The dynamo equation for the mean field ⟨B⟩ contains a random forcing term of unknown magnitude, which is therefore always omitted. The influence of this term is potentially large. To evaluate its effect, we employ ensemble averaging. If an ensemble average is used, there is no random forcing term in the dynamo equation. The effect of fluctuations is that the ensemble members get out of phase, so that ⟨B⟩→0. The damping time of ⟨B⟩ can be found by requiring that the mean energy ⟨BB⟩ remains finite. The eigenvalues к of the dynamo equation then all have negative real parts.Imк determines the period, and −Reк/Imк the relative period stability of the dynamo. We have developed a code to solve the equation for ⟨BB⟩ in a spherical shell (the convection zone), assuming axisymmetry. We report our first results, which do not yet include differential rotation. Using spherically symmetric boundary conditions, we reproduce the well known α2-dynamo, whose behaviour is known analytically. For instance, for an α2-dynamo located in a shell with inner boundary at R/2, we find that ⟨BB⟩ remains finite for R2γ/β=1.48, where β represents turbulent diffusion and γ turbulent vorticity. Taking α = ¼(βγ)½ — a factor of four below maximum helicity — implies that we have a dynamo number Cα ≡ Rα/β = 0.30. Using this value we find a damping time of 6 × 10−2R2/β for ⟨B⟩, which is a measure for the coherence time of B in a single ensemble member. This result implies that the large-scale field of this particular α2-dynamo reorganizes its structure completely on a time scale of only about one year (for solar values of R and β), and it shows the enormous influence of random forcing in general.

We discuss the methods and results of analysis of nonlinear mean-field dynamo models based on α-quenching in two asymptotic regimes, namely for weakly and highly supercritical excitation. In the former case the spatial distribution of the steady-state magnetic field is close to that given by the neutrally stable eigenfunction of the corresponding kinematic dynamo. In the latter case the magnetic field distribution within the main part of the dynamo volume is presumably determined by the balance between the Lorentz and Coriolis forces while near the boundaries boundary layers arise in which the field adjusts itself to the boundary conditions. The asymptotic behaviour of the highly supercritical αω-dynamos is sensitive to the particular form of dependence of the mean helicity on magnetic field while α2-dynamos are less sensitive to this dependence.

The small-scale solar magnetic field exceeding a given threshold forms a fractal set. A dimension of this fractal is found from magnetograms with varying linear resolution. The dimension depends on the value of the threshold magnetic field (multifractality). A simple dynamo model explaining the origin of the fractal magnetic structure is considered. The dynamo produces a magnetic field in the form of flux tubes with a fractal distribution of magnetic field across the tube. The observed dimension gives a possibility of estimating a degree of structuredness of the solar velocity field.

In contrast to αω-dynamos, where the angular velocity is arbitrarily prescribed, we consider here αΛ-dynamos, for which the differential rotation and meridional circulation are solutions of the momentum equation. The non-diffusive parts of the Reynolds stress tensor are parameterized by the Λ-effect. In earlier investigations we have shown that the turbulent magnetic diffusivity has to be much smaller than the eddy viscosity, otherwise the dynamo is not oscillatory or else the contours of constant angular velocity are cylindrical, contrary to observations. In the present paper we investigate the effects of compressibility.

The strength of fossil magnetic fields in young stars is estimated. The interaction of a fossil magnetic field with convection in stellar envelopes is investigated. The influence of such a magnetic field, transformed to a small scale, on the activity of young stars is discussed.

Well-confirmed helioseismic data from several groups using various observational techniques at different sites have allowed us to determine the differential rotation in the outer half of the Sun’s interior. The resulting rotation law is simple – the surface differential rotation persists through much of the convection zone with a transition toward solid body rotation beneath. To date there is no appealing evidence for a rapidly rotating core. There is however, weak evidence for a solar cycle dependence of the Sun’s internal rotation.

Rotation laws for the solar convection zone are produced by the Λ-effect in rotating anisotropic turbulence fields. In this paper we use the structure of the turbulence parameters introduced by Rüdiger and Kichatinov (1990), based on a simplified turbulence model. When we neglect the meridional circulation, for small inverse Rossby numbers the angular velocity isolines are spherical, while for increasing inverse Rossby number they approach more and more the helioseismologically derived shape. This simple picture becomes more complicated if the meridional circulation is allowed to act as an angular momentum transporter.

Recent results from helioseismology (see Goode, these Proceedings) have shown that the inferred contours of the solar angular velocity are more or less radial in the convection region, and the rotation becomes uniform below. These observations contradict the prevailing numerical models of Taylor columns which predict angular velocity contours parallel to the rotation axis of the Sun. Thus, an alternative explanation of solar differential rotation is called for.

Presently, it is not feasible to construct a thermally-relaxed, dynamically self-consistent numerical model of the solar convection zone (see Chan and Serizawa, these Proceedings). It is then appropriate to explore simplified models that may shed some light. A number of analytical models have been proposed for the solar differential rotation, and the reader is referred to the book by Rüdiger (1989) for a comprehensive review of this subject. Here, we report on some recent development on the convective resonance model proposed by Chan et al. (1987; hereafter referred as CSM).

The present internal rotation of the Sun as deduced from helioseismological data has many implications which concern the location of the solar cycle dynamo mechanism, core magnetism and history of the angular momentum. These aspects and their mutual connections are briefly discussed.

