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Nat. Hazards Earth Syst. Sci., 13, 3395–3403, 2013

www.nat-hazards-earth-syst-sci.net/13/3395/2013/

doi:10.5194/nhess-13-3395-2013

© Author(s) 2013. CC Attribution 3.0 License.

Natural Hazards

and Earth System

Sciences

Open Access

Toward a possible next geomagnetic transition?

A. De Santis1,2, E. Qamili1, and L. Wu3,4

1Istituto Nazionale di Geoﬁsica e Vulcanologia, Sezione Roma 2, Roma, Italy

2Università “G. D’Annunzio”, Campus Universitario, Chieti, Italy

3Northeastern University, Shenyang, China

4China University of Mining and Technology, Xuzhou, China

Correspondence to: A. De Santis (angelo.desantis@ingv.it)

Received: 12 June 2013 – Published in Nat. Hazards Earth Syst. Sci. Discuss.: 27 September 2013

Revised: – – Accepted: 28 November 2013 – Published: 23 December 2013

Abstract. The geomagnetic ﬁeld is subject to possible rever-

sals or excursions of polarity during its temporal evolution.

Considering that: (a) in the last 83 million yr the typical av-

erage time between one reversal and the next (the so-called

chron) is around 400000yr, (b) the last reversal occurred

around 780000yr ago, (c) more excursions (rapid changes

in polarity) can occur within the same chron and (d) the geo-

magnetic ﬁeld dipole is currently decreasing, a possible im-

minent geomagnetic reversal or excursion would not be com-

pletely unexpected. In that case, such a phenomenon would

represent one of the very few natural hazards that are re-

ally global. The South Atlantic Anomaly (SAA) is a great

depression of the geomagnetic ﬁeld strength at the Earth’s

surface, caused by a reverse magnetic ﬂux in the terrestrial

outer core. In analogy with critical point phenomena charac-

terized by some cumulative quantity, we ﬁt the surface ex-

tent of this anomaly over the last 400yr with power law or

logarithmic functions in reverse time, also decorated by log-

periodic oscillations, whose ﬁnal singularity (a critical point

tc)reveals a great change in the near future (2034±3yr),

when the SAA area reaches almost a hemisphere. An inter-

esting aspect that has recently been found is the possible di-

rect connection between the SAA and the global mean sea

level (GSL). That the GSL is somehow connected with SAA

is also conﬁrmed by the similar result when an analogous

critical-like ﬁt is performed over GSL: the corresponding

critical point (2033±11yr) agrees, within the estimated er-

rors, with the value found for the SAA. From this result, we

point out the intriguing conjecture that tcwould be the time

of no return, after which the geomagnetic ﬁeld could fall into

an irreversible process of a global geomagnetic transition that

could be a reversal or excursion of polarity.

1 Introduction

The magnetic ﬁeld of the Earth changes in time and space,

in an irregular fashion, including dramatic manifestations

such as the geomagnetic reversals or excursions, when the

magnetic polarities exchange in sign, so that the geomag-

netic south becomes north and vice versa (e.g., Jacobs, 1994).

Over the last 83 million years we count 184 reversals (Cande

and Kent, 1995). From the facts that: (a) the typical average

time between a reversal and another (the so-called chron)

is around 400000yr, (b) the last reversal occurred around

780000yr ago, (c) more excursions (rapid changes in polar-

ity) can occur within the same chron and (d) the geomagnetic

ﬁeld dipole is currently decreasing, a possible imminent geo-

magnetic reversal or excursion would not be completely un-

expected. Such a phenomenon would represent one of the

very few natural hazards that are really global, because it

would affect the whole globe, although the detailed conse-

quences over the planet, in general, and the biosphere, in par-

ticular, are not completely known. For instance, we recall a

presumed link with mass extinctions (Raup, 1985; Courtillot

and Besse, 1997; but see also Constable and Korte, 2006).

In the last 25yr, some papers have appeared suggesting

that an imminent reversal could occur (e.g., De Santis et

al., 2004 and the references therein). The recent dipole de-

crease is considered part of a trend that has continued for

the last 2000yr (Merrill and McElhinny, 1983) and a more

rapid poleward drift of the dipole axis in the past 50yr has

also been suggested (Amit et al., 2010). Analyzing the past

150yr of magnetic data, a more signiﬁcant decay of the ge-

omagnetic dipole intensity was found (Gubbins, 1987; Gub-

bins et al., 2006), much faster than the rate of free decay

Published by Copernicus Publications on behalf of the European Geosciences Union.

