Advances in Theory and Simulations of Large-Scale Dynamos

Recent analytical and computational advances in the theory of large-scale dynamos are reviewed. The importance of the magnetic helicity constraint is apparent even without invoking mean-field theory. The tau approximation yields expressions that show how the magnetic helicity gets incorporated into mean-field theory. The test-field method allows an accurate numerical determination of turbulent transport coefficients in linear and nonlinear regimes. Finally, some critical views on the solar dynamo are being offered and targets for future research are highlighted.     

Planetary Dynamos from a Solar Perspective

Direct numerical simulations of the geodynamo and other planetary dynamos have been successful in reproducing the observed magnetic fields. We first give an overview on the fundamental properties of planetary magnetism. We review the concepts and main results of planetary dynamo modeling, contrasting them with the solar dynamo. In planetary dynamos the density stratification plays no major role and the magnetic Reynolds number is low enough to allow a direct simulation of the magnetic induction process using microscopic values of the magnetic diffusivity. The small-scale turbulence of the flow cannot be resolved and is suppressed by assuming a viscosity far in excess of the microscopic value. Systematic parameter studies lead to scaling laws for the magnetic field strength or the flow velocity that are independent of viscosity, indicating that the models are in the same dynamical regime as the flow in planetary cores. Helical flow in convection columns that are aligned with the rotation axis play an important role for magnetic field generation and forms the basis for a macroscopic α-effect. Depending on the importance of inertial forces relative to rotational forces, either dynamos with a dominant axial dipole or with a small-scale multipolar magnetic field are found. Earth is predicted to lie close to the transition point between both classes, which may explain why the dipole undergoes reversals. Some models fit the properties of the geomagnetic field in terms of spatial power spectra, magnetic field morphology and details of the reversal behavior remarkably well. Magnetic field strength in the dipolar dynamo regime is controlled by the available power and found to be independent of rotation rate. Predictions for the dipole moment agree well with the observed field strength of Earth and Jupiter and moderately well for other planets. Dedicated dynamo models for Mercury, Saturn, Uranus and Neptune, which assume stably stratified layers above or below the dynamo region, can explain some of the unusual field properties of these planets.     

Dynamo Models for Planets Other Than Earth

Observations from planetary spacecraft missions have demonstrated a spectrum of dynamo behaviour in planets. From currently active dynamos, to remanent crustal fields from past dynamo action, to no observed magnetization, the planets and moons in our solar system offer magnetic clues to their interior structure and evolution. Here we review numerical dynamo simulations for planets other than Earth. For the terrestrial planets and satellites, we discuss specific magnetic field oddities that dynamo models attempt to explain. For the giant planets, we discuss both non-magnetic and magnetic convection models and their ability to reproduce observations of surface zonal flows and magnetic field morphology. Future improvements to numerical models and new missions to collect planetary magnetic data will continue to improve our understanding of the magnetic field generation process inside planets.     

The Origin of Mercury’s Internal Magnetic Field

Mariner 10 measurements proved the existence of a large-scale internal magnetic field on Mercury. The observed field amplitude, however, is too weak to be compatible with typical convective planetary dynamos. The Lorentz force based on an extrapolation of Mariner 10 data to the dynamo region is 10⁻⁴ times smaller than the Coriolis force. This is at odds with the idea that planetary dynamos are thought to work in the so-called magnetostrophic regime, where Coriolis force and Lorentz force should be of comparable magnitude. Recent convective dynamo simulations reviewed here seem to resolve this caveat. We show that the available convective power indeed suffices to drive a magnetostrophic dynamo even when the heat flow though Mercury’s core–mantle boundary is subadiabatic, as suggested by thermal evolution models. Two possible causes are analyzed that could explain why the observations do not reflect a stronger internal field. First, toroidal magnetic fields can be strong but are confined to the conductive core, and second, the observations do not resolve potentially strong small-scale contributions. We review different dynamo simulations that promote either or both effects by (1) strongly driving convection, (2) assuming a particularly small inner core, or (3) assuming a very large inner core. These models still fall somewhat short of explaining the low amplitude of Mariner 10 observations, but the incorporation of an additional effect helps to reach this goal: The subadiabatic heat flow through Mercury’s core–mantle boundary may cause the outer part of the core to be stably stratified, which would largely exclude convective motions in this region. The magnetic field, which is small scale, strong, and very time dependent in the lower convective part of the core, must diffuse through the stagnant layer. Here, the electromagnetic skin effect filters out the more rapidly varying high-order contributions and mainly leaves behind the weaker and slower varying dipole and quadrupole components (Christensen in Nature 444:1056–1058, 2006). Messenger and BepiColombo data will allow us to discriminate between the various models in terms of the magnetic fields spatial structure, its degree of axisymmetry, and its secular variation.     

