In the quiet Sun, convective motions form a characteristic granular pattern, with broad upflows enclosed by a network of narrow downflows. Magnetic fields tend to accumulate in the intergranular lanes, forming localised flux concentrations. One of the most plausible explanations for the appearance of these quiet Sun magnetic features is that they are generated and maintained by dynamo action resulting from the local convective motions at the surface of the Sun. Motivated by this idea, we describe high resolution numerical simulations of nonlinear dynamo action in a (fully) compressible, non-rotating layer of electrically-conducting fluid. The dynamo properties depend crucially upon various aspects of the fluid. For example, the magnetic Reynolds number (Rm) determines the initial growth rate of the magnetic energy, as well as the final saturation level of the dynamo in the nonlinear regime. We focus particularly upon the ways in which the Rm-dependence of the dynamo is influenced by the level of stratification within the domain. Our results can be related, in a qualitative sense, to solar observations.

Three-dimensional (3D) hydrodynamic simulations of shell oxygen burning by Meakin & Arnett (2007b) exhibit bursty, recurrent fluctuations in turbulent kinetic energy. These are shown to be due to a global instability in the convective region, which has been suppressed in simulations of stellar evolution which use mixing-length theory (MLT). Quantitatively similar behavior occurs in the model of a convective roll (cell) of Lorenz (1963), which is known to have a strange attractor that gives rise to random fluctuations in time. An extension of the Lorenz model, which includes Kolmogorov damping and nuclear burning, is shown to exhibit bursty, recurrent fluctuations like those seen in the 3D simulations. A simple model of a convective layer (composed of multiple Lorenz cells) gives luminosity fluctuations which are suggestive of irregular variables (red giants and supergiants, see Schwarzschild (1975). Details and additional discussion may be found in Arnett & Meakin (2011).

Apparent inconsistencies between Arnett, Meakin, & Young (2009) and Nordlund, Stein, & Asplund (2009) on the nature of convective driving have been resolved, and are discussed.

Current-driven instabilities in stellar radiation zones, to which we refer as Tayler instabilities, can lead to complex nonlinear evolutions. It is of fundamental interest whether magnetically driven turbulence can lead to dynamo action in these radiative zones. We investigate initial-value simulations in a 3D spherical shell including differential rotation. The Tayler instability is connected with a very weak kinetic helicity, stronger current helicity, and a positive αφφ in the northern hemisphere. The amplitudes are small compared to the effect of the tangential cylinder producing an eddy with negative kinetic helicity and negative αφφ in the northern hemisphere. The αφφ from the Tayler instability reaches about 1% of the rms velocity.

Recently, Silvers et al. (2009b), using numerical simulations, confirmed the existence of a double diffusive magnetic buoyancy instability of a layer of horizontal magnetic field produced by the interaction of a shear velocity field with a weak vertical field. Here, we demonstrate the longer term nonlinear evolution of such an instability in the simulations. We find that a quasi two-dimensional interchange instability rides (or “surfs”) on the growing shear-induced background downstream field gradients. The region of activity expands since three-dimensional perturbations remain unstable in the wake of this upward-moving activity front, and so the three-dimensional nature becomes more noticeable with time.

Magnetic reconnection is an important process that is prevalent in a wide range of astrophysical bodies. It is the mechanism that permits magnetic fields to relax to a lower energy state through the global restructuring of the magnetic field and is thus associated with a range of dynamic phenomena such as solar flares and CMEs. The characteristics of three-dimensional reconnection are reviewed revealing how much more diverse it is than reconnection in two dimensions. For instance, three-dimensional reconnection can occur both in the vicinity of null points, as well as in the absence of them. It occurs continuously and continually throughout a diffusion volume, as opposed to at a single point, as it does in two dimensions. This means that in three-dimensions field lines do not reconnect in pairs of lines making the visualisation and interpretation of three-dimensional reconnection difficult.

By considering particular numerical 3D magnetohydrodynamic models of reconnection, we consider how magnetic reconnection can lead to complex magnetic topologies and current sheet formation. Indeed, it has been found that even simple interactions, such as the emergence of a flux tube, can naturally give rise to ‘turbulent-like’ reconnection regions.

