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Heartbeat of the Sun from Principal Component Analysis and prediction of solar activity on a millenium timescale

Authors:
  • Universidad Bernardo O’Higgins

Abstract and Figures

We derive two principal components (PCs) of temporal magnetic field variations over the solar cycles 21-24 from full disk magnetograms covering about 39% of data variance, with λ=-0.67. These PCs are attributed to two main magnetic waves travelling from the opposite hemispheres with close frequencies and increasing phase shift. Using symbolic regeression analysis we also derive mathematical formulae for these waves and calculate their summary curve which we show is linked to solar activity index. Extrapolation of the PCs backward for 800 years reveals the two 350-year grand cycles superimposed on 22 year-cycles with the features showing a remarkable resemblance to sunspot activity reported in the past including the Maunder and Dalton minimum. The summary curve calculated for the next millennium predicts further three grand cycles with the closest grand minimum occurring in the forthcoming cycles 26-27 with the two magnetic field waves separating into the opposite hemispheres leading to strongly reduced solar activity. These grand cycle variations are probed by α-ω dynamo model with meridional circulation. Dynamo waves are found generated with close frequencies whose interaction leads to beating effects responsible for the grand cycles (350-400 years) superimposed on a standard 22 year cycle. This approach opens a new era in investigation and confident prediction of solar activity on a millenium timescale.
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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689
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Heartbeat of the Sun from
Principal Component Analysis and
prediction of solar activity on a
millenium timescale
V. V. Zharkova1,2,*, S. J. Shepherd3,*, E. Popova4,* & S. I. Zharkov5,*
We derive two principal components (PCs) of temporal magnetic eld variations over the solar
cycles 21–24 from full disk magnetograms covering about 39% of data variance, with σ = 0.67.
These PCs are attributed to two main magnetic waves travelling from the opposite hemispheres
with close frequencies and increasing phase shift. Using symbolic regeression analysis we also derive
mathematical formulae for these waves and calculate their summary curve which we show is linked
to solar activity index. Extrapolation of the PCs backward for 800 years reveals the two 350-year
grand cycles superimposed on 22 year-cycles with the features showing a remarkable resemblance
to sunspot activity reported in the past including the Maunder and Dalton minimum. The summary
curve calculated for the next millennium predicts further three grand cycles with the closest grand
minimum occurring in the forthcoming cycles 26–27 with the two magnetic eld waves separating
into the opposite hemispheres leading to strongly reduced solar activity. These grand cycle variations
are probed by α Ω dynamo model with meridional circulation. Dynamo waves are found generated
with close frequencies whose interaction leads to beating eects responsible for the grand cycles
(350–400 years) superimposed on a standard 22 year cycle. This approach opens a new era in
investigation and condent prediction of solar activity on a millenium timescale.
Solar activity is manifested in sunspot occurrence on the solar surface characterized by the smoothed
sunspot numbers, which were selected as a proxy of solar activity (see, for example, the top plot in
http://solarscience.msfc.nasa.gov/images/by.gif). e sunspot numbers show quasi-regular maxima and
minima of solar activity changing approximately every 11 years, with changing leading magnetic polarity
in a given hemisphere (or 22 years for sunspots with the same polarity) reecting changing magnetic
activity of the Sun1.
e longest direct observation of solar activity is the 400-year sunspot-number series, which depicts a
dramatic contrast between the almost spotless Maunder and Dalton minima, and the period of very high
activity in the most recent 5 cycles2,3, prior to cycle 24. Many observations indicate essential dierences
between the activity occurring in the opposite hemispheres for sunspots4 and for solar and heliospheric
magnetic elds5.
Prediction of a solar cycle through sunspot numbers has been used for decades as a way of testing
accuracy of solar dynamo models, including processes providing production, transport and disintegra-
tion of the solar magnetic eld. Cycles of magnetic activity are associated with the action of a dipole
1Northumbria University, Department of Mathematics & Information Sciences, Newcastle upon Tyne, NE2 1XE, UK.
2Institution of Space Science Research, Space Physics Department, Kiev, 03022, Ukraine. 3University of Bradford,
School of Engineering, Bradford, BD7 1DP, UK. 4Skobeltsyn Institute of Nuclear Physics, Moscow 119234, Russia.
5University of Hull, Department of Physics and Mathematics, Kingston upon Hull, HU6 7RX, UK. *These authors
contributed equally to this work. Correspondence and requests for materials should be addressed to V.V.Z. (email:
valentina.zharkova@northumbria.ac.uk)
Received: 28 April 2015
Accepted: 25 September 2015
Published: 29 October 2015
OPEN
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solar dynamo mechanism called ‘αΩ dy n a m o’ 6. It assums the action of solar dynamo to occur in a
single spherical shell, where twisting of the magnetic eld lines (α-eect) and the magnetic eld line
stretching and wrapping around dierent parts of the Sun, owing to its dierential rotation (Ω -eect),
are acting together7,8.
As a result, magnetic ux tubes (toroidal magnetic eld) seen as sunspots are produced from the solar
background magnetic eld (SBMF) (poloidal magnetic eld) by a joint action of dierential rotation
(Ω -eect) and radial shear (α-eect), while the conversion of toroidal magnetic eld into poloidal eld
is governed by the convection in the rotating body of the Sun. e action of the Coriolis force on the
expanding, rising (compressed, sinking) vortices results in a predominance of right-handed vortices in
the Northern hemisphere and le-handed vortices in the Southern hemisphere leading to the equator-
ward migration of sunspots during a solar cycle duration visible as buttery diagams (see http://solarsci-
ence.msfc.nasa.gov/images/by.gif, the bottom plot).
e last few decades were extremely fruitful in investigating the contribution of various mechanisms
to the dynamo processes including the conditions for dynamo wave generation from the mean dynamo
models with dierent properties of solar and stellar plasmas, as discussed in the recent reviews7,8.
As usual, the understanding of solar activity is tested by the accuracy of its prediction. e records
show that solar activity in the current cycle 24 is much lower than in the previous three cycles 21–23
revealing more than a two-year minimum period between cycles 23 and 24. is reduced activity in cycle
24 was very surprising because the previous ve cycles were extremely active and sunspot productive
forming the Modern Maximum2,3. Although the reduction of solar activity in cycle 24 led some authors
to suggest that the Sun is on its way towards the Maunder Minimum of activity9.