Physical Causes of Solar Activity

The magnetic fields that dominate the structure of the Sun's atmosphere are controlled by processes in the solar interior, which cannot be directly observed. Magnetic activity is found in all stars with deep convective envelopes: young and rapidly rotating stars are very active but cyclic activity only appears in slow rotators. The Sun's 11-year activity cycle corresponds to a 22-year magnetic cycle, since the sunspot fields (which are antisymmetric about the equator) reverse at each minimum. The record of magnetic activity is aperiodic and is interrupted by episodes of reduced activity, such as the Maunder Minimum in the seventeenth century, when sunspots almost completely disappeared. The proxy record from cosmogenic isotopes shows that similar grand minima recur at intervals of around 200 yr. The Sun's large-scale field is generated by dynamo action rather than by an oscillator. Systematic magnetic cycles are apparently produced by a dynamo located in a region of weak convective overshoot at the base of the convection zone, where there are strong radial gradients in the angular velocity . The crucial parameter (the dynamo number) increases with increasing and kinematic (linear) theory shows that dynamo action can set in at an oscillatory (Hopf) bifurcation that is probably subcritical. Although it has been demonstrated that the whole process works in a self-consistent model, most calculations have relied on mean-field dynamo theory. This approach is physically plausible but can only be justified under conditions that do not apply in the Sun. Still, mean-field dynamos do reproduce the butterfly diagram and other key features of the solar cycle. An alternative approach is to study generic behaviour in low-order models, which exhibit two forms of modulation, associated with symmetry-breaking and with reduced activity. Comparison with observed behaviour suggests that modulation of the solar cycle is indeed chaotic, i.e. deterministically rather than stochastically driven.     

The Solar Dynamo

It is generally accepted that the strong toroidal magnetic fields that emerge through the solar surface in sunspots and active regions are formed by the action of differential rotation on a poloidal field, and then stored in or near the tachocline at the base of the Sun’s convection zone. The problem is how to explain the generation of a reversed poloidal field from this toroidal flux—a process that can be parametrised in terms of an α-effect related to some form of turbulent helicity. Here we first outline the principal patterns that have to be explained: the 11-year activity cycle, the 22-year magnetic cycle and the longer term modulation of cyclic activity, associated with grand maxima and minima. Then we summarise what has been learnt from helioseismology about the Sun’s internal structure and rotation that may be relevant to our subject. The ingredients of mean-field dynamo models are differential rotation, meridional circulation, turbulent diffusion, flux pumping and the α-effect: in various combinations they can reproduce the principal features that are observed. To proceed further, it is necessary to rely on large-scale computation and we summarise the current state of play.     

Laboratory Dynamo Experiments

Since the turn of the century, experiments have produced laboratory fluid dynamos that enable a study of the effect in controlled conditions. We review here magnetic induction processes that are believed to underlie dynamo action, and we present results of these dynamo experiments. In particular, we detail progress that have been made through the study of von Kármán flows, using gallium or sodium as working fluids.     