Following earlier work by Hughes & Proctor (2009) on the role of velocity shear in convectively driven dynamos, we present preliminary results on the nature of dynamo action due to modified flows derived by filtration from the full convective flow. The results suggest that filtering the flow fields has surprisingly little effect on the dynamo growth rates.

Records of the solar magnetic field extend back for millennia, and its surface properties have been observed for centuries, while helioseismology has recently revealed the Sun's internal rotation and the presence of a tachocline. Dynamo theory has developed to explain these observations, first with idealized models based on mean-field electrodynamics and, more recently, by direct numerical simulation, notably with the ASH code at Boulder. These results, which suggest that cyclic activity relies on the presence of the tachocline, and that its modulation is chaotic (rather than stochastic), will be critically reviewed. Similar theoretical approaches have been followed in order to explain the magnetic properties of other main-sequence stars, whose fields can be mapped by Zeeman-Doppler imaging. Of particular interest is the behaviour of fully convective, low-mass stars, which lack any tachocline but are nevertheless extremely active.

We review recent insights into the dynamics of the solar convection zone obtained from global numerical simulations, focusing on two recent developments in particular. The first is quasi-cyclic magnetic activity in a long-duration dynamo simulation. Although mean fields comprise only a few percent of the total magnetic energy they exhibit remarkable order, with multiple polarity reversals and systematic variability on time scales of 6-15 years. The second development concerns the maintenance of the meridional circulation. Recent high-resolution simulations have captured the subtle nonlinear dynamical balances with more fidelity than previous, more laminar models, yielding more coherent circulation patterns. These patterns are dominated by a single cell in each hemisphere, with poleward and equatorward flow in the upper and lower convection zone respectively. We briefly address the implications of and future of these modeling efforts.

The understanding of fossil fields origin, topology, and stability is one of the corner stones of the stellar magnetism theory. On one hand, since they survive on secular time scales, they may modify the structure and the evolution of their host stars. On the other hand, they must have a complex stable structure since it has been demonstrated that the simplest purely poloidal or toroidal fields are unstable on dynamical time scales. In this context, the only stable stellar configurations found today are those resulting from numerical simulations by Braithwaite and collaborators who studied the evolution of an initial stochastic magnetic field, which relaxes with a selective decay of magnetic helicity and energy, on mixed stable configurations (poloidal and toroidal) that seem to be in equilibrium and then diffuse. In this talk, we report the semi-analytical investigation of such an equilibrium field in the axisymmetric case. We use variational methods, which describe selective decay of magnetic helicity and energy during MHD relaxation, and we identify a supplementary invariant due to the stable stratification of stellar radiation zones. This leads to states that generalize force-free Taylor's relaxation states studied in plasma laboratory experiments that become non force-free in the stellar case. Moreover, astrophysical applications are presented and the stability of obtained configurations is studied.

Convectively driven dynamos with rotation generating magnetic fields on scales large compared with the scale of the turbulent eddies are being reviewed. It is argued that such fields can be understood as the result of an α effect. Simulations in Cartesian domains show that such large-scale magnetic fields saturate on a time scale compatible with the resistive one, suggesting that the magnitude of the α effect is here still constrained by approximate magnetic helicity conservation. It is argued that, in the absence of shear and/or any other known large-scale dynamo effects, these simulations prove the existence of turbulent α2-type dynamos. Finally, recent results are discussed in the context of solar and stellar dynamos.

We present the effects of introducing results of 3D MHD simulations of buoyant magnetic fields in the solar convection zone in 2D mean-field Babcock-Leighton models. In particular, we take into account the time delay introduced by the rise time of the toroidal structures from the base of the convection zone to the solar surface. We find that the delays produce large temporal modulation of the cycle amplitude even when strong and thus rapidly rising flux tubes are considered. The study of a reduced model reveals that aperiodic modulations of the solar cycle appear after a sequence of period doubling bifurcations typical of non-linear systems. We also discuss the memory of such systems and the conclusions which may be drawn concerning the actual solar cycle variability.

This paper discusses nonlinear dynamos where the nonlinearity arises directly via the Lorentz force in the Navier-Stokes equation, and leads to a situation where the Lorentz force and the velocity and the magnetic field are in direct competition over substantial regions of the flow domain. Filamentary and non-filamentary dynamos are contrasted, and the concept of Alfvénic dynamos with almost equal magnetic and kinetic energies is reviewed via examples. So far these remain in the category of toy models; the paper concludes with a discussion of whether similar dynamos are likely to exist in astrophysical objects, and whether they can model the solar cycle.

