Is the Sun causing global warming?
By NASA  13. December 2019
No. The Sun can influence the Earth’s climate, but it isn’t responsible for the warming trend we’ve seen over the past few decades. The Sun is a giver of life; it helps keep the planet warm enough for us to survive. We know subtle changes in the Earth’s orbit around the Sun are responsible for the comings and goings of the ice ages. But the warming we’ve seen over the last few decades is too rapid to be linked to changes in Earth’s orbit, and too large to be caused by solar activity.
One of the “smoking guns” that tells us the Sun is not causing global warming comes from looking at the amount of the Sun’s energy that hits the top of the atmosphere. Since 1978, scientists have been tracking this using sensors on satellites and what they tell us is that there has been no upward trend in the amount of the Sun’s energy reaching Earth.
A second smoking gun is that if the Sun were responsible for global warming, we would expect to see warming throughout all layers of the atmosphere, from the surface all the way up to the upper atmosphere (stratosphere). But what we actually see is warming at the surface and cooling in the stratosphere. This is consistent with the warming being caused by a buildup of heattrapping gases near the surface of the Earth, and not by the Sun getting “hotter.”
The amount of solar energy received by the Earth has followed the Sun’s natural 11year cycle of small ups and downs with no net increase since the 1950s. Over the same period, global temperature has risen markedly. It is therefore extremely unlikely that the Sun has caused the observed global temperature warming trend over the past halfcentury.
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Could past and increasingly ongoing global warming of the lower atmosphere actually have buffered and prevented the slow onset of a miniiceage on Earth as predicted due to reduced solar activity? And could that situation suddenly flip?.
Science needs to come together to work that out.
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It’s official: we are in a “deep” solar minimum.
By Anthony Watts  16. December 2019
SUNSPOTS BREAK A SPACE AGE RECORD:
Solar physicist Dr. Tony Phillips writes via email:
Solar Minimum is officially “deep.” 2019 has just broken a Space Age record for days without sunspots. Moreover, an international panel of scientists led by NOAA and NASA predicts that Solar Minimum could deepen even further, not reaching its lowest point until April of 2020.
Solar Minimum is becoming very deep indeed. Over the weekend, the sun set a Space Age record for spotlessness. So far in 2019, the sun has been without sunspots for more than 270 days, including the last 33 days in a row. Since the Space Age began, no other year has had this many blank suns.
The previous recordholder was the year 2008, when the sun was blank for 268 days. That was during the epic Solar Minimum of 20082009, formerly the deepest of the Space Age. Now 2019 has moved into first place.
Solar Minimum is a normal part of the 11year sunspot cycle. The past two (20082009 and 20182019) have been long and deep, making them “centuryclass” Minima. To find a year with more blank suns, you have to go back to 1913, which had 311 spotless days.
Last week, the NOAA/NASA Solar Cycle Prediction Panel issued a new forecast. Based on a variety of predictive techniques, they believe that the current Solar Minimum will reach its deepest point in April 2020 (+/ 6 months) followed by a new Solar Maximum in July 2025. This means that low sunspot counts and weak solar activity could continue for some time to come.
Solar Minimum definitely alters the character of space weather. Solar flares and geomagnetic storms subside, making it harder to catch Northern Lights at midlatitudes. Space weather grows “quiet.” On the other hand, cosmic rays intensify. The sun’s weakening magnetic field allows more particles from deep space into the solar system, boosting radiation levels in Earth’s atmosphere. Indeed, this is happening now with atmospheric cosmic rays at a 5year high and flirting with their own Space Age record. It’s something to think about the next time you step on an airplane.
Source: Dr. Tony Phillips, Spaceweather.com
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Sun and IceAges on Earth
Heartbeat of the Sun from Principal Component Analysis and prediction of solar activity on a millenium timescale
By V. V. Zharkova, S. J. Shepherd, E. Popova & S. I. Zharkov 
Abstract
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 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 350year grand cycles superimposed on 22 yearcycles 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.
New Solar Research Raises Climate Questions, Triggers Attacks
[Ed.: GWPF Interview: This recent research by Professor Valentina Zharkova (Northumbria University) and colleagues has shed new light on the inner workings of the Sun. If correct, this new discovery means that future solar cycles and variations in the Sun's activity can be predicted more accurately. The research suggests that the next three solar cycles will see solar activity reduce significantly into the middle of the century, producing conditions similar to those last seen in the 1600s – during the Maunder Minimum. This may have implications for temperatures here on Earth. Future solar cycles will serve as a test of the astrophysicists’ work, but some climate scientists have not welcomed the research and even tried to supress the new findings.]