3396 A. De Santis et al.: Toward a possible next geomagnetic transition?

in the Earth’s core (Olson and Amit, 2006). Most of this

decay stems from the Southern Hemisphere, as shown by

Gubbins (1987), who also suggested a direct correlation be-

tween the dipole decrease and the westward movements of

a pair of reverse ﬂuxes under South Africa. Other studies

(e.g., Hulot et al., 2002; Constable, 2011) conﬁrm the pres-

ence, at the core mantle boundary (CMB), of two reverse

ﬂux features: in particular, one is placed inside the tangent

cylinder near the North Pole and the other is a large reverse

ﬂux patch under the Southern Atlantic that has been asso-

ciated with the rapid decay of the ﬁeld strength. Other au-

thors have concentrated their studies on understanding the

mechanism of magnetic polar reversals in dynamo numerical

models (e.g., Glatzmaier and Roberts, 1995). Flux patches

of reversed polarity appear at low or mid latitude prior to a

reversal and then migrate polewards, thus reducing the ax-

ial dipole component (Wicht and Olson, 2004; Takahashi et

al., 2005; Aubert et al., 2008; Wicht et al., 2009; Wicht and

Christensen, 2010; Christensen, 2011). All these results are

in agreement with early stages of a dipole collapse in the nu-

merical dynamo model by Olson et al. (2009). In a detailed

study of the Matuyama–Brunhes polarity reversal (Leonhardt

and Fabian, 2007) and Laschamp excursion (Leonhardt et al.,

2009) the ﬁeld instability starts when reverse ﬂux patches ap-

pear in low or mid latitude regions at the CMB and then move

poleward. In contrast, Aubert et al. (2008) found a mixed be-

havior, with reversals and excursions initiated by reversed

ﬂux generated both outside and inside the tangent cylinder.

The same authors suggested that the appearance of the South

Atlantic reversed ﬂux patch could be attributed to a reverse

magnetic anticyclone supplied by a strong equatorial mag-

netic upwelling.

The most recent geomagnetic dipole ﬁeld is decreasing

very rapidly and its temporal linear extrapolation would pre-

dict a null ﬁeld at around 1000yr from now. In some parts

of the Earth’s surface this zero value would be reached even

earlier since this ﬁeld is more complex than a pure dipolar

ﬁeld: for instance, in the polar regions the ﬁeld would be zero

in around 300yr (De Santis, 2007). Some other papers (De

Santis et al., 2004; De Santis and Qamili, 2008, 2010a) have

found clear evidence for a chaotic state of the present geo-

magnetic ﬁeld. De Santis (2007, 2008) calculated the Shan-

non information, which is a measure of the spatial order, for

the ﬁeld of the last 400yr. He found that the Shannon in-

formation started to decay from around 1690, and began to

decrease more rapidly at around 1775 and even more rapidly

after 1900, revealing that the ﬁeld is increasing its overall

complexity. The author also found that some parts of the

globe (e.g., Antarctica) contribute more than others to this

trend, in agreement with what was found by Gubbins (1987).

All these aspects can be interpreted as a sign that the

Earth’s magnetic ﬁeld might be in the early stage of a re-

versal (Hulot et al., 2002; De Santis et al., 2004; but see also

Constable and Korte, 2006). Other authors, studying the fu-

ture evolution of the ﬁeld from numerical dynamos, use more

caution in interpreting these results (Hulot et al., 2010). Ana-

lyzing the exponential growth of errors in numerical models,

these authors concluded that predictions for the next reversal

will not be possible for more than one century, although bet-

ter predictions for the evolution of the ﬁeld in the near future

could be possibly made.