However, most predictions of solar activity by various methods, such as considering linear regression
analysis10, neural network forecast11, or a modied ux-transport dynamo model calibrated with histor-
ical sunspot data from the middle-to-equator latitudes12, anticipated a much stronger cycle 2410. ere
were only a few predictions of the weaker cycle 2413 obtained with the high diusivity Babcock-Leighton
dynamo model applied to polar magnetic elds as a new proxy of solar activity. However, a dynamo
model with a single wave was shown to be unable to produce reliable prediction of solar activity for
longer than one solar cycle because of the short memory of the mean dynamo14.
Consistent disagreement between the sunspot numbers, measured averaged sunpost numbers and
the predicted ones by a large number of complex mathematical models for cycle 24, is undoubtedly the
result, which emphasizes the importance of dierent physical processes occurring in solar dynamo and
aecting complex observational appearance of sunspots on the surface.
Results
Two principal components as two dynamo waves. In order to reduce dimensionality of these
processes in observational data, Principal Component Analysis (PCA) was applied15 to low-resolution
full disk magnetograms captured by the Wilcox Solar Observatory16. is approach revealed a set of more
than 8 independent components (ICs), which seem to appear in pairs15, with two principal components
(PCs) covering about 39% of the variance of the whole magnetic eld data, or standard deviation of
σ = 0.67. e main pair of PCs is associated with two magnetic waves of opposite polarities attributed to
the poloidal eld produced by solar dynamo from a dipole source17.
e two principal components (PCs) derived from solar background magnetic eld (SBMF)15 (cycle
21–23) and predicted for cycle 24–26 are presented in Fig. 1 (the upper plot). For the rst time PCA
allowed us to detect, two magnetic waves in the SBMF15 and not a single one assumed in the mean
dynamo models. ese waves are found originating in the opposite hemispheres and travelling with an
increasing phase shi to the Northern hemisphere in odd cycles and the Southern hemisphere in even
cycles15. is can explain the well-observed North-South asymmetry in sunspot numbers, background
magnetic eld, are occurences and so on (see Zharkov et al.4 and references therein) dening the active
hemisphere for odd (North) and even (South) cycles.
e formation of magnetic ux tubes emerging on the solar surface as sunspots can be considered as
a result of interaction in the solar interior of the two magnetic waves of the solar background magnetic
eld15 when their phase shi is not very large. ese two magnetic waves of the poloidal eld can account
for the observed sunspot magnetic eld18, or averaged sunspot numbers, aer their amplitudes are added
together into the summary wave (Fig.1, bottom plot) and converted to the modulus curve by taking
modulus of the summary curve19 (Fig.2, bottom plot). e modulus curve plotted for cycles 21–23 in
Fig.2 (top plot) corresponds rather closely to the averaged sunspot numbers for cycles 21 and 22 while
being noticeably lower than the sunspot curve for cycle 23, which anticipated the recently discovered
sunspot calibration errors occurred in the past few decades20.
e maximum (or double maximum for the waves with a larger phase shi of solar activity for a given
cycle) coincides with the time when each of the waves approaches a maximum amplitude and the hem-
isphere where it happens becomes the most active one. is can account naturally for the north-south
asymmetry of solar activity oen reported in many cycles. Also the existence of two waves in the poloidal
magnetic eld instead of a single one, used in most prediction models, and the presence of a variable
phase dierence between the waves can naturally explain the diculties in predicting sunspot activity on
a scale longer than one solar cycle with a single dynamo wave14 since the sunspot activity is associated
with the modulus summary curve of the two dynamo waves19 that is a derivative from these two waves.
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Mathematical description of the observed magnetic waves. Amplitude and frequency varia-
tions of these waves, or PCs, over time are found using symbolic regression analysis21 with Euriqa so-
ware (see the Methods section for data analysis19). e wave amplitudes follow the product of two cosine
functions (cos * cos), while the frequencies folow a nested function (cos (cos)) depicting the fact that the
waves periodically change their frequency and phase with time. ese formulae are used to extract the
key parameters of the principal components of SBMF waves, which are, in turn, used for prediction of
the overall level of solar activity for solar cycles 24–26 associated with the averaged sunspot numbers19.
e accuracy of these formulae for prediction of the principal components is tested for cycle 24 show-
ing the predicted curve tting very closely (with an accuracy of about 97.5%) the PCs derived from the
observations of SBMF and sunspot numbers19.
Figure 1. Top plot: the two principal components (PCs) of SBMF (blue and red curves) obtained for
cycles 21–23 (historic data15) and predicted19 for cycles 24–26 with the Eqs. (2)–(3). e dotted lines
show the PCs derived from the data and the solid lines present the curves plotted from formulae 2 (blue)
and 3 (red). e accuracy of t of the both PC curves is better than 97%. e point A shows the current
time. e cycle lengths (about 11 years) are marked at the minima by the vertical lines. e bottom plot:
e summary PC derived from the two PCs above for the ‘historical’ (21–23) and predicted cycles (24–26)
data. e dotted curve shows PCs derived from the data and the solid line - from the the solid curves from
the top plot using formulae 2–3. e cycle lengths (about 11 years) are again marked by the vertical lines at
the cycle minima. All the plots are a courtesy of Shepherd et al.19. © AAS. Reproduced with permission.
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For the forthcoming cycles 25 and 26 (Fig.1) the two waves are found to travel between the hemi-
spheres with decreasing amplitudes and increasing phase shi approaching nearly a half period in cycle
26. is leads, in fact, to a full separation of these waves in cycle 26 into the opposite hemispheres19.
is separation reduces any possibility for wave interaction for this cycle that will result in signicantly
reduced amplitudes of the summary curve and, thus, in the strongly reduced solar activity in cycle 2619,
or the next Maunder Minimum9 lasting in 3 cycles 25–27.
