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