A dynamo model for axisymmetric and non-axisymmetric solar magnetic fields

More and more observations are showing a relatively weak, but persistent, non-axisymmetric magnetic field co-existing with the dominant axisymmetric field on the Sun. Its existence indicates that the non-axisymmetric magnetic field plays an important role in the origin of solar activity. A linear non-axisymmetric  α²– Ω  dynamo model is derived to explore the characteristics of the axisymmetric  ( m = 0)  and the first non-axisymmetric  ( m = 1)  modes and to provide a theoretical basis with which to explain the 'active longitude', 'flip-flop' and other non-axisymmetric phenomena. The model consists of an updated solar internal differential rotation, a turbulent diffusivity varying with depth, and an α-effect working at the tachocline in a rotating spherical system. The difference between the  α²–Ω  and the  α–Ω  models and the conditions that favour the non-axisymmetric modes under solar-like parameters are also presented.     

Helicity of solar active regions from a dynamo model

We calculate helicities of solar active regions based on the idea that poloidal flux lines get wrapped around a toroidal flux tube rising through the convection zone, thereby giving rise to the helicity. We use our solar dynamo model based on the Babcock-Leighton α-effect to study how helicity varies with latitude and time.     

The solar dynamo

The solar dynamo continues to pose a challenge to observers and theoreticians. Observations of the solar surface reveal a magnetic field with a complex, hierarchical structure consisting of widely different scales. Systematic features such as the solar cycle, the butterfly diagram, and Hale's polarity laws point to the existence of a deep-rooted large-scale magnetic field. At the other end of the scale are magnetic elements and small-scale mixed-polarity magnetic fields. In order to explain these phenomena, dynamo theory provides all the necessary ingredients including the effect, magnetic field amplification by differential rotation, magnetic pumping, turbulent diffusion, magnetic buoyancy, flux storage, stochastic variations and nonlinear dynamics. Due to advances in helioseismology, observations of stellar magnetic fields and computer capabilities, significant progress has been made in our understanding of these and other aspects such as the role of the tachocline, convective plumes and magnetic helicity conservation. However, remaining uncertainties about the nature of the deep-seated toroidal magnetic field and the effect, and the forbidding range of length scales of the magnetic field and the flow have thus far prevented the formulation of a coherent model for the solar dynamo. A preliminary evaluation of the various dynamo models that have been proposed seems to favor a buoyancy-driven or distributed scenario. The viewpoint proposed here is that progress in understanding the solar dynamo and explaining the observations can be achieved only through a combination of approaches including local numerical experiments and global mean-field modeling.Received: 5 May 2003, Published online: 15 July 2003     

On the possibility of a bimodal solar dynamo

A simple way to couple an interface dynamo model to a fast tachocline model is presented, under the assumption that the dynamo saturation is due to a quadratic process and that the effect of finite shear layer thickness on the dynamo wave frequency is analogous to the effect of finite water depth on surface gravity waves. The model contains one free parameter which is fixed by the requirement that a solution should reproduce the helioseismically determined thickness of the tachocline. In this case it is found that, in addition to this solution, another steady solution exists, characterized by a four times thicker tachocline and 4–5 times weaker magnetic fields. It is tempting to relate the existence of this second solution to the occurrence of grand minima in solar activity. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On global solar dynamo simulations

Global dynamo simulations solving the equations of magnetohydrodynamics (MHD) have been a tool of astrophysicists who try to understand the magnetism of the Sun for several decades now. During recent years many fundamental issues in dynamo theory have been studied in detail by means of local numerical simulations that simplify the problem and allow the study of physical effects in isolation. Global simulations, however, continue to suffer from the age‐old problem of too low spatial resolution, leading to much lower Reynolds numbers and scale separation than in the Sun. Reproducing the internal rotation of the Sun, which plays a crucial role in the dynamo process, has also turned out to be a very difficult problem. In the present paper the current status of global dynamo simulations of the Sun is reviewed. Emphasis is put on efforts to understand how the large‐scale magnetic fields, i.e. whose length scale is greater than the scale of turbulence, are generated in the Sun. Some lessons from mean‐field theory and local simulations are reviewed and their possible implications to the global models are discussed. Possible remedies to some current issues of solar simulations are put forward (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flux-dominated solar dynamo model with a thin shear layer

Flux-dominated solar dynamo models have demonstrated to reproduce the main features of the large scale solar magnetic cycle, however the use of a solar like differential rotation profile implies in the the formation of strong toroidal magnetic fields at high latitudes where they are not observed. In this work, we invoke the hypothesis of a thin-width tachocline in order to confine the high-latitude toroidal magnetic fields to a small area below the overshoot layer, thus avoiding its influence on a Babcock-Leighton type dynamo process. Our results favor a dynamo operating inside the convection zone with a tachocline that essentially works as a storage region when it coincides with the overshoot layer. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The role of magnetic buoyancy in a Babcock-Leighton type solar dynamo

We study the effects of incorporating magnetic buoyancy in a model of the solar dynamo—which draws inspiration from the Babcock-Leighton idea of surface processes generating the poloidal field. We present our main results here.     