Prediction of solar activity on millennium scale. By far the most impressive achievement to-date
of this approach is its ability to make very long term predictions of solar activity with high accuracy
over the timescales of many centuries. e summary curve of the two principal components (magnetic
waves) expressed by the formulae (2 and 3) in the Method of data analysis19 is calculated backwards and
forwards for the period 1200–3200 years as shown in Fig.3.
Remarkably, our current prediction of the summary curve backwards by 800 years shown in the le
(from oval) part of Fig.3, corresponds very closely to the sunspot data observed in the past 400 years as
indicated by the brackets in Fig.3, with the black oval marking the data used to derive Eq. (2) and (3)
dening the wave variations. We predict correctly many features from the past, such as: 1) an increase in
solar activity during the Medieval Warm period; 2) a clear decrease in the activity during the Little Ice
Figure 2. Top plot: Comparison of the modulus summary curve (black curve) obtained from the
summary curve in Fig. 1 with averaged sunspot numbers (brown curve) and magnetic el (blue curve)
for cycles 21–23. Bottom plot: e modulus summary curve associated with the sunspot numbers derived
for cycles 21–23 (plotted in the top plot) and calculated for cycles 24–26 using the mathematical formulae
(2–3). e plots are a courtesy of Shepherd et al.19. © AAS. Reproduced with permission.
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Age, the Maunder Minimum and the Dalton Minimum; 3) an increase in solar activity during a modern
maximum in 20th century.
is visual correspondence in the features between the summary curve and the averaged sunspot
numbers is most surprising, given the fact that the principal components are derived from the solar back-
ground magnetic eld, and they are not linked directly to the sunspot numbers (see Methods for data
analysis) besides the modulus summary curve derived from the principal components as shown in Fig.2.
e summary curve reveals a superposition of the amplitudes of the two dynamo waves, or a ‘beating’
eect creating two resulting waves: one of higher frequency (corresponding to a classic 22-year cycle) and
a second wave of lower frequency (corresponding to a period of about 350–400 years), which modulates
the amplitude of the rst wave. It appears that this grand cycle has a variable length from 320 years (in
18–20 centuries) to 400 (in 2300–2700) predicted for the next millennium. Amplitudes in the shorter
grand cycles are much higher than the amplitudes in the longer ones.
is long-term ‘grand’ cycle was previously postulated in 1876 by Clough22 as a 300-year cycle super-
imposed on the 22 year cycle using the observations of aurorae, periods of grape harvests etc, which was
later suggested to have a period of about 205 years23. ese periods are close to those reported for the last
800 years in the summary curve plotted in Fig.3 derived from the observed magnetic eld variations.
e spectacular accuracy of the historical t in the past 800 years gave us the condence to extrap-
olate the data into the future for a similar epoch of 1200 years (Fig. 3, right part of the curve) clearly
Figure 3. e predicted summary wave (the sum of two principal components) calculated from 1200 to
3200 years from the ‘historical’ period (cycles 21–23) marked with a black oval. e historical maxima
and minima of the solar activity in the past are marked by the horizontal brackets.
Figure 4. e schematic dynamo model with two cells in the solar interior having the opposite
meridional circulation as derived from HMI/SDO observations by Zhao et al.24. © AAS. Reproduced with
permission.
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showing, as expected, several 350–400-year grand cycles. We note, in particular, a decreasing activity
for solar cycles 25 and 26 coinciding with the end of the previous 350–400-year grand cycle and then
increase of the solar activity again from cycle 27 onwards as the start of a new grand cycle with an unu-
sually weak cycle 30. Hence, cycles 25–27 marks a clear end of the modern grand period that can have
signicant implications for many aspects of solar activity in human lives including the current debate
on climate change.
Discussion
Preliminary interpretation with the two layer α Ω dynamo model. Now let us attempt some
preliminary interpretation of the two principal components, or two magnetic waves of solar poloidal
eld, generated by the solar dynamo in two dierent cells, similar to those derived by Zhao et al.24
from helioseismological observations (Fig.4), in order to t the background magnetic eld observations
(Figs1 and 3). is can be achieved with the modied Parker’s non-linear two layers dynamo model for
two dipoles17 with meridional circulation: in the layer 1 of the top cell and layer 2 of the bottom cell from
Fig.4 (see Methods section for the model description) tested for the interpretation of latitudinal waves
in the solar background magnetic eld for cycles 21–2317 derived with PCA15.
e simulation results presenting the toroidal magnetic eld are plotted in Fig.5 (bottom plot) derived
from the poloidal eld (Fig.1, top plot) for a period of six 11-year cycles using the dynamo equations
(16–19) from Popova et al.17. e curves for poloidal (derived with PCA) and toroidal elds (simulated
with the dynamo model) are found to have similar periods of oscillations whilst having opposite polar-
ities (or having the phase shi of a half of the period), being in anti-phase every 11 years as previously
reported4,25. e amplitude of generated toroidal magnetic eld is plotted versus the dynamo number
=⋅
α
DRR
in Fig.5 (top plot).
Furthermore, in cycles 25–27 and, especially, in cycle 26, the toroidal magnetic eld waves generated
in these two layers become fully separated into the opposite hemispheres, similar to the two PC waves
attributed to poloidal eld (Fig.1, top plot), that makes their interaction minimal. is will signicantly
reduce the occurance of sunspots in any hemisphere, that will result in a very small solar activity index
for this cycle, resembling the Maunder Minimum occurred in the 17th century.
Figure 5. Top plot: Dependence of the solar dynamo-number D = RαRΩ on a magnitude of the toroidal
magnetic eld (for detials of the parameters see the text). Bottom plot: Variations of the toroidal magnetic
eld simulated for cycles 21–26 with two layer αΩ dynamo model (see Methods section) for the inner (red
line) and upper (blue line) layers. One arbitrary unit corresponds to 1–1.5 Gauss (see text for details).
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Using the same dynamo parameters derived from the observed principal components for these 6
cycles, let us extend the calculation (see the Methods for details) to a longer period of two millennia
shown in Fig.6 for both poloidal (top plot) and toroidal (bottom plot) elds. According to the dynamo
theory and analysis of observational data7,27 the generated toroidal eld is much stronger than the poloi-
dal. Although, exact values of the amplitudes of these elds in the solar convection zone are unknown
and estimated from dynamo models. In our simple model the amplitude of toroidal eld at the maximum
is about 1000 Gauss, and of the poloidal one is of the order of several tens of Gauss. Hence, in Figs5 and
6 one arbitrary unit approximately corresponds to 1–1.5 Gauss.