Recently, De Santis and Qamili (2010b) focused their at-

tention on the South Atlantic Anomaly (SAA), which is a

great depression of the geomagnetic ﬁeld at the Earth’s sur-

face. These authors proposed a simple model to represent the

dynamics of this feature in terms of an apparent monopo-

lar magnetic source moving at the top of the outer core. In

practice, under the frozen ﬂux approximation in the outer

core (strong coupling between material motions and mag-

netic ﬁelds), this would represent the magnetic expression

of a vortex in the outer core, as a component of a strong

magnetic ﬂux with reversed polarities with respect to the

surroundings. The origin of the SAA can be either due to

a decrease in the whole geomagnetic ﬁeld strength or in the

dipole ﬁeld, which are two typical ingredients for a possible

geomagnetic reversal. Another cause could be an increase in

the ﬁeld complexity, i.e., an increase (decrease) in the corre-

sponding Shannon entropy (information), which has been re-

cently revealed as another important ingredient for a possible

polarity change (De Santis et al., 2004; De Santis and Qamili,

2008). Therefore, we could postulate that a possible immi-

nent reversal would be preceded by a signiﬁcant increase in

the reversed magnetic ﬂux in the CMB, and in turn at the

Earth’s surface, in the SAA area. In this paper we study the

surface extension of this anomaly over the last 400yr. In par-

ticular we analyze the variation in space and time of the area

included by the 32000nT isoline as deduced from GUFM1

(1590–1990; Jackson et al., 2000) and IGRF-11 (1900–2010;

Finlay et al., 2010) global models of the geomagnetic ﬁeld.

The combined time series was obtained with a point every

5 yr taking the values from GUFM1 in the period 1590–1955,

and IGRF-11 afterward (at 1960 the two models agree quite

well). Then we will ﬁt it with a nonlinear function usually

characterizing a system under a signiﬁcant change of state,

the so-called “critical” or “tipping point”. The 32 000nT iso-

line was chosen as a reference because it is the lowest value

in the time interval of study, so it is easy to follow the in-

crease in the SAA extent with time. In the GUFM1 model

this isoline appears at the beginning of the interval of the

model validity; some recent papers (Gubbins et al., 2006;

Finlay, 2008) have cast some doubts on the validity of the

back linear extrapolation in time of the g0

1coefﬁcient (related

to the axial dipole of the ﬁeld) before 1840, which is just af-

ter the time Gauss introduced an absolute method to measure

the geomagnetic ﬁeld intensity in 1832 (Malin, 1982). In the

following we will denote with GUFM1-G and GUFM1-F the

two models derived from the suggestions given by Gubbins

et al. (2006) and Finlay (2008), respectively, whose models

differ from GUFM1 mainly in the values of g0

1before 1840

(the other Gauss coefﬁcients are just rescaled appropriately).

Nat. Hazards Earth Syst. Sci., 13, 3395–3403, 2013 www.nat-hazards-earth-syst-sci.net/13/3395/2013/

A. De Santis et al.: Toward a possible next geomagnetic transition? 3397

In the next section we will introduce some concepts related to

critical point processes, i.e., dynamical systems coping with

dramatic changes of state, and then we will apply these con-

cepts to the temporal evolution of SAA area extension over

the last 400yr, together with the changes in the mean global

sea level (GSL) as provided by Jevrejeva et al. (2008) and

Church and White (2011) (but for an alternative view on GSL

please see Mörner, 2004, 2010). The comparison between

SAA and GSL is important because an unexpectedly close

correlation between these quantities has recently been found

(De Santis et al., 2012). Our joint analysis will conﬁrm the

existence of a tipping point for both time series. Finally we

will present some conclusions and discussions.

2 Critical point processes and critical time

Many complex systems have “critical” thresholds (the so-

called critical or tipping points) at which the system moves

abruptly from one state to another, i.e., shifts toward a crit-

ical transition (Scheffer, 2009); the corresponding times are

also called critical times. In the literature we can ﬁnd differ-

ent methods for scientiﬁc predictions of catastrophic events

based on the concepts of non-linear physics (e.g., Bunde et

al., 2002; Dakos et al., 2012). A way to attempt to recognise

these critical transitions is to detect some early warnings that

may anticipate them (Scheffer et al., 2009). This strategy has

been applied in ecology, medicine and global ﬁnance (May

et al., 2008). Another approach is related to the critical point

hypothesis for processes usually characterized by some cu-

mulative critical quantity. This approach has also found ap-

plications in such different ﬁelds as: climate dynamics, seis-

mology, material rupture, ﬁnancial crashes, etc. (Sornette,

2003). It is important to note that the critical point hypoth-

esis can be used when the system is close to or moving to-

ward a critical state, in analogy with a phase transition (e.g.,

Stanley, 1971), and the capability to predict the critical point

generally improves as the latter is more approaching. With

the term “critical” we denote the state of a system between

order and disorder, and which is strongly inﬂuenced by ex-

ternal and internal factors. Examples of systems that respond

to such characteristics are some cases of liquids and mag-

nets, but many others can be found in different disciplines

(Sornette, 2006; Scheffer et al., 2012).

