Photospheric processes and magnetic flux tubes

New high-resolution observations reveal that small-scale magnetic flux concentrations have a delicate substructure on a spatial scale of 0.1′′. Their basic structure can be interpreted in terms of a magnetic flux sheet or tube that vertically extends through the ambient weak-field or fieldfree atmosphere with which it is in mechanical equilibrium. A more refined interpretation comes from new three-dimensional magnetohydrodynamic simulations that are capable of reproducing the corrugated shape of magnetic flux concentrations and their signature in the visible continuum. Faculae are another manifestation of small-scale magnetic flux concentrations. It is shown that the characteristic asymmetric shape of the contrast profile of faculae is an effect of radiative transfer across the rarefied atmosphere of the magnetic flux concentration. Also discussed are three-dimensional radiation magnetohydrodynamic simulations of the integral layers from the top of the convection zone to the mid-chromosphere. They show a highly dynamic chromospheric magnetic field, marked by rapidly moving filaments of stronger than average magnetic field that form in the compression zone downstream and along propagating shock fronts. The simulations confirm the picture of flux concentrations that strongly expand through the photosphere into a more homogeneous, space filling chromospheric field. Future directions in the simulation of small-scale magnetic fields are indicated with a few examples from recent reports. The second part of these lecture notes is devoted to a few basic properties of magnetic flux tubes that can be considered to be an abstraction of the more complicated flux concentrations known from observations and numerical simulations. By analytical means we will find that an electrical current flows in a sheet at the surface of a flux-tube for which location we also derive the mechanical equilibrium condition. The equations for constructing a magnetohydrostatic flux tube embedded in a gravitationally stratified atmosphere are derived. It is shown that the expansion of a flux tube with height sensibly depends on the difference in the thermal structure between the atmosphere of the flux tube and the surrounding atmosphere. Furthermore, we will find that radiative equilibrium produces a smaller temperature gradient within the flux tube compared to that in the surrounding atmosphere. The condition for interchange stability is derived and it is shown that small-scale magnetic flux concentrations are liable to the interchange instability.