Coronal structures as tracers of sub-surface processes

The solar corona – one of the most spectacular celestial shows and yet one of the most challenging puzzles – exhibits a spectrum of structures related to both the quiet Sun and active regions. In spite of dramatic differences in appearance and physical processes, all these structures share a common origin: they are all related to the solar magnetic field. The origin of the field is beneath the turbulent convection zone, where the magnetic field is not a master but a slave, and one can wonder how much the coronal magnetic field “remembers” its dynamo origin. Surprisingly, it does. We will describe several observational phenolmena that indicate a close relationship between coronal and sub-photospheric processes.     

Mechanism of cyclically polarity reversing solar magnetic cycle as a cosmic dynamo

We briefly describe historical development of the concept of solar dynamo mechanism that generates electric current and magnetic field by plasma flows inside the solar convection zone. The dynamo is the driver of the cyclically polarity reversing solar magnetic cycle. The reversal process can easily and visually be understood in terms of magnetic field line stretching and twisting and folding in three-dimensional space by plasma flows of differential rotation and global convection under influence of Coriolis force. This process gives rise to formation of a series of huge magnetic flux tubes that propagate along iso-rotation surfaces inside the convection zone. Each of these flux tubes produces one solar cycle. We discuss general characteristics of any plasma flows that can generate magnetic field and reverse the polarity of the magnetic field in a rotating body in the Universe. We also mention a list of problems which are currently being disputed concerning the solar dynamo mechanism together with observational evidences that are to be constraints as well as verifications of any solar cycle dynamo theories of short and long term behaviors of the Sun, particularly time variations of its magnetic field, plasma flows, and luminosity.     

Quenching of Meridional Circulation in Flux Transport Dynamo Models

Guided by the recent observational result that the meridional circulation of the Sun becomes weaker at the time of the sunspot maximum, we have included a parametric quenching of the meridional circulation in solar dynamo models such that the meridional circulation becomes weaker when the magnetic field at the base of the convection zone is stronger. We find that a flux transport solar dynamo tends to become unstable on including this quenching of meridional circulation if the diffusivity in the convection zone is less than about 2×10¹¹ cm² s⁻¹. The quenching of α, however, has a stabilizing effect and it is possible to stabilize a dynamo with low diffusivity with sufficiently strong α-quenching. For dynamo models with high diffusivity, the quenching of meridional circulation does not produce a large effect and the dynamo remains stable. We present a solar-like solution from a dynamo model with diffusivity 2.8×10¹² cm² s⁻¹ in which the quenching of meridional circulation makes the meridional circulation vary periodically with solar cycle as observed and does not have any other significant effect on the dynamo.     

On the interaction between differential rotation and magnetic fields in the Sun

We have performed 3-D numerical simulations of compressible convection under the influence of rotation and magnetic fields in spherical shells. They aim at understanding the subtle coupling between convection, rotation and magnetic fields in the solar convection zone. We show that as the magnetic Reynolds number is increased in the simulations, the magnetic energy saturates via nonlinear dynamo action, to a value smaller but comparable to the kinetic energy contained in the shell, leading to increasingly strong Maxwell stresses that tend to weaken the differential rotation driven by the convection. These simulations also indicate that the mean toroidal and poloidal magnetic fields are small compared to their fluctuating counterparts, most of the magnetic energy being contained in the non-axisymmetric fields. The intermittent nature of the magnetic fields generated by such a turbulent convective dynamo confirms that in the Sun the large-scale ordered dynamo responsible for the 22-year cycle of activity can hardly be located in the solar convective envelope.     

Fluid Dynamics and MHD of the Solar Convection Zone and Tachocline: Current Understanding and Unsolved Problems – (Invited Review)

We review recent progress and define unanswered scientific questions in five related topics: granulation- to supergranulation-scale convection and magnetic structures; global convection and circulation; the rise of magnetic flux tubes to the photosphere, and their injection into the base of the convection zone; tachocline fluid dynamics and MHD; and the solar dynamo. We close with a set of observational `targets' for helioseismologists to aim for.     