Introduction
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 https://solarscience.msfc.nasa.gov/images/bfly.gif). The sunspot numbers show quasiregular 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) reflecting changing magnetic activity of the Sun^{1}.
The longest direct observation of solar activity is the 400year sunspotnumber 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 cycles^{2,3}, prior to cycle 24. Many observations indicate essential differences between the activity occurring in the opposite hemispheres for sunspots^{4} and for solar and heliospheric magnetic fields^{5}.
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 disintegration of the solar magnetic field. Cycles of magnetic activity are associated with the action of a dipole solar dynamo mechanism called ‘α − Ω dynamo’^{6}. It assums the action of solar dynamo to occur in a single spherical shell, where twisting of the magnetic field lines (αeffect) and the magnetic field line stretching and wrapping around different parts of the Sun, owing to its differential rotation (Ωeffect), are acting together^{7,8}.
As a result, magnetic flux tubes (toroidal magnetic field) seen as sunspots are produced from the solar background magnetic field (SBMF) (poloidal magnetic field) by a joint action of differential rotation (Ωeffect) and radial shear (αeffect), while the conversion of toroidal magnetic field into poloidal field is governed by the convection in the rotating body of the Sun. The action of the Coriolis force on the expanding, rising (compressed, sinking) vortices results in a predominance of righthanded vortices in the Northern hemisphere and lefthanded vortices in the Southern hemisphere leading to the equatorward migration of sunspots during a solar cycle duration visible as butterfly diagams (see https://solarscience.msfc.nasa.gov/images/bfly.gif, the bottom plot).
The 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 different properties of solar and stellar plasmas, as discussed in the recent reviews^{7,8}.
As usual, the understanding of solar activity is tested by the accuracy of its prediction. The 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 twoyear minimum period between cycles 23 and 24. This reduced activity in cycle 24 was very surprising because the previous five cycles were extremely active and sunspot productive forming the Modern Maximum^{2,3}. 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 activity^{9}.
However, most predictions of solar activity by various methods, such as considering linear regression analysis^{10}, neural network forecast^{11}, or a modified fluxtransport dynamo model calibrated with historical sunspot data from the middletoequator latitudes^{12}, anticipated a much stronger cycle 24^{10}. There were only a few predictions of the weaker cycle 24^{13} obtained with the high diffusivity BabcockLeighton dynamo model applied to polar magnetic fields 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 dynamo^{14}.
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 different physical processes occurring in solar dynamo and affecting 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 applied^{15} to lowresolution full disk magnetograms captured by the Wilcox Solar Observatory^{16}. This approach revealed a set of more than 8 independent components (ICs), which seem to appear in pairs^{15}, with two principal components (PCs) covering about 39% of the variance of the whole magnetic field data, or standard deviation of σ = 0.67. The main pair of PCs is associated with two magnetic waves of opposite polarities attributed to the poloidal field produced by solar dynamo from a dipole source^{17}.
The two principal components (PCs) derived from solar background magnetic field (SBMF)^{15} (cycle 21–23) and predicted for cycle 24–26 are presented in Fig. 1 (the upper plot). For the first time PCA allowed us to detect, two magnetic waves in the SBMF^{15} and not a single one assumed in the mean dynamo models. These waves are found originating in the opposite hemispheres and travelling with an increasing phase shift to the Northern hemisphere in odd cycles and the Southern hemisphere in even cycles^{15}. This can explain the wellobserved NorthSouth asymmetry in sunspot numbers, background magnetic field, flare occurences and so on (see Zharkov et al.^{4} and references therein) defining the active hemisphere for odd (North) and even (South) cycles.
Top plot: the two principal components (PCs) of SBMF (blue and red curves) obtained for cycles 21–23 (historic data^{15}) and predicted^{19} for cycles 24–26 with the Eqs. (2)–(3).