Planetary Magnetic Fields: Achievements and Prospects

The past decade has seen a wealth of new data, mainly from the Galilean satellites and Mars, but also new information on Mercury, the Moon and asteroids (meteorites). In parallel, there have been advances in our understanding of dynamo theory, new ideas on the scaling laws for field amplitudes, and a deeper appreciation on the diversity and complexity of planetary interior properties and evolutions. Most planetary magnetic fields arise from dynamos, past or present, and planetary dynamos generally arise from thermal or compositional convection in fluid regions of large radial extent. The relevant electrical conductivities range from metallic values to values that may be only about one percent or less that of a typical metal, appropriate to ionic fluids and semiconductors. In all planetary liquid cores, the Coriolis force is dynamically important. The maintenance and persistence of convection appears to be easy in gas giants and ice-rich giants, but is not assured in terrestrial planets because the quite high electrical conductivity of an iron-rich core guarantees a high thermal conductivity (through the Wiedemann-Franz law), which allows for a large core heat flow by conduction alone. This has led to an emphasis on the possible role of ongoing differentiation (growth of an inner core or “snow”). Although planetary dynamos mostly appear to operate with an internal field that is not very different from (2ρΩ/σ)^1/2 in SI units where ρ is the fluid density, Ω is the planetary rotation rate and σ is the conductivity, theoretical arguments and stellar observations suggest that there may be better justification for a scaling law that emphasizes the buoyancy flux. Earth, Ganymede, Jupiter, Saturn, Uranus, Neptune, and probably Mercury have dynamos, Mars has large remanent magnetism from an ancient dynamo, and the Moon might also require an ancient dynamo. Venus is devoid of a detectable global field but may have had a dynamo in the past. Even small, differentiated planetesimals (asteroids) may have been capable of dynamo action early in the solar system history. Induced fields observed in Europa and Callisto indicate the strong likelihood of water oceans in these bodies. The presence or absence of a dynamo in a terrestrial body (including Ganymede) appears to depend mainly on the thermal histories and energy sources of these bodies, especially the convective state of the silicate mantle and the existence and history of a growing inner solid core. As a consequence, the understanding of planetary magnetic fields depends as much on our understanding of the history and material properties of planets as it does on our understanding of the dynamo process. Future developments can be expected in our understanding of the criterion for a dynamo and on planetary properties, through a combination of theoretical work, numerical simulations, planetary missions (MESSENGER, Juno, etc.) and laboratory experiments.     

The Solar Dynamo: The Role of Penetration, Rotation and Shear on Convective Dynamos

In this paper I discuss the importance of turbulence, rotation, penetration and shear for solar dynamos (both local and global). An understanding of these processes is vital for progress towards a self-consistent theory for the generation of solar magnetic activity. I discuss the difficulties for large-scale field generation and suggest that large-scale solar magnetic activity may be driven by dynamos that arise owing to instabilities, with these dynamos modified by the presence of turbulence.     

The Solar Dynamo

Observations relevant to current models of the solar dynamo are presented, with emphasis on the history of solar magnetic activity and on the location and nature of the solar tachocline. The problems encountered when direct numerical simulation is used to analyse the solar cycle are discussed, and recent progress is reviewed. Mean field dynamo theory is still the basis of most theories of the solar dynamo, so a discussion of its fundamental principles and its underlying assumptions is given. The role of magnetic helicity is discussed. Some of the most popular models based on mean field theory are reviewed briefly. Dynamo models based on severe truncations of the full MHD equations are discussed.     