A SOLAR α - ω DYNAMO STUDY AND ITS TRANSITION TO CHAOS

In this contribution we present a nonlinear dynamo model, described by an infinite dimensional system of differential equations, whose solutions depend on the essential parameter D, the dynamo number. The solutions and the bifurcation points of the system are determined with the help of a new developed computer code. We show that, depending on D, stationary, oscillatory and chaotic solutions, which are characterized by Lyapunov exponents, result. We find that the solar dynamo may operate either in the chaotic or in the stable limit cycle domain, depending on the characteristic value of the dynamo number or the motion of the convection zone.     

Kinematic Numerical Simulations of the Solar Dynamo: Dependence on α and Ω Values

We investigated the generation of dynamo waves in the solar convection zone through a numerical simulation. We integrated the axisymmetric α–Ω kinematic dynamo equations in a spherical geometry, where the α- and Ω-profiles depend on the spatial coordinates. The model results show that the fundamental parameter that determines the behavior of the system is the product between the characteristic intensities of the α and Ω contributions. In particular, we found three different regimes in which the system exhibits different behaviors: a regime without a dynamo effect, one with an exponential amplification of the magnetic field, and one with dynamo waves.     

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)     

Electric current helicity in the solar atmosphere

In the theories of solar magnetism, kinetic and magnetic helicities, which arise as a consequence of the rotation of the Sun, play a key role. The dynamo for the main field is assumed to operate in the convection zone. The solar rotation also may be the ultimate cause for the generation of dc electric currents in the atmosphere, needed as the energy source for flares. Then in the atmosphere the electric current helicity, H _C = B · × B, which is a pseudo-scalar quantity, should be antisymmetric about the equatorial plane. An inspection of 16 active regions, for which H _C has been estimated by using extrapolation of measured photospheric magnetic fields, leads to the result that the electric current helicity is predominantly negative in the northern and positive in the southern hemisphere. The helicity of the large-scale currents generated according to standard dynamo theory by the alpha effect in the convection zone is just opposite in sign. Current generation due to rotational motions of sunspots and other magnetic elements in accordance with the global differential rotation, i.e., counter-clockwise in the northern and clockwise in the southern hemisphere, however, can explain the rule found. Also in some alternative dynamo models for the global field, in which the dynamo operates at the base of the convection zone, the large-scale current helicity generated by the alpha effect has the sign needed.     

On the Connection Between Mean Field Dynamo Theory and Flux Tubes

Mean field dynamo theory deals with various mean quantities and does not directly throw any light on the question of existence of flux tubes. We can, however, draw important conclusions about flux tubes in the interior of the Sun by combining additional arguments with the insights gained from solar dynamo solutions. The polar magnetic field of the Sun is of order 10 G, whereas the toroidal magnetic field at the bottom of the convection zone has been estimated to be 100000 G. Simple order-of-magnitude estimates show that the shear in the tachocline is not sufficient to stretch a 10 G mean radial field into a 100000 G mean toroidal field. We argue that the polar field of the Sun must get concentrated into intermittent flux tubes before it is advected to the tachocline. We estimate the strengths and filling factors of these flux tubes. Stretching by shear in the tachocline is then expected to produce a highly intermittent magnetic configuration at the bottom of the convection zone. The meridional flow at the bottom of the convection zone should be able to carry this intermittent magnetic field equatorward, as suggested recently by Nandy and Choudhuri (2002). When a flux tube from the bottom of the convection zone rises to a region of pre-existing poloidal field at the surface, we point out that it picks up a twist in accordance with the observations of current helicities at the solar surface.     

