Investigations of spherical kinematic mean-field dynamo models

The paper supplements an earlier one on the mean-field approach to spherical kinematic dynamo models (Rädler 1980a) by results of numerical investigations. A number of dynamo models working on the basis of the α²-mechanism are considered. Cases of pure α²-mechanism are studied, which includes only the simplest form of α-effect and no other induction effect, as well as cases with several additional effects due to fluctuating or mean motions. By the pure α²-mechanism axisymmetric and non-axisymmetric fields, can be excited and maintained with nearly equal ease. Part of the additional induction effects, however, clearly favour axisymmetric fields, and others non-axisymmetric fields. The non-axisymmetric fields are waves which travel in azimuthal direction, eastward or westward, depending on the models. For special dynamo models the transition from α² to αω-mechanism and properties of the latter are investigated. The results support the presumption that the αω-mechanism is able to maintain only axisymmetric but never non-axisymmetric fields. Conditions for the occurrence of non-oscillatory or oscillatory fields are discussed, and again the influence of additional induction effects is studied. There are further presented a model with βω-mechanism maintaining an axisymmetric non-oscillatory field, and models with two kinds of δω-mechanisms allowing axisymmetric non-oscillatory and oscillatory fields. Some ideas concerning dynamo models for the Earth, the Sun and magnetic stars are discussed. It seems possible to construct dynamo models for the Earth, on the basis of the α²-mechanism which explain not only the presence of a magnetic field with a strong dipole part but also the inclination of the dipole axis against the axis of rotation, the occurrence of higher multipoles and the westward drift of the non-axisymmetric parts. Models with αω, βω or δω-mechanism, which have to be considered in the case of a strong differential rotation inside the core, provide an explanation at first only of the axisymmetric parts of the field, and the non-axisymmetric parts have then to be interpreted, for example, as MAC-waves. As far as dynamo models for the Sun are concerned, in addition to the possibility of an αω-mechanism also that of a βω or δω-mechanism is discussed, which, however, does not look not very promising. In the models developed so far, which work with the αω-mechanism, only a few of the induction effects of fluctuating motions have been included; it seems necessary to investigate also influences of other effects. The sectorial structure of the solar magnetic field can hardly be understood in terms of the traditional mean-field concept. The magnetic stars possess fields which strongly deviate from symmetry with respect to the axis of rotation. The occurrence of such fields seems understandable only if there is no noticeable differential rotation. They can be maintained by an α²-mechanism but hardly by αω, βω or δω-mechanisms.     

A dynamo driven by zonal jets at the upper surface: Applications to giant planets

We present a dynamo mechanism arising from the presence of barotropically unstable zonal jet currents in a rotating spherical shell. The shear instability of the zonal flow develops in the form of a global Rossby mode, whose azimuthal wavenumber depends on the width of the zonal jets. We obtain self-sustained magnetic fields at magnetic Reynolds numbers greater than 10³. We show that the propagation of the Rossby waves is crucial for dynamo action. The amplitude of the axisymmetric poloidal magnetic field depends on the wavenumber of the Rossby mode, and hence on the width of the zonal jets. We discuss the plausibility of this dynamo mechanism for generating the magnetic field of the giant planets. Our results suggest a possible link between the topology of the magnetic field and the profile of the zonal winds observed at the surface of the giant planets. For narrow Jupiter-like jets, the poloidal magnetic field is dominated by an axial dipole whereas for wide Neptune-like jets, the axisymmetric poloidal field is weak.     

A dynamo model for axisymmetric and non-axisymmetric solar magnetic fields

More and more observations are showing a relatively weak, but persistent, non-axisymmetric magnetic field co-existing with the dominant axisymmetric field on the Sun. Its existence indicates that the non-axisymmetric magnetic field plays an important role in the origin of solar activity. A linear non-axisymmetric  α²– Ω  dynamo model is derived to explore the characteristics of the axisymmetric  ( m = 0)  and the first non-axisymmetric  ( m = 1)  modes and to provide a theoretical basis with which to explain the 'active longitude', 'flip-flop' and other non-axisymmetric phenomena. The model consists of an updated solar internal differential rotation, a turbulent diffusivity varying with depth, and an α-effect working at the tachocline in a rotating spherical system. The difference between the  α²–Ω  and the  α–Ω  models and the conditions that favour the non-axisymmetric modes under solar-like parameters are also presented.     

Global MHD modeling of Mercury's magnetosphere with applications to the MESSENGER mission and dynamo theory

We use a global magnetohydrodynamic (MHD) model to simulate Mercury's space environment for several solar wind and interplanetary magnetic field (IMF) conditions in anticipation of the magnetic field measurements by the MESSENGER spacecraft. The main goal of our study is to assess what characteristics of the internally generated field of Mercury can be inferred from the MESSENGER observations, and to what extent they will be able to constrain various models of Mercury's magnetic field generation. Based on the results of our simulations, we argue that it should be possible to infer not only the dipole component, but also the quadrupole and possibly even higher harmonics of the Mercury's planetary magnetic field. We furthermore expect that some of the crucial measurements for specifying the Hermean internal field will be acquired during the initial fly-bys of the planet, before MESSENGER goes into orbit around Mercury.     

