward; (iii) the influence of South America RFP contraction by downwelling on the SAA motion reaches an asymptote for highly concentrated patches; (iv) RFPs intensification or dissipation will not affect the longitude of the SAA; (v) RFPs expulsion is the most efficient mechanism to reduce minimum surface intensity and to increase the field roughness.

# 4 Application to geomagnetic field models

We used the historical geomagnetic field model gufm1 (Jackson et al., 2000) for the period 1840-1990 and the field model CHAOS5 based on high quality global coverage satellite data (Finlay et al., 2015) for the period 1997-2015 to monitor the South Atlantic Anomaly (SAA). In addition we used the IGRF-12 model for the period 1980-2010 (Thébault et al., 2015) to fill the gap between gufm1 and CHAOS5. The gufm1, CHAOS5 and IGRF-12 models were truncated at spherical harmonic degree $n_{max} = 14, 14$ and $13$, respectively, although obviously different regularizations were applied by the modelers depending on the data coverage and quality.

Fig. 6a shows the SAA minimum tracking over the historical and modern periods. Persistent westward drift is observed. A much slower poleward drift is also found. Between 1900-1940 the SAA minimum turned relatively more to the poleward direction. In the past 20 years the SAA minimum has been moving in a nearly purely westward direction. From 1840 to 2015, the SAA minimum has drifted from the mid Atlantic (17°W, 21°S) to inland
South Brazil (53°W, 28°S), and its intensity has been gradually decreasing (see colors in Fig. 6a). The tracking for the IGRF-12 model connects well with those of the gufm1 and CHAOS5 models. The time evolution of the SAA minimum in longitude and latitude is also shown in Fig. 6. The longitude curve (Fig. 6b) shows roughly two trends, ~ -0.38°/yr before and ~ -0.16°/yr after 1900. The latitude curve (Fig. 6c) shows alternating faster/slower trends, with the latest transition of ~ -0.054°/yr before and ~ -0.004°/yr after 1950, clearly getting nearly constant in latitude at recent times. At present the SAA minimum drifts in a nearly purely westward direction (Fig. 6). Overall between 1840-2015 the SAA minimum moved an angular distance of ~ 33° from the South Atlantic to inland South Brazil.

Fig. 7a compares the time evolutions of the intensity of the SAA minimum and the absolute value of the axial dipole Gauss coefficient $|g_1^0|$. Although these two quantities are distinctive, both curves show approximately linear decreasing trends, with the SAA intensity decreasing much faster. The intensities of the SAA minimum and $g_1^0$ have been reduced by 25% and 9% respectively from 1840 to 2015. It is worth noting that the non-linear parts of the curves in Fig. 7a seem at some times out of phase. We removed the linear fits to these curves and then plotted the non-linear parts of the intensities of the SAA minimum and $g_1^0$ (Fig. 7b). The resulting two curves indeed have maxima/minima at different times, with a time-dependent delay. This delay is larger at earlier times than at recent times.

Fig. 8 shows five snapshots of the kernel functions centered at the time-dependent position of the SAA minimum (Fig. 6a). Because the SAA minimum changes its position with time, the geometric contribution of the radial field at the CMB to the surface field components also changes with time as reflected by the patterns of the kernel functions $G_\theta$, $G_\phi$ and $G_r$. The radial kernel function $G_r$ has the strongest values and accounts mostly for field structures in the region of the CMB right beneath the surface point. In the case of the SAA minimum, $G_r$ is most sensitive to $B_r$ at the CMB below mid Atlantic in 1850 to below inland south Brazil in
2010. The $G_{\theta}$ regions of sensitivity to $B_r$ at the CMB are below the South Atlantic (negative values) and equatorial Atlantic (positive values) in 1850 to below Patagonia (negative values) and Amazon (positive values) in 2010. The $G_{\phi}$ sensitive regions to $B_r$ at the CMB are below west South Africa (negative values) and east Brazil (positive values) in 1850 to below mid Atlantic (negative values) and west South America (positive values) in 2010. According to (8) $G_{\theta}$, $G_{\phi}$ and $G_r$ represent weights of the intensity kernel function $G_F$. The monopolar $G_r$ and the positive surface $B_r$ in the Southern Hemisphere give a monopolar positive contribution to $G_F$ in the Southern Hemisphere. The bipolar $G_{\theta}$ and the negative surface $B_{\theta}$ give a bipolar contribution to $G_F$ (negative/positive in the Northern/Southern hemisphere, respectively). The resulting $G_F$ is the sum of these two dominant structures.

