This paper is concerned with limits on kinetic and magnetic energies and dissipation rates in forced flows that lead to dynamo action and a finite amplitude magnetic field. Rigorous results are presented giving upper and lower limits on the values of these quantities, in a simple cubic geometry with periodic boundary conditions, using standard inequalities. In addition to the general case, results in the special case of the Archontis dynamo are presented, in which fields and flows are closely similar in much of the domain.

This paper is a detailed report on a programme of simulations used to settle a long-standing issue in the dynamo theory and demonstrate that the fluctuation dynamo exists in the limit of large magnetic Reynolds number Rm>>1 and small magnetic Prandtl number Pm<<1. The dependence of the critical Rm_c vs. the hydrodynamic Reynolds number Re is obtained for 11. The stability curve Rm_c(Re) (and, it is argued, the nature of the dynamo) is substantially different from the case of the simulations and liquid-metal experiments with a mean flow. It is not as yet possible to determine numerically whether the growth rate is ~Rm^{1/2} in the limit Re>>Rm>>1, as should be the case if the dynamo is driven by the inertial-range motions. The magnetic-energy spectrum in the low-Pm regime is qualitatively different from the Pm>1 case and appears to develop a negative spectral slope, although current resolutions are insufficient to determine its asymptotic form. At 1

Direct numerical simulations of incompressible nonhelical randomly forced MHD turbulence are used to demonstrate for the first time that the fluctuation dynamo exists in the limit of large magnetic Reynolds number Rm>>1 and small magnetic Prandtl number Pm<<1. The dependence of the critical Rmc for dynamo on the hydrodynamic Reynolds number Re is obtained for 1 less than or similar Re less than or similar 6700. In the limit Pm<<1, Rmc is about 3 times larger than for the previously well-established dynamo at large and moderate Prandtl numbers: Rmc less than or similar 200 for Re greater than or similar 6000 compared to Rmc approximately 60 for Pm>or=1. It is not yet possible to determine numerically whether the growth rate of the magnetic energy is proportional, Rm1/2 in the limit Rm-->infinity, as it should be if the dynamo is driven by the inertial-range motions at the resistive scale.

The last thirty years have seen great leaps forward in the subject of magnetoconvection. Computational techniques can now explain exotic nonlinear behaviour, transition to chaos and the formation of structures that can be observed on the surface of the Sun. Here, two leading experts present the current state of knowledge of the subject. They provide a mathematical and numerical treatment of the interactions between electrically conducting fluids and magnetic fields that lead to the complex structures and rich behaviour observed on the Sun and other stars, as well as in the interiors of planets like the Earth. The authors' combined analytical and computational approach provides a model for the study of a wide range of related problems. The discussion includes bifurcation theory, chaotic behaviour, pattern formation in two and three dimensions, and applications to geomagnetism and to the properties of sunspots and other features at the solar surface.

Through multiple-scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady-state pattern-forming instabilities, in the case that the wavelengths of the instabilities are nearly in the ratio 1:2. In the case of exact 1:2 resonance the amplitude equations are ODEs; here they are PDEs. We discuss the stability of spatially periodic solutions to long-wavelength disturbances. By including these modulational effects we are able to explore the relevance of the exact 1:2 results to spatially extended physical systems for parameter values near to this codimension-two bifurcation point. These new instabilities can be described in terms of reduced ‘normal form’ PDEs near various secondary codimension-two points. The robust heteroclinic cycle in the ODEs is destabilised by long-wavelength perturbations and a stable periodic orbit is generated that lies close to the cycle. An analytic expression giving the approximate period of this orbit is derived.

Thermal convection is the most significant driver of time-dependent patterns of motion within the Sun. Observations of magnetic and convection phenomena in the Sun, together with an understanding of the basic physical processes involved, provide a basis for large numerical simulations. Models of solar convection produced in this way and using considerable computer power can investigate such problems as the role of stratification within the Sun, in two or three dimensions. The models are now sufficiently complex to produce results that can be compared with observations, in some cases. In future this technique offers the chance to replicate surface solar phenomena such as sunspots, while uncovering the complex magnetoconvection beneath.

We respond to the comment of V.P. Zhdanov on our earlier Letter [Chem. Phys. Lett. 377 (2003) 69]. We give an explicit account of the derivation of our coarse-grained equations for the spatial variation of CO coverages. (C) 2004 Elsevier B.V. All rights reserved.

We demonstrate a close analogy between a viscoelastic medium and an electrically conducting fluid containing a magnetic field. Specifically, the dynamics of the Oldroyd-B fluid in the limit of large Deborah number corresponds to that of a magnetohydrodynamic (MHD) fluid in the limit of large magnetic Reynolds number. As a definite example of this analogy, we compare the stability properties of differentially rotating viscoelastic and MHD flows. We show that there is an instability of the Oldroyd-B fluid that is physically distinct from both the inertial and elastic instabilities described previously in the literature, but is directly equivalent to the magnetorotational instability in MHD. It occurs even when the specific angular momentum increases outwards, provided that the angular velocity decreases outwards; it derives from the kinetic energy of the shear flow and does not depend on the curvature of the streamlines. However, we argue that the elastic instability of viscoelastic Couette flow has no direct equivalent in MHD.

We extend a detailed kinetic model for CO+O2 on Pt{100} to describe pattern formation. The model includes: (i) a non-linear power law to describe the phase transition, (ii) trapping and untrapping processes explicitly considered, and (iii) experimentally determined coverage-dependent sticking probabilities and rate constants. This model is extended to include diffusion and gas global coupling. Diffusion is included through a mass-balance equation which couples the migration of CO with the phase transition. Gas global coupling is introduced considering realistic values of the pumping flow, the reactor volume and the size of the crystal.

