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名古屋大学年代測定総合研究センターシンポジウム報告

Observations show that Coronal Mass Ejections (CMEs) are associated with twisted magnetic flux configurations. Conventionally, CMEs are modeled by shearing and twisting the footpoints of a certain distribution of magnetic flux at the solar surface and letting it evolve at the surface. Of course, the surface velocities and magnetic field patterns should ultimately be obtained from realistic simulations of the solar convection zone where the field is generated by dynamo action. Therefore, a unified treatment of the convection zone and the CMEs is needed. Numerical simulations of turbulent dynamos show that the amplification of magnetic fields can be catastrophically quenched at magnetic Reynolds numbers typical of the interior of the Sun. A strong flux of magnetic helicity leaving the dynamo domain can alleviate this quenching. In this sense, a realistic (magnetic) boundary condition is an important ingredient of a successful solar dynamo model. Using a two-layer model developed in this thesis, we combine a dynamo-active region with a magnetically inert but highly conducting upper layer which models the solar corona. In four steps we improve this setup from a forced to a convectively driven dynamo and from an isothermal to a polytropic stratified corona. The simulations show magnetic fields that emerge at the surface of the dynamo region and are ejected into the coronal part of the domain. Their morphological form allows us to associate these events with CMEs. Magnetic helicity is found to change sign in the corona to become consistent with recent helicity measurements in the solar wind. Our convection-driven dynamo model with a coronal envelope has a solar-like differential rotation with radial (spoke-like) contours of constant rotation rate, together with a solar-like meridional circulation and a near-surface shear layer. The spoke-like rotation profile is due to latitudinal entropy gradient which violates the Taylor--Proudman balance through the baroclinic term. We find mean magnetic fields that migrate equatorward in models both with and without the coronal layer. One remarkable result is that the dynamo action benefits substantially from the presence of a corona becoming stronger and more realistic. The two-layer model represents a new approach to describe the generation of coronal mass ejections in a self-consistent manner. On the other hand, it has important implications for solar dynamo models as it admits many magnetic features observed in the Sun. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript.</p>

The Sun, by the virtue of its proximity to Earth, serves as an excellent astrophysical laboratory for testing our theoretical ideas. The Sun displays a plethora of visually awe-inspiring phenomena including flares, prominences, sunspots, corona, CMEs and uncountable others. It is now known that it is the magnetic field of the Sun which governs all these and also the geomagnetic storms at the Earth, which owes its presence to the interaction between the geomagnetic field and the all-pervading Solar magnetic field in the interplanetary medium. Since the solar magnetic field affects the interplanetary space around the Earth in a profound manner, it is absolutely essential that we develop a comprehensive understanding of the generation and manifestation of magnetic fields of the Sun. This thesis aims at developing a state-of-the-art dynamo code SURYA1taking into account important results from helioseismology and magnetohydrodynamics. This dynamo code is then used to study various phenomenon associated with solar activity including evolution of solar parity, response to stochastic fluctuations, helicity of active regions and prediction of future solar cycles. Within last few years dynamo theorists seem to have reached a consensus on the basic characteristics of a solar dynamo model. The solar dynamo is now believed to be comprised of three basic processes: (i)The toroidal field is produced by stretching of poloidal field lines primarily inside the tachocline – the region of strong radial shear at the bottom of the convection zone. (ii) The toroidal field so formed rises to the surface due to magnetic buoyancy to form active regions. (iii) Poloidal field is generated at the surface due to decay of tilted active regions – an idea attributed to Babcock (1961) & Leighton (1969). The meridional circulation then carries the poloidal field produced near the surface to the tachocline. The profile of the solar differential rotation has now been mapped by helioseismology and so has been the poleward branch of meridional circulation near the surface. The model I describe in this thesis is a two-dimensional kinematic solar dynamo model in a full sphere. Our dynamo model Surya was developed over the years in stages by Prof. Arnab Rai Choudhuri, Dr. Mausumi Dikpati, Dr. Dibyendu Nandy and myself. We provide all the technical details of our model in Chap. 2 of this thesis. In this model we assume the equatorward branch of the meridional circulation (which hasn’t been observed yet), to penetrate slightly below the tachocline (Nandy & Choudhuri 2002, Science, 296, 1671). Such a meridional circulation plays an important role in suppressing the magnetic flux eruptions at high latitudes. The only non-linearity included in the model is the prescription of magnetic buoyancy. Our model is shown to reproduce various aspects of observational data, including the phase relation between sunspots and the weak, efficient. An important characteristic of our code is that it displays solar-like dipolar parity (anti-symmetric toroidal fields across equator) when certain reasonable conditions are satisfied, the most important condition being the requirement that the poloidal field should diffuse efficiently to get