Magnetic Fields in Massive Stars, Their Winds, and Their Nebulae

Massive stars are crucial building blocks of galaxies and the universe, as production sites of heavy elements and as stirring agents and energy providers through stellar winds and supernovae. The field of magnetic massive stars has seen tremendous progress in recent years. Different perspectives—ranging from direct field measurements over dynamo theory and stellar evolution to colliding winds and the stellar environment—fruitfully combine into a most interesting and still evolving overall picture, which we attempt to review here. Zeeman signatures leave no doubt that at least some O- and early B-type stars have a surface magnetic field. Indirect evidence, especially non-thermal radio emission from colliding winds, suggests many more. The emerging picture for massive stars shows similarities with results from intermediate mass stars, for which much more data are available. Observations are often compatible with a dipole or low order multi-pole field of about 1 kG (O-stars) or 300 G to 30?kG (Ap/Bp stars). Weak and unordered fields have been detected in the O-star ζ Ori A and in Vega, the first normal A-type star with a magnetic field. Theory offers essentially two explanations for the origin of the observed surface fields: fossil fields, particularly for strong and ordered fields, or different dynamo mechanisms, preferentially for less ordered fields. Numerical simulations yield the first concrete stable (fossil) field configuration, but give contradictory results as to whether dynamo action in the radiative envelope of massive main sequence stars is possible. Internal magnetic fields, which may not even show up at the stellar surface, affect stellar evolution as they lead to a more uniform rotation, with more slowly rotating cores and faster surface rotation. Surface metallicities may become enhanced, thus affecting the mass-loss rates.     

Polarity Reversals from Paleomagnetic Observations and Numerical Dynamo Simulations

Recent advances in the study of geomagnetic field reversals are reviewed. These include studies of the transitional field during the last geomagnetic reversal and the last geomagnetic excursion based on paleomagnetic observations, and analysis of reversals in self-consistent 3D numerical dynamo simulations. Field models inferred from observations estimate reversal duration in the range of 1–10 kyr (depending on site location). The transitional fields during both the Matuyama/Brunhes reversal and the Laschamp excursion are characterized by low-latitude reversed flux formation and subsequent poleward migration. During both events the dipole as well as the non-dipole field energies decrease. However, while the non-dipole energy dominates the dipole energy for a period of 2 kyr in the reversal, the non-dipole energy merely exceeds the dipole energy for a very brief period during the excursion. Numerical dynamo simulations show that stronger convection, slower rotation, and lower electrical conductivity provide more favorable conditions for reversals. A non-dimensional number that depends on the typical length scale of the flow and represents the relative importance of inertial effects, termed the local Rossby number, seems to determine whether a dynamo will reverse or not. Stable polarity periods in numerical dynamos may last about 1 Myr, whereas reversals may last about 10 kyr. Numerical dynamo reversals often involve prolonged dipole collapse followed by shorter directional instability of the dipole axis, with advective processes governing the field variation. Magnetic upwellings from the equatorial inner-core boundary that produce reversed flux patches at low-latitudes of the core-mantle boundary could be significant in triggering reversals. Inferences from the observational and modeling sides are compared. We summarize with an outlook on some open questions and future prospects.     

Navier–Stokes analysis methods for turbulent jet flows with application to aircraft exhaust nozzles

This article presents the current status of computational fluid dynamics (CFD) methods as applied to the simulation of turbulent jet flowfields issuing from aircraft engine exhaust nozzles. For many years, Reynolds-averaged Navier–Stokes (RANS) methods have been used routinely to calculate such flows, including very complex nozzle configurations. RANS methods replace all turbulent fluid dynamic effects with a turbulence model. Such turbulence models have limitations for jets with significant three-dimensionality, compressibility, and high temperature streams. In contrast to the RANS approach, direct numerical simulation (DNS) methods calculate the entire turbulent energy spectrum by resolving all turbulent motion down to the Kolmogorov scale. Although this avoids the limitations associated with turbulence modeling, DNS methods will remain computationally impractical in the foreseeable future for all but the simplest configurations. Large-Eddy simulation (LES) methods, which directly calculate the large-scale turbulent structures and reserve modeling only for the smallest scales, have been pursued in recent years and may offer the best prospects for improving the fidelity of turbulent jet flow simulations. A related approach is the group of hybrid RANS/LES methods, where RANS is used to model the small-scale turbulence in wall boundary layers and LES is utilized in regions dominated by the large-scale jet mixing. The advantages, limitations, and applicability of each approach are discussed and recommendations for further research are presented.     