It can be seen that variations of the model magnetic elds (Fig.6) generated by the two dipole sources
located in diferent layers reproduce the main features discovered in Fig.3, e.g modulation of the ampli-
tude of 22 year cycle by much slower oscillations of about 350 years, dierent duration (320–400) and
amplitudes of dierent grand cycles. ese variations are governed by dierent dynamo parameters as
discussed below.
Beating eect of two dynamo waves with close frequencies. e waves generated by a dynamo
mechanism in each layer are found to have similar (but not equal) frequencies caused by a dierence
in the meridional ow amplitudes in the two layers (Fig. 5, bottom plot). In order to reproduce the
summary curve in Fig.3 from the two original waves, or PCs, the dynamo waves generated in dierent
layers with an amplitude A0 have to have close but not equal frequencies ω1 and ω2 (or periods varying
between 20 and 24 years), similar to Gleissberg’s cycle7,26.
e interference of these waves enabled by diusion of the waves in the solar interior from the bottom
to the top layer27 leads to formation of the resulting envelope of waves Y(t), or beating eect (see Fig.3
and theoretical plots in Fig.6), showing oscillations of a higher frequency
ωω(+ )/2
12
within the enve-
lope and those of the envelope itself with a lower frequency of
ωω(− )/2
12
(or in a grand cycle) as
follows:
ωω
ωω ωω
()(−)
+
,
()
Yt A
ktt
2sin2sin2
1
0
1
2
2
2
12 12
Figure 6. Variations of the summary poloidal (top plot) and toroidal (bottom plot) magnetic elds
simulated for 2000 years with the two layer αΩ-dynamo model (see Methods section) with the
parameters derived from the two PCs from Fig. 1 using mathematical formulae (2–3). One arbitrary unit
corresponds to 1–1.5 Gauss (see text for details).
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where k is some parameter dening properties of the solar interior where the waves propagate, e.g. dif-
fusivity, dynamo number (α and Ω eects) and meridional circulation.
Frequency and period variations. e beating eect between these frequencies can easily explain seem-
ingly sporadic variations of high frequency amplitudes and the period of the low-frequency envelope
wave in the resulting grand cycles seen in both the observational curve (Fig. 3 and theoretical curves
(Fig.6) reproducing the observational one. e higher the dierence of frequencies the larger is the fre-
quency, or a shorter period, of the grand cycle (350 years) and the smaller is a number of high frequency
waves ( 22 year period) within this grand cycle. is eect is clearly seen in Figs 3 and 6, where the
grand periods with a lower number of 22 year cycles are shorter (300–340 years, 2nd, 3rd and 5th grand
cycles in Fig. 3), while those with higher number of 22-year cycles are longer (360–400 years, the 1st
and 4th in Fig.3).
e dierence in frequencies of the dynamo waves in two layers is governed by the variations of
velocities of meridional circulations in the very top and the very bottom zones of these two layers (see
the Method section) (schematically presented in Fig.4 from Zhao et al.24). e frequency of a wave is
reduced (or its period is increased) when the meridional circulation has higher velocities and this fre-
quency is increased (or its period is decreased) when the meridional circulation is slower. It means that
the meridional circulation acts as a drag force for dynamo waves generated in each layer altering their
natural frequencies that would occur without the circulation.
For example, within the two layers model considered, and taking into account that the low frequency
cycles can have length Tg from 20 to 24 years (variations within Gleissberg’s cycle7), in order to produce
the grand cycle with a beating period of 350 years, the periods of the dynamo waves in two layers should
vary as follows: for the sunspot activity period Tg = 20 years -for the inner layer wave 1 − T1 = 18.9 years
(corresponding to the velocity of meridional circulation about V = 7–8 m/s), for the upper layer wave
2 T2 = 21 years (V = 9–10 m/s); for the activity period Tg = 24 years: the inner layer wave 1 T1 = 22.46
years (V = 10–11 m/s), the upper layer wave 2 T2 = 25.8 years (V = 13–14 m/s).
If the grand cycle is 400 years, then the dynamo wave periods in two layers would slightly change;
e.g. for the cycle period Tg = 20 years - for the inner layer wave 1 T1 = 19 years (V = 7–8 m/s), for the
upper layer wave 2 T2 = 21 years (V = 9–10 m/s); for the period of Tg = 24 years: the inner layer wave
1 T1 = 22.6 years (V = 10–11 m/s), the upper layer wave 2 T2 = 25.53 years (V = 13–14 m/s).
It can be seen that the period of the wave 1 generated in the inner layer (at the bottom of the convec-
tive zone) remains more or less stable at about T1 = 19 years (for generation of the low frequency activity
period Tg = 20 years) or T1 = 22.6 year (for Tg = 24 years). While the period of the wave 2 generated in
the upper layer should have larger uctuations (e.g. T2 = 25.8 years for 350 grand cycle versus T2 = 25.53
years for 400 years grand cycle). ese uctutation are likely to be aected by the physical conditions
in the solar interior, where the wave 2 is formed and the wave 1 has to travel through and to interact
with the wave 2 to cause the beating eect combining the grand (ranging in 300–400 years) and short
(ranging in 20–24 years) cycles seen in Fig.3 as reproduced with the dynamo model in Fig.6 for both
poloidal and toroidal magnetic elds.
Of course, estimations of the wave beating above are rather preliminary, given the fact that the PCs
(or dynamo waves) in each layers comprise at least 5 waves with close frequencies as discussed in the
Method section (Eqs. 2 and 3). is results in much more complex beating eects derived from PCA as
presented in Fig.3. e dynamo calculations only partially reproduced the long cycle with a period of
about 350 years, which is the same for the whole millennium. However, in order to reproduce the full
summary curve with the variable long-term period in Fig.3 more detailed dynamo simulations including
quadruple magnetic sources in all the three layers (shown in Fig.4) are required.