In analogy with standard critical phenomena of solid state

physics, it is thought that the precursory seismicity of large

events may follow power laws or alternative diverging func-

tions in time. This approach has found more applications

in the attempt to predict large earthquakes, although mostly

from a retrospective point of view. In particular, Bufe and

Varnes (1993) and Bowman et al. (1998) suggested that the

time tcof the largest main shock of a seismic sequence is

the critical time of the seismic sequence, i.e., the time when

the system drastically changes its dynamical regime. Since

the seismological phenomena are mainly earthquakes, which

are large ruptures or failures of a part of the crust, this ap-

proach has also been called the “time-to-failure” approach.

In a broad sense, also in other occasions and ﬁelds when a

general system shifts to a critical transition, the latter event

could be considered as a failure of the system to maintain its

previous typical state; thus the term failure must be taken in

this general meaning, not implying necessarily that there is a

physical failure or rupture in the system under study. Then, a

measure y(t ) of the seismic release (e.g., the seismic defor-

mation) at any preceding time treasonably close to the time

tccan be described by a power law relation of the form:

y(t ) =k(tc−t)−n(1)

where k > 0 and 1 >n>0 are appropriate parameters.

Equation (1) is characterized to have a singularity at t=tc

because y(tc)= ∞. In practice, in seismology it is preferred

to integrate Eq. (1) in time to use a cumulative function s(t)

of y(t ), in order to have a ﬁnite value for s(tc), its time deriva-

tive being singular, i.e., the slope of the function s(t ) at tcis

vertical. In this way, we have:

s(t ) =Zy(t)dt=a−k

m(tc−t)m=a+b(tc−t )m(2)

where a>0 is the constant of integration; b= −k/m < 0,

and m=1−n > 0 are constant parameters that are found by

means of a nonlinear least regression on the available data;

m, normally 0.2<m<0.6 (Mignan, 2011), is a critical expo-

nent that represents the degree of accelerating energy release

(De Santis et al., 2010). It is clear that ais the value of the

measure related to the cumulative seismic release at the crit-

ical time, i.e., a=s(tc). In addition to the accelerating strain

release in Eqs. (1) or (2), Sornette and Sammis (1995) pro-

posed an extension of this method, ﬁnding a better ﬁt to the

time of occurrence of large seismic events by ﬁtting a func-

tion that included a log-periodic ﬂuctuation:

s(t ) =a+b(tc−t)m·{1+d·cos[2πf ln(tc−t ) +ϕ]}(3)

where dis the magnitude of the ﬂuctuations around the ac-

celeration growth, fis the frequency of the ﬂuctuations, ϕis

the phase shift, and tcis the critical time. Note that for d=0

we have the simple power law as in Eq. (2). The equations

from Eqs. (1) to (3) have also been applied in analyzing ﬁ-

nancial crises (Sornette, 2003).

An alternative form of diverging functions in time is that

of considering just a logarithmic function in (reversed) time

(e.g., Vandewalle et al., 1998):

s(t ) =A+Bln(tc−t) (4)

where A>0 and B<0 (and tc)are parameters to be found

from the experimental data, thus reducing the unknown pa-

rameters from four of Eq. (2) to only three. With respect to

Eq. (2), but as for Eq. (1), the price to pay of Eq. (4) is that

we have s(tc)= ∞ at t=tcand A=s(tc−1). Since in our

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3398 A. De Santis et al.: Toward a possible next geomagnetic transition?