Corresponding snapshots of the surface field intensity, the intensity kernel function $G_F$ with respect to the SAA minimum and the radial field $B_r$ at the CMB with identified RFPs are shown in Fig. 9. In general, the SAA minimum is not found at the exact position of the $G_F$ maximum. Moreover, at most snapshots the SAA is not found in the center of mass of the identified RFPs or the reversed flux region (Fig. 9). At 1850, the SAA minimum and the $G_F$ maximum are close to two identified RFPs, but the SAA minimum is north of the $G_F$ maximum and east of the RFPs. At 1890, a magnetic equator intrusion prevents RFPs identification (Terra-Nova et al., 2015). The SAA minimum is north of the intrusion and north of the $G_F$ maximum. At 1930, 1970 and 2010, the SAA minimum and the $G_F$ maximum are close to one RFP below Patagonia. However, both quantities are far from another RFP below Africa.

The distance between the SAA minimum and the $G_F$ maximum is thus time-dependent. Fig. 10 shows this great-circle distance $\Delta D$ as a function of time. The SAA minimum is about $15^\circ \pm 3^\circ$ away from the $G_F$ maximum. The $\Delta D$ value before 1880 exhibits a roughly constant trend oscillating between $\sim 15.5^\circ$ and $\sim 16.5^\circ$. In contrast, between 1885 and 1980
ΔD decreases monotonously from ~ 17° to ~ 13°. ΔD has stable values of 12.1° – 12.4° from 1997 to 2015, although at two snapshots it deviates from this range. For the IGRF-12 model the ΔD value varies strongly between ~ 12° and ~ 13.5°.

In Fig. 10 we also plot the ratio between the powers of the axial dipole (AD) and the non-axial dipole (NAD) fields (e.g. Christensen et al., 2010):

$$AD/NAD = \frac{2(g_1^0)^2}{2((g_1^1)^2 + (h_1^1)^2) + \sum_{n=2}^{n_{max}} \left( (n+1) \left(\frac{a}{c}\right)^{2n-2} \sum_{m=0}^{n} (g_n^m)^2 + (h_n^m)^2 \right)} \quad (13)$$

where n is degree, m is order, and the sets $g_n^m$ and $h_n^m$ are the Gauss coefficients. The truncation degree $n_{max}$ affects the absolute values of (13), but its temporal trend is less affected by $n_{max}$ (compare Fig. 10 with Fig. 1 of Christensen et al., 2010). The AD/NAD ratio and ΔD are correlated after 1885 (see Fig. 10). Prior to 1885 reversed flux regions are less prominent thus the location of the SAA minimum is more uncertain, as evident in the oscillating behavior of ΔD at that time. Later the relative NAD power rapidly increases, more RFPs appear and consequently the SAA minimum becomes more robust in terms of its relation to the field morphology at the CMB. The two quantities display similar trends even with the fall in the AD/NAD ratio for the CHAOS5 model due to its increased spatial resolution (Finlay et al., 2015). The correlation between ΔD and AD/NAD after 1885 observed in gufm1 and CHAOS5 is not found in IGRF-12. It is worth noting that IGRF-12 is the mean of several independent candidate field models (Thébault et al., 2015) and as such it is considered somewhat less robust.

Next, we compare the positions of RFPs at the CMB with the position of the SAA minimum at the surface (Fig. 11). In longitude, the SAA minimum is close to RFPs in the Southern Atlantic before 1860. The SAA minimum cannot be related to RFPs between 1860 and 1900 due to a magnetic equator intrusion in this period (see snapshot 1890 in Fig. 9) that prevents
RFPs identification (Terra-Nova et al., 2015). After 1900, the SAA minimum exhibits westward drift similar to the Patagonia RFP (Terra-Nova et al., 2015). However, additional RFPs appear close to Africa, also exhibiting westward drift, especially after 1997. In latitude, both the SAA minimum and the Patagonia RFP move poleward whereas the Africa RFP moves equatorward. Note that the latitudes of the SAA minimum and the RFPs become nearly constant with time.