Helioseismology and the solar interior dynamics

The inversion of helioseismic modes leads to the sound velocity inside the Sun with a precision of about 0.1 per cent. Comparisoons of solar models with the “seismic sun” represent powerful tools to test the physics: depth of the convection zone, equation of state, opacities, element diffusion processes and mixing inside the radiative zone. We now have evidence that microscopic diffusion (element segregation) does occur below the convection zone, leading to a mild helium depletion in the solar outer layers. Meanwhile this process must be slowed down by some macroscopic effect, presumably rotation-induced mixing. The same mixing is also responsible for the observed lithium depletion. On the other hand, the observations of beryllium and helium 3 impose specific constraints on the depth of this mildly mixed zone. Helioseismology also gives information on the internal solar rotation: while differential rotation exists in the convection zone, solid rotation prevails in the radiative zone, and the transition layer (the so-called “tachocline”) is very small. These effects are discussed, together with the astrophysical constraints on the solar neutrino fluxes.     

Dynamical behavior of strong magnetic fields in the solar convection zone

Magnetic buoyancy is thought to play an important role in the dynamical behavior of the Sun's magnetic field in the convection zone. Magnetic buoyancy is commonly thought to cause inescapable rapid loss of toroidal flux from much of the convection zone, thereby suppressing effective operation of a solar dynamo. This paper re-examines the detailed character of magnetic buoyancy, especially as it is influenced by the magnetic field's effect on heat transport and temperature gradients in the convection zone. It is suggested that suppression of convective heat transport across strong magnetic flux tubes can alter the temperature within the tubes and can subdue, or even reverse, the effect of magnetic buoyancy.     

Invisible dynamo in mean-field models

The inverse problem in a spherical shell to find the two-dimensional spatial distributions of the α-effect and differential rotation in a mean-field dynamo model has been solved. The derived distributions lead to the generation of a magnetic field concentrated inside the convection zone. The magnetic field is shown to have no time to rise from the region of maximum generation located in the lower layers to the surface in the polarity reversal time due to magnetic diffusion. The ratio of the maximum magnetic energy in the convection zone to its value at the outer boundary reaches two orders of magnitude or more. This result is important in interpreting the observed stellar and planetary magnetic fields. The proposed method of solving the inverse nonlinear dynamo problem is easily adapted for a wide class of mathematical-physics problems.     

The Kinematic Theory of Solar Dynamo

Generation of the Sun‘s magnetic fields by self-inductive processes in the solar electrically conducting interior, the solar dynamo theory, is a fundamentally important subject in astrophysics. The kinematic dynamo theory concerns how the magnetic fields are produced by kinematically possible flows without being constrained by the dynamic equation. We review a number of basic aspects of the kinematic dynamo theory, including the magnetohydrodynamic approximation for the dynamo equation, the impossibility of dynamo action with the solar differential rotation, the Cowling‘s anti-dynamo theorem in the solar context, the turbulent alpha effect and recently constructed three-dimensional interface dynamos controlled by the solar tachocline at the base of the convection zone.     

Differential rotation and meridional flow of Arcturus

The spectroscopic variability of Arcturus hints at cyclic activity cycle and differential rotation. This could provide a test of current theoretical models of solar and stellar dynamos. To examine the applicability of current models of the flux transport dynamo to Arcturus, we compute a mean‐field model for its internal rotation, meridional flow, and convective heat transport in the convective envelope. We then compare the conditions for dynamo action with those on the Sun. We find solar‐type surface rotation with about 1/10th of the shear found on the solar surface. The rotation rate increases monotonically with depth at all latitudes throughout the whole convection zone. In the lower part of the convection zone the horizontal shear vanishes and there is a strong radial gradient. The surface meridional flow has maximum speed of 170 m/s and is directed towards the equator at high and towards the poles at low latitudes. Turbulent magnetic diffusivity is of the order 10¹⁵–10¹⁶ cm²/s. The conditions on Arcturus are not favorable for a circulation‐dominated dynamo (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A two-layer αω-dynamo model with dynamic feedback on the ω-effect