Alleviation of catastrophic quenching in solar dynamo model with nonlocal alpha‐effect

The nonlocal alpha‐effect of Babcock‐Leighton type is not prone to the catastrophic quenching due to conservation of magnetic helicity. This is shown with a dynamo model, which jointly applies the nonlocal alpha‐effect, the diamagnetic pumping, and dynamical equation for the magnetic alpha‐effect. The same model shows catastrophic quenching when the alpha‐effect is changed to its local formulation. The nonlocal model shows a preferred excitation of magnetic fields of dipolar symmetry, which oscillate with a period of about ten years and have a toroidal‐to‐polar fields ratio of about a thousand (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar activity forecast with a dynamo model

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. This suggests that the Babcock–Leighton mechanism of poloidal field generation from decaying sunspots involves randomness, whereas the other aspects of the dynamo process must be reasonably ordered and deterministic. Only if the magnetic diffusivity within the convection zone is assumed to be high (of order  10¹² cm² s⁻¹  ), we can explain the correlation between the polar field at 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. To model actual solar cycles with a dynamo model having such high diffusivity, we have to feed the observational data of the poloidal field at the minimum into the theoretical model. We develop a method of doing this in a systematic way. Our model predicts that cycle 24 will be a very weak cycle. Hemispheric asymmetry of solar activity is also calculated with our model and compared with observational data.     

Evidence for a long-term variation of the dynamo action responsible for the solar magnetic cycle

By using the sunspot time series as a proxy, we have made a detailed analysis of the mean solar magnetic field over the last two and half centuries, by means of a reconstruction of its phase space. We find evidence of a long-term trend variation of some of the solar physical processes (over a few decades) that might be responsible for the apparent erratic behaviour of the solar magnetic cycle. The analysis is done by means of a careful study of the axisymmetric dynamo model equations, where we show that the temporal counterpart of the magnetic field can be described by a self-regulated two-dimensional dynamic system, usually known as a Van der Pol–Duffing oscillator. Our results suggest that during the last two and half centuries, the velocity of the meridional flow, v _p, and the efficiency of the α mechanism responsible for the conversion of toroidal magnetic field into poloidal magnetic field might have suffered variations that can explain the observed variability in the solar cycle.     

Non-axisymmetric solar magnetic fields

A simple non-linear, non-axisymmetric mean field dynamo model is applied to a differentially rotating spherical shell. Two approximations are used for the angular velocity, to represent what is now believed to be the solar rotation law. In each case, stable solutions are found which possess a small non-axisymmetric field component. Although the model has a number of obvious shortcomings, it may be relevant to the problem of the solar active longitudes.     

Mode enslavement in a two-layer stellar dynamo

We consider a conventional stellar  α²ω  -dynamo with dynamo generators localized in two spherical shells separated by a passive layer. The signs of the α-effect as well as rotational shear in the dynamo active layers can be chosen to give dynamo waves that propagate in opposite directions (poleward and equatorward) if the layers are considered separately in the framework of the Parker migratory dynamo. In a sequence of numerical experiments we show that the variety of dynamo-generated magnetic configurations in the system under discussion is quite rich. We identify the possibility of almost independent dynamo waves existing in the two layers as well as enslavement of one layer by the other, and of activity waves generated by a joint action of the two layers. We suggest some qualitative explanations of the behaviour and discuss also the limited nature of these explanations. This variety of phenomena suggests previously underexploited freedoms to understand how predictions of dynamo theory may accommodate the observed solar and stellar activity phenomenology.     

The direction of propagation of the solar dynamo wave

We propose a solution to one of the oldest problems in the solar-dynamo theory: explaining the equatorward drift of magnetic activity in the solar cycle. The well-known suggestion that the dynamo waves propagate along the surfaces of constant angular velocity is shown to be restricted to an isotropic medium. Allowance for the rotation-induced anisotropy in turbulent diffusion leads to an equatorward deviation of the wave phase velocity from the isorotational surface. Estimates for the dynamo waves are illustrated with two-dimensional numerical models in a spherical geometry. The model with anisotropic diffusion also shows an equatorward drift of the toroidal magnetic field when the rotation is radially uniform.     

Mean-field dynamo in partially ionized plasmas – I

There are several astrophysical situations where one needs to study the dynamics of magnetic flux in partially ionized turbulent plasmas. In a partially ionized plasma, the magnetic induction is subjected to the ambipolar diffusion and the Hall effect in addition to the usual resistive dissipation. In this paper, we initiate the study of the kinematic dynamo in a partially ionized turbulent plasma. The Hall effect arises from the treatment of the electrons and the ions as two separate fluids and the ambipolar diffusion due to the inclusion of neutrals as the third fluid. It is shown that these non-ideal effects modify the so-called α effect and the turbulent diffusion coefficient β in a rather substantial way. The Hall effect may enhance or quench the dynamo action altogether. The ambipolar diffusion brings in an α which depends on the mean magnetic field. The new correlations embodying the coupling of the charged fluids and the neutral fluid appear in a decisive manner. The turbulence is necessarily magnetohydrodynamic with new spatial and time-scales. The nature of the new correlations is demonstrated by taking the Alfvénic turbulence as an example.     

The heating of the solar corona

The heating of the solar corona has been a fundamental astrophysical issue for over sixty years. Over the last decade in particular, space-based solar observatories (Yohkoh, SOHO and TRACE) have revealed the complex and often subtle magnetic-field and plasma interactions throughout the solar atmosphere in unprecedented detail. It is now established that any energy release mechanism is magnetic in origin - the challenge posed is to determine what specific heat input is dominating in a given coronal feature throughout the solar cycle. This review outlines a range of possible magnetohydrodynamic (MHD) coronal heating theories, including MHD wave dissipation and MHD reconnection as well as the accumulating observational evidence for quasi-periodic oscillations and small-scale energy bursts occurring in the corona. Also, we describe current attempts to interpret plasma temperature, density and velocity diagnostics in the light of specific localised energy release. The progress in these investigations expected from future solar missions (Solar-B, STEREO, SDO and Solar Orbiter) is also assessed.Received: 6 February 2003, Published online: 14 November 2003 Correspondence to: R. W. Walsh