The concept of a sequence of large sunspot groups is defined. From distribution functions of large groups from the period of sequences and that outside sequences it has been found that there are relatively more groups with a smaller maximum area in the period of sequences than in the period outside sequences. With an increased frequency of the occurrence of large sunspot groups on the solar disc their size decreases.

The most characteristic feature of the solar activity is its cyclic change in the course of time. The basic cycle is the 11-year one.

The theory of convection in a rotating spherical shell, when applied to the Sun, should ultimately satisfy at least three broad constraints. The radial heat flux by the convection must not vary significantly with latitude; an equatorial acceleration must be produced; and the convection must give the right dynamo action to produce the gross features of a solar cycle.

Important quantities to look for in the observations with which a convection theory can be compared include evidence of global velocities of giant convection cells, differences in motion features between low latitudes and high, persistence of motion features over successive rotations, evidence of excess brightness and variations in total solar luminosity, correlations of velocities between northern and southern hemispheres, and between north-south and east-west motions, and time periodic changes in motion fields.

The general theory of convective motions in a rotating spherical shell such as the convection zone of the Sun has developed rapidly over the last several years, but is still a long way from providing quantitative models which agree in a satisfactory way with the main solar observations. Most work has (for mathematical reasons) employed the Boussinesq approximation, and most has been either linear or nearly so. Early work demonstrated the basic ability of global convection to transport momentum toward the equator, but whether an equatorial acceleration results depends on the effects of competing angular momentum transport processes, namely transport in the radial direction by convection, and transport by axisymmetric meridional circulations. The end result can be assessed only by nonlinear calculations.

Some recent calculations directed at this question by the author indicate that, at least at Prandtl number of unity, convection growing from an initial state of solid rotation produces equatorial acceleration only when equatorial modes dominate and the rotational constraint is sufficiently strong, which results in a convective heat flux which is strongly dependent on latitude. When the rotational constraint is broken enough to give convection at all latitudes and therefore more nearly uniform heat flux, equatorial deceleration and angular momentum mixing in latitude results. A few calculations for Prandtl number substantially smaller than one give results which suggest it is possible to produce equatorial acceleration and nearly uniform heat flux, but these solutions appear not to be unique. In particular, the amount of differential rotation present in the initial conditions appears to be important in determining the final state.

The structure of solar magnetic fields is dominated by the effects of convection, which should be incorporated in any model of the solar cycle. Although mixing length theory is adequate for calculating the structure of main sequence stars, a better description of convection is needed for any detailed dynamo model. Recent work on nonlinear convection at low Prandtl numbers is reviewed. There has been some progress towards a theory of compressible convection, though there is still no firm theoretical evidence for cells with scales less than the depth of the converting layer. However, it remains likely that the pattern of solar convection is dominated by granules, supergranules and giant cells. The effects of rotation on these cells are briefly considered.

The main theories of solar rotation are critically reviewed.

The interaction of large-scale convection with rotation gives rise to a transport of angular momentum towards the equator and therefore to differential rotation with equatorial acceleration. (Large-scale convection is defined as follows: in a highly turbulent fluid, the small-scale turbulence acts as a viscosity and organizes fluid motions on a much larger scale.) This transport of angular momentum towards the equator arises because of the highly non-axisymmetric character of the large-scale convective motions in the presence of rotation. These motions tend to be concentrated near the equator. It is not surprising, therefore, that for magnitudes of large-scale convection which are needed to generate the observed solar differential rotation, large and unobserved pole-equator differences in flux appear in the Boussinesq approximation.

It is important, therefore, to take the variations in density into account. Studies of large-scale convection in a compressible rotating medium are still in a very early stage; these studies suggest, however, that the surface layers must indeed rotate differentially.

The interaction of rotation with convection appears to be especially efficient in generating a pole-equator difference in flux, Such a drives meridional motions, and the action of Coriolis forces on these motions gives rise to differential rotation. In the ‘large-viscosity’ approximation the problem separates; the meridional motions can be determined first (from the radial and latitudinal equations of motions, and the energy equation) and the angular velocity can be determined next from the azimuthal equation of motion. Since very little is known about compressible large-scale convection, it has been assumed in the development of this theory that the stabilizing effect of rotation on turbulent convection depends on the polar angle θ and on depth. The solution for the angular velocity in the large viscosity approximation gives a differential rotation that varies slowly with depth. As a consequence, the large viscosity approximation is not valid over most of the convection zone, the Coriolis term being larger than the viscous term; a thin layer at the top excepted. (It appears, however, that if the angular velocity, ω, is a slowly varying function of depth and the azimuthal stresses vanish at both ends of the convection zone, then the general behavior of ω will be very much like that predicted by the large viscosity approximation.)