The dotted lines show the PCs derived from the data and the solid lines present the curves plotted from formulae 2 (blue) and 3 (red). The accuracy of fit of the both PC curves is better than 97%. The point A shows the current time. The cycle lengths (about 11 years) are marked at the minima by the vertical lines. The bottom plot: The summary PC derived from the two PCs above for the ‘historical’ (21–23) and predicted cycles (24–26) data. The 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. The 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.
The formation of magnetic flux 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 field^{15} when their phase shift is not very large. These two magnetic waves of the poloidal field can account for the observed sunspot magnetic field^{18}, or averaged sunspot numbers, after 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 curve^{19} (Fig. 2, bottom plot). The 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 decades^{20}.
Top plot: Comparison of the modulus summary curve (black curve) obtained from the summary curve inFig. 1 with averaged sunspot numbers (brown curve) and magnetic fiel (blue curve) for cycles 21–23.
Bottom plot: The 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). The plots are a courtesy of Shepherd et al.^{19}. © AAS. Reproduced with permission.
The maximum (or double maximum for the waves with a larger phase shift of solar activity for a given cycle) coincides with the time when each of the waves approaches a maximum amplitude and the hemisphere where it happens becomes the most active one. This can account naturally for the northsouth asymmetry of solar activity often reported in many cycles. Also the existence of two waves in the poloidal magnetic field instead of a single one, used in most prediction models and the presence of a variable phase difference between the waves can naturally explain the difficulties in predicting sunspot activity on a scale longer than one solar cycle with a single dynamo wave^{14} since the sunspot activity is associated with the modulus summary curve of the two dynamo waves^{19} that is a derivative from these two waves.
Mathematical description of the observed magnetic waves
Amplitude and frequency variations of these waves, or PCs, over time are found using symbolic regression analysis^{21} with Euriqa software (see the Methods section for data analysis^{19}). The 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. These 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 numbers^{19}. The accuracy of these formulae for prediction of the principal components is tested for cycle 24 showing the predicted curve fitting very closely (with an accuracy of about 97.5%) the PCs derived from the observations of SBMF and sunspot numbers^{19}.
For the forthcoming cycles 25 and 26 (Fig. 1) the two waves are found to travel between the hemispheres with decreasing amplitudes and increasing phase shift approaching nearly a half period in cycle 26. This leads, in fact, to a full separation of these waves in cycle 26 into the opposite hemispheres^{19}. This separation reduces any possibility for wave interaction for this cycle that will result in significantly reduced amplitudes of the summary curve and, thus, in the strongly reduced solar activity in cycle 26^{19}, or the next Maunder Minimum^{9} lasting in 3 cycles 25–27.
Prediction of solar activity on millennium scale
By far the most impressive achievement todate of this approach is its ability to make very long term predictions of solar activity with high accuracy over the timescales of many centuries. The summary curve of the two principal components (magnetic waves) expressed by the formulae (2 and 3) in the Method of data analysis^{19} is calculated backwards and forwards for the period 1200–3200 years as shown in Fig. 3.
The 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.
The historical maxima and minima of the solar activity in the past are marked by the horizontal brackets.
Remarkably, our current prediction of the summary curve backwards by 800 years shown in the left (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) defining 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 Age, the Maunder Minimum and the Dalton Minimum; 3) an increase in solar activity during a modern maximum in 20th century.
This 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 background magnetic field 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.
The summary curve reveals a superposition of the amplitudes of the two dynamo waves, or a ‘beating’ effect creating two resulting waves: one of higher frequency (corresponding to a classic 22year cycle) and a second wave of lower frequency (corresponding to a period of about 350–400 years), which modulates the amplitude of the first 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.
This longterm ‘grand’ cycle was previously postulated in 1876 by Clough^{22} as a 300year cycle superimposed 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 years^{23}. These periods are close to those reported for the last 800 years in the summary curve plotted in Fig. 3 derived from the observed magnetic field variations.
The spectacular accuracy of the historical fit in the past 800 years gave us the confidence to extrapolate the data into the future for a similar epoch of 1200 years (Fig. 3, right part of the curve) clearly showing, as expected, several 350–400year grand cycles. We note, in particular, a decreasing activity for solar cycles 25 and 26 coinciding with the end of the previous 350–400year grand cycle and then increase of the solar activity again from cycle 27 onwards as the start of a new grand cycle with an unusually weak cycle 30. Hence, cycles 25–27 marks a clear end of the modern grand period that can have significant 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 field, generated by the solar dynamo in two different cells, similar to those derived by Zhao et al.^{24} from helioseismological observations (Fig. 4), in order to fit the background magnetic field observations (Figs 1 and 3). This can be achieved with the modified Parker’s nonlinear two layers dynamo model for two dipoles^{17} 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 field for cycles 21–23^{17} derived with PCA^{15}.
