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 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 field for cycles 21–2317 derived with PCA15.

Figure 4 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. Full size image

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 11-year 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 anti-phase every 11 years as previously reported4,25. The amplitude of generated toroidal magnetic field is plotted versus the dynamo number in Fig. 5 (top plot).

Figure 5 Top plot: Dependence of the solar dynamo-number 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). Full size image

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 data7,27 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.

Figure 6 Variations of the summary poloidal (top plot) and toroidal (bottom plot) magnetic fields simulated for 2000 years with the two layer αΩ-dynamo 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). Full size image

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 cycle7,26.

The interference of these waves enabled by diffusion 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 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 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. 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 22-year 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 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 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 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

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 long-term prediction of solar activity with impressive accuracy.