Plate tectonics provides a remarkably accurate kinematic description of the motion of the earth's crust but a fully dynamical theory requires an understanding of convection in the mantle. Thus the properties of plates and of the mantle must be related to a systematic study of convection. This paper reviews both the geophysical information and the fluid dynamics of convection in a Boussinesq fluid of infinite Prandtl number. Numerical experiments have been carried out on several simple two-dimensional models, in which convection is driven by imposed horizontal temperature gradients or else by heating either internally or from below. The results are presented and analysed in terms of simple physical models. Although the computations are highly idealized and omit variation of viscosity and other major features of mantle convection, they can be related to geophysical measurements. In particular, the external gravity field depends on changes in surface elevation; this suggests an observational means of investigating convection in the upper mantle.

The empirical relation between rotation period, spectral type, and activity cycle period is investigated for a sample of 13 slowly rotating lower main-sequence stars, including the Sun, all of which show long-term chromospheric variability like that of the solar cycle. It is found that for slowly rotating stars of similar spectral type, the cycle period P cyc and rotation period P rot are related by P cyc oc P rot n , where n & 1.25. When the stars, whose individual spectral types range from G2 to K7, are considered as a group, their cycle periods are found to be consistent with the relation P cyc ae (P rot /T c)", where t c is the convective turnover time near the bottom of the convection zone appropriate to each spectral type, and n is the same as before. These relations are interpreted in terms of simple nonlinear dynamo models. The increase of P cyc with increasing P rot disagrees with models in which the magnetic field is limited by quenching of the a effect or of differential rotation; however, it is consistent with models in which dynamo action is limited by losses due to magnetic buoyancy.

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Measurements of 10 Be concentration in the Dye 3 ice core show that magnetic cycles persisted throughout the Maunder Minimum, although the Sun's overall activity was drastically reduced and sunspots virtually disappeared. Thus the dates of maxima and minima can now be reliably estimated. Similar behaviour is shown by a nonlinear dynamo model, which predicts that, after a grand minimum, the Sun's toroidal field may switch from being antisymmetric to being symmetric about the equator. The presence of cyclic activity during the Maunder Minimum limits estimates of the solar contribution to climatic change.

In the outer layers of the Sun and other late-type stars thermal convection is affected by the presence of magnetic fields. This review deals principally with the interaction between convection and an externally imposed magnetic field in a Boussinesq fluid.Further effects of compressibility are considered briefly at the end. Magnetoconvection exhibits a particularly rich variety of behaviour when the ratio of the magnetic to the thermal diffusivity is small; the governing equations then allow both steady and oscillatory solutions. Our account provides a unified description of linear theory and of the interesting new results that are found in the non-linear regime. Recent work has classified the behaviour of isolated flux tubes and the relationship between time-dependent and steady solutions. Convection in a magnetic field is now the best-studied example of double-diffusive convection and serves as a guide to the behaviour of related systems.

This paper examines the effects of dynamical and resistive instabilities on magnetic redistribution of angular momentum within a star. It is tentatively concluded that if significant differential rotation survives in a stably stratified radiative zone over a stellar evolution time, then the poloidal field, cannot exceed an upper limit of order 3x 10" 2 G and is probably less than 10" 3 G. In the radiative core of the Sun B p is estimated to be at least of order 5xlO _2 G and probably 100 G or more. Values greater than 0.1 G cannot easily be reconciled with the rotational shear inferred from frequency splitting of solar oscillations.Angular momentum within a star 125

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Two-dimensional convection in a Boussinesq fluid confined between free boundaries is studied in a series of numerical experiments. Earlier calculations by Fromm and Veronis were limited to a maximum Rayleigh number R 50 times the critical value R, for linear instability. This range is extended to 1000Rc. Convection in water, with a Prandtl number p = 6·8, is systematically investigated, together with other models for Prandtl numbers between 0·01 and infinity. Two different modes of nonlinear behaviour are distinguished. For Prandtl numbers greater than unity there is a viscous regime in which the Nusselt number $N \approx 2(R/R_c)^{\frac{1}{3}}$, independently of p. The heat flux is a maximum for cells whose width is between 1·2 and 1·4 times the layer depth. This regime is found when $5 \leqslant R/R_c \lesssim p^{\frac{3}{2}}$. At higher Rayleigh numbers advection of vorticity becomes important and N ∞ R0·365. When p = 6·8 the heat flux is a maximum for square cells; steady convection is impossible for wider cells and finite amplitude oscillations appear instead, with periodic fluctuations of temperature and velocity in the layer. For p < 1 it is also found that N ∞ R0·365, with a constant of proportionality equal to 1·90 when p [Lt ] 1 and decreasing slowly as p is increased. The physical behaviour in these regimes is analysed and related to astrophysical convection.