coupled across the equator. When the magnetic coupling between the hemispheres is enhanced by either increasing the diffusion or introducing an α ff distributed throughout the convection zone, we find that the solutions in the two hemispheres evolve together with a single period even when we make the meridional circulation or the α effect different in the two hemispheres. The effect of diffusive coupling in our model is investigated in Chap. 3. After having explored the regular behaviour of the solar cycle using the dynamo code we proceed to study the irregularities of the Solar cycle.We introduce stochastic fluctuations in the poloidal source term at the solar surface keeping the meridional circulation steady for all the numerical experiments. The dynamo displays oscillatory behaviour with variable cycle amplitudes in presence of fluctuations with amplitudes as large as 200%. We also find a statistically significant correlation between the strength of polar fields at the endofone cycle and the sunspot number of the next cycle. In contrast to this there exist a very poor correlation between the sunspot number of a cycle and the polar field formed at its end. This suggests that during the declining phase of the sunspot cycle poloidal field generation from decaying spots takes place via the Babcock-Leighton mechanism which involves randomness and destroys the correlation between sunspot number of a cycle and the polar at its end. In addition to this we also see that the time series of asymmetries in the sunspot activity follows the time series of asymmetries in the polar field strength with a lag of 5 years. We also compare our finding with available observational data. Although systematic measurements of the Sun’s polar magnetic field exist only from mid-1970s, other proxies can be used to infer the polar field at earlier times. The observational data indicate a strong correlation between the polar field at a sunspot minimum and the strength of the next cycle, although the strength of the cycle is not correlated well with the polar field produced at its end. We use these findings about the correlation of polar fields with sunspots to develop an elegant method for predicting future solar cycles. We feed observational data for polar fields during the minima of cycle n into our dynamo model and run the code till the next minima in order to simulate the sunspot number curve for cycle n+1. Our results fit the observed sunspot numbers of cycles 21-23 reasonably well and predict that cycle 24 will be about 30–35% weaker than cycle 23. We fit that the magnetic diffusivity in the model plays an important role in determining the magnetic memory of the Solar dynamo. For low diffusivity, the amplitude of a sunspot cycle appears to be a complex function of the history of the polar field of earlier cycles. Only if the magnetic diffusivity within the convection zone is assumed to be high (of order 1012cms−1), we are able to explain the correlation between the polar fiat a minimum and the next cycle. We give several independent arguments that the diffusivity must be of this order. In a dynamo model with diffusivity like this, the poloidal field generated at the mid-latitudes is advected toward the poles by the meridional circulation and simultaneously diffuses towards the tachocline, where the toroidal field for the next cycle is produced. The above ideas are put forward in Chap. 6. We next come to an important product of the dynamo process namely the magnetic helicity. It has been shown independently by many research groups that the mean value of the normalized current helicity αp= B (Δ×B)/B2in solar active regions is of the order of 10−8m−1, predominantly negative in the northern hemisphere, positive in the southern hemisphere. Choudhuri (2003, Sol. Phys., 215, 31)developed a model for production of the helicity of the required sign in a Babcock-Leighton Dynamo by wrapping of poloidal field lines around a fluxtube rising through the convection zone. In Chap. 7 we calculate helicities of solar active regions based on this idea. Rough estimates based on this idea compare favourably with the observed magnitude of helicity. We use our solar dynamo model to study how helicity varies with latitude and time. At the time of solar maximum, our theoretical model gives negative helicity in the northern hemisphere and positive helicity in the south, in accordance with observed hemispheric trends. However, we fit that during a short interval at the beginning of a cycle, helicities tend to be opposite of the preferred hemispheric trends. After calculating the sign and magnitude of helicity of the sunspots we worry about the distribution of helicity inside a sunspot. In Chap. 8 we model the penetration of a wrapped up background poloidal field into a toroidal magnetic flux tube rising through the solar convective zone. The rise of the straight, cylindrical flux tube is followed by numerically solving the induction equation in a comoving Lagrangian frame, while an external poloidal magnetic field is assumed to be radially advected onto the tube with a speed corresponding to the rise velocity. One prediction of our model is the existence of a ring of reverse current helicity on the periphery of active regions. On the other hand, the amplitude of the resulting twist depends sensitively on the assumed structure (ffvs. concentrated/intermittent) of the active region magnetic field right before its emergence, and on the assumed vertical profile of the poloidal field. Nevertheless, in the model with the most plausible choice of assumptions a mean twist comparable to the observational results. Our results indicate that the contribution of this mechanism to the twist can be quite find under favourable circumstances it can potentially account for most of the current helicity observed in active regions.