Why is the SAA minimum close to the Patagonia RFP but far from the Africa RFPs? Why do the SAA minimum and the RFPs latitudes become stable in latitude nearly constant with time? To address these questions we calculated the integrated normal flux in the Southern Hemisphere (i.e. $B_r > 0$) at the CMB $<B_r^N(\phi)>$ vs. longitude:

$$<B_r^N(\phi)> = \frac{2}{\pi} \int_{\frac{\pi}{2}}^{\pi} B_r(\phi, \theta) \sin\theta d\theta \quad (14)$$

Fig. 12 shows five snapshots of $<B_r^N(\phi)>$. The two high-latitude normal flux patches (NFPs) are evident by the two peaks between 90° W and 130° W and between 80° E and 120° E. The former is weaker and loses strength with time, as reported by Amit et al. (2011). The peak between 10° E and 60° E that intensifies with time and drifts westward corresponds to a low-latitude NFP (see Fig. 9) below Africa (Jackson, 2003). We argue that the presence of this intense NFP at low latitudes causes the SAA minimum to be away from Africa. In contrast, no intense low-latitude NFP is observed around South America (Figs. 9 and 12). Following the insights gained from the synthetic tests (see e.g. synthetic cases 8-12 in Figs. 2-3 and Table 1), this may suggest that the longitude of the SAA minimum is determined by the Patagonia RFP. Likewise, the southwestward migration of the SAA minimum is possibly determined by the motion of the Patagonia RFP. Indeed the westward drift of the SAA minimum correlates with the westward drift of the Patagonia RFP. The poleward drift of the SAA minimum is probably associated to both the weakening of the high-latitude NFP below South Pacific (Fig. 12) as
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well as the migration of the Patagonia RFP poleward (see Fig. 11), both of which reduce the ADM.

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# 5 Discussion

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The intensity of the SAA minimum has been decreasing much faster than the $g_1^0$ intensity since 1840 (Fig. 7a). Thus the collapse of $|g_1^0|$ is not enough to explain the weakening in the SAA intensity and an intensification of non-dipolar field structures must be invoked (e.g. Hartmann and Pacca, 2009). The non-linear parts of the intensities of the SAA minimum and $g_1^0$ (Fig. 7b) are shifted, with this delay being larger for the earlier period and smaller more recently.

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The morphology of $G_F$ shows that the SAA minimum is mostly sensitive to field structures beneath it (Fig. 9). However, there are considerable differences between the positions of the SAA minimum and the $G_F$ maximum. We showed that these differences are related to the relative axial dipolarity of the field (Fig. 10). In a hypothetical pure axial dipole field configuration the minimum surface intensity would be located along the entire geographic equator. Additional non-axial dipole field contributions lead to more complex morphologies resulting in a localized surface intensity minimum. Weaker axial dipole therefore gives more robust and stable minimum intensity surface point with more localized influence of the CMB field just below it.

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The synthetic tests shed light on the relation between magnetic flux patches at the CMB and the intensity at the Earth's surface. These tests clearly demonstrate that RFPs prescribe the longitude of the minimum intensity at the surface. The latitude of $F_{min}$ is a consequence of the competition between the latitude of the RFPs and the axial dipolarity of the field. If the axial dipole is strong the minimum intensity point is located closer to the equator, whereas if the RFPs are strong $F_{min}$ appears at latitudes closer to the RFPs. The synthetic field model with
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the closest $F_{min}$ coordinates to those of the SAA in CHAOS5 has a weaker high-latitude NFP below South Pacific and the RFP below South America is at higher latitudes than the Africa RFP, in agreement with the present geomagnetic field (as in e.g. Finlay et al., 2015). This SAA recovery demonstrates that indeed the interplay among the most intense patches determines the SAA position while the impact of weaker smaller-scale patches is likely secondary.

