Role of rotational radial magnetic advection in possible explaining a cycle with two peaks

Highlights

• Double maximum of the 11-year solar cycle.

• Turbulent effects of macroscopic magnetohydrodynamics.

• Negative magnetic buoyancy.

• Turbulent pumping of a large-scale magnetic field.

• Reconstructing of the toroidal magnetic field of the Sun.

Abstract

To explain the observed phenomenon of 11-year sunspot cycle with two peaks, we draw attention to the possible role of the turbulent magnetic pumping effects in the solar convection zone (SCZ) at the reconstructing of the toroidal magnetic field that generates the sunspots. Turbulent radial pumping of magnetic field has both kinetic (diamagnetism) and magnetic (vertical advection) contributions. We take into account five physical processes for the combined magnetic reconstructing in deep layers of the SCZ. They are the Ω effect near the tachocline, magnetic buoyancy of the smoothed field, macroscopic diamagnetism, rotational magnetic advection, and deep equatorward meridional circulation. It is the effect of advection with taking into account Sun's rotation that plays a key role in the proposed scenario of magnetic reconstructing. We found that reconstructing of deep toroidal field develops differently in the near-polar and equatorial domains of the SCZ. In the lower half of the near-polar domain, two downward pumping effects (diamagnetism and advection) act against magnetic buoyancy and, as a result, they neutralize magnetic buoyancy and thereby block the toroidal field near the tachocline. At the same time, in the deep layers of the equatorial domain, the rotational advection due to the latitudinal convection anisotropy changes its direction to the opposite (from downward to upward), thereby helping magnetic buoyancy. As a result, this upward magnetic pumping here contributes to the transport of the toroidal field to the solar surface. We call this transport as first upward magnetic advection surge. It ensures a transfer of the deep strong fields to the surface in the latitudes belt of sunspots. Meanwhile, a deep equatorward meridional flow transports the blocked near pole toroidal field in the tachocline from high latitudes to low ones where are favourable conditions for floating up of the magnetic field. Here this belated field is transported upward to the solar surface (the second upward magnetic advection surge). Ultimately, two time-delayed upward magnetic surges may cause on the surface in the “royal zone” the first and second peaks of the sunspots cycle.

Introduction

The double-peak phenomenon in the sunspot cycle was discovered by Mstislav Gnevyshev in 1963 during his studies of coronal emission and areas of active formations in the 11-year cycle 19 (Gnevyshev, 1963). After that Antalova and Gnevyshev (1965) studied the time curves of the latitudinal index of the total area of sunspot groups for 12–18 cycles (from 1874 to 1962). They found that there were two maxima in each sunspot cycle. The first maximum was manifested simultaneously at all the latitudes of sunspots belt (which Christopher Scheiner, one of the four telescopic discoverers of the sunspots, called as “royal zone”; Scheiner's “Rosa Ursina sive Sol”, published in 1630). It was centred at the latitude 25° in the total areas of the sunspots, while the second maximum appeared later and at low latitudes only (10–15°). Subsequently, these features were also confirmed for cycle 20, including for the Sun’s northern and southern hemispheres separately (Gnevyshev, 1967). In this regard, there was a need to search for physical mechanisms underlying the double maxima.

Gnevyshev (1967) was the first to suggest that the two maxima in the total sunspot group area are caused by two different mechanisms of excitation of magnetism occurring at two different latitude bands. However, the question about the nature of these physical processes and the time delay of activity between them remained open. Over time, the most widespread opinion among researchers was that the mechanisms of the double maximum should be based on the effects of magnetized plasma enveloped in the turbulent motions in the solar convection zone (SCZ). Researchers have focused on the capabilities of the mean field magnetohydrodynamics and the αΩ dynamo models (Krause and Rädler, 1980, Brandenburg, 2018).

The solar αΩ dynamo is believed to be driven by two main processes: the winding of the global toroidal field from the poloidal one by differential rotation (the Ω effect) and conversion of the toroidal field back to the poloidal configuration by helical motions (the α effect). According to Parker, 1955, Parker, 1979, the solution of the αΩ dynamo equations can be represented in the form of dynamo waves, which in observations appear as waves of magnetic activity, migrating in the meridional direction from mid-latitudes to the equator (Parker’s migration dynamo). Yoshimura (1975) specified that dynamo wave propagates along the lines of constant angular velocity. The direction of the meridional migration of dynamo waves (to the equator or pole) depends on the sign (negative or positive) of the product α∂Ω/∂r (α – is the helicity parameter of turbulent convection, ∂Ω/∂r – is the radial gradient of angular velocity of the Sun).