Our Sun is a variable star. The magnetic fields in the Sun play an important role for the existence of a wide variety of phenomena on the Sun. Among those, sunspots are the slowly evolving features of the Sun but solar ares and coronal mass ejections are highly dynamic phenomena. Hence, the solar magnetic fields could affect the Earth directly or indirectly through the Sun's open magnetic flux, solar wind, solar are, coronal mass ejections and total solar irradiance variations. These large scale magnetic fields originate due to Magnetohydrodynamic dynamo process inside the solar convection zone converting the kinetic energy of the plasma motions into the magnetic energy. Currently the most promising model to understand the large scale magnetic fields of the Sun is the Flux Transport Dynamo (FTD) model. FTD models are mostly axisymmetric models, though the non-axisymmetric 3D FTD models are started to develop recently. In these models, we assume the total magnetic fields of the Sun consist of poloidal and toroidal components and solve the magnetic induction equation kinematicaly in the sense that velocity fields are invoked motivated from the observations. Differential rotation stretches the poloidal field to generate the toroidal field. When toroidal eld near the bottom of the convection zone become magnetically buoyant, it rises through the solar convection zone and pierce the surface to create bipolar sunspots. While rising through the solar convection zone, the Coriolis force keeps on acting on the flux tube, which introduces a tilt angle between bipolar sunspots. Since the sunspots are the dense region of magnetic fields, they diffuse away after emergence. The leading polarity sunspots (close to equator) from both the hemisphere cancel each other across the equator and trailing polarity sunspots migrate towards the pole to generate effective poloidal fields. This mechanism for generation of poloidal field from the decay of sunspots is known as Babcock-Leighton process. After the poloidal field is generated, the meridional flow carries this field to the pole and further to the bottom of the convection zone where differential rotation again acts on it to generate toroidal field. Hence the solar dynamo goes on by oscillation between the poloidal field and toroidal field, where they can sustain each other through a cyclic feedback process. Just like other physical models, FTD models have various assumptions and approximations to incorporate these different processes. Some of the assumptions are observationally verified and some of them are not. Considering the availability of observed data, many approximations have been made in these models on the theoretical basis. In this thesis, we present various studies leading to better understanding of the different processes and parameters of FTD models, which include magnetic buoyancy, meridional circulation and Babcock-Leighton process. In the introductory Chapter 1, we first present the observational features of the solar magnetic fields, theoretical background of the FTD models and motivation for investigating different processes. Most of the results of our work are presented in Chapters 2 - 7. In the Chapters 2 - 5, we explain various important issues regarding the treatment of magnetic buoyancy, irregularities of the solar cycle during descending phase, effect of different spatial structure of meridional flow on the dynamo and how dynamo generated fields would a ect the meridional ow using 2D axisymmetric Flux Transport Dynamo model. In the Chapters 6 & 7, the build up of polar fields from the decay of sunspots and a proper treatment of Babcock-Leighton process by invoking realistic convective flows, are presented using 3D Flux Transport Dynamo model. Finally the conclusions and future works are given in the Chapter 8. In 2D axisymmetric Flux Transport Dynamo models, the rise of the toroidal magnetic field through the convection zone due to magnetic buoyancy and then the generation of the poloidal magnetic field from these bipolar sunspots, has been treated mainly in two ways|a non-local method and a local method. In Chapter 2, we have analyzed the advantages and disadvantages of both the methods. We find that none of them are satisfactory to depict the correct picture of magnetic buoyancy because it is an inherently 3D process. Unless we go to the 3D framework of Flux Transport Dynamo models, we have to treat the magnetic buoyancy in such simplistic way. We find that the non-local treatment of magnetic buoyancy is very robust for a large span of parameter space but it does not take into account the depletion of flux from the bottom of the convection zone which has a significant importance in irregularity study of the solar cycle. The local treatment of magnetic buoyancy includes the flux depletion from the bottom of the convection zone and treats the magnetic buoyancy