Current Status of Turbulent Dynamo Theory

Several recent advances in turbulent dynamo theory are reviewed. High resolution simulations of small-scale and large-scale dynamo action in periodic domains are compared with each other and contrasted with similar results at low magnetic Prandtl numbers. It is argued that all the different cases show similarities at intermediate length scales. On the other hand, in the presence of helicity of the turbulence, power develops on large scales, which is not present in non-helical small-scale turbulent dynamos. At small length scales, differences occur in connection with the dissipation cutoff scales associated with the respective value of the magnetic Prandtl number. These differences are found to be independent of whether or not there is large-scale dynamo action. However, large-scale dynamos in homogeneous systems are shown to suffer from resistive slow-down even at intermediate length scales. The results from simulations are connected to mean field theory and its applications. Recent work on magnetic helicity fluxes to alleviate large-scale dynamo quenching, shear dynamos, nonlocal effects and magnetic structures from strong density stratification are highlighted. Several insights which arise from analytic considerations of small-scale dynamos are discussed.     

Electromagnetic Induction Effects and Dynamo Action in the Hermean System

Embedded in a large mass density and strong interplanetary magnetic field solar wind environment and equipped with a magnetic field of minor strength, planet Mercury exhibits a small magnetosphere vulnerable to severe solar wind buffeting. This causes large variations in the size of the magnetosphere and its associated currents. External fields are of far more importance than in the terrestrial case and of a size comparable to any internal, dynamo-generated field. Induction effects in the planetary interior, dominated by its huge core, are thought to play a much more prominent role in the Hermean magnetosphere compared to any of its companions. Furthermore, the external fields may cause planetary dynamo amplification much as discussed for the Galilean moons Io and Ganymede, but with the ambient field generated by the dynamo and its magnetic field-solar wind interaction.     

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.     

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.     

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.     

Daily variations of the geomagnetic field

The atmospheric dynamo theory of the daily magnetic variations (S) has received substantial support from recent observational and theoretical work. In particular, several features of the variations, such as their remarkable enhancement close to the dip equator and other effects indicating a strong control by the main geomagnetic field, are well explained by the dynamo theory. Also the detection of ionospheric currents by instrumental rockets has confirmed an essential part of the theory.Considerable impetus was given to their study by the acquirement of much new data on magnetic variations during the IGY-IQSY period. Additional observations in the Pacific area were obtained during the IQSY by the establishment of four island stations equipped with newly developed magnetometers. A major advance at other stations was the development of automatic standard observatories using nuclear magnetometers.Several methods for the world-wide analysis of the S-field have been developed. A possibility now being studied is the completely automatic evaluation and construction by computers of ionospheric current charts for any day and any epoch UT.Some theoretical and statistical papers are briefly reviewed. These include discussions of the day-to-day variability of S, seasonal changes of the S-field, the nature of the equatorial electrojet, the possibility of solar wind effects contributing to the daily variations, and the modification of the dynamo theory to take account of the possible flow of electric current from the ionosphere along magnetic lines of force in the magnetosphere.Finally, an attempt to extend the dynamo theory of S by treating the ionosphere as a three-dimensional medium, instead of regarding it as a thin shell, has revealed that, although the relations between the horizontal components of electric field and current density in the dynamo layer are given with reasonable accuracy by the well-known layer equations, the assumption, implicit in the thin shell treatment, that the horizontal currents are non-divergent is not in fact true. Hence a revision of some earlier theoretical work on S appears necessary.     

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.     

Plasma-physical aspects of the solar cycle

Mass motions below the photosphere drive the solar cycle which is associated with variations in the magnetic field structure and accompanying phenomena. In addition to semi-empirical models, dynamo theories have been used to explain the solar cycle. The emergence of magnetic field generated by these mechanisms and its expansion into the corona involves many plasma physical processes. Magnetic buoyancy aids the expulsion of magnetic flux. The corona may respond dynamically or by continually adjusting to a quasi-static force-free or pressure-balanced equilibrium. The formation and disruption of current sheets is significant for the overall structure of the coronal magnetic field and the physics of quiescent prominences. The corona has a fine structure consisting of magnetic loops. The structure and stability of these are important as they are one of the underlying elements which make up the corona.     

Solar Weather Event Modelling and Prediction

Key drivers of solar weather and mid-term solar weather are reviewed by considering a selection of relevant physics- and statistics-based scientific models as well as a selection of related prediction models, in order to provide an updated operational scenario for space weather applications. The characteristics and outcomes of the considered scientific and prediction models indicate that they only partially cope with the complex nature of solar activity for the lack of a detailed knowledge of the underlying physics. This is indicated by the fact that, on one hand, scientific models based on chaos theory and non-linear dynamics reproduce better the observed features, and, on the other hand, that prediction models based on statistics and artificial neural networks perform better. To date, the solar weather prediction success at most time and spatial scales is far from being satisfactory, but the forthcoming ground- and space-based high-resolution observations can add fundamental tiles to the modelling and predicting frameworks as well as the application of advanced mathematical approaches in the analysis of diachronic solar observations, that are a must to provide comprehensive and homogeneous data sets.     

Theory and Modeling of Planetary Dynamos

Numerical dynamo models are increasingly successful in modeling many features of the geomagnetic field. Moreover, they have proven to be a useful tool for understanding how the observations connect to the dynamo mechanism. More recently, dynamo simulations have also ventured to explain the surprising diversity of planetary fields found in our solar system. Here, we describe the underlying model equations, concentrating on the Boussinesq approximations, briefly discuss the numerical methods, and give an overview of existing model variations. We explain how the solutions depend on the model parameters and introduce the primary dynamo regimes. Of particular interest is the dependence on the Ekman number which is many orders of magnitude too large in the models for numerical reasons. We show that a minor change in the solution seems to happen at $\mbox {E}=3\mbox {$\times 10^{-6}$}$ whose significance, however, needs to be explored in the future. We also review three topics that have been a focus of recent research: field reversal mechanisms, torsional oscillations, and the influence of Earth’s thermal mantle structure on the dynamo. Finally we discuss the possibility of tidally or precession driven planetary dynamos.     

