The solar chromosphere is identified with the atmosphere inside magnetic flux tubes. In the quiet sun, the layers of the low and middle chromosphere are heated by compressive waves with periods mainly between 2 min and 4 min. These long-period waves probably supply all the energy required for the heating of the quiet solar chromosphere.

As a prelude to discussing the interaction of magnetic fields with convection, we first review some general properties of convection in a stratified medium. Granulation, which is the surface manifestation of the major energy carrying convection scales, is a shallow phenomenon. Below the surface, the topology changes to one of filamentary cool downdrafts, immersed in a gently ascending isentropic background. The granular downflows merge into more widely separated downdrafts, on scales of mesogranulation and super-granulation.

The local topology and time evolution of the small scale, kilo Gauss, network and facular magnetic field elements are controlled by convection on the scale of granulation. The topology and time evolution of larger scale magnetic field concentrations are controlled by the hierarchical structure of the horizontal components of the large scale velocity field. In sunspots, the small scale magnetic field structure determines the energy balance, the systematic flows and the waves. Below the surface, the small scale structure of the magnetic field may change drastically, with little observable effect at the surface. We discuss results of some recent numerical simulations of sunspot magnetic fields, and some mechanisms that may be relevant in determining the topology of the sub-surface magnetic field. Finally, we discuss the role of active region magnetic fields in the global solar dynamo.

We have carried out detailed numerical simulations of solar granular convection cells of different horizontal dimension. The calculations account for the basic physics of compressible convection, including the ionization of H I, He I and He II, and H2 molecule formation as well as non-local, multi-dimensional radiative transfer (grey approximation in LTE). A more detailed description of the simulations has been given by Steffen and Muchmore (1988) and by Steffen et al. (1989).

The solar photosphere is the outer boundary of the cavity which defines the spectrum of the global p-modes. It is also the lowest layer that permits direct observation of the boiling mixture of motions which eventually create the complex patterns apparent in the chromosphere and corona. Studies of the dynamical behaviour of this layer are therefore of paramount importance for an overall understanding of the dynamic sun.

According to observational data solar magnetic fields have a pronounced filamentary structure. Theoretical investigations of plasmas containing structured magnetic fields, including the study of the properties of these structures and their interactions with associated gas flows, are of great importance for our understanding of the basic processes in the solar atmosphere, whose structure and dynamics are dominated by magnetic fields. In the present review theoretical models of thin magnetic fluxtubes and their behaviour in the ambient plasma are discussed.

The 5-minute oscillations are standing waves trapped in subphotospheric cavities, whose evanescent part affect the profiles of photospheric lines.

We examine the wave modes in a sunspot umbra. Assuming a stratification, based on a model atmosphere in a sunspot, the normal mode spectrum is determined. The modes are classified using a scheme based on a Helmholtz decomposition of the displacements into l(longitudinal) and t(transverse) components. In certain cases these can be related to the usual fast and slow MHD waves. We compute the theoretical eigenfrequencies and note the existence of umbral oscillations with periods in the range 2-3 min, which are interpreted as slow and mixed modes. The frequencies of the Alfvén waves are also calculated. It is suggested that these modes might also have been observed.

The influence of undamped linear longitudinal tube waves on Stokes V profiles is considered. A rough upper limit is set on the energy flux transported by such waves through the photosphere. It is found that this upper limit is larger than the flux in the quiet sun. However, due to the small filling factor of the magnetic elements, the total luminosity of flux tube waves is unlikely to be larger than that of acoustic waves when averaged over the whole sun. Therefore, probably both kinds of waves contribute to chromospheric heating. However, the derived upper limit does not rule out that flux tube waves can significantly enhance the chromospheric brightness in active regions and the supergranular network where the magnetic filling factor is large.

Siphon flows along arched, isolated magnetic flux tubes, connecting photospheric footpoints of opposite magnetic polarity, cause a significant increase in the magnetic field strength of the flux tube due to the decreased internal gas pressure associated with the flow (the Bernoulli effect). These siphon flows offer a possible mechanism for producing intense, inclined, small-scale magnetic structures in the solar photosphere.

The structure of the magnetic field, proper motions of sunspots and electric currents have been studied and related to the evolution of sunspot groups. Further the height variations of the magnetic fluxes and electric currents in active regions have been explored.

Velocities of electrons, ions and neutrals are computed in the three-fluid approximation for an axisymmetrical magnetic field. By prescribing a radial dependence of the velocity of neutrals in agreement with a downflow, the radial dependence of the magnetic field energy density is derived for a given set of values of the magnetic field at the central and external boundaries. Flux-tube cooling by advection of ionization energy is found to be significant. Vortices in the low photosphere could produce significant electric power and DC current intensity along the coronal magnetic lines of forces. The velocities of neutrals, the size and the number of flux-tubes required to power flares in plage regions, are estimated.

