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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689

www.nature.com/scientificreports

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 eects responsible for the grand cycles

(350–400 years) superimposed on a standard 22 year cycle. This approach opens a new era in

investigation and condent 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/by.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) reecting 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 dierences

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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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689

solar dynamo mechanism called ‘α − Ω 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-eect) and the magnetic eld line

stretching and wrapping around dierent parts of the Sun, owing to its dierential rotation (Ω -eect),

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 dierential rotation

(Ω -eect) and radial shear (α-eect), 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 buttery diagams (see http://solarsci-

ence.msfc.nasa.gov/images/by.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 dierent 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 modied 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 diusivity 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 dierent physical processes occurring in solar dynamo and

aecting 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) dening 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, aer 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 oen 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 dierence between the waves can naturally explain the diculties 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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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689

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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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689

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 signicantly

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)

dening 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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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689

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’

eect 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 condence 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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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689

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

signicant 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 dierent cells, similar to those derived by Zhao et al.24

from helioseismological observations (Fig.4), in order to t the background magnetic eld observations

(Figs1 and 3). is can be achieved with the modied 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

=⋅

αΩ

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 signicantly

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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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689

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 Figs5 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, dierent duration (320–400) and

amplitudes of dierent grand cycles. ese variations are governed by dierent dynamo parameters as

discussed below.

Beating eect 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 dierence

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 dierent

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 diusion 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 eect (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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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689

where k is some parameter dening properties of the solar interior where the waves propagate, e.g. dif-

fusivity, dynamo number (α and Ω eects) and meridional circulation.

Frequency and period variations. e beating eect 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 dierence 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 eect 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 dierence 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 aected 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 eect 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 eects 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 aected by the variations of both α

and Ω eects, 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 eect 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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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689

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 dened 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 dened 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 dening 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 dening 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 classication of the derived PCs we apply the symbolic regression approach based on the

Hamiltonian principle implemented in the Euriqa soware21. is allows us to derive the exact mathe-

matical formulae for the amplitude variations and phase shis 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 dene the corresponding wave frequencies and φ dene their phase shis.

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 (reected to the positive amplitudes only) linked to the averaged sunspot numbers cur-

rently used for denition 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 dierent17.

is results in the simultaneous existence of two magnetic waves with dierent periods and phase

shis17, 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 eects 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 veried analytically by the low-mode approach32.

In these equations the dynamo number

=αΩ

DRR

is dened by the parameters Rα and RΩ describing,

respectively, the intensity of α-eect, and the dierential rotation, or Ω -eect. e latitudinal prole of

the poloidal magnetic eld is assumed for simplicity to be proportional cos(θ) where θ is the solar

latitude measured from the equator.

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SCIENTIFIC RepoRts | 5:15689 | DOI: 10.1038/srep15689

We consider the α-eect 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 eect, 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. redening

α

=

αθ

ξ

()

(+ )B1

0

22

,

where

αθθ()=cos

0

is the helicity in unmagnetized medium and

ξ

=

−

B0

1

is the magnetic eld, for

which the

α

-eect is considerably suppressed.

e contribution of dierential rotation into the generation of magnetic eld is dened27 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

=αΩ

DRR

also includes an amplitude of

dierential rotation RΩ, which can vary in dierent layers.

Since the frequencies of magnetic waves generated by dynamo mechanism are known to be mostly

aected 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 dierent layers with dierent dynamo

numbers and slightly dierent 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 conrmed 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 aects 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 dierent 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 aecting 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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