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Starting from a synthetic field model with surface intensity minimum location close to the SAA, we explored $F_{min}$ path predictions based on some simple synthetic SV scenarios (Fig. 4). Westward drift of low- and mid-latitude patches move $F_{min}$ westward, but at a rate about 2/3 slower. Expansion of the South America RFP moves $F_{min}$ equatorward and slightly eastward component whereas its contraction may give some westward drift if the concentration is not yet at the asymptote. Intensification of the Southern Hemisphere RFPs by radial diffusion does not affect the $F_{min}$ longitude. In contrast, intensification of RFPs due to radial diffusion is the most efficient mechanism to reduce the intensity of the surface field minimum (Fig. 5). We conclude that while the SAA trajectory can be explained by a frozen-flux SV, the decrease in its intensity requires the presence of radial diffusion SV.

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Our analysis of geomagnetic field models highlights the different roles of specific RFPs and NFPs in the localization of the SAA minimum. Before 1900 the SAA minimum was close to the center of mass of the reversed flux region in the Southern Hemisphere (see Fig. 9). However, at more recent times, the SAA minimum moved farther from this center of mass towards the longitude of an identified RFP below Patagonia. We argue that the reason for this SAA minimum location is that the intense low-latitude NFPs cause the SAA minimum to be away from the Africa RFPs, while low-latitude NFPs are absent below South America (as in synthetic cases 9-12). In addition, the South Pacific high-latitude NFP causes the SAA minimum to reside north of the Patagonia RFP (as in synthetic cases 4 and 5). The geomagnetic field models indicate a prominent westward drift as well as some poleward drift of the SAA
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minimum (Fig. 6). Comparable westward drift is observed for the Patagonia RFP (Terra-Nova et al., 2015). The poleward drifts of the Patagonia RFP and the SAA minimum are also similar.

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In addition, the poleward drift of the SAA minimum is also influenced by the weakening of the South Pacific NFP at high latitudes (Amit et al., 2011).

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The westward drift of the SAA minimum is time-dependent (Fig. 6b). Prior to 1900 the SAA drifted westward relatively fast whereas after 1900 the drift was more than twice slower.

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Poleward drift shows alternating faster/slower trends: recently, prior to 1950 the SAA moved poleward relatively fast whereas after 1950 the poleward drift significantly diminished. The westward drift of the SAA minimum may have decelerated because of the emergence of the

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Africa RFP at 1920 (Figs. 9 and 11). The poleward deceleration of the SAA minimum may have occurred due to the poleward deceleration of the Patagonia RFP (Fig. 11) and the decrease in the rate of intensity drop of the high-latitude NFP below South Pacific with time (see Fig. 12).

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The rate of westward drift of geomagnetic CMB flux patches (Finlay and Jackson, 2003;

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Aubert et al., 2013) is comparable to the recent SAA westward drift in Fig. 6b. This is in contrast to our synthetic advective SV scenario in which the SAA drifts at a rate of ~ 2/3 of

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the patches drift rate (Fig. 4a). One possible explanation is uniform drift over all latitudes, but this is in contradiction to most core flow models (Holme, 2015). Another possibility is a combined advection and stretching SV, but RFP expansion actually gives eastward drift (Fig.

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4b). Radial diffusion has a minor impact on the SAA longitude (Fig. 4c). The historical SAA drift rate was probably caused by a somewhat more complex SV scenario involving several prominent patches as suggested in our analysis of the geomagnetic field.

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Aubert (2015) performed a forecast of the SAA minimum until 2115 predicting a faster

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SAA westward drift than in recent times and slight northward drift. Such faster SAA westward drift may arise due to faster westward drifting low- and mid-latitude patches, further weaken-

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ing of the NFP below south Pacific and intensification of the low-latitude NFP below Africa.

The predicted northward SAA drift by Aubert (2015) could originate from RFP expansion by

stretching effects (Fig. 4b).

We argue that the SAA minimum is indeed related to RFPs, but also to the positions of
low- and high-latitude NFPs, as we demonstrated in the synthetic tests. A persistent extension
of a reversed flux region coupled with an NFP at one border would cause the SAA minimum
to drift away from the center of reversed flux possibly causing the westward drift of the SAA
minimum. If the axial dipole eventually increases due to shrinking of the reversed flux region
below the South Atlantic, the surface minimum may lose its robustness and appear at sporadic
longitudes near the magnetic equator.