The 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.
The simulation results presenting the toroidal magnetic field are plotted in Fig. 5 (bottom plot) derived from the poloidal field (Fig. 1, top plot) for a period of six 11year cycles using the dynamo equations (16–19) from Popova et al.^{17}. The curves for poloidal (derived with PCA) and toroidal fields (simulated with the dynamo model) are found to have similar periods of oscillations whilst having opposite polarities (or having the phase shift of a half of the period), being in antiphase every 11 years as previously reported^{4,25}. The amplitude of generated toroidal magnetic field is plotted versus the dynamo number in Fig. 5 (top plot).
Top plot: Dependence of the solar dynamonumber D = R_{α}R_{Ω} on a magnitude of the toroidal magnetic field (for detials of the parameters see the text).
Bottom plot: Variations of the toroidal magnetic field 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).
Furthermore, in cycles 25–27 and, especially, in cycle 26, the toroidal magnetic field waves generated in these two layers become fully separated into the opposite hemispheres, similar to the two PC waves attributed to poloidal field (Fig. 1, top plot), that makes their interaction minimal. This will significantly 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.
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) fields. According to the dynamo theory and analysis of observational data^{7,27} the generated toroidal field is much stronger than the poloidal. Although, exact values of the amplitudes of these fields in the solar convection zone are unknown and estimated from dynamo models. In our simple model the amplitude of toroidal field 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.
Variations of the summary poloidal (top plot) and toroidal (bottom plot) magnetic fields simulated for 2000 years with the two layer αΩdynamo model (see Methods section) with the parameters derived from the two PCs fromFig. 1 using mathematical formulae (2–3).
One arbitrary unit corresponds to 1–1.5 Gauss (see text for details).
It can be seen that variations of the model magnetic fields (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 amplitude of 22 year cycle by much slower oscillations of about 350 years, different duration (320–400) and amplitudes of different grand cycles. These variations are governed by different dynamo parameters as discussed below.
Beating effect of two dynamo waves with close frequencies
The waves generated by a dynamo mechanism in each layer are found to have similar (but not equal) frequencies caused by a difference in the meridional flow 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 different layers with an amplitude A_{0} have to have close but not equal frequencies ω_{1} and ω_{2} (or periods varying between 20 and 24 years), similar to Gleissberg’s cycle^{7,26}.
The interference of these waves enabled by diffusion of the waves in the solar interior from the bottom to the top layer^{27} leads to formation of the resulting envelope of waves Y(t), or beating effect (see Fig. 3 and theoretical plots in Fig. 6), showing oscillations of a higher frequency within the envelope and those of the envelope itself with a lower frequency of (or in a grand cycle) as follows:
where k is some parameter defining properties of the solar interior where the waves propagate, e.g. diffusivity, dynamo number (α and Ω effects) and meridional circulation.
Frequency and period variations
The beating effect between these frequencies can easily explain seemingly sporadic variations of high frequency amplitudes and the period of the lowfrequency envelope wave in the resulting grand cycles seen in both the observational curve (Fig. 3 and theoretical curves (Fig. 6) reproducing the observational one. The higher the difference of frequencies the larger is the frequency, 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. This effect 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 22year cycles are longer (360–400 years, the 1st and 4th in Fig. 3).