much realistically than the non-local treatment. But this local treatment of magnetic buoyancy is not so robust. We also pointed out that the long-standing issue about appearance of sunspots in the low-latitudes needs to be studied carefully. In Chapter 3, we have studied various irregularities of the solar cycle during its decaying phase. We have reported that the decay rate of the cycle is strongly correlated with amplitude of the same cycle as well as the amplitude of the next cycle from different sunspot proxies like sunspot number, sunspot area and 10.7 cm radio flux data. We explain these correlation from flux transport dynamo models. We nd that the correlations can only be reproduced if we introduce stochastic fluctuations in the meridional circulations. We also reproduced most of the correlation found in ascending and descending phase of the solar cycle from century long sunspot area data (Mandal et al., 2017) from Kodaikanal observatory, India which are in great agreement with the correlations found earlier from Greenwich sunspots data. In most of the FTD models, a single cell meridional circulation is assumed within the solar convection zone, with the equatorward return flow at its bottom. But with recent development in helioseismology, plenty of results have come out about various spatial structure of meridional circulation (Zhao et al., 2013; Schad et al., 2013; Rajaguru & Antia, 2015; Jackiewicz et al., 2015). Some helioseismology group (Zhao et al., 2013) reported that the meridional circulation has a double cell structure in solar convection zone and some groups (Schad et al., 2013; Jackiewicz et al., 2015) have reported a multi-cellular structure of meridional circulation in the convection zone. By probing the supergranular motion Hathaway (2012) estimated that the meridional ow has an equatorward return ow at the upper convection zone 70 Mm below the surface. In view of the above observed results, we have discussed in Chapter 4 what would happen to Flux Transport Dynamo model if we consider other structure of meridional circulation instead of single cell meridional circulation encompassing whole convection zone. We nd that the our dynamo model works perfectly ne as long as there is an equatorward propagation at the bottom of the convection zone. Our model also works with shallow meridional circulation as found by Hathaway (2012), if we consider the latitudinal pumping in our model. The temporal variation of meridional circulation on the surface is also observed from various measurement techniques. Chou & Dai (2001) rst observed a variation of meridional circulation with the solar cycle from their helioseismic measurements. Hathaway & Rightmire (2010) also found a variation up to 5 m s 1 for the solar cycle 23 by measuring the magnetic elements on the surface of the Sun. Recently Komm et al. (2015) have analyzed MDI and HMI Dopplergram data and reported a solar cyclic variation with detail latitudinal dependence. To explain this variation of the meridional circulation with the solar cycle, we construct a theoretical model by coupling the equation of the meridional circulation (the component of the vorticity equation within the solar convection zone) with the equations of the flux transport dynamo model in Chapter 5. We consider the back reaction due to the Lorentz force of the dynamo-generated magnetic fields and study the perturbations produced in the meridional circulation due to it. This enables us to model the variations of the meridional circulation without developing a full theory of the meridional circulation itself. We obtain results which reproduce the observational data of solar cycle variations of the meridional circulation reasonably well. We get the best results on assuming the turbulent viscosity acting on the velocity field to be comparable to the magnetic diffusivity (i.e. on assuming the magnetic Prandtl number to be close to unity). We have to assume an appropriate bottom boundary condition to ensure that the Lorentz force cannot drive a flow in the sub-adiabatic layers below the bottom of the tachocline. Our results are sensitive to this bottom boundary condition. We also suggest a hypothesis how the observed inward flow towards the active regions may be produced. In Chapter 6 and Chapter 7, we have studied some of the aspects of solar magnetic eld generation process using 3D dynamo model that were not possible to study earlier using axisymmetric 2D Flux Transport dynamo models. We have used the 3D dynamo model developed by Mark Miesch (Miesch & Dikpati, 2014; Miesch & Teweldebirhan, 2016) and study how polar fields build up from the decay of sunspots more realistically in Chapter 6. We first reproduce the observed butter y diagram and periodic solution considering higher diffusivity value than earlier reported results and use it as a reference model to study the build up polar