The investigation of magnetic phenomena on the Sun is confronted with the problem of turbulently moving electrically conducting media in rotating objects. One way of attacking this complicated problem, which was at first successful, was the development of mean-field magnetohydrodynamics, where the cooperative action of the small-scale turbulence was taken into account by certain average effects. The development of more and more powerful computers now offers possibilities of calculating the small-scale phenomena in a direct way. This paper is an attempt to compare these approaches, at least as far as there are comparable results.

It has become a cliché to say that convection lies at the foundation of the activities, magnetism, and dynamos of the Sun and cool stars – the title of this colloquium. However, it has not been possible, until recently, to study theoretically these phenomena and their connections with quantitative rigor. The new development owes much to the rapid growth in the power of computers and the spread of sophisticated numerical techniques.

Since the last decade numerical studies of solar and stellar convection problems thrive. The trend towards a greater use of computers in these studies will likely become stronger. Now is a good time to reflect on the numerical approach itself, so that we may understand it better and make better use of it in the future. Here we look at two aspects of this methodology. First (Section 2), we examine its relationships to the analytical approach and observation, for which we make a differentiation between numerical experimentation and simulation. It is obvious that the ultimate test of any theory, numerical or not, is to confront observations, but not generally realized is that the connection to analytical studies is an important requisite for the healthy development of numerical studies. This point is to be elaborated. Second (Section 3), we discuss the prevalence of pitfalls in the numerical approach; two examples of current interest in convection computations are analysed in detail. In Section 4, we go on to suggest that one of these pitfalls might have caused the discrepancy between the results of numerical modelling of the solar differential rotation and helioseismology.

We use the latest models of turbulence to compute a new expression for the turbulent convective flux, Fc. The new values of Fc are up to ten times larger than those given by the mixing length theory, MLT. Astrophysical considerations indicate that the new model fares better with observational data than the MLT.

We show that, representing the descending fluid in a convection zone by a porous medium, the differential rotation of the (rising) fluid is very close to that in an axisymmetric model of the convection zone with anisotropic viscosity.

In addition to the well-known granulation and supergranulation of the solar convection zone (the “SCZ”), the presence of so-called “giant cells” has been postulated. These are supposed span the entire thickness of the SCZ and to stretch from pole to pole in a sequence of elongated cells like a “cartridge belt” or a bunch of “bananas” strung uniformly round the Sun. Conclusive evidence for the existence of such giant cells is still lacking, despite strenuous observational efforts to find them. After analyses of sunspot motion, Ribes and others believe that convective motions near the solar surface occurs in a pattern that is the antithesis of the cartridge belt: a system of “toroidal” or “doughnut” cells, girdling the Sun in a sequence that extends from one pole to the other. Galloway, Jones and Roberts have recently tried to meet the resulting theoretical challenge, with the mixed success reported in this paper.

Thermal convection in the Sun and cool stars is often modeled with the assumption of an effective Prandtl number σ ≃ 1. Such a parameterization results in masking of the presence of internal shear layers which, for small σ, might control the large scale dynamics. In this paper we discuss the relevance of such layers in turbulent convection. Implications for heat transport – i.e. for the Nusselt number power law – are also discussed.

In order to understand what happens in the Sun when convection overshoots into the radiation zone an idealized model of penetrative convection with rotation is studied. Here we highlight two properties of the model which occur with parameters relevant to the Sun. Firstly rotation allows motions to persist far beneath the convection zone, and secondly the profile of helicity with depth is dominated by a local maxima just beneath the convection zone. This second result has consequences in dynamo theory.

Problems associated with topological pumping in the lower overshoot layer suggest a strongly turbulent and strongly differentially rotating upper radiative zone as the seat of the dynamo and as flux reservoir.

A new method for the calculation of kinetic coefficients is presented. This method allows us to obtain the distribution of scalar and vector fields (such as the temperature, the admixture particle number density and the magnetic field) in turbulent cosmic media with any value of S = u0т0/R0. The explicit expression for the “turbulent” diffusivity DT is obtained. In some cases DT becomes negative, implying the clustering of the admixture particles in patches (a local increase of the temperature and magnetic fields). The magnetic α-effect is considered for the case S ~ 1.

A direct numerical simulation at resolution 2403 is used to obtain a statistically stationary three-dimensional homogeneous and isotropic turbulent field at a Reynolds number around 1000 (Rλ ≈ 150). The energy spectrum displays an inertial subrange extending over more than one decade. In the physical space, the strongest vorticity is organized in very elongated thin tubes. The typical length of a tube is around the integral scale of the flow and its thickness is a few dissipation scales.

