We examine an idealized αω-dynamo model in which the magnetic fields depend only on latitude and time. The solutions that bifurcate from the field-free state are either symmetric or antisymmetric about the equator (quadrupolar or dipolar respectively). Nonlinear steady and periodic solutions, whether stable or unstable, can be followed numerically as the dynamo number is varied, revealing a rich bifurcation structure with mixed-mode solutions (lacking symmetry about the equator) appearing at secondary bifurcations. These results show how stable asymmetric fields can occur in the sun and illustrate the formation of complicated spatial structure in more active stars.

After the very suggestive results of the early days, the theory of the solar dynamo has now entered a period of re-evaluation. It is clear that our initial expectations have been too high. I shall review some of the recent attempts to formulate nonlinear and stochastic mode excitation theoretically. We now have evidence from synoptic observations that the solar dynamo features many periods. Periods both shorter and longer than the fundamental 22 yr cycle have been claimed. The phase stability of any of these periods is uncertain. The phase memory of the 22 yr period may be as short as ~ 10 cycles, but could also be much longer. Linear mean field theories permit only one marginally stable mode; they predict one period with an infinitely long phase memory. Attempts to explain multiperiodicity and finite phase memory effects fall in two categories:

(1). Nonlinear models. These feature a few nonlinearly coupled variables and may exhibit a multiperiodic or chaotic behaviour; (2). If the number of relevant variables is very high, then the dynamo behaves stochastically. It has been argued that this takes the form of stochastic excitation of many dynamo modes (overtones).

Spherical-harmonic-fourier (SHF) analysis of sun's ‘nominal toroidal magnetic field’, computed using sunspot data during 1874-1976, shows that sunspot activity can be considered as possibly originating in interference of sun's axi-symmetric ‘hydromagnetic’ oscillations of odd degrees up to ℓ ≈ 21 and periods ≈ 22y. The relative amplitudes and relative phases of these oscillations remain fairly constant, leading to butterfly diagrams with stable latitude-time correlations on all latitude scales ≳9°. The amplitudes and phases do however undergo slow coherent variations. The main modes (η<11) represent an approximately standing oscillation.

We have studied axisymmetric nonlinear αω-dynamo models taking the interaction between even and odd parities fully into account. It turns out that the dominating type of symmetry is not always determined uniquely, but it can vary on a very long time scale compared to the period of the magnetic cycle. In some cases the frequency of this long term variation is close to the beat frequency of the two solutions with purely dipolar and purely quadrupolar parity. The occurrence of a second frequency is typical of solutions whose trajectory describes a torus in the phase space. We argue that this finding is of relevance for understanding secular variations observed in the Sun. For example measurements of sunspots indicate that the spot number on the northern hemisphere at present exceeds the number on the southern hemisphere. The reverse seems to have been the case at the end of last century.

We model the evolution of photospheric field elements by treating them as mean field structures undergoing a nonlinear self-interaction mediated by much smaller-scale, convectively driven plasma turbulence. Distributed fields can gather into discrete, strong elements of a minimum permitted scale. Also studied are the transport of flux from dissolving elements and to growing elements via weak intermediate fields and the cancellation of adjacent emements of opposite polarity.

We discuss the observational results of cyclic variations of solar rotation and how these can be used as a means of probing the solar dynamo. We shortly describe two examples of dynamo models where the α-effect has been modified, and compare the resulting flows to the observations.

The behaviour of the magnetic field during the formation and evolution of the Sun is investigated. It is shown that an internal poloidal magnetic field of the order of 104 − 105 G near the core of the Sun may be compatible with differential rotation and with torsional waves, travelling along the magnetic field lines (Dudorov et al., 1989).

Numerical simulations of the three-dimensional structure and time evolution of stellar surface convection are now possible, at least for solar-type stars. Using the output from such simulations as sets of spatially and temporally varying model atmospheres, synthetic granulation images and spectral line profiles are computed, and compared to observations. Thus obtained disk-integrated data agree with observed lineshapes and bisector patterns in different stars, and also permit stellar rotation to be determined. Such simulations represent the first generation of models that are free from the classical ad hoc parameters of ‘mixing-length’, ‘micro-’ or ‘macro-turbulence’, parameters which in the past have characterized and limited stellar astrophysics.

To infer the surface structure also in more exotic stars, simpler and parametrized models must still be used to interpret observed line asymmetries. Such models suggest that rapidly rising ‘granules’ cover only a small surface fraction on early-type stars, a situation opposite to that in solar-type ones, and one likely to affect magnetic fields and stellar activity. Theoretical challenges for the future include detailed modeling also of early-type, giant, and other non-solar type stars of different rotational velocities; the hydrodynamics of entire stellar envelopes (including the interaction with global oscillations); and the interaction with magnetic fields (including their generation). Greatly increased computing power will be needed for such detailed modeling throughout the Herzsprung-Russell diagram, possibly requiring custom-designed computers.

Signatures of stellar granulation are primarily observed as asymmetries and wavelength shifts in photospheric absorption lines. Observational challenges include achieving sufficient spectral resolution to fully resolve such asymmetries; identifying granulation signatures throughout the HR-diagram (including the blended spectra of cool stars); observing how line asymmetries for a given spectral type depend on stellar rotational velocity, measuring wavelength shifts between groups of different lines in the same star, and between different stars; monitoring lineshift variations during stellar activity cycles; and ultimately high-resolution spectroscopy of spatially resolved granulation structures across stellar disks. The latter will require active optics on future very large telescopes, or the use of long-baseline optical interferometers.