The difference 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}). The frequency of a wave is reduced (or its period is increased) when the meridional circulation has higher velocities and this frequency 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 T_{g} from 20 to 24 years (variations within Gleissberg’s cycle^{7}), 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 T_{g} = 20 years for the inner layer wave 1 − T_{1} = 18.9 years (corresponding to the velocity of meridional circulation about V = 7–8 m/s), for the upper layer wave 2 − T_{2} = 21 years (V = 9–10 m/s); for the activity period T_{g} = 24 years: the inner layer wave 1 − T_{1} = 22.46 years (V = 10–11 m/s), the upper layer wave 2 − T_{2} = 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 T_{g} = 20 years  for the inner layer wave 1 −T_{1} = 19 years (V = 7–8 m/s), for the upper layer wave 2 − T_{2} = 21 years (V = 9–10 m/s); for the period of T_{g} = 24 years: the inner layer wave 1 − T_{1} = 22.6 years (V = 10–11 m/s), the upper layer wave 2 − T_{2} = 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 convective zone) remains more or less stable at about T_{1} = 19 years (for generation of the low frequency activity period T_{g} = 20 years) or T_{1} = 22.6 year (for T_{g} = 24 years). While the period of the wave 2 generated in the upper layer should have larger fluctuations (e.g. T_{2} = 25.8 years for 350 grand cycle versus T_{2} = 25.53 years for 400 years grand cycle). These fluctutation are likely to be affected 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 effect 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 fields.
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). This results in much more complex beating effects derived from PCA as presented in Fig. 3. The 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 longterm 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
The amplitudes of dynamo waves are affected by the variations of both α and Ω effects, 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 field amplitude (see Fig. 5, top plot).
This effect 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 amplitudes of the high frequency wave (T_{g} = 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.
Therefore, the derived mathematical laws in cyclic variations of principal components of the observed solar magnetic field, which fit 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 longterm prediction of solar activity with impressive accuracy.
Methods
Derivation of parameters of the observed magnetic waves
In order to distill the main parameters of the waves present in the observational solar magnetic data, one needs to reduce their dimensionality with the Principal Component Analysis (PCA)^{28}. PCA is an orthogonal linear transformation allowing a vector space to be transformed to a new coordinate system, reducing the multidimensional data to lower dimensions for analysis, so that the greatest variance by any projection of the data lies on the first coordinate called the Principal Component (PC) with the second PC orthogonal to the first is defined by the second largest variance. This technique simultaneously (i) reduces the data dimensionality, (ii) increases the signaltonoise 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). The PCA is an exact method and its accuracy defined only by the noise of measurements, , of the original vector^{29}.
PCA was applied to lowresolution full disk solar background magnetic field (associated with the poloidal magnetic field) 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 . We derive the dominant eigenvalues (0.1 and 1.0) covering the maximum variance of 39%^{15} defining the eigenfunctions, or Principal Components (PCs), which came as a pair of waves. These PCs are considered as the main (dipole) dynamo waves of the solar poloidal magnetic field.
By applying a 3year running averaging filter, any shortterm (<3 years) fluctuations of magnetic field data are removed allowing us to keep the accuracy of PCA not worse than the measurement error (Wentzell and Lohnes^{30}). The overall PCA accuracy of defining 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 field 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 PCA^{15}.
For classification of the derived PCs we apply the symbolic regression approach based on the Hamiltonian principle implemented in the Euriqa software^{21}. This allows us to derive the exact mathematical formulae for the amplitude variations and phase shifts of both principal components as follows^{19}:
for wave 1:
for wave 2:
where the parameters with ω define the corresponding wave frequencies and ϕ define their phase shifts. 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 field data (before deriving PCs) were unsuccessful indicating the very complex nature of the original magnetic field waves.
These two PCs are used for calculation of the summary wave (a sum of amplitudes) and the modulus summary wave (reflected to the positive amplitudes only) linked to the averaged sunspot numbers currently used for definition of solar activity.
Nonlinear αΩ dynamo model in a twolayer medium with meridional circulation
In order to understand the basic features of the derived PCs, let us use Parker’s αΩdynamo model with two layers with meridional circulation^{17} updated by considering a nonlinear dynamo process. It is assumed that dynamo waves are generated by the dipole sources only located in two layers: one dipole in the subsurface layer and the other dipole deeply in the solar convection zone (see Fig. 4); and the parameters (dynamo number and meridional circulation) of magnetic field generation in each layer are different^{17}.
This results in the simultaneous existence of two magnetic waves with different periods and phase shifts^{17}, 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 independent components^{17}, which are shown to slightly modify the overall appearance of magnetic waves that will be considered in the forthcoming paper.
The dynamo equations describing the generation and evolution of the solar magnetic field in a twolayer medium, are obtained from a system of electrodynamic equations for mean fields in the assumption that the dynamo wave propagates in a thin spherical shell^{17}. In this case, the magnetic field is averaged along the radius within a certain spherical shell and the terms describing the curvature effects near the pole are excluded. In addition, in this approximation, we assume that the magnetic field is generated independently 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 lines^{31} and verified analytically by the lowmode approach^{32}.