Wave amplitude variations. e amplitudes of dynamo waves are aected by the variations of both α
and Ω eects, or by the dynamo number D, i.e. a decrease of the negative dynamo number D (or its
increase in absolute value) leads to an increase of toroidal eld amplitude (see Fig.5, top plot).
is eect can be observed in both the observational (Fig.3) and theoretical (Fig.6) plots. In shorter
grand cycles (with periods of 300–340 years), e.g. in 1800–2000 years and 2100–2350 years, the ampli-
tudes of the high frequency wave (Tg = 20–24 years) are much higher than in longer cycles (periods of
350–400 years) in 1300–1650 years or 2400–2800 years. Although, in order to reproduce more closely
the whole variety of observational features on a longer timescale, more detailed 3D model simulations
are required.
erefore, the derived mathematical laws in cyclic variations of principal components of the observed
solar magnetic eld, which t closely most of the observational features of solar activity in the past as
shown in Fig.3 and reproduced by the dynamo model in Fig. 6 opens a new era in the investigation
of solar activity on millennium scale. By combining the observational curve with simulations of solar
dynamo waves in two layers, it is possible to derive better understanding of the processes governing solar
activity and produce long-term prediction of solar activity with impressive accuracy.
Methods
Derivation of parameters of the observed magnetic waves. In order to distill the main param-
eters of the waves present in the observational solar magnetic data, one needs to reduce their
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dimensionality with the Principal Component Analysis (PCA)28. PCA is an orthogonal linear transfor-
mation allowing a vector space to be transformed to a new coordinate system, reducing the multi-di-
mensional data to lower dimensions for analysis, so that the greatest variance by any projection of the
data lies on the rst coordinate called the Principal Component (PC) with the second PC orthogonal to
the rst is dened by the second largest variance. is technique simultaneously (i) reduces the data
dimensionality, (ii) increases the signal-to-noise ratios and (iii) orthogonalises the resulting components
so that they can be ascribed to separate physical processes (see Zharkova et al.15 for more details). e
PCA is an exact method, and its accuracy dened only by the noise of measurements,
σx
2
, of the original
vector29.
PCA was applied to low-resolution full disk solar background magnetic eld (associated with the
poloidal magnetic eld) only become available from cycle 21 to cycle 24 as measured by the Wilcox Solar
Observatory (with accuracy better than 0.5 Gauss, or the measurement error
σ
=.
)
00025
x
2
. We derive
the dominant eigenvalues (0.1 and 1.0) covering the maximum variance of 39%15 dening the eigenfunc-
tions, or Principal Components (PCs), which came as a pair of waves. ese PCs are considered as the
main (dipole) dynamo waves of the solar poloidal magnetic eld.
By applying a 3-year running averaging lter, any short-term (< 3 years) uctuations of magnetic
eld data are removed allowing us to keep the accuracy of PCA not worse than the measurement error
(Wentzell and Lohnes30). e overall PCA accuracy of dening its eigen values from the WSO data with
known measurement error (see Faber et al.29) is not worse than 0.2%. Running PCA on a combination
of magnetic eld measurements for any two cycles, or for all four cycles21–24 produces, within the error
of 0.2%, the same eigenvalues as for the three cycles used in PCA15.
For classication of the derived PCs we apply the symbolic regression approach based on the
Hamiltonian principle implemented in the Euriqa soware21. is allows us to derive the exact mathe-
matical formulae for the amplitude variations and phase shis of both principal components as follows19:
for wave 1:
ωφ ωφ()=(+)((+)),
()
=
,,,,
ft At Btcoscos cos2
k
N
kk
kkkk
1
1122
for wave 2:
ωφ ωφ()=(+)((+)),
()
=
,,,,
ft Ct Dtcoscos cos
3
k
N
kkkkkk
1
3344
where the parameters with ω dene the corresponding wave frequencies and φ dene their phase shis.
Shepherd et al.19 found that the approximations with only N = 5 terms in the series above allow them
to capture the functions describing the waves of PCs for the cycles 21–24 with an accuracy better than
97%19. As expected, any attempts to distill the parameters from the original magnetic eld data (before
deriving PCs) were unsuccessful indicating the very complex nature of the original magnetic eld waves.
ese two PCs are used for calculation of the summary wave (a sum of amplitudes) and the modulus
summary wave (reected to the positive amplitudes only) linked to the averaged sunspot numbers cur-
rently used for denition of solar activity.
Non-linear αΩ dynamo model in a two-layer medium with meridional circulation. In order
to understand the basic features of the derived PCs, let us use Parker’s αΩ -dynamo model with two
layers with meridional circulation17 updated by considering a non-linear dynamo process. It is assumed
that dynamo waves are generated by the dipole sources only located in two layers: one dipole in the sub-
surface layer and the other dipole deeply in the solar convection zone (see Fig. 4); and the parameters
(dynamo number and meridional circulation) of magnetic eld generation in each layer are dierent17.
is results in the simultaneous existence of two magnetic waves with dierent periods and phase
shis17, similar to those derived with PCA (see Fig.1). For the sake of simplicity this approach excludes
the dynamo waves generated by quadruple sources in both layers accounting for the other six independ-
ent components17, which are shown to slightly modify the overall appearance of magnetic waves that will
be considered in the forthcoming paper.
e dynamo equations describing the generation and evolution of the solar magnetic eld in a two-layer
medium, are obtained from a system of electrodynamic equations for mean elds in the assumption that
the dynamo wave propagates in a thin spherical shell17. In this case, the magnetic eld is averaged along
the radius within a certain spherical shell and the terms describing the curvature eects near the pole
are excluded. In addition, in this approximation, we assume that the magnetic eld is generated inde-
pendently in either layer. As a result, the equations take the form of equations (16–19) by Popova et al.17
solved numerically using the method of lines31 and veried analytically by the low-mode approach32.
In these equations the dynamo number
=α
DRR
is dened by the parameters Rα and RΩ describing,
respectively, the intensity of α-eect, and the dierential rotation, or Ω -eect. e latitudinal prole of
the poloidal magnetic eld is assumed for simplicity to be proportional cos(θ) where θ is the solar
latitude measured from the equator.
www.nature.com/scientificreports/
10
SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689
We consider the α-eect the amplitude F(t, coordinates) and widely used algebraic quenching7 in a
form:
αξ
αθ
ξ
=
(, )
+( )=
()
+( ).