fields by putting a single sunspot pair in one hemisphere and two sunspot pairs in both the hemispheres. The build up of the polar fields from the decay of sunspots are studied earlier using Surface Flux Transport model (Wang et al., 1989; Baumann et al., 2004; Cameron et al., 2010) which solve only radial component of the induction equation on the surface of the Sun ( | plane). But these 2D SFT models have some inherent limitation for not considering the 3D vectorial nature of the magnetic fields and subsurface processes. We have shown that not considering the vectorial nature and subsurface process has an important effect on the development of the polar fields. We have also studied the effect of a few large sunspot pairs violating Hale's law on the strength of the polar field in this Chapter. We nd that such ant-Hale sunspot pairs do produce some effect on the polar fields, if they appear at higher latitudes during the mid-phase of the solar cycle|but the effect is not dramatic. In Chapter 7, we have incorporated observed surface convective ows directly in our 3D dynamo model. As we know that the observed convective flows on the photosphere (e.g., supergranulation, granulation) play a key role in the Babcock-Leighton (BL) process to generate large scale polar fields from sunspots fields. In most surface flux transport (SFT) and BL dynamo models, the dispersal and migration of surface fields is modeled as an effective turbulent diffusion. Recent SFT models have incorporated explicit, realistic convective flows in order to improve the fidelity of convective transport but, to our knowledge, this has not yet been implemented in previous BL models. Since most Flux-Transport (FT)/BL models are axisymmetric, they do not have the capacity to include such flows. We present the first kinematic 3D FT/BL model to explicitly incorporate realistic convective flows based on solar observations. Though we describe a means to generalize these flows to 3D, we find that the kinematic small-scale dynamo action they produce disrupts the operation of the cyclic dynamo. Cyclic solution is found by limiting the convective flow to surface flux transport. The results obtained are generally in good agreement with the observed surface flux evolution and with non-convective models that have a turbulent diffusivity on the order of 3 1012 cm 2 s 1 (300 km2 s 1). However, we nd that the use of a turbulent diffusivity underestimates the dynamo efficiency, producing weaker mean fields than in the convective models. Also, the convective models exhibit mixed polarity bands in the polar regions that have no counterpart in solar observations. Also, the explicitly computed turbulent electromotive force (emf) bears little resemblance to a diffusive flux. We also find that the poleward migration speed of poloidal flux is determined mainly by the meridional flow and the vertical diffusion.

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Avec les récents développements obtenus grâce aux modèles globaux magnétohydrodynamiques en trois dimensions de la convection solaire, il est désormais possible de simuler des champs magnétiques structurés à grande échelle et présentant des inversions de polarité bien synchronisées dans chaque hémisphère. Ces modèles qui n'incluent, pour la plupart, aucune modélisation de la surface du Soleil génèrent donc leur dynamo avec l'action de la force électromotrice turbulente (FEM) et de la rotation différentielle uniquement. À partir de cette FEM, différentes techniques peuvent être utilisées pour extraire les coefficients de transport turbulent. Notamment, différents auteurs ont obtenu un tenseur alpha (coefficient du premier ordre) dont les 9 composantes présentent des amplitudes du même ordre, remettant en doute l'approximation faite dans le cas des modèles dynamo de type alphaOmega qui ne tient en compte qu’une de composante du tenseur. À partir d'un code d'analyse par décomposition en valeurs singulières pour évaluer les coefficients du tenseur alpha, nous avons généralisé la procédure pour extraire 18 des composantes du tenseur de deuxième ordre (tenseur beta). Les tenseurs alpha et beta obtenus par cette nouvelle procédure tel qu'appliquée aux sorties du modèle global EULAG-MHD, sont similaires aux tenseurs alpha et beta équivalant obtenus en utilisant l'approximation « Second Order Correlation Approximation ». À l'aide des coefficients de transport turbulent du premier ordre introduit dans un modèle dynamo en champ moyen, nous avons ensuite étudié certaines solutions magnétiques présentant des doubles dynamos. Cette analyse avait pour but de comparer les résultats obtenus par ce modèle simplifié caractérisé par la FEM provenant de EULAG-MHD aux résultats de EULAG-MHD directement. Cette preuve de concept nous a permis de démontrer que l'oscillation observée dans le champ magnétique en surface de EULAG-MHD pouvait provenir