In these equations the dynamo number is defined by the parameters R_{α} and R_{Ω} describing, respectively, the intensity of αeffect and the differential rotation, or Ωeffect. The latitudinal profile of the poloidal magnetic field is assumed for simplicity to be proportional cos(θ) where θ is the solar latitude measured from the equator.
We consider the αeffect the amplitude F(t, coordinates) and widely used algebraic quenching^{7} in a form:
For calculations the amplitude of α effect, R_{α} is moved to the dynamo number D, while the algebraic quenching of the helicity is used for stabilization of a magnetic field growth, e.g. redefining , where is the helicity in unmagnetized medium and is the magnetic field, for which the effect is considerably suppressed.
The contribution of differential rotation into the generation of magnetic field is defined^{27} by the terms for one layer or for another layer, following the general trend of being maximum at the equator and minimal at the poles. The dynamo number also includes an amplitude of differential rotation R_{Ω}, which can vary in different layers.
Since the frequencies of magnetic waves generated by dynamo mechanism are known to be mostly affected by meridional circulation velocities^{17,32} while their amplitudes are governed by the variations of dynamo number D, then the PC waves can be reproduced in different layers with different dynamo numbers and slightly different 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. , so that it vanishes at the poles and is maximal at the middle latitudes approaching amplitudes of 9–15 m/s^{24,33}. Also, to comply with the material conservation rule, the meridional circulation the multicellular 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 Dikpati^{34,35} and Popova et al.^{17} that was recently confirmed 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 affects the magnetic field: 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 different velocities and 3 the middle layer where cells have a boundary and their the circulation has the opposite direction, complying with the mass conservation law. In the current model we consider the top and bottom layers (1–2) only to reproduce the longterm oscillations produced in them, while oscillations in the middle layer 3 affecting the shortterm biennial^{32} are out of scope of the current study.
For each layer the principal components (PCs) of poloidal magnetic field were substituted into the dynamo equations (17 and 19) of Popova et al.^{17}, from which the corresponding toroidal magnetic field components are derived fitting these PCs. Then 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.
Additional Information
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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Acknowledgements
The authors wish to thank the Directorate and staff of the Wilcox Solar Observatory (Stanford) for providing online the synoptic magnetic field 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 IST200132409.
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Author notes
Zharkova V. V., Shepherd S. J., Popova E. and Zharkov S. I. contributed equally to this work.
Affiliations
 Department of Mathematics & Information Sciences, Northumbria University, Newcastle upon Tyne, NE2 1XE, UK
 V. V. Zharkova
 Space Physics Department, Institution of Space Science Research, Kiev, 03022, Ukraine
 V. V. Zharkova
 University of Bradford, School of Engineering, Bradford, BD7 1DP, UK
 S. J. Shepherd
 Skobeltsyn Institute of Nuclear Physics, Moscow, 119234, Russia
 E. Popova
 Department of Physics and Mathematics, University of Hull, Kingston upon Hull, HU6 7RX, UK
 S. I. Zharkov
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.
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Zharkova, V., Shepherd, S., Popova, E. et al. Heartbeat of the Sun from Principal Component Analysis and prediction of solar activity on a millenium timescale. Sci Rep 5, 15689 (2015) doi:10.1038/srep15689
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But did the paidfor Hollywood fearmongers have a point?
The storyline has it that when global warming triggers the onset of a new Ice Age, tornadoes flatten Los Angeles, a tidal wave engulfs New York City and the entire Northern Hemisphere begins to freeze solid, as fullscale, massive evacuations to the South sets in.
Could past and increasingly ongoing global warming of the lower atmosphere actually have buffered and prevented the slow onset of a miniiceage on Earth as predicted due to reduced solar activity? And could that situation suddenly flip?.
Science needs to come together to work that out.
Nikolov and Zeller refuted: Planetary surfaces are not kept hot by the pressure of the atmosphere.
Groundbreaking Atmospheric Physics shatters the Greenhouse conjecture
I've noticed that many science students and graduates from recent decades become "formula" people without understanding the limitations and conditions under which such expressions are applicable. This has led to scientists like Drs Jelbring, Nikolov and Zeller all publishing papers in which they point out a kind of correlation (not linear) between pressure and temperature in planetary tropospheres, but then they incorrectly deduce that it is high pressure that is maintaining high temperatures such as at the surface of Venus. They think this is a result of the Ideal Gas Law (IGL) but they confuse cause and effect.
