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 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.

Abstract. Principle component analysis (PCA) of the solar background magnetic field (SBMF) measured from Wilcox Solar Observatory (WSO) magnetograms revealed the following principal components (PCs) in latitudes: two main symmetric components, which are the same for all cycles 21–23, and three pairs of asymmetric components, which are unique for each cycle. These SBMF variations are assumed to be those of poloidal magnetic field travelling slightly off-phase from pole to pole while crossing the equator. They are assumed to be caused by a joint action of dipole and quadruple magnetic sources in the Sun. In the current paper, we make the first attempt to interpret these latitudinal variations in the surface magnetic field with Parker's two-layer dynamo model. The latitudinal distributions of such waves are simulated for cycles 21–23 by the modified Parker's dynamo model taking into account both α and ω effects operating simultaneously in the two (upper and lower) layers of the solar convective zone (SCZ) and having opposite directions of meridional circulation. The simulations are carried out for both dipole and quadruple magnetic sources with the dynamo parameters specifically selected to provide the curves fitting closely the PCs derived from SBMF variations in cycles 21–23. The simulations are optimised for matching the positions of maximums in latitude, the number of equator crossings and the phase difference between the two dynamo waves operating in the two layers. The dominant pair of PCs present in each cycle is found to be fully asymmetric with respect to the magnetic poles and produced by a magnetic dipole. This pair is found to account for the two main dynamo waves operating between the two magnetic poles. There are also three further pairs of waves unique to each cycle and associated with multiple magnetic sources in the Sun. For the odd cycle 21 the simulated poloidal field fits the observed PCs, only if they are produced by magnetic sources with a quadruple symmetry in both layers, while for the even cycle 22 the fit to the observed PCs is achieved only in the case of quadruple magnetic sources in the upper layer and dipole sources in the inner layer. For the other odd cycle 23 the fit to observation is obtained for the quadruple magnetic sources in the inner layer and the dipole sources in the upper layer. The magnitudes of dynamo numbers D defining the conditions (depth and latitude) of a magnetic flux formation and the numbers N of zeros (equator crossings by the waves) are found to increase and the meridional circulation speed to decrease with a cycle number increase (D = −700, N = 3 for cycle 21 and D = −104, N = 9 for cycle 23). The phase delays between the waves in each unique pairs are also found to increase with the cycle number from ~9° in cycle 21 to ~13° in cycle 23.

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Recently discovered long-term oscillations of the solar background magnetic field associated with double dynamo waves generated in inner and outer layers of the Sun indicate that the solar activity is heading in the next three decades (2019–2055) to a Modern grand minimum similar to Maunder one. On the other hand, a reconstruction of solar total irradiance suggests that since the Maunder minimum there is an increase in the cycle-averaged total solar irradiance (TSI) by a value of about 1–1.5 Wm −2 closely correlated with an increase of the baseline (average) terrestrial temperature. In order to understand these two opposite trends, we calculated the double dynamo summary curve of magnetic field variations backward one hundred thousand years allowing us to confirm strong oscillations of solar activity in regular (11 year) and recently reported grand (350–400 year) solar cycles caused by actions of the double solar dynamo. In addition, oscillations of the baseline (zero-line) of magnetic field with a period of 1950 ± 95 years (a super-grand cycle) are discovered by applying a running averaging filter to suppress large-scale oscillations of 11 year cycles. Latest minimum of the baseline oscillations is found to coincide with the grand solar minimum (the Maunder minimum) occurred before the current super-grand cycle start. Since then the baseline magnitude became slowly increasing towards its maximum at 2600 to be followed by its decrease and minimum at ~3700. These oscillations of the baseline solar magnetic field are found associated with a long-term solar inertial motion about the barycenter of the solar system and closely linked to an increase of solar irradiance and terrestrial temperature in the past two centuries. This trend is anticipated to continue in the next six centuries that can lead to a further natural increase of the terrestrial temperature by more than 2.5 °C by 2600.