A model for stellar convection zones based on linear convective modes using a nonlocal mixing length theory is developed to study the spectral line asymmetries and the line shifts resulting from convective motions in the stellar photospheric region. The amplitudes of these linear convective modes is estimated by requiring the convective flux due to a linear superposition of such modes to reproduce the convective flux in the mixing length model. To study the spectral line asymmetries the convective mode with the largest amplitude in the photospheric line formation region is chosen to represent the stellar velocity field and the accompanying intensity fluctuations. Synthetic spectral line profiles are obtained by summing locally symmetric profiles over the stellar disk according to the local Doppler velocity and intensity fluctuations. The resulting line bisector shapes and the line shifts are compared with observations for α-Cen B. It is found that while the simple model proposed here can explain either the line shifts or the line bisector shape reasonably well, it fails to explain both these characteristics simultaneously.

An analysis is made of the velocity field determined from spectral absorption line profiles as well as inhomogeneous element sizes obtained from stochastic theory. The results of three-dimensional simulations of convective motions in the photospheres of the Sun and Procyon are presented. It is concluded that penetrative convection covers a considerable fraction of the Procyon photosphere.

I review the progress made over the past decade in the measurement of magnetic fields on solar–like stars. I describe the evolution of magnetic analysis techniques, summarize our current understanding of stellar magnetic properties, and outline some future research directions.

We present results of nearly simultaneous polarimetric and Ca II(H+K) emission observations taken over a full rotation period for a sample of six solar-type stars. Significant variations are seen for two stars in polarization and four stars in Ca emission, which we interpret as due to rotational modulation of magnetic areas on the stellar surface. The irregularity of the variations suggest changes of the area size and/or distribution in time scales of the rotation period. Preliminary models (including rough surface maps of two stars) and future prospects are discussed.

A simple mean-field model of a nonlinear stellar α-ω dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the main nonlinearity retained is the influence of the Lorentz force on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.

Based on an extended list of lower main sequence stars from Rutten (1987), the relation between chromospheric activity and Rossby number has been revised. The increased statistics changes the shape of the curve as compared with that of Noyes et al. (1984). The saturation at small Rossby numbers has disappeared. The dependence on Rossby number in the range of very large Rossby numbers has weakened. The standard deviation of the activity indices from the mean curve is about 40%. This scatter of individual stars is not due to differences in the spectral type or age of the stars.

This paper sets about a brief review of the current knowledge of ground based interferometry, and then, using the relevant information from this up-to-date technique, its tentative application to the study of solar granular structures. As a matter of fact, most of the fundamental processes in the photosphere take place on very small spatial scales and understanding the formation of the solar granulation, pores as well as tiny photospheric faculae, requires insight into the magnetic network where the thermal instabilities are initiated. The study of fine scale structures, typically of about 15 milliarcsec (11 km) by interferometry, i.e. to a resolution not accessible through classical optical telescopes, is certainly the key to open the door to new scenarios in solar physics. By the way, similar phenomena are likely to be accountered in other type of stars. Particular attention is paid to the role of imaging reconstruction at very high angular resolution. Some of the thus achievable solar programmes are listed, mainly with solar granulation. The new type of instrumentation involved should be in the 90's commissioned. At last, it is pointed out that solar interferometry could be also performed in space, as earlier suggested by J.L. DAME and C. AIME to ESA, for the study of flares, prominences or coronal loops, which are closely linked to the photosphere, and for the 2000's horizon. Some problems inherent to this whole prospect (ground and space) are listed, and discussion is welcomed for determining adequate programmes, and also to highlight promising directions for future investigations.

The requirements and conditions for high resolution imaging and polarimetry of the Sun are reviewed. Various methods and techniques are discussed for image stabilization and sharpening in solar observations. The new solar facilities in the Canary Islands in particular are frequently reaching diffraction limited resolution and yield new insight in the structure and dynamics of the solar atmosphere. Future ground based telescopes like THEMIS and LEST, as well as planned solar missions in space will trigger a next advance in solar physics.

The development of model atmospheres from the ‘classical’ static but deductive models to present-day dynamic but inductive models is sketched. The main problems facing theory are defined in terms of the need to produce a post-classical inductive model. Attention is focused on the most promising tool available today, the computers able to realize simulations of astrophysical systems. The direction of progress in the areas of radiative transfer, convective transport, waves and oscillations, and MHD is reviewed. It is concluded that the major outstanding radiative and hydrodynamic problems are likely to be elucidated in the foreseeable future, especially if there is a suitable commitment by the international community. However, the understanding of the behaviour of magnetic fields and their associated activity will require a longer, but no less urgent, programme.

I have found four models in the literature for doing conference summaries:

1. 1. The Literal Summary.

   One summarizes all t h a t has been presented, preferably interspersed with comments as “of particular interest was…”.

2. 2. The Historical Perspective.

   One places all (or some) presentations within a historical background, preferably implying that science progresses smoothly in well-planned, orderly fashion.

3. 3. The Future Perspective.

   One points out the way to go, preferably in overly optimistic vein.

4. 4. The personal Impressions.

   One concedes lack of wisdom to forego balanced summarizing, prefering to discuss primarily one's own interests.