Solar Cycle Forecasting

Predicting the behavior of a solar cycle after it is well underway (2–3 years after minimum) can be done with a fair degree of skill using auto-regression and curve fitting techniques that don’t require any knowledge of the physics involved. Predicting the amplitude of a solar cycle near, or before, the time of solar cycle minimum can be done using precursors such as geomagnetic activity and polar fields that do have some connection to the physics but the connections are uncertain and the precursors provide less reliable forecasts. Predictions for the amplitude of cycle 24 using these precursor techniques give drastically different values. Recently, dynamo models have been used directly with assimilated data to predict the amplitude of sunspot cycle 24 but have also given significantly different predictions. While others have questioned both the predictability of the solar cycle and the ability of current dynamo models to provide predictions, it is clear that cycle 24 will help to discriminate between some opposing dynamo models.     

Origin of Sunspots

Sunspots, seen as cool regions on the surface of the Sun, are a thermal phenomenon. Sunspots are always associated with bipolar magnetic loops that break through the solar surface. Thus to explain the origin of sunspots we have to understand how the magnetic field originates inside the Sun and emerges at its surface. The field predicted by mean-field dynamo theories is too weak by itself to emerge at the surface of the Sun. However, because of the turbulent character of solar convection the fields generated by dynamo are intermittent – i.e., concentrated into ropes or sheets with large spaces in between. The intermittent fields are sufficiently strong to be able to emerge at the solar surface, in spite of the fact that their mean (average) value is weak. It is suggested here that magnetic fields emerge at the solar surface at those random times and places when the total magnetic field (mean field plus fluctuations) exceeds the threshold for buoyancy. The clustering of coherently emerged loops results in the formation of a sunspot. A non-axisymmetric enhancement of the underlying magnetic field causes in the clustering of sunspots forming sunspot groups, clusters of activity and active longitudes. The mean field, which is not directly observable, is also important, being responsible for the ensemble regularities of sunspots, such as Hale's law of sunspot polarities and the 11-year periodicity.     

Modulation of Cosmic Rays in the Heliosphere From Solar Minimum to Maximum: a Theoretical Perspective

The modulation of galactic cosmic rays in the heliosphere seems to be dominated by four major mechanisms: convection, diffusion, drifts (gradient, curvature and current sheet), and adiabatic energy losses. In this regard the global structure of the solar wind, the heliospheric magnetic field (HMF), the current sheet (HCS), and that of the heliosphere itself play major roles. Individually, the four mechanisms are well understood, but in combination, the complexity increases significantly especially their evolvement with time - as a function of solar activity. The Ulysses observations contributed significantly during the past solar minimum modulation period to establish the relative importance of these major mechanisms, leading to renewed interest in developing more sophisticated numerical models, and in the underlying physics, e.g., what determines the diffusion tensor. With increased solar activity, the relative contributions of the mentioned mechanisms change, but how they change and what causes these changes over an 11-year solar cycle is not well understood. It can therefore be expected that present and forthcoming observations during solar maximum activity will again produce very important insights into the causes of long-term modulation. In this paper the basic theory of solar modulation is reviewed for galactic cosmic rays. The influence of the Ulysses observations on the development of the basic theory and numerical models are discussed, especially those that have challenged the theory and models. Model-based predictions are shown for what might be encountered during the next solar minimum. Lastly, modulation theory and modelling are discussed for periods of maximum solar activity when a global reorganization of the HMF, and the HCS, occurs. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Vlasov theory of the equilibrium structure of tangential discontinuities in space plasmas

Extensive theoretical work has been performed on the equilibrium structure of tangential discontinuities (TDs) in collisionless plasmas. This paper reviews kinetic models based on steady-state solutions of the Vlasov equation. It is shown that most of the existing models are special cases of a generalized multi-species model. In this generalized model all particle populations -from both outer regions and from inside the layer — are described using a unique formalism for the velocity distribution functions. Because of their historical importance, the Harris and Sestero models are reviewed and deduced from the generalized model. The Lee and Kan model is also a special case of the generalized model. The generalized model, however, is also able to describe TDs with velocity shear and large angles of magnetic field rotation. Such a multi-species model with a large number of free parameters and different gradient scales illustrates many observable features of TDs, including their multiscale fine structure. Particular attention is paid to the magnetopause. Observed magnetopause crossings are simulated. The effects of the relative flow velocity and asymmetrical magnetic field profiles on the structure of the magnetopause and on its stability with respect to tearing perturbations are discussed. We also present calculations that demonstrate the potential of the generalized model in explaining the origin of discrete auroral arcs. Numerical simulations of solar wind TDs with heavy ions and a large spectrum of thicknesses are also feasible. This indicates that such a model is of fundamental importance for understanding the detailed structure of solar wind TDs, like those observed by the interplanetary spacecraft ULYSSES. The problems associated with the one-dimensional, time-independent Vlasov approach are discussed and a variational principle is suggested to reduce the arbitrariness resulting from the large number of free parameters.     

Flux-Transport Solar Dynamos

Large-scale solar dynamo models were first built by Parker (1955). Over the past half a century these models have evolved significantly. We discuss here the development of a class of large-scale dynamo models which include, along with the α-effect and Ω-effect, an important third process, flux transport by meridional circulation. We present the properties of this ‘flux-transport’ dynamo, including the crucial role meridional circulation plays in giving this dynamo predictive power.     

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.     

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.