de l'action inductive d'une seconde dynamo. Une oscillation biennale est également observée dans plusieurs indices d'activité solaire dont l'origine n'est toujours pas établie. Il est évident que les deux modèles décrits ci-haut et le Soleil opèrent dans des régimes physiques différents. Toutefois, malgré leurs différences, le fait qu'il soit relativement facile de produire une seconde dynamo dans EULAG-MHD et dans le modèle en champ moyen indique que l'action inductive de la FEM peut facilement générer deux dynamos. Finalement, dans le but d'étudier les périodes de grands minima, phénomène encore non reproduit par les modèles globaux, nous avons ajouté une rétroaction magnétique sur l'écoulement azimutal au modèle dynamo cinématique en champ moyen décrit ci-haut. En analysant les solutions de ce modèle dynamo de type alpha2Omega non cinématique, nous avons pu reproduire la tendance observée jusqu'ici uniquement dans les modèles de type alphaOmega selon laquelle le nombre de nombre de Prandtl magnétique contrôle le rapport des périodes générées. De plus, en analysant une solution sur 50 000 ans présentant des périodes de grands minima et maxima non périodiques, nous avons obtenu une distribution de temps de séparation des grands minima presque exponentielle, caractéristique observée dans les reconstructions de l'activité solaire. La rotation différentielle associée à ces périodes de grands minima présente un niveau de fluctuation de 1% par rapport au profil moyen. Ce niveau de fluctuation est d'ailleurs comparable avec les reconstructions historiques de la rotation différentielle en surface obtenues lors du grand minimum de Maunder. / The recent developments achieved by tri-dimensionals magnetohydrodynamic (3D-MHD) global simulations of solar convection allow us to generate an organized large-scale magnetic fields with well-synchronized hemispheric polarity reversal. Because the vast majority of these simulations do not include a modelization of the Sun's surface layer, the generation of their dynamo is thus solely due to the action of the turbulent electromotive force (EMF) in conjunction with differential rotation. From this EMF, different methods can be used to extract the turbulent transport coefficients. In particular, various authors found a full 9 component alpha-tensor (first order coefficients) where all the components are of the same order of magnitude. This finding calls into question the alphaOmega approximation made by the vast majority of mean field dynamo models. We generalized a first order (alpha-tensor) singular value decomposition (SVD) analysis procedure to extract the 18 additional components of the second order tensor (beta-tensor). The alpha and beta tensors obtained by this new procedure as applied to the EULAG-MHD outputs, are similar to the equivalent alpha and beta tensors obtained using the second order correlation approximation (SOCA). By introducing the first order turbulent transport coefficients in a mean field dynamo model, we study the magnetic solutions where double dynamo modes were observed. This analysis allows us to compare the mean field dynamo solutions produced with the EMF, as extracted from EULAG-MHD, with the real magnetic output of EULAG-MHD. This proof of concept demonstrated that the quasi-biennal oscillation observed in the surface toroidal magnetic field in EULAG-MHD can be produced by the inductive action of a secondary dynamo. A similar quasi-biennal oscillation signal is also observed in multiple proxies of the solar activity whose origin is still not confirmed. Although the physical set of properties under which the two numerical models described above operate are different from the Sun, the fact that both models can reproduce a secondary dynamo shows us that the inductive action of the EMF can easily produce two dynamos. Finally, in order to study epochs of grand minima that still cannot be reproduced in global 3D-MHD simulations of convection, we added a magnetic feedback on the mean azimutal flow in our kinematic mean field model. This non-kinematic alpha2Omega model was able to reproduce the tendency of the Prandtl number (Pm) to control the ratio of the modulation period. More specifically, we found an inverse relation between Pm and the ratio of the main magnetic cycle period to the grand minima occurrence period. Moreover, by analyzing a simulation of a length of 50,000 years, where aperiodic periods of grand minima and maxima are observed, we found a waiting time distribution (WTD) of the grand minima close to an exponential, a characteristic also observed in the reconstruction of the solar activity. Finally, the level of fluctuation in the surface differential rotation associated with epochs of grand minima is ~1%. This level of fluctuation was also observed in historical reconstructions of the surface differential rotation during the Maunder minimum.