Maps of advective sources of ADM changes (Olson and Amit, 2006; Finlay et al., 2016) showed that below Asia equatorward advection of an NFP decreases the ADM while below North America poleward drift of an NFP increases the ADM. The opposite motions of these two Northern Hemisphere NFPs cancel out their contributions to ADM changes. Below the Indian Ocean the equatorward drift of an NFP decreases the ADM, while below South America no significant advective ADM source is observed. Thus, the Patagonia RFP breaks the symmetry of ADM changes and is responsible to the decrease of the ADM. Our results show that the same field structure, the Patagonia RFP, is also the geomagnetic feature at the CMB that mostly determines the location and mobility of the SAA minimum.

Several archeointensity studies based on archeological materials from Africa and South America claimed that the geomagnetic field variations in these regions are attributed to the SAA time evolution (e.g. Hartmann et al., 2010, 2011; Goguitchaichvili et al., 2011, 2012, 2015; Roperch et al., 2014, 2015; Osete et al., 2015; Tarduno et al., 2015; Poletti et al., 2016; Shah et al., 2016). Tarduno et al. (2015) argued for persistent low field intensity associated to the SAA. They hypothesized that the SAA influences the magnetic field in Africa as early as
∼ 1200. Aubert (2015) predicted continued SAA westward drift until reaching eastern Pacific Ocean at 2115. According to these two studies there is ~ 1000 yr of minimum field intensity in the Southern Hemisphere, possibly from Africa to eastern Pacific Ocean. Tarduno et al. (2015) also proposed that topographic heterogeneities in the lowermost mantle at the edge of a low shear wave velocity province located below Africa are responsible for the emergence of RFPs at the CMB and the low intensity antique field there. Terra-Nova et al. (2016) used various archeomagnetic field models to show that RFPs have statistically preferred positions, and that these positions may be prescribed by heat flux heterogeneities in the lowermost mantle. If the SAA minimum has indeed preferred positions its current westward drift must cease at some point.

In summary, combining the tracking of the SAA minimum at Earth's surface, the identification of RFPs at the CMB, the use of kernels and accounting for NFPs at the CMB, we investigated the relation between CMB geomagnetic flux patches and the SAA minimum. We showed that the level of axial dipolarity of the field determines the stability of the relation between the SAA and RFPs. The position and motion of the SAA minimum are highly influenced by the interplay among several robust geomagnetic flux patches at the CMB: an RFP below Patagonia, the South Pacific high-latitude NFP and the low-latitude intense NFPs near Africa. Simple SV scenarios applied to a synthetic field model with some realistic features allowed to speculate possible future paths of the SAA.

New geomagnetic measurements may improve the identification and tracking of RFPs and SAA at present, in particular at South America. Overall, further monitoring of the surface intensity of the geomagnetic field and its morphology at the CMB by ground observatories and dedicated satellites (e.g. the currently orbiting Swarm mission) as well as reliable archeomagnetic data from Africa and South America for periods before the observatories era will shed light on the relation between the SAA minimum and kinematic processes originating in the
## Acknowledgements

We are grateful to Phil Livermore and an anonymous reviewer for their constructive comments that improved the manuscript. The Gauss coefficients of the gufm1, CHAOS5 and IGRF-12 field models used in this paper are available respectively at http://jupiter.ethz.ch/cfindlay/gufm1.html, www.spacecenter.dk/files/magnetic-models/CHAOS-5/ and http://www.ngdc.noaa.gov/IAGF. T-N. acknowledges The National Council for Scientific and Technological Development (CNPq/Brazil) for grant 206997/2014-0. F. T-N., H. A. and K. J. P. were partly supported by the Centre National des Études Spatiales (CNES). G. A. H. thanks CAPES (grant AUXPE 2043/2014) and CNPq/Brazil (grant 454609/2014-0). R. I. F. T. is supported by CNPq/Brazil PQ (grant 304934/2014-3). K. J. P. is also supported by la Région des Pays de la Loire and FAPERJ (grant E-26/2002.803/2015).

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**Figure captions**

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**Figure 1** (a): Kernels $G_r$ and $G_h$ (black and red lines, respectively) as functions of $\mu = \cos\alpha$. (b) and (c): Kernels $G_{\theta}$ and $G_{\phi}$, respectively, with black diamonds indicating the surface point for which the kernels are calculated.