The stabilizing effect of rotation on turbulent convection is neglected; if differential rotation is significant over the entire convection zone, and if the meridional and large-scale convective velocities are not too large, then in the radial and latitudinal equations of motion, the main balance of forces is between pressure gradients, buoyancy and Coriolis forces. If rotation is not constant along cylinders, then the differential rotation gives rise to latitudinal variations in the convective flux which are proportional to (where T is the temperature and g is gravity). In the lower part of the convection zone, is of the order of the superadiabatic gradient itself. Therefore large pole-equator differences in flux, will be present unless the angular velocity is constant along cylinders. The meridional velocities associated with this rotation law are not small, however, and could generate a significant It could well be that large s can be avoided only if rotation is uniform in the lower part of the convection zone. (To be certain of these results, however, it is important to estimate the magnitude of the stabilizing effect of rotation on turbulent convection.)

Turbulent convection is driven by the buoyancy force which thus introduces a preferred direction: gravity. In consequence, the turbulence in the sun should be anisotropic and if this is the case the convection zone cannot rotate uniformly. The degree of anisotropy is not known and must be determined from the observed solar differential rotation. The anisotropy is such that the horizontal exchange of momentum is larger than the vertical.

The normal mode of vibrations and the inner rotation of the Sun are briefly discussed.

Taking into account effects produced by the convective motions, the equilibrium problem for a rotating star becomes greatly complicated. We consider this problem for the case of slow rotation in the following approximations.

We treat the convection zone as a medium with turbulent viscosity and turbulent thermal conductivity. However, we take into account the nonlinear effects produced by the most rapidly growing perturbations. The corresponding nonlinear terms are calculated by using the solution of linear perturbed equations. Each independent convective mode is supposed to have initially the same amount of kinetic energy.

In the limit of small turbulent viscosity, we show that unstable convective perturbations produce a mean azimuthal force due to which rigid rotation appears not to be in the equilibrium. For the case of small-scale perturbations and latitudinal differential rotation, this force is analogous to the viscous force, but the coefficient of viscosity is negative.

We suggest that such a force maintains the differential rotation of the solar convection zone. Note that in the case under consideration the latitudinal dependence of the solar heat flux is small. However, difficulties arise due to different conditions at different depths in the convection zone. In this connection, a hypothesis is put forward that magnetic fields are also necessary to get balance in full. A model of the solar cycle is discussed which is similar in some respects to the well-known Babcock model. We propose, however, that the field reversal takes place in the lower layers of that zone where fields are intensified.

In this paper we study the dependence of the solar angular velocity on depth and latitude as produced by a meridional circulation in the convection zone.

For dealing analytically with the problem of differential rotation we investigate the spatial dependence of the angular velocity in a rotating turbulent fluid. The original turbulence unaffected by the rotation is assumed to be two-dimensional, where the stochastic motions completely lie in the horizontal planes. From the expression describing the relation between the correlations of rotating and nonrotating turbulent fields the meridional flux of momentum is derived. The resulting rotational law is determined by using Bochner's theorem for homogeneous turbulence as well as the characteristic scales of the turbulence field considered. The conclusions are:

1. (a) The angular velocity ω is increasing toward the outer layers.

2. (b) For 2 Ω ≪ ωc (ωc frequency of turbulent mode) the Biermann-Kippenhahntheory of anisotropic viscosity is deduced. An equatorial acceleration is only caused by a meridional circulation.

3. (c) For 2 Ω ≲ ωc a latitudinal dependence of ω is possible without any meridional circulation. If the two-dimensional eddy viscosity is negative the equatorial regions are accelerated. The expression for the two-dimensional eddy viscosity which has been derived earlier allows negativity in contrast to that for three-dimensional eddy viscosity. The scale length and the scale time of supergranulation as well as of giant cells lead to negative two-dimensional eddy viscosity.

Observations of the solar surface show that some of the physical quantities, especially the velocity field and the magnetic field, show random character.

One of the most striking features of both the magnetic field and the motions observed at the Sun is their highly irregular or random character which indicates the presence of rather complicated magnetohydrodynamic processes. Of great importance in this context is a comprehension of the behaviour of the large scale components of the magnetic field; large scales are understood here as length scales in the order of the solar radius and time scales of a few years. Since there is a strong relationship between these components and the solar 22-years cycle, an insight into the mechanism controlling these components also provides for an insight into the mechanism of the cycle. The large scale components of the magnetic field are determined not only by their interaction with the large scale components of motion. On the contrary, a very important part is played also by an interaction between the large and the small scale components of magnetic field and motion so that a very complicated situation has to be considered.