()
α
Ftcoordinates
BRB11 4
toroidal toroid al
22
0
22
For calculations the amplitude of α eect, Rα is moved to the dynamo number D, while the algebraic
quenching of the helicity is used for stabilization of a magnetic eld growth, e.g. redening
α
=
αθ
ξ
()
(+ )B1
0
22
,
where
is the helicity in unmagnetized medium and
ξ
=
B0
1
is the magnetic eld, for
which the
α
-eect is considerably suppressed.
e contribution of dierential rotation into the generation of magnetic eld is dened27 by the terms
θ()
θ
co
s
A
for one layer or θ()
θ
co
s
a
for another layer, following the general trend of being maximum at
the equator and minimal at the poles. e dynamo number
=α
DRR
also includes an amplitude of
dierential rotation RΩ, which can vary in dierent layers.
Since the frequencies of magnetic waves generated by dynamo mechanism are known to be mostly
aected by meridional circulation velocities17,32 while their amplitudes are governed by the variations
of dynamo number D, then the PC waves can be reproduced in dierent layers with dierent dynamo
numbers and slightly dierent meridional circulation velocities.
In each layer we consider a 1D dynamo model with meridional circulation V being a function
dependent only on θ e.g.
θ=()Vvsin2
, so that it vanishes at the poles and is maximal at the middle
latitudes approaching amplitudes of 9–15 m/s24,33. Also, to comply with the material conservation rule,
the meridional circulation the multi-cellular meridional circulation has to have the opposite directions
in upper and inner layers of the cells in as shown in Fig. 4 suggested earlier by Dikpati34,35 and Popova
et al.17 that was recently conrmed from the helioseismic observations with HMI by Zhao et al.24.
In general, there are three layers (see Fig.4), in which the meridional circulation aects the magnetic
eld: 1- the very top layer in the upper cell and 2- the very bottom layer in the inner cell where the
meridional circulation has the same direction but dierent velocities and 3- the middle layer where cells
have a boundary and their the circulation has the opposite direction, complying with the mass conser-
vation law. In the current model we consider the top and bottom layers (1–2) only to reproduce the
long-term oscillations produced in them, while oscillations in the middle layer 3 aecting the short-term
biennial32 are out of scope of the current study.
For each layer the principal components (PCs) of poloidal magnetic eld were substituted into the
dynamo equations (17 and 19) of Popova et al.17, from which the corresponding toroidal magnetic eld
components are derived tting these PCs. en we substitute the toroidal and poloidal components into
the dynamo equations (16 and 18) of Popova et al.17 for corresponding layers and derive their dynamo
numbers D.
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Acknowledgements
e authors wish to thank the Directorate and sta of the Wilcox Solar Observatory (Stanford) for pro-
viding online the synoptic magnetic eld data of the Sun for the past 4 solar cycles. VZ and SZ wish to
acknowledge that this research was started during the EU Framework 5 grant ‘European Grid of Solar
Observations’, grant IST-2001-32409.
Author Contributions
V.Z. and S.Z. conceived the experiment and analysed the data with PCA, S.S. conducted the data
prediction with Euriqa, E.P. developed the model, V.Z. and E.P. analysed the results. All authors reviewed
the manuscript.
Additional Information
Competing nancial interests: e authors declare no competing nancial interests.
How to cite this article: Zharkova, V. V. et al. Heartbeat of the Sun from Principal Component
Analysis and prediction of solar activity on a millennium timescale. Sci. Rep. 5, 15689; doi: 10.1038/
srep15689 (2015).
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... Recently, Zharkova et al. (2012Zharkova et al. ( , 2015 suggested to use an additional proxy of solar activity -the eigen vectors of the solar background magnetic field (SBMF) obtained from the Wilcox Solar Observatory low resolution synoptic magnetic maps. . By applying the principal component analysis (PCA) to the synoptic magnetic data for cycles 21-23 (Zharkova et al. 2012) and recently for 21-24 (Zharkova & Shepherd 2022) the authors identified and confirmed a number of eigen values and eigen vectors from the SBMF representing magnetic waves of the solar surface generated by different magnetic sources in the solar interior. ...
... The first pair of eigen vectors covered by the largest amount of the magnetic data by variance, or principal components (PCs), reflects the primary waves of solar magnetic dynamo produced by the dipole magnetic sources (Zharkova et al. 2015). The temporal features of the summary curve of these two PCs shown a remarkable resemblance to the sunspot index of solar activity (representing toroidal magnetic field of active regions) for cycles 21-23 (Zharkova et al. 2015) and cycles 21-24 (Zharkova & Shepherd 2022). ...
... The first pair of eigen vectors covered by the largest amount of the magnetic data by variance, or principal components (PCs), reflects the primary waves of solar magnetic dynamo produced by the dipole magnetic sources (Zharkova et al. 2015). The temporal features of the summary curve of these two PCs shown a remarkable resemblance to the sunspot index of solar activity (representing toroidal magnetic field of active regions) for cycles 21-23 (Zharkova et al. 2015) and cycles 21-24 (Zharkova & Shepherd 2022). This correspondence occurs despite the PCs and sunspot indices represent very different entities of solar activity: poloidal magnetic field for PCs and toroidal magnetic field for sunspot numbers. ...