Fig. 1. Extension of the SAA over the last 400yr and the best non-

linear ﬁt of the function indicated in the text as Eq. (5). The “critical

time” tcwould be 2034±3 yr,where the curve will have a singular-

ity, i.e., where the curve is tangent to the vertical dashed line drawn

at the critical time in the smaller picture. Our interpretation is that

this time will represent the time of no return for a great change in the

geomagnetic ﬁeld, possibly going toward a reversal or excursion. In

the inset table, DoF are the degrees of freedom and ris the correla-

tion coefﬁcient of the nonlinear ﬁt; for the other ﬁtting parameters

see the text.

calculations the time is in years, the value of Ais a good ap-

proximation of the actual value that the quantity under study

will take close to its critical time (i.e., just one year before).

Equation (4) is the time integral of the limiting case of Eq. (1)

with n=1, and Ais the constant term of integration. The cor-

responding log-periodic form can be written as (e.g., Vande-

walle et al., 1998):

s(t ) =A+Bln(tc−t)·{1+D·cos[2πf ln(tc−t ) +ϕ]}.(5)

Note that for D=0 we have simple logarithmic divergence

as in Eq. (4).

It is clear that the “integral” Eqs. (2)–(5) are more appro-

priate than Eq. (1) for SAA and GSL, because they are all

cumulative processes as the seismic deformation for which

some of those equations had been introduced.

The quality of the acceleration toward the critical point can

be evaluated by the Cfactor (Bowman et al., 1998), which

measures the ratio between the root mean squares (rms) of

the diverging function (rmsdf)and the rms of the best ﬁt line

(rmsline):

C=rmsdf

rmsline =s1−r2

df

1−r2

line (6)

where ris the correlation coefﬁcient of the corresponding

ﬁt. The lower than 1 the Cfactor is, the greater (and more

signiﬁcant) the acceleration toward the critical point is.

In the next section we will analyze the SAA at the Earth’s

surface, because the geomagnetic ﬁeld is known there and

Fig. 2. Global sea level (GSL) rise and its best log-periodic ﬁt with

Eq. (5). The critical time (2033±11 yr indicated by the vertical

dashed line in the smaller picture) within the given error is the same

as that estimated for the SAA. In the inset table, DoF are the degrees

of freedom and ris the correlation coefﬁcient of the nonlinear ﬁt;

for the other ﬁtting parameters see the text.

any global model is more reliable at the Earth’s surface than

at that extrapolated at the CMB, where the main sources of

the geomagnetic ﬁeld are placed (e.g., Merrill and McEl-

hinny, 1983): the higher harmonics, which are typically mea-

sured at the surface with a low signal-to-noise ratio, are

greatly ampliﬁed together with their errors, when extrapo-

lated downward to the CMB, contaminating any ﬁnal rep-

resentation of the ﬁeld at that depth (Lowes, 1974). In the

Appendix we show that the critical time tcfor the SAA is as

important at the Earth’s surface as at the CMB.

3 Application to SAA and GSL data and interpretation:

a great planetary change?

We applied all possible functions given by Eqs. (1)–(5) over

the SAA and GSL data. In our study the log-periodic ap-

proach Eq. (5) has shown the best ﬁt over the available data

with respect to the other possible functions in terms of the

lowest χ2and the highest correlation coefﬁcient r. Figures 1

and 2 show the corresponding results for SAA and GSL, re-

spectively. A low Cfactor (0.18 and 0.48 for SAA and GSL,

respectively) conﬁrms a signiﬁcant acceleration toward the

critical point. When we compare the couples of the same ﬁt-

ting parameters with each other, the agreement is astonishing

for most of them: in particular, the critical time tcis prac-

tically the same (around 2034±3yr and 2033±11yr, for

SAA and GSL, respectively; please note that the indicated

errors are only statistical because they could be up to two

times greater, Gross and Rundle, 1998); when the ﬁt is ap-

plied to GUFM1-G and GUFM1-F the results change a little,

with a critical time ranging from 2014 to 2027. In the above

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A. De Santis et al.: Toward a possible next geomagnetic transition? 3399

analyses we did not consider any error in the SAA area esti-

mates. Deﬁning an accurate error budget for the area of the

SAA is not possible.