An important role in the solar αΩ dynamo is played by the value of the turbulent diffusion coefficient ν_T ≈ (1/3) ul ≈ (1/3) u²τ (u, l and τ are the rms velocity, the mixing length and the characteristic time of the convective turbulent motions) (Köhler, 1973). The 11-year period of solar cycles can be reproduced only if the turbulent magnetic diffusivity is reduced much below its usual mixing-length estimation ν_T ≈ 10¹³ cm²/s. A cyclic dynamo has to regenerate magnetic fields faster than diffusion destroys them. The cycle period should, therefore, be shorter than the diffusion time t_d = L²/ν_T, where L is the SCZ thickness (L ≈ 200 Mm). Therefore, a small diffusivity, ν_T < 10¹² cm²/s, is required to reproduce the 11-year period of the cycle (Kitchatinov, 2014). In this regard, we note that our calculations in the mixing length approximation showed (Krivodubskii, 1982, Krivodubskii, 1984a) that the radial profile of turbulent diffusivity is a smooth convex function with a maximum value, ν_T ≈ 10¹³ cм²/c, approximately in the middle of the SCZ (Fig. 1). At the same time, near the bottom of the SCZ, where the turbulent convection velocity drops sharply, the value of the coefficient ν_T decreases by an order of magnitude (see also Kitchatinov and Rüdiger, 2008, Pipin and Kosovichev, 2011a, Pipin and Kosovichev, 2011b; and calculations by Pipin (2021) for the solar analogs). The low turbulent diffusion in deep layers of turbulent convection allows us to remove the noted difficulty in matching the diffusion time t_d with the cycle duration, since the near-bottom region in the SCZ has long been recognized as a favorite place for the solar dynamo. In addition, the radial inhomogeneity of turbulent diffusion in the lower half of the SCZ promotes the downward macroscopic diamagnetic pumping, which acts against magnetic buoyancy and thus can store the strong magnetic fields in this area (see Section 5.1 for more details).

The stratification, rotation, and strong magnetic fields cause the turbulent viscosity, ν_T, to be anisotropic and quenched (Kitchatinov et al., 1994, Rogachevskii and Kleeorin, 2001, Rogachevskii et al., 2011, Brandenburg and Sokoloff, 2002, Kitchatinov and Nepomnyashchikh, 2016). Rogachevskii and Kleeorin (2001) shown that the toroidal and poloidal magnetic fields have different nonlinear turbulent magnetic diffusion coefficients. However, the anisotropy of the turbulent diffusivity is less important for the time-latitude evolution of the toroidal field when the meridional circulation is included in the model (Pipin and Kosovichev, 2013). When the rotation of the Sun is taken into account, the diamagnetic properties of the turbulized plasma manifest themselves differently with respect to the toroidal and poloidal fields (Kleeorin and Rogachevskii, 2003, Rogachevskii et al., 2011, Kitchatinov and Nepomnyashchikh, 2016). Even so, we will take into account the turbulent magnetic diffusion and the diamagnetic pumping only for the toroidal field, since the reconstructing of this field is the subject of our study.