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We decompose the monthly cosmic-ray data, using several neutron monitor count rates, of Cycles 19-24 with principal component analysis (PCA). We show using different cycle limits that the first and second PC of cosmic-ray (CR) data explain 77-79% and 13-15% of the total variation of the Oulu CR Cycles 20-24 (C20- C24), 73-77% and 13-17% of the variation of Hermanus C20-C24, and 74-78% and 17-21% of the Climax C19-C22, respectively. The PC1 time series of the CR Cycles 19-24 has only one peak in its power spectrum at the period 10.95 years, which is the average solar cycle period for the interval SC19-SC24. The PC2 time series of the same cycles has a clear peak at period 21.90 (Hale cycle) and another peak at 1/3 of that period with no peak at the solar cycle period. We show that the PC2 of the CR is essential in explaining the differences in the intensities of the even and odd cycles of the CR. The odd cycles have positive phase in the first half and negative phase in the second half of their PC2. This leads to slow decrease of the intensity in the beginning of the cycle and at minimum for the odd cycles. On the contrary, for the even cycles the phases are vice versa and this leads to faster decrease and more rapid recovery in the CR intensity of the cycle. As a consequence the even cycles have more peak-like structure. The only exceptions of this rule are Cycles 23 and 24 such the former has almost zero line PC2, and the latter has similar PC2 than the earlier odd cycles. These results are confirmed with skewness-kurtosis (S-K) analysis. Furthermore, S-K shows that other even and odd cycles, except Cycle 21, are on the regression line with correlation coefficient 0.85. The Cycles 21 of all calculated eight stations are compactly located in the S -K coordinate system and have smaller skewnesses and higher kurtoses than the odd Cycles 23.
... PCA finds combinations of variables, that describe major trends in the data. PCA has earlier been applied, e.g., to studies of the geomagnetic field (Bhattacharyya and Okpala, 2015), geomagnetic activity Takalo, 2021b), ionosphere (Lin, 2012), the solar background magnetic field (Zharkova et al., 2015), variability of the daily cosmic-ray count rates (Okpala and Okeke, 2014), and atmospheric correction to cosmic-ray detectors (Savi et al., 2019). As far as we know, this is the first time to study and compare cosmic-ray cycles using PCA. ...
Article
Full-text available
We decompose the monthly cosmic-ray data, using several neutron monitor count rates, of Cycles 19-24 with principal component analysis (PCA). We show using different cycle limits that the first and second PC of cosmic-ray (CR) data explain 77-79% and 13-15% of the total variation of the Oulu CR Cycles 20-24(C20-C24), 73-77% and 13-17% of the variation of Hermanus C20-C24, and 74-78% and 17-21% of the Climax C19-C22, respectively. The PC1 time series of the CR Cycles 19-24 has only one peak in its power spectrum at the period 10.95 years, which is the average solar cycle period for the interval SC19-SC24. The PC2 time series of the same cycles has a clear peak at period 21.90 (Hale cycle) and another peak at 1/3 of that period with no peak at the solar cycle period. We show that the PC2 of the CR is essential in explaining the differences in the intensities of the even and odd cycles of the CR. The odd cycles have positive phase in the first half and negative phase in the second half of their PC2. This leads to slow decrease of the intensity in the beginning of the cycle and flat minimum for the odd cycles. On the contrary, for the even cycles the phases are vice versa and this leads to faster decrease and more rapid recovery in the CR intensity of the cycle. As a consequence the even cycles have more peak-like structure. The only exceptions of this rule are Cycles 23 and 24 such the former has almost zero line PC2, and the latter has similar PC2 than the earlier odd cycles. The reason for this may be that the aforementioned rule is only valid for grand solar maximum cycles or that there is a phase shift going on in the CR overall shape. These results are confirmed with skewness-kurtosis (S-K) analysis. Furthermore, S-K shows that other even and odd cycles, except Cycle 21, are on the regression line with correlation coefficient 0.85. The Cycles 21 of all calculated eight stations are compactly located in the S-K coordinate system and have smaller skewnesses and higher kurtoses than the odd Cycles 23.
... PCA finds combinations of variables, that describe major trends in the data. PCA has earlier been applied, e.g., to studies of the geomagnetic field (Bhattacharyya and Okpala, 2015), geomagnetic activity Takalo, 2021b), ionosphere (Lin, 2012), the solar background magnetic field (Zharkova et al., 2015), variability of the daily cosmic-ray count rates (Okpala and Okeke, 2014), and atmospheric correction to cosmic-ray detectors Savić et al. (2019). ...
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Homogeneous coronal data set (HCDS) of the green corona (Fe XIV) and coronal index of the solar activity (CI) have been used to study time-latitudinal distribution in solar cycles 18-24 and compared with similar distribution of sunspots, the magnetic fields and the solar radio flux 10.7 cm. The most important results are: (a) distribution of coronal intensities related to the cycle maximum are different for individual cycles, (b) the poleward migration of the HCDS from mid latitudes in each cycle exists, even in extremely weak Cycle 24, and the same is valid for the equatorward migration (c) the overall values of HCDS are slightly stronger for the northern hemisphere than for the southern one, (d) distribution of the HDCS are in coincidence with strongest photospheric magnetic fields (B>50 Gauss) and histogram of the sunspot groups, (e) Gnevy-shev gap was confirmed with at least 95 % confidence in the CI, however, with different behavior for odd and even cycles. Principal component analysis (PCA) showed that the first and second component account for 87.7 % and 7.3 % of the total variation of the CI. Furthermore, the PC2 of the green corona was quite different for cycle 21, compared with other cycles.
... It is known that it is rather difficult to predict even the next 11-year cycle (Pesnell, 2012) or the envelope of 10-year averages (Volobuev and Makarenko, 2008;Mordvinov et al., 2018): different methods may yield opposite prognostic tendencies. At the same time, there are forecasts for 500 (Steinhilber and Beer, 2013) and even for 1000 years (Zharkova et al., 2015). The former is an entirely empirical forecast, while the latter uses a model hypothesis, which remains highly controversial (Usoskin, 2017). ...
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In this study, the results from the analysis of annual ring widths (‘Dm’) time series of two “very sensitive” to the climate and solar–climate relationships of long lived European beech (Fagus sylvatica) samples (on age of 209 ± 1 and 245 ± 5 years correspondingly) are discussed. Both series are characterized by very good expressed and relating to the solar magnetic Hale cycle 20–22-year oscillations. A good coincidence between the changes of ‘Dm’ and the growth or fading of the solar magnetic cycle is found. The transition effects at the beginning and ending of the grand Dalton (1793–1833) and Gleissberg minima (1898–1933) are very clearly visible in the annual tree ring width data for the one of beech samples. Some of these effects are also detected in the second sample. The problem for the possible “lost” sunspot cycle at the end of 18th century is also discussed. A prediction for a possible “phase catastrophe” during the future Zurich sunspot cycles 26 and 27 between 2035–2040 AD as well as for general precipitation upward and temperature fall tendencies in Central Bulgaria, more essential after 2030 AD, are brought forth.