Not only has one to ﬁnd what the accuracy of the Gauss

coefﬁcients is, but one also has to estimate what the contri-

butions of the unknown small scales of the magnetic ﬁeld

are. One also has to estimate what effect the regularization

process (if present) applied for deriving magnetic ﬁeld mod-

els from geomagnetic data has on the SAA area. Neverthe-

less we expect that the greatest contribution will come from

the Gauss coefﬁcient errors, so we try to take them into ac-

count in a simple way. Likely, errors in the Gauss coefﬁ-

cients change with time, say from 10% at the beginning of

the considered time interval and 1% in more recent times,

so we cannot be too wrong in supposing an average crude

error budget of 5% to propagate with the same percentage

to the SAA area values. When these errors are considered in

a weighted log-periodic ﬁt the results (not shown here) are

not signiﬁcantly different from those above (in particular, we

ﬁnd a critical time of 2042). Therefore, in all cases a criti-

cal process is still compatible with model data. This means

that the overall trend that underlies both quantities (SAA and

GSL) is something real and not an artefact. This conﬁrms

the choice of De Santis et al. (2012) to make the compari-

son of SAA and GSL (in terms of Spearman rank correlation

and relative entropy) without removing any trend (although,

when removing a trend and normalizing both time series to

unitary standard deviation, correlation still remains signiﬁ-

cant, with the Pearson correlation coefﬁcient r=0.62 and

P < 0.0001; this correlation increases much more when we

consider more recent data after 1800, reaching r=0.94 and

P < 0.0001). The low values of χ2/DoF (degrees of free-

dom) and the high values of the correlation coefﬁcient r(for

both quantities r>0.98), with respect to the corresponding

ﬁt, indicate that the acceleration of both SAA and GSL is un-

likely to be a mere coincidence, and that they are, rather, in-

dications of some physical underlying critical point process.

Also, the Dand fparameters are very similar in both SAA

and GSL, indicating that the ﬂuctuations affect the acceler-

ation in almost the same way in both physical quantities. In

addition, it is interesting to note that the critical time of the

SAA will be almost the time at which the SAA area, i.e., the

parameter A, will cover a hemisphere: because of the valid-

ity of Eqs. (A2) and (A3), this is limited not only to the ﬁeld

at the Earth’s surface, but would also be at the CMB, where

A0of Eq. (A4) will cover more than half of the core sur-

face. Since the SAA is usually considered the manifestation

at the Earth’s surface of a reversal magnetic ﬂux produced at

the CMB (e.g., Hulot et al., 2002), the epoch when the SAA

may reach the area corresponding to the surface of half the

planet is a critical moment for the present geomagnetic ﬁeld.

This time is not the time of the eventual geomagnetic rever-

sal, but we interpret it as the time of the point of no return,

after which the geomagnetic ﬁeld could fall in the process

of a global geomagnetic transition, which could be a rever-

sal or excursion of polarities. How long after the critical time

tcthis transition will occur cannot be fully established, be-

cause what we predict is a time when the dynamical system

reaches its critical state, after which any successive time is

a potential candidate for the actual start of the reversal or

excursion. Why GSL also shows the same overall trend with

similar parameters is a question that deserves further scrutiny

and is left to future work. What we can speculate now is that

when GSL reaches its critical point it will correspond to a

signiﬁcant coverage of many present coasts, implying a big

change in the land–ocean system. In addition, the similarities

found in both SAA and GLS conﬁrm that the two quantities

are really closely related, and, if the interpretation of an im-

minent geomagnetic ﬁeld reversal is correct, this would once

more support the internal hypothesis indicated among other

possibilities in De Santis et al. (2012).

4 Conclusions

In this work we analyze both SAA and GSL overall trends

in the last few centuries, ﬁnding an astonishing similarity,

further conﬁrming previous results (De Santis et al., 2012).

These similar trends can be explained by the theory of the

critical point processes for which each dynamical system is

close to or is going toward a critical point, when the system

will undergo a dramatic change in its macroscopic proper-

ties. This interpretation comes from the analysis of the SAA

behavior, for which the critical time tcwould correspond to

practically the time at which the SAA area will exceed the

extent of a hemisphere. Since SAA is a superﬁcial manifes-

tation of a reverse magnetic ﬂux at the CMB, this time will

be the time of no return after which the geomagnetic ﬁeld

will go to a signiﬁcant transition reverse in polarity, such as a

geomagnetic excursion or a complete geomagnetic reversal.

A similar dramatic change would have to occur in the

oceans, although no clear information can be obtained from

the present work. Regarding this, only some questions can

be asked: would the entire Earth or most of it be ﬂooded?