For example, people know from first year Physics that the IGL tells us that pressure is proportional to the product of temperature and density. So, if we have a sealed, perfectly insulated cylinder full of gas and, using an inserted electric element, we raise the temperature (by adding kinetic energy to the gas molecules and making them move faster between collisions) then, since the density remains constant, the pressure will indeed increase in proportion to the absolute (K) temperature.
But if we have a piston in our cylinder and slowly force it down to the halfway point we do of course double the density. But this is the important point: whenever we do work against a force the energy is mostly stored as potential energy, not kinetic energy. For example, if we raise an object the energy we used in doing so adds to the gravitational potential energy. We can get the energy back when the object falls. Likewise when we compress a spring we can get that energy back when the spring expands again. And similarly, compressing the gas in the cylinder increases the pressure (and the density) with that new energy able to be retrieved when the extra pressure is allowed to push the cylinder back to its original position.
But I ask, how could the slow moving of the piston at, say, 0.1 m/sec have any significant impact on the speed of the molecules which move at about 450 to 530 m/sec? So clearly all that happens is that the pressure and density each double and so, by the Ideal Gas Law, the temperature remains constant, just as this simple application of the Kinetic Theory of Gases indicates. The Ideal Gas Law is easily understood and derived from this Kinetic Theory.
The relevance of this is that we see many attempts to explain why the surface temperature of planets is greater than that which direct solar radiation to the surface could achieve. So some people say the high pressure is causing the temperature to be hotter. That is simply not the case. Correlation does not imply cause.
What actually happens occurs at the molecular level in every small parcel of air at every altitude, as was explained by the brilliant physicist Josef Loschmidt in 1876 but totally ignored by climatologists. As a direct result of the Second Law of Thermodynamics which says entropy will tend towards a maximum (by diminishing unbalanced energy potentials) we find that gravity forms a stable density gradient in the troposphere of every planet. Simultaneously it forms a temperature gradient, this being represented by the same state of maximum entropy which in physics is called thermodynamic equilibrium. That density gradient is thermodynamic equilibrium because entropy is a function of all forms of internal energy including gravitational potential energy. At maximum entropy in any small region the sum of molecular kinetic energy + gravitational potential energy is constant so that there are no unbalanced energy potentials.
Finally, it is because this temperature gradient is the state of thermodynamic equilibrium that the nonradiative heat transfer process described in my 2013 paper occurs, transferring new thermal energy absorbed from solar radiation in the middle troposphere and above down to the base of the troposphere and into any solid surface. If this did not happen, then the temperature at the base of the 350Km high nominal troposphere of Uranus would not be hotter than Earth's surface and, indeed, would be close to absolute zero, and likewise for the Venus surface. Earth's mean surface temperature would be colder than that for the Moon which is probably colder than 15°C.
So, it is not back radiation which climatologists have just guessed must be supplying about twice as much thermal energy to Earth's surface as does the direct solar radiation, but rather it is this nonradiative process that I have explained in detail in this 2013 paper.
https://www.researchgate.net/publication/318008633_Planetary_Core_and_Surface_Temperatures
THE SUN IS NOT GASEOUS – 15mins Solar Physics
By Richard Small  02.
SuspiciousObservers have released a new video detailing the work of Dr. Pierre Marie Robitaille. He has pioneered the MRI field back in the 90’s by showing the scientific community they we’re wrong in their assumptions and proved it. He’s had a track record of not going with the general consensus and as such is a remarkable rouge physicist that continues to seek and reveal the actual evidence based truth.
In this 15 minute video presentation from Observing The Frontier, Pierre gives us a wonderful example in to the erroneous assumptions and blatent lies and decipt that has entered our texts throughout the years. This man is only happy when something is proven and based on experimental observations, anything which is not, is nothing but an unsubstantiated theory at best usually masking the aweful hubris of a complacent establishment.
If you think that ONE man can’t redefine 100’s if not thousands of years of science, your
A) forgetting the scientific method and
B) Galileo Galilei, who spent his life persecuted by the church for speaking the truth and never giving up.