The generation of magnetic field by a conducting, compressible fluid inside a spherical shell is studied by direct numerical simulations. A pseudo-spectral method is used in order to resolve accurately all the scales present in the problem. The range of parameters considered is the following: a unit Prandtl number, Rayleigh numbers up to 100 times critical, Taylor number 625, an aspect ratio of 2, a Mach number slightly less than 1, and pressure and temperature scale heights of the order of the thickness of the shell. A dynamo effect is observed for magnetic Prandtl numbers larger than 1. We present the properties of the turbulent flow, the role of the helicity and of the differential rotation in the enhancement of the magnetic field, and the spectral properties of the flow fields.

We investigate convective overshoot in a layer of electrically conducting fluid. The radiative conductivity is assumed to be larger in the lower part of the layer which makes it stable to convective motions, yet penetrative convection from the upper layer can occur. The numerical resolution is 633 gridpoints. We observe a dynamo effect for magnetic Reynolds numbers around one thousand when a magnetic seed field is rapidly concentrated to form flux tubes. Later the average magnetic field is expelled from the convectively unstable regions, but it accumulates in the interface between the convection zone and the radiative interior.

Recent observations (Tarbell et al., 1990; see also Ruzmaikin et al. these Proceedings) show that the surface distribution of magnetic fields on the solar surface is selfsimilar, with an approximately constant fractal dimension over a large range of horizontal scales. Also, recent ideas about flare energy release assumes a “fibrous” corona, with the energy release consisting of many small scale events (Vlahoz, 1989) Prompted partly by these observations, we investigate the properties of self-similar three-dimensional magnetic fields.

A magnetic tube is introduced into turbulent compressible penetrative convection. After being strongly advected, most of the magnetic flux is stored in the overshoot region. With rotation there are meridional travelling waves.

The condition for homogeneous radiative stellar models to be marginally stable to convection at the centre is investigated for the family of models where the opacity к and energy generation ε are given by power laws in temperature and density κ = κo ρα T−β, ε = εo ρ Tη. The Naur-Osterbrock (1953) condition 6η > 6 + 10β − 15α is a necessary but not sufficient condition. A better estimate is obtained by taking the effective polytropic index n = dlogP/dlogT - 1 to be a linear function of temperature T throughout the model. This gives the condition

The predictions of this condition agree well with results for a set of stellar models 0 ≤ α ≤ 1, 0≤ β ≤ 5.

Three-dimensional hydrodynamic simulations are carried out in a rectangular box. The angle between gravity and rotation axis is kept as an external parameter in order to study the latitude-dependence of convection. Special attention is given to the horizontal Reynolds stress and the ∧-effect (Rüdiger, 1989). The results of the simulations are compared with observations and theory and a good agreement is found.

A local potential approach to nonlinear dynamo models which allows the use of variational techniques to investigate the problem of stability is introduced. The method applies at least to quasi-kinematic dynamo models, i.e. to models which include the back-reaction of the magnetic field on the fluid motion in a simplified way. A special application leads to a previously investigated one-dimensional dynamo model which shows a coexistence of a periodic solution (limit cycle) with two stable steady solutions of opposite polarities. The inclusion of some small-amplitude noise leads to interesting transition phenomena which may be of relevance to explain the behaviour of astrophysical dynamos. A simple dynamical system with a two-dimensional phase-space is used for illustration.

Recent observations seem to have detected large scale nonaxisymmetric structures on active giant stars. These structures are plausibly associated with underlying, dynamo generated, nonaxisymmetric magnetic fields. Such developments have motivated the development of a computer code to solve the nonlinear mean field dynamo equation in spherical geometry with no imposed geometrical symmetries. The nonlinearity is a simple α-quenching. The nature of the stable solutions found depends quite sensitively on the assumed spatial distribution of both α-eflect and differential rotation, and also on the degree of supercriticality of the dynamo. Such a dynamo model with stable, purely nonaxisymmetric solutions is described in this paper.

We develop a theory of the sunspot cycle predicated on the assumption that the observed bands of activity are packets of dynamo waves. An approximate equation is proposed to describe the dynamics of these packets, using standard ideas from bifurcation theory. We show that in a certain limit the system can be described in terms of a slowly-evolving solitary wave, and that periodic behavior, like that of the observed butterfly diagram, can easily be found. Generalizations of the theory are discussed.