The main features concerning the evolution of the large scale photospheric magnetic fields derived from synoptic maps as well as from H-alpha synoptic charts are reviewed. The significance of a variety of observations that indicate the presence of a high latitude component as a counterpart to the sunspot phenomenon at lower latitudes is reviewed. It is argued that these two components describe the global magnetic field on the sun. It is demonstrated that this scenario is able to link many phenomena observed on the sun (coronal emission, ephemeral active regions, geomagnetic activity, torsional oscillations, polar faculae and global modes in the magnetic field pattern) with the global magnetic activity.

The present status of the Sun's differential rotation theories is reviewed. Attention is mainly focused on mechanisms for differential rotation based on the anisotropic viscosity concept and their modern developments within the framework of the mean-field hydrodynamics. The models with latitude-dependent heat transport and non-axisymmetric numerical simulations are briefly discussed.

The pattern of solar magnetic fields has been used as a tracer to determine how the sun's rotation rate varies with latitude and time. Two distinctly different rotation laws emerge from such an analysis, one agreeing with the surface Doppler rotation rate, the other corresponding to much more rigid rotation with a small polar spin-up. Detailed analysis shows that this second law cannot be explained in terms of flux redistribution on the solar surface, but that it represents the rotation properties of the sources of magnetic flux, which are likely to be located at the bottom of the convection zone.

The rotational phase velocity of the source pattern is found to be constant with time, which suggests that the depth at which the magnetic flux is stored and amplified inside the sun does not vary with the solar cycle, and that the phase velocity also represents the plasma velocity.

We interpret the helioseismological results for the solar rotation law as implying a uniform Ω0 in depth, if the angular velocity Ω(θ) is expanded in series of orthogonal polynomials. One of the possibilities to ensure uniformity of Ω0 is to exclude any anisotropy from the generating turbulence field, so that only the rotationally originated part of the heat conductivity tensor survive as a generator of meridional flow and consequently equatorial acceleration is produced. In addition to this simplest possible turbulence model, we discuss some possible forms of the nondiffusive parts of the Reynolds stresses which may be compatible with the uniformity of Ω0.

Observational and theoretical arguments in favor of a supposed serious asymmetry of magnetic flux loops emerging through the convective zone are briefly summarized. Results from numerical models of flux tubes moving through a differentially rotating upper convective zone are presented (plane parallel geometry, SFT approximation). In most models, especially in the most realistic ones, a remarkable asymmetry of the flux loop is found. It is concluded that in the future observational effects caused by the asymmetry may be used to put quantitative constraints on subphotospheric rotation.

The problem of the existence, evolution, and stability of spatial structures in convection is of considerable importance to astrophysics as well as to geophysical hydrodynamics. The Boussinesq approximation will be used because the considered motions in stars are sufficiently slow. The system of hydrodynamic equations describing convection in a rotating inhomogeneous medium has the form: Here Dt is the total time derivative, U the velocity, P, T, and C the deviations of the pressure, temperature, and helium abundance (by mass) from the basic equilibrium values, ρm, νm, χm, and Dm the values averaged over the considered layer of the density, viscosity, thermal and helium diffusivities, βT and βc the averaged coefficients of the thermal and helium expansions, g and Ω the gravitational acceleration and angular velocity, ∇Tb, and ∇Cb the values of the basic equilibrium temperature and helium gradients, and ñTad the adiabatic temperature gradient.

Self-organization i.e. the formation of large ordered structures in a turbulent medium is a consequence of inverse cascade where energy preferentially transfers towards large spatial scales. It is envisaged that this may be one way of explaining solar granulation at various scales.

The hypothesis is considered that the torsional wave observed on the Sun is an eigenmode oscillation excited in the presence of a weak poloidal magnetic field. We derive asymptotic linear equations for a perturbation with a large number of nodes along the radius, assuming the rotation to be slow and the characteristic perturbation period to be much longer than the rotational period. The results of a preliminary numerical study of the stability of the torsional mode indicate that the superadiabaticity of the solar convection may contribute to the excitation of this mode. In the present work the approximation of harmonic radial dependence of the perturbation has been used.

Solar activity varying with an 11-year cycle is chaotic at large time scales. The evidence comes from an analysis of observations of the sunspot number and of radioactive carbon. Thereby an estimate of the dimension of the solar attractor can be obtained.

The origin of the sunspots can be associated with the interactions of the regular, large-scale, chaotic, and intermittent magnetic fields.