Dynamo Scaling Laws and Applications to the Planets

Scaling laws for planetary dynamos relate the characteristic magnetic field strength, characteristic flow velocity and other properties to primary quantities such as core size, rotation rate, electrical conductivity and heat flux. Many different scaling laws have been proposed, often relying on the assumption of a balance of Coriolis force and Lorentz force in the dynamo. Their theoretical foundation is reviewed. The advent of direct numerical simulations of planetary dynamos and the ability to perform them for a sufficiently wide range of control parameters allows to test the scaling laws. The results support a magnetic field scaling that is not based on a force balance, but on the energy flux available to balance ohmic dissipation. In its simplest form, it predicts a field strength that is independent of rotation rate and electrical conductivity and proportional to the cubic root of the available energy flux. However, rotation rate controls whether the magnetic field is dipolar or multipolar. Scaling laws for velocity, heat transfer and ohmic dissipation are also discussed. The predictions of the energy-based scaling law agree well with the observed field strength of Earth and Jupiter, but for other planets they are more difficult to test or special pleading is required to explain their field strength. The scaling law also explains the very high field strength of rapidly rotating low-mass stars, which supports its rather general validity.     

Astrophysical Hydromagnetic Turbulence

Recent progress in astrophysical hydromagnetic turbulence is being reviewed. The physical ideas behind the now widely accepted Goldreich–Sridhar model and its extension to compressible magnetohydrodynamic turbulence are introduced. Implications for cosmic ray diffusion and acceleration is being discussed. Dynamo-generated magnetic fields with and without helicity are contrasted against each other. Certain turbulent transport processes are being modified and often suppressed by anisotropy and inhomogeneities of the turbulence, while others are being produced by such properties, which can lead to new large-scale instabilities of the turbulent medium. Applications of various such processes to astrophysical systems are being considered.     

Coronal magnetic field evolution under reconnective relaxation

The nonlinear evolution of a partially open coronal magnetic configuration is considered, assuming that corona responds to photospheric footpoint motions by small-scale reconnection events that produce a relaxed lower-energy state while conserving the global magnetic helicity of the system. The results of numerical calculations for such a relaxed equilibrium show an essential role of the amount of helicity injected to the closed-field region. If photospheric perturbations are incoherent (small-scale shearing with inefficient helicity injection), the relaxed state becomes close to an initial potential field. In this case reconnective relaxation does not result in a substantial global evolution, just providing heating of the corona (Vekstein et al, 1993). On the contrary, sufficient injection of the magnetic helicity can lead to a considerable restructuring of the coronal magnetic configuration, with possible change of its topology (formation of magnetic islands), and even catastrophic loss of equilibrium (Wolfson et al, 1994)     

Planetary magnetism in the outer solar system

A brief review of the salient considerations which apply to the existence of magnetic fields in connection with planetary and subplanetary objects in the outer solar system is given. Consideration is given to internal dynamo fields, fields which might originate from interaction with the solar wind or magnetospheres (externally driven dynamos) and lastly fossil magnetic fields such as have been discovered on the Moon. Where possible, connection is made between magnetism, means of detection, and internal body properties.This is one of the publications by the Science Advisory Group.     

Paleomagnetic Records of Meteorites and Early Planetesimal Differentiation

The large-scale compositional structures of planets are primarily established during early global differentiation. Advances in analytical geochemistry, the increasing diversity of extraterrestrial samples, and new paleomagnetic data are driving major changes in our understanding of the nature and timing of these early melting processes. In particular, paleomagnetic studies of chondritic and small-body achondritic meteorites have revealed a diversity of magnetic field records. New, more sensitive and highly automated paleomagnetic instrumentation and an improved understanding of meteorite magnetic properties and the effects of shock, weathering, and other secondary processes are permitting primary and secondary magnetization components to be distinguished with increasing confidence. New constraints on the post-accretional histories of meteorite parent bodies now suggest that, contrary to early expectations, few if any meteorites have been definitively shown to retain records of early solar and protoplanetary nebula magnetic fields. However, recent studies of pristine samples coupled with new theoretical insights into the possibility of dynamo generation on small bodies indicate that some meteorites retain records of internally generated fields. These results indicate that some planetesimals formed metallic cores and early dynamos within just a few million years of solar system formation.