We review some of the recent results obtained in MHD turbulence, as encountered in many astrophysical objects. We focus attention on the lack of universality in such flows, including in the simplest case (no externally imposed magnetic field, no forcing, unit magnetic Prandtl number). Several parameters can foster such a breakdown of classical Kolmogorov scaling, such as the presence of velocity-magnetic field correlations, or of magnetic helicity and the role of the interplay between nonlinear eddies and Alfvén waves. A link with avalanche processes is also discussed. These findings have led to the conjecture of the emergence of a new paradigm for MHD turbulence, as a possibly unsettled competition between several dynamical phenomena.

As compressible convection has inherent up/down asymmetry, overshooting above and below a convection zone behave differently. In downward overshooting, the narrow down-flow columns dynamically play an important role. It is customary, and reasonable, to use the downward flux of kinetic energy as a proxy for overshooting. In the upward situation, the flux of kinetic energy can take on different signs near the upper boundary of the convection zone, and its magnitude is generally small. It cannot make a good proxy for overshooting. This paper discusses the results of a set of numerical experiments that investigate the problem of overshooting above a convection zone. Particle tracing and color advection are used to follow the mixing process. The overshoot region above a convection zone is found to contain multiple counter cell layers.

We discuss the problem of turbulent dynamo and illustrate it through numerical simulations and few results from the VKS experiment.

Thermal convection plays a very important role in the structure and evolution of stars, as it is one of the main physical processes that transport heat from their interior where it is released, to the surface where it is radiated into space. Much progress has been achieved in modeling that process during the past 60 years, and I shall recall here how Juri Toomre has greatly contributed to it.

We reflect upon a few of the research challenges in stellar convection and dynamo theory that are likely to be addressed in the next five or so years. These deal firstly with the Sun and continuing study of the two boundary layers at the top and bottom of its convection zone, namely the tachocline and the near-surface shear layer, both of which are likely to have significant roles in how the solar dynamo may be operating. Another direction concerns studying core convection and dynamo action within the central regions of more massive A, B and O-type stars, for the magnetism may have a key role in controlling the winds from these stars, thus influencing their ultimate fate. Such studies of the interior dynamics of massive stars are becoming tractable with recent advances in codes and supercomputers, and should also be pursued with some vigor.

Recent work has shown a relationship between between the equations of Reduced Magnetohydrodynamics (RMHD), used to model magnetic fusion laboratory experiments, and incompressible magnetoconvection (IMC), employed in the simulation of astrophysical fluid dynamics (AFD), which means that the two systems are mathematically equivalent in certain geometries. Limitations on the modelling of RMHD, which were found over twenty years ago, are reviewed for an AFD audience, together with hitherto unpublished material on the role of finite-time singularities in the discrete equations used to model fluid dynamical systems. Possible implications for turbulence modelling are mentioned.

Main-sequence massive stars possess convective cores that likely harbor strong dynamo action. To assess the role of core convection in building magnetic fields within these stars, we employ the 3-D anelastic spherical harmonic (ASH) code to model turbulent dynamics within a 10 M⊙ main-sequence (MS) B-type star rotating at 4 Ω⊙. We find that strong (900 kG) magnetic fields arise within the turbulence of the core and penetrate into the stably stratified radiative zone. These fields exhibit complex, time-dependent behavior including reversals in magnetic polarity and shifts between which hemisphere dominates the total magnetic energy.

Internal gravity waves are excited at the interface of convection and radiation zones of a solar-type star, by the tidal forcing of a short-period planet. The fate of these waves as they approach the centre of the star depends on their amplitude. We discuss the results of numerical simulations of these waves approaching the centre of a star, and the resulting evolution of the spin of the central regions of the star and the orbit of the planet. If the waves break, we find efficient tidal dissipation, which is not present if the waves perfectly reflect from the centre. This highlights an important amplitude dependence of the (stellar) tidal quality factor Q′, which has implications for the survival of planets on short-period orbits around solar-type stars, with radiative cores.