Alexander G. Kosovichev1 & Valery V. Pipin2
1. Center for Computational Heliophysics, New Jersey Institute of Technology, Newark, NJ 07102, USA
2. Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, Irkutsk, 664033, Russia
In 1955, Eugene Parker showed that differential rotation and helical turbulence in the solar convection zone can result in dynamo action in the form of migrating dynamo waves that transport magnetic flux from the deep interior to the solar surface, creating the sunspot butterfly diagram. However, in the absence of observational evidence of the dynamo waves, alternative scenarios, called the flux-transport models, were developed. These models suggest that the cyclic evolution of the magnetic field is controlled by meridional circulation which, similarly to a conveyor belt, transports magnetic flux at the bottom of the convection zone from high to low latitudes where it emerges in the form of sunspots.
Although the magnetic field in the convection zone cannot be measured directly, its structure can still be tracked through its effects on large-scale zonal flows that have been historically called “torsional oscillations” because of their cyclic variations synchronized with the magnetic activity cycles. Magnetic forces slow down the flows, which can be observed using techniques in helioseismology to measure the deceleration of zonal flows. The flow speed is about 8 m/s, and is measured in frequency shifts of solar oscillations. The measurements have been performed over the last 22 years by two NASA space missions, Solar and Heliospheric Observatory (1996-2010) and Solar Dynamics Observatory (2010-current).
Figure 1| a) The magnetic “butterfly” diagram showing the evolution of the radial component of magnetic field during the last two solar cycles as a function of time and latitude. b) The zonal flow velocity near the solar surface as a function of latitude and time. c) The zonal flow acceleration calculated after applying a Gaussian filter to smooth noise and small-scale variations. d) Overlay of the zonal acceleration (color image) and the radial magnetic field (gray-scale) reveals that the regions of magnetic field emergence coincide with the zones of flow deceleration.
The torsional oscillation pattern in the near-surface layers obtained by subtracting the mean differential rotation (separately for Solar Cycles 23 and 24), and combining the residuals in the time-latitude diagram is shown in Figure 1b. For comparison with the evolution of solar magnetic field, in Figure 1a we present a superposition of longitudinally averaged magnetic synoptic maps (so-called “magnetic butterfly diagram”). The migration towards the equator mid-latitude branches represent zones of emerging active regions and sunspot formation. At high latitudes the magnetic field reverses polarity in the middle of the activity cycles when the sunspot number reaches its maximum. The first butterfly structure, lasting from 1997 to 2009, represents Solar Cycle 23, the second one, from 2010 to 2018, is Solar Cycle 24. The corresponding zonal acceleration is shown in Figure 1c. It clearly reveals the torsional oscillation patterns of both Cycles 23 and 24. By overlying the zonal acceleration and magnetic field diagrams Figure 1d, we find that the active region zones coincide with the flow deceleration zone (blue color). In the polar regions the deceleration zones correspond to the periods of strong polar magnetic field. This confirms that the torsional oscillations are due to back reaction of solar magnetic fields.
Figure 2| Image of the zonal acceleration pattern beneath the solar surface shows zone of flow acceleration (red) and deceleration (blue). The flow deceleration is caused by the magnetic field. The inner sphere shows the bottom of the convection zone (the solar tachocline). The image shows a snapshot of the zonal deceleration in 2001 near the maximum of the sunspot number of Cycle 23, when the generation of magnetic field of Cycle 24 already started in the tachocline.
The evolution of zonal acceleration in the convection zone can be identified from a time series of radius-latitude images as shown in the movie, a representative sample of which is shown in Figure 2.
The results provide strong evidence of dynamo waves and reveal their migration pattern in a form of two branches: migrating towards the poles and the equator. The polar branch reaches the surface in 1-2 years, while it takes about 8-9 years for the equatorial branch to reach the solar surface (Figure 3a), and form the sunspot butterfly diagram. Since the polar branch reaches the surface quicker, the strength of the polar magnetic field can help predict the following sunspot maximum. These dynamo wave patterns explain the extended solar cycle phenomenon and why the maximum polar magnetic field strength is a good predictor of the next sunspot maximum.
Figure 3| a) Time-radius diagrams of the zonal acceleration at 15, 30 and 60 degrees latitude after applying the Principal Component Analysis. Inclined lines marks regions of zonal deceleration corresponding to Cycles 23 (solid), 24 (dashed) and 25 (dotted). b) Evolution of the sunspot number (solid curve) and the zonal acceleration (dashed) in the region of initiation of torsional oscillations, located near the base of the convection zone and 60 degrees latitude.
The presented analysis suggests the following scenario of the solar dynamo, generally consistent with the Parker’s theory. The poloidal magnetic field, generated by helical turbulence in a high-latitude zone, around 60° latitude, near the bottom of the convection zone, and quickly (during 1-2 years) migrates to the surface. In the low latitude zone (<45°) the poloidal field is stretched and converted to the toroidal field by differential rotation. It migrates much slower towards the surface and lower latitudes in the form of a dynamo wave. The latitudinal migration of the toroidal field, forming the butterfly diagram, is due to a horizontal gradient of angular velocity and radial rotational shear in the subsurface layer. The emerging toroidal magnetic field forms active regions. After their decay the remnants of the magnetic field of active regions are transported by turbulent diffusion and meridional circulation to high latitudes where they migrate downwards, and, probably, contribute to the seed field of the solar cycle which follows after the next cycle that is already in progress.
Recent measurements show a significant decrease of zonal deceleration in the tachocline indicating that the upcoming sunspot cycle may be even weaker than the current sunspot cycle (Figure 3b). Thus, the long-term trend of the declining magnetic field of the Sun is likely to continue.
For more details, please see Ref. .
 Parker, E. N. 1955, ApJ, 122, 293
 Dikpati, M., & Gilman, P. A. 2009, Space Sci. Rev., 144, 67
 Howard, R., & Labonte, B. J. 1980, ApJ Lett, 239, L33
 Larson, T. P., & Schou, J. 2018, Solar Phys, 293, 29
 Kosovichev, A. G., & Pipin, V. V. 2019, ApJ Lett, 871, L20