This seems not to be the case, because from a simple calcula-

tion (Woo, 2011), the predicted sea level rise of around 0.5m

higher than the present value (at time tc−1) will cause about

3km of present coasts to be covered by water. Nevertheless,

if this is the case, the consequences will be very dramatic as

well (let us think of the many cities and mega-cities that are

close to the coasts). Or would the GSL suddenly collapse? Or

would the GSL’s abrupt increase imply an enormous change

in the land–ocean system? Or what else? In this sense, if the

model we propose for both SAA and GSL is correct, what is

in preparation will be a really global change, and many more

parts of the planet could be involved, humankind included.

Of course, this interpretation must be taken with some cau-

tion at least for three reasons, so that it will need further in-

vestigation in the coming years. First, the SAA surface in the

most recent years seems to deviate slightly from the overall

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3400 A. De Santis et al.: Toward a possible next geomagnetic transition?

ﬁt, although this could simply be due to some edge effects in

the IGRF-11 model construction that we used for the most

recent years. Second, because of its intrinsic chaotic char-

acteristics (De Santis et al., 2004), the time of predictability

of the geomagnetic ﬁeld is comparable with the remaining

time to the predicted tc(e.g., De Santis et al., 2004; Hulot et

al., 2010). Thus, the prediction of the critical time should be

updated again as soon as more SAA and GSL data become

available, since any prediction based on a log-periodic func-

tion such as Eq. (5) is not stable when we are far from the

critical time, but improves its quality of prediction as soon as

we are closer to tc(e.g., Brée and Joseph, 2013). The study

of the diverging function parameters at successive predic-

tions/times together with the use of the Cfactor will also

allow one to investigate any deviation of the real behavior

from the prediction, and possibly to detect a change from the

present “catastrophic” trend: any departure from the behavior

predicted so far would be seen in terms of signiﬁcant increase

of both tcand the Cfactor. Third, both SAA and GSL can

also be well ﬁtted by some higher degree polynomial: for ex-

ample, a quintic polynomial (containing the same number of

unknown coefﬁcients of our log-periodic function) provides,

in terms of r2and χ2/DoF, a ﬁtting quality similar to that

obtained by the log-periodic function, although, of course,

the found polynomial behaves unrealistically outside the data

range, thereby excluding its use for forecasting purposes.

Now one might ask why we are able to predict the point

of no return from just a few hundred years of a phenomenon

that usually lasts several thousand years (Jacobs, 1994; but

see also Nowaczyk et al., 2012, where the Laschamp excur-

sion seems to change the geomagnetic polarity in a few hun-

dred years), i.e., with some analysis based on data taken over

a temporal window much shorter than the typical timescales

of the reversal or excursion process. A simple answer is that

we are analyzing a sufﬁcient (although short) time before the

eventual critical transition: we believe that the recent accel-

eration of both SAA and GSL is nothing casual, but probably

uncovers important physical information regarding the future

of our planet in the near future, such as a possible precursor

to the eventual close critical transition of the geomagnetic

ﬁeld.

Appendix A

This appendix has the aim of showing that the results ob-

tained by means of analyses made on the SAA at the Earth’s

surface are equivalent to those made at the CMB, where the

main sources of the geomagnetic ﬁeld are placed, but where

any extrapolation is difﬁcult or even impossible.

However complicated the geomagnetic ﬁeld may be at

the SAA within the 32000nT isoline, we can deﬁne a frus-

tum of quasi-cone that is conﬁned by the lower surface

S(rCMB)at the CMB, i.e., at r=rCMB =3485 km and the

upper surface S(r0)of the SAA at the Earth’s surface, i.e., at

Fig. A1. The magnetic ﬂux crossing both the core mantle bound-

ary (CMB) and South Atlantic Anomaly (SAA) is conserved. Thus

analyses on the SAA at the Earth surface are equivalent to those

made at the CMB.

r=r0=6371 km (Fig. A1); the lateral surface is here called

Sl. The lower surface S(rCMB)at the CMB is representative

of some typical isoline enclosing the reverse magnetic ﬂux

(we will come back to this in the ﬁnal part of the Appendix).