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A comprehensive spectral analysis of both the solar background magnetic field (SBMF) in cycles 21-23 and the sunspot magnetic field in cycle 23 reported in our recent paper showed the presence of two principal components (PCs) of SBMF having opposite polarity, e.g., originating in the northern and southern hemispheres, respectively. Over a duration of one solar cycle, both waves are found to travel with an increasing phase shift toward the northern hemisphere in odd cycles 21 and 23 and to the southern hemisphere in even cycle 22. These waves were linked to solar dynamo waves assumed to form in different layers of the solar interior. In this paper, for the first time, the PCs of SBMF in cycles 21-23 are analyzed with the symbolic regression technique using Hamiltonian principles, allowing us to uncover the underlying mathematical laws governing these complex waves in the SBMF presented by PCs and to extrapolate these PCs to cycles 24-26. The PCs predicted for cycle 24 very closely fit (with an accuracy better than 98%) the PCs derived from the SBMF observations in this cycle. This approach also predicts a strong reduction of the SBMF in cycles 25 and 26 and, thus, a reduction of the resulting solar activity. This decrease is accompanied by an increasing phase shift between the two predicted PCs (magnetic waves) in cycle 25 leading to their full separation into the opposite hemispheres in cycle 26. The variations of the modulus summary of the two PCs in SBMF reveals a remarkable resemblance to the average number of sunspots in cycles 21-24 and to predictions of reduced sunspot numbers compared to cycle 24: 80% in cycle 25 and 40% in cycle 26.
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Our knowledge of the long-term evolution of solar activity and of its primary modulation, the 11-year cycle, largely depends on a single direct observational record: the visual sunspot counts that retrace the last 4 centuries, since the invention of the astronomical telescope. Currently, this activity index is available in two main forms: the International Sunspot Number initiated by R. Wolf in 1849 and the Group Number constructed more recently by Hoyt and Schatten (Sol. Phys. 179:189–219, 1998a, 181:491–512, 1998b). Unfortunately, those two series do not match by various aspects, inducing confusions and contradictions when used in crucial contemporary studies of the solar dynamo or of the solar forcing on the Earth climate. Recently, new efforts have been undertaken to diagnose and correct flaws and biases affecting both sunspot series, in the framework of a series of dedicated Sunspot Number Workshops. Here, we present a global overview of our current understanding of the sunspot number calibration. After retracing the construction of those two composite series, we present the new concepts and methods used to self-consistently re-calibrate the original sunspot series. While the early part of the sunspot record before 1800 is still characterized by large uncertainties due to poorly observed periods, the more recent sunspot numbers are mainly affected by three main inhomogeneities: in 1880–1915 for the Group Number and in 1947 and 1980–2014 for the Sunspot Number. After establishing those new corrections, we then consider the implications on our knowledge of solar activity over the last 400 years. The newly corrected series clearly indicates a progressive decline of solar activity before the onset of the Maunder Minimum, while the slowly rising trend of the activity after the Maunder Minimum is strongly reduced, suggesting that by the mid 18th century, solar activity had already returned to levels equivalent to those observed in recent solar cycles in the 20th century. We finally conclude with future prospects opened by this epochal revision of the Sunspot Number, the first one since Wolf himself, and its reconciliation with the Group Number, a long-awaited modernization that will feed solar cycle research into the 21st century.
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The aim of this paper is to derive the principal components (PCs) in variations of (i) the solar background magnetic field (SBMF), measured by the Wilcox Solar Observatory with low spatial resolution for solar cycles 21-23, and (ii) the sunspot magnetic field (SMF) in cycle 23, obtained by SOHO/MDI. For reduction of the component dimensions, principal component analysis (PCA) is carried out to identify global patterns in the data and to detect pairs of PCs and corresponding empirical orthogonal functions (EOFs). PCA reveals two main temporal PCs in the SBMF of opposite polarities originating in opposite hemispheres and running noticeably off-phase (with a delay of about 2.5 yr), with their maxima overlapping in the most active hemisphere for a given cycle. Their maximum magnitudes are reduced by a factor of 3 from cycle 21 to 23, and overlap in the Northern hemisphere for cycle 21, in the Southern one in cycle 22 and in the Northern one again in cycle 23. The reduction of magnitudes and slopes of the maxima of the SBMF waves from cycle 21 to cycle 23 leads us to expect lower magnitudes of the SBMF wave in cycle 24. In addition, PCA allowed us to detect four pairs of EOFs in the SBMF latitudinal components: the two main latitudinal EOFs attributed to symmetric types and another three pairs of EOFs assigned to asymmetric types of meridional flows. The results allow us to postulate the existence of dipole and quadruple (or triple-dipole) magnetic structures in the SBMF, which vary from cycle to cycle and take the form of two waves travelling off-phase, with a phase shift of one-quarter of the 11 yr period. Similar PC and EOF components were found in temporal and latitudinal distributions of the SMF for cycle 23, revealing polarities opposite to the SBMF polarities, and a double maximum in time or maxima in latitude corresponding to the maxima of the SBMF PC residuals or minima in the SBMF EOFs, respectively. This suggests that the SBMF waves modulate the occurrence and magnitude of the SMF in time and latitude.
Chapter
Principal component analysis has often been dealt with in textbooks as a special case of factor analysis, and this tendency has been continued by many computer packages which treat PCA as one option in a program for factor analysis—see Appendix A2. This view is misguided since PCA and factor analysis, as usually defined, are really quite distinct techniques. The confusion may have arisen, in part, because of Hotelling’s (1933) original paper, in which principal components were introduced in the context of providing a small number of ‘more fundamental’ variables which determine the values of the p original variables. This is very much in the spirit of the factor model introduced in Section 7.1, although Girschick (1936) indicates that there were soon criticisms of Hotelling’s method of PCs, as being inappropriate for factor analysis. Further confusion results from the fact that practitioners of ‘factor analysis’ do not always have the same definition of the technique (see Jackson, 1981). The definition adopted in this chapter is, however, fairly standard.
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