Analysis of the fine structure of the solar oscillations has enabled us to determine the internal rotation of the Sun and to estimate the magnitude of the large-scale magnetic field inside the Sun. According to the data of Duvall et al. (1984), the core of the Sun rotates about twice as fast as the solar surface. Recently Dziembowski et al. (1989) have showed that there is a sharp radial gradient in the Sun’s rotation at the base of the convection zone, near the boundary with the radiative interior. It seems to us that the sharp radial gradients of the angular velocity near the core of the Sun and at the base of the convection zone, acting on the relict poloidal magnetic field Br, must excite an intense toroidal field Bф, that can compensate for the loss of the magnetic field due to magnetic buoyancy.

The influence of rotation on the transfer of the mean magnetic field of the Sun, caused by the radial inhomogeneity of the solar turbulent plasma density, is investigated. It turns out that the transfer directions of the poloidal and toroidal magnetic fields do not coincide.

The solar surface represents for the solar dynamo an outer boundary that is directly accessible to observations. The evolutionary and rotational properties of the magnetic fields at this boundary can be empirically determined using existing synoptic magnetograph data. The derived properties provide detailed constraints on the underlying theory, such that an inversion approach to the dynamo problem becomes feasible. Ambiguities in the interpretation may be removed using the independent constraints from helioseismology.

The global solar cycle is considered as an interaction of 3 types of activity: at low-latitude (sunspots), at high-latitude (polar faculae) and the weak magnetic field. The properties of single and 3-fold reversals of the polar magnetic field are considered. The variation spectrum of the large-scale magnetic field of the Sun is analyzed in the range of 1–30 nHz. A dependence between the rate of a poleward meridional flow and phase of the global cycle is discussed.

This paper will discuss issues relating to the detailed numerical simulation of solar magnetic fields, those on the small scale which are directly observable on the surface, and those on larger scales whose properties must be deduced indirectly from phenomena such as the sunspot cycle. Results of simulations using the ADISM technique will be presented to demonstrate the importance of the treatment of Alfvén waves, the boundary conditions, and the statistical evolution of small scale convection with magnetic fields. To study the large scale fields and their time dependence, the magnetic resistivity plays an important role; its use will be discussed in the paper.

The traditional αΩ-dynamo as a model for the solar cycle has been successful in explaining the butterfly diagram, phase relations between poloidal and toroidal field, and polar branch migration features. Observational and theoretical achievements in recent years have however shaken this picture. The current trend is towards dynamos operating in the overshoot region of the convection zone. Nevertheless, there are many open questions and a consistent picture has not been established. In this paper we compare recent approaches and discuss remaining problems.

The phenomenon of a three-fold reversal of the solar polar magnetic field in both hemispheres has not been observed during the last 115 years. Such three-fold reversals took place in the southern hemisphere alone in the even cycles Nos 12 (1885.8), 14 (1908.4) and in the northern hemisphere alone in solar cycles Nos 16 (1928.5), 18 (1949.0), 20 (1970.6). The single reversal took place in the odd cycles, the only exception is the solar cycle No 19 (Fig. 1).

There are periods of 1.7-2.5 years in the variation of background magnetic fields (Makarov et al., 1985). It determines the quasi-period of the high-frequency component and corresponds to a time interval between the zones of alternating polarity of the magnetic field. This enables us to show topologically that single and three-fold polarity reversals of the solar magnetic fields can result from interaction of two types of magnetic fields: a low-frequency component with period of the order of 20 years and a high frequency component with period of order of 1.7-2.5 years (Benevolenskaya and Makarov, 1990).

We suggest that the latitude distribution of solar activity belts and the related equatorward or poleward migration of different tracers of the solar cycle are a natural consequence of the internal radial profile of angular velocity via the working of a dynamo in the boundary layer beneath the convection zone. This has been confirmed by the results of a non-linear dynamo model in a very thin spherical shell which show that dynamo action may reasonably take place in the boundary layer and reproduce the observed surface phenomenology.

Extending the argument to late main-sequence stars, it is reasonable to think that observations of the latitude distribution and migration of stellar active regions by current sophisticated techniques may make it possible to infer their internal rotation profile in a simple and direct way.

Having measured the rotation rate of sunspots through solar cycle 21, from 1977 to 1983, we have found that the mean differential rotation averaged over this seven year record is similar to the grand average differential rotation determined by Howard et al. (1984) over the period 1921-1982. However, the rotation rate does change from year to year. These changes are evidenced by a steepening or a flattening of the mean differential rotation profile, as well as significant changes in the equatorial rate. The presence of a time-dependent pattern of azimuthal rolls inferred from the meridional circulation pattern of the sunspots offers a qualitative explanation of the observed rotation rates. The amplitude of the changes is almost one order of magnitude larger than that of the torsional oscillations found by Howard and LaBonte (1981).

From May 1980 to April 1990 we have observed several times a year the solar irradiance spectrum from 5000Å to 6300Å using the 1-m Fourier Transform Spectrometer. During the ascending phase of the solar cycle, spectrum line asymmetry diminishes about 15% as the total magnetic field encreases. At some epoch near the time of total magnetic field maximum, line asymmetry ceases to track surface magnetism and abruptly assumes a value appropriate to solar minimum.

The behaviour of the HeI λ10830 Å line profile in spectra of solar features is reviewed. The connection of line characteristics with various phenomena on the solar disc, such as plages (faculae), sunspots, jets, prominences, coronal holes, flares, fine structure and emission above the solar limb, is discussed. We present a short summary of the mechanisms for helium excitation in the solar chromosphere. Time variations of the line profile, observed in quiet and active regions, and HeI λ10830 Å absorption in the Sun as a star are discussed and compared with other indexes of solar activity. It is shown that the equivalent width of the helium line varies in an 11 year cycle and reflects the 27 day periodicity of the rotation of the Sun. It is also shown that the HeI line is a more sensitive index of activity than K-index.