It is well know that the ‘dynamo’ theory has a number of vetoes; e.g. axisymmetric, two-dimensional, central-symmetric, etc. dynamo are impossible. In principle, the problem is essentially three-dimensional in any coordinate system. This is the main difficulty of both the theory itself and its possible applications. In fact, one prefers to believe that, for example, a non-rigid body-rotating star or convection in the Earth's nucleus possesses axis symmetry. However, due to the above vetoes one has to add finer effects (Coriolis strength, density, inhomogeneity) to create asymmetrical convection. On the other hand, the authors try to find the most simple movements with minimum deviations from axial symmetry. Thus, the Herzenberg's dynamo (Herzenberg, 1958) is realized by two rotating cylinders, axes of which are parallel to each other (see also Galaitis, 1973; Galaitis and Freinberg, 1974), the Lortz's dynamo-spiral movement (Lortz, 1968; Ponomarenko, 1973). Nevertheless, the mentioned vetoes possess a common feature, the assumption regarding the symmetry extends both to the movement and to the field. Hence, it makes sense to raise a question whether symmetric movements are able to generate an asymmetric field. A positive answer to this question, in particular, is given by Tverskoy's model (Tverskoy, 1966) – the toroidal vortex. The latter possesses axial symmetry. Nevertheless, the toroidal vortex is a complex motion; we will proceed along the path of a minimum simplification.

The first model for ‘fast’ magnetic field reconnection at speeds comparable with the Alfvén speed was put forward by Petschek (1964). It involves one shock wave in each quadrant radiating from a central diffusion region and leads to a maximum reconnection rate dependent on the electrical conductivity but typically of order 10-1 or 10-2 of the Alfvén speed. Sonnerup (1970) and Yeh and Axford (1970) then looked for similarity solutions of the magnetohydrodynamic equations, valid at large distances from the diffusion region; by contrast with Petschek's analysis, their models have two waves in each quadrant and produce no sub-Alfvénic limit on the reconnection rate.

Our approach has been, like Yeh and Axford, to look for solutions valid far from the diffusion region, but we allow only one wave in each quadrant, since the second is externally generated and so unphysical for astrophysical applications. The result is a model which qualitatively supports Petschek's picture; in fact it can be regarded as putting Petschek's model on a firm mathematical basis. The differences are that the shock waves are curved rather than straight and the maximum reconnection rate is typically a half of what Petschek gave. The paper is a summary of a much larger one (Soward and Priest, 1976).

In this paper solutions of the mean field induction equation in a spherical geometry are discussed. In particular, the 22-year solar magnetic cycle is considered to be governed by an axisymmetric, periodic solution which is antisymmetric with respect to the equatorial plane. This solution essentially describes flux tubes travelling as waves from mid-latitudes towards the equator. In a layer of infinite extent the period of such dynamo waves solely depends on the strength of the two induction effects, differential rotation and α-effect (cyclonic turbulence). In a spherical shell, however, mean flux must be destroyed by turbulent diffusion, so the latter process might actually control the time scale of the solar cycle.

A special discussion is devoted to the question of whether the angular velocity increases with increasing depth, as the dynamo waves seem to require, or whether it decreases, as many theoretical models concerned with the Sun's differential rotation predict. Finally, theories for the sector structure of the large scale photospheric field are reviewed. These describe magnetic sectors as a consequence of the sectoral pattern in the underlying large scale convection, as non-axisymmetric solutions of the mean field induction equation, or as hydromagnetic waves, modified by rotational effects.

We review the primordial field theory of solar magnetic fields (Papers I–VI) whose three main features are, first, a permanent dipole-like magnetic field, second a mainly toroidal field formed by shearing and rolling into individual, helically twisted ropes as suggested by Babcock, and third a mechanism for reversing the toroidal field. The theory explains numerous observational effects where the dynamo theory fails.

(i) An active region forms when a rope section emerges and expands layer by layer to form a rotating arch filament system and then spots. Only a rope model explains the radial inflow of magnetic elements to build up a spot, as well as the spiral structure and other features pointed out by Babcock. (ii) The model explains umbral and penumbral structures, the Wilson depression, Evershed flow, the sunspot energy deficit and the very slow loss of flux fragments by some sunspots. (iii) The model is then extended to background magnetic fields to show that surface magnetic fields are like the upppermost branches of a magnetic ‘tree’ whose trunk is a flux rope. This explains unipolar magnetic regions, ‘pepper and salt’, and ephemeral active regions. Tension in the submerged flux ropes accounts for the observed migrations of magnetic regions, active longitudes and magnetic longitudes. (iv) On a smaller scale spicules, mottles and other network elements are explained in terms of the tree structure. (v) The mechanism of reversal of the toroidal field system is explained and the manner of disposal of old toroidal fields. (vi) The basic error in the dynamo theory is discussed briefly. We point out that radical changes in dynamo theory have been suggested by Parker, but appear to have been ignored by others.