This chapter reviews recent research on the interaction of magnetic fields with MHD turbulence, with particular application to the question of the influence of Lorentz forces on the efficiency of large-scale field generation.

Three-dimensional non-linear magnetoconvection in a strongly stratified compressible layer exhibits different patterns as the strength of the imposed magnetic field is reduced. There is a transition from a magnetically dominated regime, with small-scale convection in slender hexagonal cells, to a convectively dominated regime, with clusters of broad rising plumes that confine the magnetic flux to narrow lanes where fields are locally intense. Both patterns can coexist for intermediate field strengths, giving rise to flux separation: clumps of vigorously convecting plumes, from which magnetic flux has been excluded, are segregated from regions with strong fields and small-scale convection. A systematic numerical investigation of these different states shows that flux separation can occur over a significant parameter range and that there is also hysteresis. The results are related to the fine structure of magnetic fields in sunspots and in the quiet Sun.

We consider the dynamics of convection in a strong vertical magnetic field, and in the presence of rapid rotation. In both these cases, in circumstances which can be realized in the laboratory, the onset of convection is in the form of tall thin cells. Because of this, the dynamics near onset is characterized by an interaction between the cellular modes and the horizontally averaged temperature profile. The effects on the dynamics are slight in the case of a Boussinesq fluid. However in both cases, when the layer is stratified (non-Boussinesq), the convection can lose stability to oscillations close to onset. Properties of the oscillations and their stability to long-wavelength modulation are extensively investigated.

Properties of the instability of steady periodic patterns induced by nonlinear interactions with a conserved quantity (the Matthews–Cox instability) are considered for very long wavelength modes. It is shown that such modes are generically subcritically unstable, in contrast to the situation in smaller domains. A simple front-type model is developed, giving a guide to the range of existence of nonlinearly modulated patterns.

Numerical experiments on three-dimensional magnetoconvection in a stratified compressible layer reveal a range of different patterns, depending on the strength of the imposed magnetic field. As the field is decreased there is a transition from small-scale plumes, in the magnetically dominated regime, to large-scale vigorous plumes when the field is dominated by the motion. In the intermediate regime magnetic flux separates from the motion, so that there are almost field-free regions, with clusters of vigorous plumes, surrounded by regions where the Lorentz force is strong enough to control the dynamics. There is a range of field strengths where either small-scale plumes or flux-separated solutions can persist, depending on initial conditions for the computation. These results can be related to magnetic features at the surface of the Sun.

A systematic computational survey of magnetoconvection in a square box with periodic lateral boundary conditions is described. This investigation demonstrates the way in which numerical experiments can be combined with techniques of nonlinear dynamics and applied to specific fluid mechanical problems. Considerations of symmetry and associated group theory can be exploited in order to explain the sequence in which the relevant bifurcations must occur. The interaction between magnetic fields and convection was chosen because of its astrophysical importance and because the nonlinear Lorentz force leads to an especially rich and interesting range of behaviour. As the solutions become progressively more nonlinear, there is a transition from an ordered pattern with a simple planform to disordered spatiotemporal behaviour via an intermediate state with intermittent bursts. The solutions are sensitive not only to variations in the key physical parameters (such as diffusivity ratios and field strengths) but also to changes in the aspect ratio of the computational box. In wide boxes a new physical effect - flux separation - appears as a consequence of long wavelength modulation.

The interaction between magnetic fields and convection is interesting both because of its astrophysical importance and because the nonlinear Lorentz force leads to an especially rich variety of behaviour. We present several sets of computational results for magnetoconvection in a square box, with periodic lateral boundary conditions, that show transitions from steady convection with an ordered planform through a regime with intermittent bursts to complicated spatiotemporal behaviour. The constraints imposed by the square lattice are relaxed as the aspect ratio is increased. In wide boxes we find a new regime, in which regions with strong fields are separated from regions with vigorous convection. We show also how considerations of symmetry and associated group theory can be used to explain the nature of these transitions and the sequence in which the relevant bifurcations occur.

Most new magnetic flux arrives at the surface of the Sun through the emergence of bipolar regions. These regions, which are often associated with a pair of sunspots, are generally thought to correspond to Ω-shaped flux tubes breaking through the solar surface. It seems likely that these flux tubes originate in the convective overshoot layer below the convection zone; an outstanding problem for theorists, therefore, is to account for this rise. Here we present results of a large number of two- and three-dimensional numerical simulations of buoyant magnetic flux tubes. Untwisted magnetic flux tubes are severely deformed as a consequence of the formation of regions of strong vorticity within the tube. Interactions between tubes can lead to trapping of coherent regions of field. For tubes constrained to rise in a two-dimensional manner (no variation along the tube) this deformation and loss of identity of the tube can be reduced if the magnetic field of the tube is twisted. However, in three dimensions, strongly twisted tubes may, depending on the precise form of the twist, be susceptible to the kink instability. We have also considered the influence of background rotation on the rise and breakup of magnetic flux tubes—restricting attention to untwisted tubes. The vertical component of rotation inhibits the breakup of flux tubes and the horizontal component decreases the rate at which tubes rise. With an oblique rotation vector the reflectional symmetry of the tube is broken and, although the tube initially breaks up, after some time the rotation causes the fragments to cluster together.