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**Figure 2** Synthetic radial magnetic fields at the CMB (left) and intensity at Earth's surface (right) for cases 1, 2, 7 and 8 (see Table 1). The $F_{min}$ location is denoted by green diamonds (right). The area of weak intensity (delimited by the $1.2F_{min}$ contour) is denoted by a white dashed line.

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**Figure 3** As in Fig. 2 for cases 9, 10, 11 and 12 (see Table 1).

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**Figure 4** Changes of $F_{min}$ longitude $\phi_{Fmin}$ (black, left axis) and latitude $\lambda_{Fmin}$ (red, right axis) due to (a) 0.2°/yr westward drift of low- and mid-latitude patches over 100 yr, (b) expansion/concentration of the South America RFP and (c) intensification/dissipation of the RFPs compensated by dissipation/intensification of Northern Hemisphere high-latitude NFPs, respectively. Case 12 (Table 1) is used as an initial field. Longitude and latitude are given in degrees. The vertical black dashed line in (b) separates expansion (left) and contraction (right) SV scenarios. The vertical black dashed line in (c) separates dissipation (left) and intensification (right) SV scenarios.

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**Figure 5** As in Fig. 4 for changes of the ratio $F_{min}/F_{min}^D$ (black, left axis) and the area $A$ (red, right axis).

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**Figure 6** Tracking of the South Atlantic Anomaly (SAA) minimum. Time-dependence of the SAA minimum at Earth's surface (a), in longitude (b) and latitude (c) based on the gufm1 (diamonds, Jackson et al., 2000), CHAOS5 (circles, Finlay et al., 2015) and IGRF-12 (crosses, Thébault et al., 2015) field models. Geomagnetic field intensities in nT are denoted by colors for gufm1 and CHAOS5.

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**Figure 7** (a): The time dependence of the intensity of the SAA minimum (dashed lines, left axis) and $|g_1^0|$ (solid lines; right axis) for the gufm1 (black), CHAOS5 (blue) and IGRF-12 (red) geomagnetic field models. (b): The non-linear part ($\delta$) of the intensities of the SAA minimum (solid line) and $g_1^0$ (dashed line) for the gufm1 field model obtained by removing the respective linear fits from (a). Note the different scales on the right axes of (a) and (b).

**Figure 8** Kernels $G_{\theta}$, $G_{\phi}$ and $G_{r}$ for the tracked positions of the SAA minimum in Fig. 6 in the years 1850, 1890, 1930, 1970 (gufm1) and 2010 (CHAOS5). The SAA minimum is denoted by green diamonds.

**Figure 9** Field intensity at Earth's surface (left), intensity kernel $G_F$ (middle) and radial field at the CMB (right) for 1850, 1890, 1930, 1970 (gufm1) and 2010 (CHAOS5). The SAA minimum is denoted by green diamonds (left and middle) and the identified reversed flux patches (RFPs) and normal flux patches (NFPs) are denoted by purple diamonds (middle and right) and white diamonds (right), respectively. Dashed lines denote the identified magnetic equator (right). Both CMB and surface fields are in nT.

**Figure 10** Time-dependence of the great-circle distance $\Delta D$ between the SAA minimum and the $G_F$ maximum (left axis), and of the ratio of axial dipole power to non-axial dipole power $AD/NAD$ (right axis). $\Delta D$ is denoted by diamonds, circles and triangles for the gufm1, CHAOS5 and IGRF-12 field models, respectively. A few points (three in IGRF-12 and one in gufm1) have values that are out of range. The ratio $AD/NAD$ is denoted by dotted, solid and dashed red lines for gufm1, CHAOS5 and IGRF-12, respectively.

**Figure 11** As in Figs. 6b and c including identified reversed flux patches (RFPs). The RFPs are denoted by non-filled diamonds, circles and triangles for the gufm1, CHAOS5 and IGRF-12 field models, respectively. A few RFPs (located in the Northern Hemisphere
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or/and further east of Africa in gufm1 and CHAOS5 field models and only in the North-
ern Hemisphere for the IGRF-12 field model) have values out of range.

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**Figure 12** Integrated normal flux < $B_r^N$> at the core-mantle boundary calculated over the Southern Hemisphere vs. longitude. Each colored line denotes another snapshot (see legend). Diamonds denote the longitudes of the SAA minimum.