Extensive numerical studies of the dynamo equations due to the global convection are presented to simulate the solar cycle and to open the way to study general stellar magnetic cycles. The dynamo equations which represent the longitudinally-averaged magnetohydrodynamical action (mean magnetohydrodynamics) of the global convection under the influence of the rotation in the solar convection zone are considered here as an initial boundary-value problem. The latitudinal and radial structure of the dynamo action consisting of a generation action due to the differential rotation and a regeneration action due to the global convection is parameterized in accordance with the structure of the rotation and of the global convection. This is done especially in such a way as to represent the presence of the two cells of the regneration action in the radial direction in which the action has opposite signs, which is typical of the regeneration action of the global convection. The effects of the dynamics of the global convection (e.g., the effects of the stratification of the physical conditions in the solar convection zone) are presumed to be all included in those parameters used in the model and they are presumed not to alter the results drastically since these effects are only to change the structure of the regeneration action topologically.

The angular velocity distribution as a function of heliographic latitude is directly observed at the solar surface. Also the dependence of the angular velocity on the distance from the solar centre can be studied observationally for the outermost layers of the Sun. But within the Sun the study of angular velocity depends very much on assumptions. One possibility to study the problem is given by the fact that in a bipolar magnetic group the leader is compact while the follower is dispersed over a large area. In Leighton's theory, following Babcock, this phenomenon is explained, in principle, in the following way: Slide 1. When a magnetic rope, as a result of magnetic buoyancy, rises to the surface, it is twisted by differential rotation. Dashed lines in the figure represent isotachial surfaces. When the rope has risen to the surface, the two ends are twisted in opposite directions. The surface differential rotation continues to twist one of the now free ends of the rope, while it untwists the other free end. Both twisting and untwisting are, in fact, very slow. They are fastest at the latitude of 35°, where the magnetic rope is rotated once in 26 days. We understand that the distribution of angular velocity within the Sun is involved in this picture. The twisting in the interior can be written equal to the untwisting of the follower spot produced by the surface differential rotation. Of course there are many solutions. Slide 2 gives two of them. The lines represent calculated isotachial surfaces and the numbers give the observed angular velocities in radians per day. In the left-hand side distribution the velocity decreases inwards, while in the right-hand side distribution it increases inwards. The right-hand side distribution is also consistent with the conclusion derived from observations that the tilt of sunspot axes from the vertical is relatively small, and with the observation that at the equatorial plane, near the surface, the rotational velocity does not vary with depth.

Statistical analysis of the photospheric magnetic field averaged over 27 day time intervals shows periodical variations. The results of such an analysis are usually considered to be a pure mathematical representation of the observation material and additional analysis is necessary to infer the physical meaning of the results. Some ideas for this analysis are given by the mathematical method of the ‘non-axisymmetric dynamo’ theories which suggest the representation of the field by the series

i.e. the field variation is equivalent to the interference of progressive wave moving parallel to the equator with velocity

If n = 1 the variation is equivalent to a progressive dipole field circulating over the photosphere by a time 2π/Ω and this period can be derived from the 27-day averages. The superimposition of the circulation to the rotation will result in the apparent ‘field rotation’ being observable as recurrence time of the largest magnetic areas. Observational evidences are found for a progressive dipole circulating in a great circle with a period of 4⅙ yr.

The higher multipole field components (if n ≥ 2) have no characteristic variations but their mean effect results in statistical spread in the ‘multipole field rotation’.

By the expression ‘stellar activity of the solar type’ we mean the manifestations of stellar activity determined by a mechanism of the same kind as that producing the solar activity.

The work described here is still in progress; hence this paper will be a brief sampling of the results to date.

Our understanding of the internal structure of the Sun and solar type stars has been undermined by recent observations. In this paper we consider some of the puzzles and possible resolutions; the solar neutrino problem, lithium and beryllium abundance, rotation and calcium emission, variation of the solar constant, solar oscillations and the solar convective zone. The picture that emerges is one of confusion, and so it should be since we have no idea of what is going on inside the Sun and a fortiori of what is going on inside other stars.