The divergence-free condition of the geomagnetic ﬁeld

imposes a null ﬂux through the surfaces bounding the vol-

ume :

8[S(rCMB)] − 8[S(r0)] − 8[Sl] = 0 (A1)

where 8[Si]is the magnetic ﬂux across the surface Si(where

Siis S(r0)or S (rCMB)or Sl)that can be expressed as follows:

8[Si] = ZBi·ndSi=BicosδSi

where Bicosδis the mean component of the ﬁeld perpendic-

ular to the surface Si(δis the angle between the vector ﬁeld

Band the vector nnormal to Si). For the geometry of the

quasi-conical volume, Bicosδwill be Zfor the upper SAA

and the lower surface at the CMB, while it will be a compo-

nent in the horizontal plane for the lateral surface Sl. We can

safely neglect the ﬂux across the lateral surface Slof ; thus

Eq. (A1) becomes:

Z(rCMB)S(rCMB)=Z(r0)S(r0)(A2)

and

S(rCMB)=Z(r0)

Z(rCMB)S(r0)=γ S (r0),

where the γratio

γ=Z(r0)/Z(rCMB)=S (rCMB)/S(r0)(A3)

Nat. Hazards Earth Syst. Sci., 13, 3395–3403, 2013 www.nat-hazards-earth-syst-sci.net/13/3395/2013/

A. De Santis et al.: Toward a possible next geomagnetic transition? 3401

Fig. A2. Surface enclosed by the isoline 200000 nT at the CMB

for an expansion of GUFM1 up to the spherical harmonic degree

N=3 and N=4. Both trends are almost monotonic and diverging

in time. A log-periodic function with critical (a priori ﬁxed) time of

2034 yr is a reasonable ﬁt for both cases.

can be taken as constant in time. This means that an equation

of the same form as Eq. (4) (but this would also be valid for

Eq. 5) can also be written for s0(t) of S(rCMB):

s0(t) =A0+B0ln(tc−t), (A4)

with A0=γ A and B0=γ B. Thus, the results we ﬁnd at the

Earth’s surface are also representative of the deep dynamics

of the geomagnetic ﬁeld; in particular, the critical time tces-

timated at the Earth’s surface will also be the same for the

CMB.

Unfortunately it is difﬁcult to verify the constancy of γ

with any global model (such as GUFM1) that is based on ob-

servational data taken at the Earth’s surface. This difﬁculty is

twofold: (i) the area S(rCMB)is impossible to determine, and

(ii) it is difﬁcult or even impossible to estimate Z(rCMB)be-

cause of the eventual explosion of errors when continuing the

vertical component from the Earth’s surface to the CMB, be-

cause of their multiplication by a factor [r0/rCMB]n+2(nis

here the spherical harmonic degree of the geomagnetic ﬁeld

expansion).

To reasonably circumvent most of the problems, we can

simply look at an isoline at the CMB that could act as the

32 000 nT at the Earth’s surface. By applying a simple dipolar

downward continuation of the 32 000nT isoline to CMB (just

multiplying by [r0/rCMB]3)we obtain about 200 000 nT.

Therefore, looking at the surface enclosed by the latter iso-

line at the CMB for an expansion of the GUFM1 model up to

N=3 and N=4 (Fig. A2), we notice an almost monotonic

trend for both cases where a log-periodic behavior pointing

to a critical (a priori ﬁxed) time of 2034 is something really

possible (Cfactor is always much less than 1 for both cases:

0.52 and 0.23 for N=3 and 4, respectively). By the way, the

area of the enclosed surfaces 1yr before the critical time for

N=3 and N=4 are 64 % and 84%, respectively, so in both

cases the surface of this isoline at the critical time will cover

more than half of the entire core surface. We limit our anal-

ysis made at the CMB to N=4, because for larger values of

N, the expected downward continuation errors would be too

large to reliably detect the 200000 nT (or any other) isoline.

Acknowledgements. Part of this work has been realised in the

frame of the SAGA-4-EPR project co-funded by the Italian Foreign

Ofﬁce, the Istituto Nazionale di Geoﬁsica e Vulcanologia (Italy)

and Northeastern University of Shenyang (China). We thank

G. Hulot, G. Balasis and P. Lurcock, whose constructive comments

helped us to improve a preliminary version of this manuscript.

We also thank two anonymous referees for their comments and

suggestions.

Edited by: R. Lasaponara

Reviewed by: two anonymous referees

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