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<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 1:</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 2:</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 3:</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>(b) Stretching</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(c) Diffusion</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 4:</figcaption></figure>
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>Figure 5:</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure 6:</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure 7:</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure 8:</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Figure 9:</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Figure 10:</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>Figure 11:</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Figure 12:</figcaption></figure>
<table><caption>Table 1: Synthetic fields setup and minimum intensity point at the Earth's surface ($F_{min}$).</caption><thead><tr><th rowspan="2">Case</th><th colspan="4">Setup</th><th colspan="5">Results</th></tr><tr><th>φ<sub>p</sub></th><th>λ<sub>p</sub></th><th>Amp.</th><th>Flux</th><th>m<sub>z</sub>/m<sub>z</sub><sup>D</sup></th><th>φ<sub>F<sub>min</sub></sub></th><th>λ<sub>F<sub>min</sub></sub></th><th>F<sub>min</sub>/F<sub>min</sub><sup>D</sup></th><th>A</th></tr></thead><tbody><tr><td rowspan="2">1</td><td>-120</td><td>-30</td><td>A<sub>0</sub></td><td>R</td><td rowspan="2">0.95</td><td>-120</td><td>-9</td><td>0.85</td><td>25.7%</td></tr><tr><td>120</td><td>30</td><td>A<sub>0</sub></td><td>R</td><td>120</td><td>9</td><td>0.85</td><td>25.7%</td></tr><tr><td rowspan="2">2</td><td>-120</td><td>-30</td><td>A<sub>0</sub></td><td>R</td><td rowspan="2">1.00</td><td>-118</td><td>-8</td><td>0.88</td><td>17.2%</td></tr><tr><td>120</td><td>-30</td><td>A<sub>0</sub></td><td>N</td><td>-118</td><td>-8</td><td>0.88</td><td>17.2%</td></tr><tr><td rowspan="2">3</td><td>-120</td><td>-60</td><td>A<sub>0</sub></td><td>N</td><td rowspan="2">1.02</td><td>118</td><td>-8</td><td>0.89</td><td>19.1%</td></tr><tr><td>120</td><td>-30</td><td>A<sub>0</sub></td><td>R</td><td>118</td><td>-8</td><td>0.89</td><td>19.1%</td></tr><tr><td rowspan="2">4</td><td>-120</td><td>-30</td><td>A<sub>0</sub></td><td>R</td><td rowspan="2">1.02</td><td>-120</td><td>-4</td><td>0.94</td><td>30.0%</td></tr><tr><td>-120</td><td>-60</td><td>A<sub>0</sub></td><td>N</td><td>-120</td><td>-4</td><td>0.94</td><td>30.0%</td></tr><tr><td rowspan="2">5</td><td>-120</td><td>-45</td><td>A<sub>0</sub></td><td>R</td><td rowspan="2">1.01</td><td>-119</td><td>-3</td><td>0.97</td><td>34.6%</td></tr><tr><td>-120</td><td>-60</td><td>A<sub>0</sub></td><td>N</td><td>-119</td><td>-3</td><td>0.97</td><td>34.6%</td></tr><tr><td rowspan="2">6</td><td>-120</td><td>-30</td><td>A<sub>0</sub></td><td>N</td><td rowspan="2">1.05</td><td>0</td><td>0</td><td>1.02</td><td>31.6%</td></tr><tr><td>120</td><td>30</td><td>A<sub>0</sub></td><td>N</td><td>0</td><td>0</td><td>1.02</td><td>31.6%</td></tr><tr><td rowspan="2">7</td><td>-120</td><td>-30</td><td>2A<sub>0</sub></td><td>R</td><td