In an attempt to produce a simple representation of an interface dynamo, I examine a dynamo model composed of two one-dimensional (radially averaged) pseudo-spherical layers, one in the convection zone and possessing an α-effect, and the other in the tachocline and possessing an ω-effect. The two layers communicate by means of an analogue of Newton's law of cooling, and a dynamical back-reaction of the magnetic field on ω is provided. Extensive bifurcation diagrams are calculated for three separate values of η, the ratio of magnetic diffusivities of the two layers. I find recognizable similarities to, but also dramatic differences from, the comparable one-layer model examined by Roald &38; Thomas. In particular, the solar-like dynamo mode found previously is no longer stable in the two-layer version; in its place there is a sequence of periodic, quasi-periodic and chaotic modes probably created in a homoclinic bifurcation. These differences are important enough to provide support for the view that the solar dynamo cannot be meaningfully modelled in one dimension.     

Variations of helioseismic parameters due to magnetic field generated by a flux transport model

The change of sound speed has been found at the base of the convection during the solar cycles,which can be used to constrain the solar internal magnetic field.We aim to check whether the magnetic field generated by the solar dynamo can lead to the cyclic variation of the sound speed detected through helioseismology.The basic configuration of magnetic field in the solar interior was obtained by using a Babcock-Leighton(BL) type flux transport dynamo.We reconstructed one-dimensional solar models by assimilating magnetic field generated by an established dynamo and examined their influences on the structural variables.The results show that magnetic field generated by the dynamo is able to cause noticeable change of the sound speed profile at the base of the convective zone during a solar cycle.Detailed features of this theoretical prediction are also similar to those of the helioseismic results in solar cycle 23 by adjusting the free parameters of the dynamo model.     

The Alpha Effect and the Observed Twist and Current Helicity of Solar Magnetic Fields

We present a straightforward comparison of model calculations for the α-effect, helicities, and magnetic field line twist in the solar convection zone with magnetic field observations at atmospheric levels. The model calculations are carried out in a mixing-length approximation for the turbulence with a profile of the solar internal rotation rate obtained from helioseismic inversions. The magnetic field data consist of photospheric vector magnetograms of 422 active regions for which spatially-averaged values of the force-free twist parameter and of the current helicity density are calculated, which are then used to determine latitudinal profiles of these quantities. The comparison of the model calculations with the observations suggests that the observed twist and helicity are generated in the bulk of the convection zone, rather than in a layer close to the bottom. This supports two-layer dynamo models where the large-scale toroidal field is generated by differential rotation in a thin layer at the bottom while the α-effect is operating in the bulk of the convection zone. Our previous observational finding was that the moduli of the twist factor and of the current helicity density increase rather steeply from zero at the equator towards higher latitudes and attain a certain saturation at about 12 – 15^∘. In our dynamo model with algebraic nonlinearity, the increase continues, however, to higher latitudes and is more gradual. This could be due to the neglect of the coupling between small-scale and large-scale current and magnetic helicities and of the latitudinal drift of the activity belts in the model.     

A new look at the relationship between activity, dynamo number and Rossby number in late-type stars

The correlation between stellar activity, as measured by the indicator Δ R _HK, and the Rossby number Ro in late-type stars is revisited in light of recent developments in solar dynamo theory. Different stellar interior models, based on both mixing-length theory and the full spectrum of turbulence, are used in order to see to what extent the correlation of activity with Rossby number is model dependent, or otherwise can be considered universal. Although we find some modest model dependence, we find that the correlation of activity with Rossby number is significantly better than with rotation period alone for all the models we consider. Dynamo theory suggests that activity should scale with the dynamo number. A current model of the solar dynamo, the so-called interface dynamo, proposes that the amplification of the toroidal magnetic field by differential rotation (the ω -effect) and the production of the poloidal magnetic field from toroidal by helical turbulence (the α -effect) take place in different, adjacent layers near the base of the convection zone. A new scale analysis based on the interface dynamo shows that the appropriate dynamo number does not depend on the Rossby number alone, but also depends on an additional dimensionless factor related to the differential rotation. This leads to a new interpretation of the correlation between activity and Rossby number, which in turn leads to some conclusions about the magnitude of differential rotation in the dynamo layers of late-type main-sequence stars.