Theoretical Models of Sunspot Structure and Dynamics

Recent progress in theoretical modeling of a sunspot is reviewed. The observed properties of umbral dots are well reproduced by realistic simulations of magnetoconvection in a vertical, monolithic magnetic field. To understand the penumbra, it is useful to distinguish between the inner penumbra, dominated by bright filaments containing slender dark cores, and the outer penumbra, made up of dark and bright filaments of comparable width with corresponding magnetic fields differing in inclination by some 30 degrees and strong Evershed flows in the dark filaments along nearly horizontal or downward-plunging magnetic fields. The role of magnetic flux pumping in submerging magnetic flux in the outer penumbra is examined through numerical experiments, and different geometric models of the penumbral magnetic field are discussed in the light of high-resolution observations. Recent, realistic numerical MHD simulations of an entire sunspot have succeeded in reproducing the salient features of the convective pattern in the umbra and the inner penumbra. The siphon-flow mechanism still provides the best explanation of the Evershed flow, particularly in the outer penumbra where it often consists of cool, supersonic downflows.Comment: To appear in "Magnetic Coupling between the Interior and the Atmosphere of the Sun", eds. S.S. Hasan and R.J. Rutten, Astrophysics and Space Science Proceedings, Springer-Verlag, Heidelberg, Berlin, 200

Recent Evidence for Convection in Sunspot Penumbrae

Whereas penumbral models during the last 15 years have been successful in explaining Evershed flows and magnetic field inclination variations in terms of flux tubes, the lack of contact between these models and a convective process needed to explain the penumbral radiative heat flux has been disturbing. We report on recent observational and theoretical evidence that challenge flux tube interpretations and conclude that the origin of penumbral filamentary structure is overturning convection.Comment: Accepted for publication in Space Science Review

Magnetic Coupling in the Quiet Solar Atmosphere

Three kinds of magnetic couplings in the quiet solar atmosphere are highlighted and discussed, all fundamentally connected to the Lorentz force. First the coupling of the convecting and overshooting fluid in the surface layers of the Sun with the magnetic field. Here, the plasma motion provides the dominant force, which shapes the magnetic field and drives the surface dynamo. Progress in the understanding of the horizontal magnetic field is summarized and discussed. Second, the coupling between acoustic waves and the magnetic field, in particular the phenomenon of wave conversion and wave refraction. It is described how measurements of wave travel times in the atmosphere can provide information about the topography of the wave conversion zone, i.e., the surface of equal Alfv\'en and sound speed. In quiet regions, this surface separates a highly dynamic magnetic field with fast moving magnetosonic waves and shocks around and above it from the more slowly evolving field of high-beta plasma below it. Third, the magnetic field also couples to the radiation field, which leads to radiative flux channeling and increased anisotropy in the radiation field. It is shown how faculae can be understood in terms of this effect. The article starts with an introduction to the magnetic field of the quiet Sun in the light of new results from the Hinode space observatory and with a brief survey of measurements of the turbulent magnetic field with the help of the Hanle effect.Comment: To appear in "Magnetic Coupling between the Interior and the Atmosphere of the Sun", eds. S.S. Hasan and R.J. Rutten, Astrophysics and Space Science Proceedings, Springer-Verlag, Heidelberg, Berlin, 200

Modeling the Subsurface Structure of Sunspots

While sunspots are easily observed at the solar surface, determining their subsurface structure is not trivial. There are two main hypotheses for the subsurface structure of sunspots: the monolithic model and the cluster model. Local helioseismology is the only means by which we can investigate subphotospheric structure. However, as current linear inversion techniques do not yet allow helioseismology to probe the internal structure with sufficient confidence to distinguish between the monolith and cluster models, the development of physically realistic sunspot models are a priority for helioseismologists. This is because they are not only important indicators of the variety of physical effects that may influence helioseismic inferences in active regions, but they also enable detailed assessments of the validity of helioseismic interpretations through numerical forward modeling. In this paper, we provide a critical review of the existing sunspot models and an overview of numerical methods employed to model wave propagation through model sunspots. We then carry out an helioseismic analysis of the sunspot in Active Region 9787 and address the serious inconsistencies uncovered by \citeauthor{gizonetal2009}~(\citeyear{gizonetal2009,gizonetal2009a}). We find that this sunspot is most probably associated with a shallow, positive wave-speed perturbation (unlike the traditional two-layer model) and that travel-time measurements are consistent with a horizontal outflow in the surrounding moat.Comment: 73 pages, 19 figures, accepted by Solar Physic

Photospheric processes and magnetic flux tubes

In the first part of these lecture notes, new high-resolution observations of small-scale magnetic flux concentrations are presented and compared to results from new three-dimensional magnetohydrodynamic simulations. Special attention is paid to the physics of faculae and to new three-dimensional radiation magnetohydrodynamic simulations of the integral layers from the top of the convection zone to the mid-chromosphere. The second part is devoted to a few basic properties of magnetic flux tubes, which can be considered to be an abstraction of the more complicated flux concentrations known from observations and numerical simulations. We treat electrical current sheets, the mechanical equilibrium condition at magnetic interfaces, the equations for constructing a magnetohydrostatic flux tube embedded in a gravitationally stratified atmosphere, the condition of radiative equilibrium, and the condition for interchange stability.Comment: 48 pages, 33 figures, Lecture notes, download pdf-file of superior quality figures from http://www3.kis.uni-freiburg.de/~steiner/kodai.pd

Anisotropic Diffusion in Mesh-Free Numerical Magnetohydrodynamics

We extend recently-developed mesh-free Lagrangian methods for numerical magnetohydrodynamics (MHD) to arbitrary anisotropic diffusion equations, including: passive scalar diffusion, Spitzer-Braginskii conduction and viscosity, cosmic ray diffusion/streaming, anisotropic radiation transport, non-ideal MHD (Ohmic resistivity, ambipolar diffusion, the Hall effect), and turbulent 'eddy diffusion.' We study these as implemented in the code GIZMO for both new meshless finite-volume Godunov schemes (MFM/MFV). We show the MFM/MFV methods are accurate and stable even with noisy fields and irregular particle arrangements, and recover the correct behavior even in arbitrarily anisotropic cases. They are competitive with state-of-the-art AMR/moving-mesh methods, and can correctly treat anisotropic diffusion-driven instabilities (e.g. the MTI and HBI, Hall MRI). We also develop a new scheme for stabilizing anisotropic tensor-valued fluxes with high-order gradient estimators and non-linear flux limiters, which is trivially generalized to AMR/moving-mesh codes. We also present applications of some of these improvements for SPH, in the form of a new integral-Godunov SPH formulation that adopts a moving-least squares gradient estimator and introduces a flux-limited Riemann problem between particles.Comment: 18 pages, 17 figures, MNRAS (replaced with accepted version). A public version of the GIZMO code is available at http://www.tapir.caltech.edu/~phopkins/Site/GIZMO.htm

The physics of twisted magnetic tubes rising in a stratified medium: two dimensional results

The physics of a twisted magnetic flux tube rising in a stratified medium is studied using a numerical MHD code. The problem considered is fully compressible (no Boussinesq approximation), includes ohmic resistivity, and is two dimensional, i.e., there is no variation of the variables in the direction of the tube axis. We study a high plasma beta case with small ratio of radius to external pressure scaleheight. The results obtained can therefore be of relevance to understand the transport of magnetic flux across the solar convection zone.Comment: To be published in ApJ, Vol. 492, Jan 10th, 1998; 25 pages, 16 figures. NEW VERSION: THE PREVIOUS ONE DIDN'T PRINT CORRECTLY. The style file overrulehere.sty is include