rowspan="2">0.90</td><td>-119</td><td>-14</td><td>0.75</td><td>14.0%</td></tr><tr><td>120</td><td>30</td><td>2A<sub>0</sub></td><td>R</td><td>-119</td><td>-14</td><td>0.75</td><td>14.0%</td></tr><tr><td rowspan="3">8</td><td>-120</td><td>-30</td><td>2A<sub>0</sub></td><td>R</td><td rowspan="3">1.08</td><td>117</td><td>-11</td><td>0.84</td><td>7.8%</td></tr><tr><td>-120</td><td>-60</td><td>4A<sub>0</sub></td><td>N</td><td>117</td><td>-11</td><td>0.84</td><td>7.8%</td></tr><tr><td>120</td><td>-30</td><td>2A<sub>0</sub></td><td>R</td><td>117</td><td>-11</td><td>0.84</td><td>7.8%</td></tr><tr><td rowspan="4">9</td><td>-60</td><td>-30</td><td>A<sub>0</sub></td><td>R</td><td rowspan="4">1.31</td><td>-46</td><td>-6</td><td>1.08</td><td>15.7%</td></tr><tr><td>60</td><td>-30</td><td>A<sub>0</sub></td><td>R</td><td>-46</td><td>-6</td><td>1.08</td><td>15.7%</td></tr><tr><td>-120</td><td>-60</td><td>2A<sub>0</sub></td><td>N</td><td>-46</td><td>-6</td><td>1.08</td><td>15.7%</td></tr><tr><td>120</td><td>60</td><td>2A<sub>0</sub></td><td>N</td><td>-46</td><td>-6</td><td>1.08</td><td>15.7%</td></tr><tr><td rowspan="4">10</td><td>-60</td><td>-30</td><td>A<sub>0</sub></td><td>R</td><td rowspan="4">2.04</td><td>-37</td><td>-9</td><td>1.35</td><td>8.7%</td></tr><tr><td>60</td><td>-30</td><td>A<sub>0</sub></td><td>R</td><td>-37</td><td>-9</td><td>1.35</td><td>8.7%</td></tr><tr><td>-120</td><td>-60</td><td>A<sub>0</sub></td><td>N</td><td>-37</td><td>-9</td><td>1.35</td><td>8.7%</td></tr><tr><td>120</td><td>60</td><td>2A<sub>0</sub></td><td>N</td><td>-37</td><td>-9</td><td>1.35</td><td>8.7%</td></tr><tr><td rowspan="4">11</td><td>-120</td><td>60</td><td>2A<sub>0</sub></td><td>N</td><td rowspan="4">1.88</td><td>-54</td><td>-13</td><td>1.24</td><td>4.1%</td></tr><tr><td>120</td><td>-60</td><td>2A<sub>0</sub></td><td>N</td><td>-54</td><td>-13</td><td>1.24</td><td>4.1%</td></tr><tr><td>120</td><td>60</td><td>2A<sub>0</sub></td><td>N</td><td>-54</td><td>-13</td><td>1.24</td><td>4.1%</td></tr><tr><td>60</td><td>0</td><td>2A<sub>0</sub></td><td>N</td><td>-54</td><td>-13</td><td>1.24</td><td>4.1%</td></tr><tr><td rowspan="4">12</td><td>-60</td><td>-45</td><td>A<sub>0</sub></td><td>R</td><td rowspan="4">1.86</td><td>-57</td><td>-22</td><td>1.11</td><td>3.3%</td></tr><tr><td>60</td><td>-30</td><td>A<sub>0</sub></td><td>R</td><td>-57</td><td>-22</td><td>1.11</td><td>3.3%</td></tr><tr><td>-120</td><td>-60</td><td>A<sub>0</sub></td><td>N</td><td>-57</td><td>-22</td><td>1.11</td><td>3.3%</td></tr><tr><td>120</td><td>60</td><td>2A<sub>0</sub></td><td>N</td><td>-57</td><td>-22</td><td>1.11</td><td>3.3%</td></tr><tr><td rowspan="4">Earth</td><td>CHAOS5 - 2003</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>8.4%</td></tr></tbody></table>

$\phi_p$ is patch longitude, $\lambda_p$ is patch latitude, Amp. is amplitude and R and N indicate if the flux patch is reversed or normal, respectively. $m_z$ is the axial dipole moment, whilst $m_z^D$ is the axial dipole moment of the background field. $\phi_{F_{min}}$ is $F_{min}$ longitude, $\lambda_{F_{min}}$ is $F_{min}$ latitude, $F_{min}/F_{min}^D$ is the minimum field intensity normalized by the minimum due only to the background field. A is the percentage of Earth's surface area that contains intensity values lower than $1.2F_{min}$. In cases 10-12 the background axial dipole field is reduced by 70%.