Benevolenskaya (1998) was the first to describe the simultaneous presence in the SCZ of two periods of the magnetic field oscillations based on the two-layer model of the αΩ dynamo. Georgieva (2011) analyzed several dynamo regimes depending on the ratio of the contributions from turbulent diffusion and meridional circulation to the magnetic flux-transport dynamo model. It was shown that some of the dynamo regimes allow the explanation of the double maximum of the sunspot cycle. However, no modeled double-peak solar cycle was presented. In a series of papers (Zharkova et al., 2012, Shepherd et al., 2014, Popova et al., 2015), the idea put forward that there are two principal components of the background magnetic field on the Sun that have opposite orientations in the northern and southern hemispheres. In the framework of the constructed scheme of the two-layer αΩ dynamo model the simultaneous presence of 22 years and quasi-biennial oscillations of magnetic fields leading to a double sunspot cycle was simulated. Zolotova and Ponyavin (2012) proposed a model of the local minimum during of the activity cycle which is based on the hypothesis of M. Gnevyshev about spot formation pulses associated with two different physical mechanisms of magnetism excitation (Antalova and Gnevyshev, 1965, Gnevyshev, 1967, Gnevyshev, 1977). Using the magnetohydrodynamics shallow-water model, Dikpati et al. (2018) have shown that quasiperiodic non-linear oscillations in the tachocline (energy exchange among magnetic fields, Rossby waves, and differential rotation) could be a possible cause of the double maximum of the solar cycle, although a detailed model is needed. Karak et al. (2018) proposed the idea that fluctuations in the surface Babcock - Leighton alpha effect may be responsible for the double peaks in the observed solar cycle. By means of the simulation, it was demonstrated that fluctuations in polar field are propagated to a new toroidal field. Ultimately, they can promote the double peaks in the next sunspot cycle. Kitchatinov (2020) examined the instability of a large-scale toroidal magnetic field caused by the magnetic suppression of convective heat transport as a candidate for the flux tube forming mechanism. The instability tends to produce regions of increased field strength with spatial scales of an order of 100 Mm at the base of the SCZ. The threshold field strength for the onset of the instability increases from several hundred Gauss in the vicinity of the equator to some kilo-Gauss at middle latitudes. Growth rates of unstable disturbances decrease with latitude. These latitudinal trends can be the reason for the observed confinement of the sunspot activity to a near-equatorial belt. Wang et al., 1991, Choudhuri et al., 1995, Durney, 1995 firstly demonstrated that a sufficiently strong equatorward meridional circulation in the deep layers of the SCZ forces the toroidal field belts (which are responsible for the surface sunspot activity) to move equatorward. Käpylä et al. (2006) proposed several kinematic solar dynamo models that included the rotation profile obtained from helioseismology, the α effect, turbulent pumping, and meridional circulation. It has been found that these models can correctly reproduce many features of the solar activity cycle. Irregular fluctuations in the meridional circulation are important in explaining many aspects of the irregularities in the solar cycle (Choudhuri, 2018, Choudhuri, 2020), in making comprehensive models of grand minima (Choudhuri and Karak, 2012) and in predicting future cycles (Hazra and Choudhuri, 2019). Guerrero and de Gouveia Dal Pino (2008) had explored the role of turbulent magnetic pumping in the solar dynamo process. The results reveal the importance of the pumping mechanism for solving current limitations in the mean field dynamo modeling such as storage of the magnetic flux and latitudinal distributions of the sunspots.

Theoretical studies show that, along with macroscopic diamagnetism, in turbulent plasma one more macroscopic transport effect can arise due to the inhomogeneity of the matter density (Drobyshevski, 1977, Kitchatinov, 1982, Kitchatinov, 1991, Vainshtein, 1983). Since the matter density varies by almost six orders of magnitude over a vertical extent of the SCZ, a very strong downward magnetic pumping should be expected here. Furthermore, it is very important, that the Sun’s rotation substantially modifies this effect (Kitchatinov, 1991). Our calculations (Krivodubskij, 2005) had shown that magnetic field radial transport due this modified effect has the opposite directions in the near-polar domain and in the near-equatorial deep region. In the high-latitudes domain, this effect acts downward, that is, against magnetic buoyancy, while in the lower part of the near-equatorial domain, it reverses direction, and therefore here it already helps magnetic buoyancy. We believe that this change in the direction of the magnetic transport in the near-equatorial domain is of fundamental importance for the radial transport of the toroidal field in the SCZ (see Section 5.2 for more details).

Here we attempted to outline the restricted pattern of the radial transport of the toroidal field, which, on our opinion, may help to understand why the magnetic fields rise to the solar surface in the sunspot belt (Section 6). We draw the attention to the possible role of the macroscopic effects of turbulent pumping, which could contribute to explaining the two peaks of the 11-year sunspot cycle (Section 5). We propose a simplified scenario for reconstructing of the toroidal magnetic field, which involves following physical processes: the Ω effect in the tachocline; macroscopic turbulent diamagnetism by Zeldovich-Rädler; turbulent magnetic pumping by Kithatinov, which takes into account the rotation of the Sun; deep equatorward meridional circulation; and magnetic buoyancy of the smoothed field. We note that other effects (turbulent diffusion, the α effect, dynamo waves) can act in the SCZ. Therefore, when studying the complete picture of the reconstruction of the solar magnetism, it is necessary to take into account all turbulent processes together. Thus, this study suggests that proposed simplified scenario should be thoroughly investigated in future taking into account the additional turbulent processes.

To explain the physics behind the proposed scenario, we briefly review the effects involved in this reconstructing scheme.