Determination of the properties of Mercury's magnetic field by the MESSENGER mission

The MESSENGER mission to Mercury, to be launched in 2004, will provide an opportunity to characterize Mercury's internal magnetic field during an orbital phase lasting one Earth year. To test the ability to determine the planetary dipole and higher-order moments from measurements by the spacecraft's fluxgate magnetometer, we simulate the observations along the spacecraft trajectory and recover the internal field characteristics from the simulated observations. The magnetic field inside Mercury's magnetosphere is assumed to consist of an intrinsic multipole component and an external contribution due to magnetospheric current systems described by a modified Tsyganenko 96 model. Under the axis-centered-dipole approximation without correction for the external field the moment strength is overestimated by ∼4% for a simulated dipole moment of , and the error depends strongly on the magnitude of the simulated moment, rising as the moment decreases. Correcting for the external field contributions can reduce the error in the dipole term to a lower limit of ∼1-2% without a solar wind monitor. Dipole and quadrupole terms, although highly correlated, are then distinguishable at the level equivalent to an error in the position of an offset dipole of a few tens of kilometers. Knowledge of the external magnetic field is therefore the primary limiting factor in extracting reliable knowledge of the structure of Mercury's magnetic field from the MESSENGER observations.     

Possible relations between the rotational axis of the Earth's inner core and the magnetic dipole axis

The geomagnetic field is maintained by amagnetohydrodynamic dynamo process within the liquid outer core. The distribution of the associated electric currents is modified if the outer core is bounded by electrically conducting material. Then, eddy currents and the related magnetic fields are generated within these regions. In particular, the relative rigid rotation of the inner core produces a secondary magnetic field, which is superimposed on the dynamo field. The angle between the dipole axis of the total field and the rotational axis of the inner core is an important quantity needed for the theory of polar motion of the Earth. This angle is investigated for a broad spectrum of angular velocities of the inner core. To simplify the mathematical procedure, we model the dynamo field using an axisymmetric field generated by a system of electric currents within the outer core. The conductivity of the mantle is neglected. We find that the position of the dipole axis depends on the angular velocity of the inner core as well as on the distribution of the current system within the outer core. Coincidence of both axes can be reached if the angular velocity is high enough and if the current system is concentrated within a thin sheet near the outer core-inner core boundary.     

Dynamo action in an ambient field

The origin of global magnetic fields in celestial bodies is generally ascribed to dynamo action by fluid motions in their electrically conducting interiors. Some objects – e.g. close‐in extra‐solar planets or the moons of some giant planets – are embedded in ambient magnetic fields which modify the generation of the internal field in these bodies. Recently, the feedback of the magnetospheric field by Chapman‐Ferraro currents in the magnetopause onto the interior dynamo has been proposed to explain the observed weakness of the intrinsic magnetic field of planet Mercury. We study a simplified mean‐field dynamo model which allows us to analytically address various issues like positive and negative feedback situations, stationary versus time‐dependent solutions, and the stability of weak and strong field branches. We discuss the influence of the response function on the solutions when the external field depends on the strength of the intrinsic field like in the situation of the feedback dynamo of Mercury. We find that the feedback mechanism works only for a narrow range of dynamo numbers in the case of Mercury which makes him unique in our solar system. We conclude with some implications for extra‐solar planets (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The solar dynamo

The basic features of the solar activity mechanism are explained in terms of the dynamo theory of mean magnetic fields. The field generation sources are the differential rotation and the mean helicity of turbulent motions in the convective zone. A nonlinear effect of the magnetic field upon the mean helicity results in stabilizing the amplitude of the 22-year oscillations and forming a basic limiting cycle. When two magnetic modes (with dipole and quadrupole symmetry) are excited nonlinear beats appear, which may be related to the secular cycle modulation.The torsional waves observed may be explained as a result of the magnetic field effect upon rotation. The magnetic field evokes also meriodional flows.Adctual variations of the solar activity are nonperiodic since there are recurrent random periods of low activity of the Maunder minimum type. A regime of such a magnetic hydrodynamic chaos may be revealed even in rather simple nonlinear solar dynamo models.The solar dynamo gives rise also to three-dimensional, non-axisymmetric magnetic fields which may be related to a sector structure of the solar field.     

Behavior of planetary dynamos under the influence of external magnetic fields: Application to Mercury and Ganymede

For planets with strong intrinsic magnetic fields such as Earth and Jupiter, an external magnetic field is unlikely to affect the internal dynamo, but for bodies with weak intrinsic fields in appropriate environments, such as Mercury and Ganymede, the interaction with nearby field sources may determine the internal dynamics and overall behavior of their liquid iron cores. On the basis of simulations of such interactions using numerical models for fluid flow and dynamo generation, the parameter regimes for stable dipolar and multipolar reversing dynamo magnetic fields established for isolated systems can be substantially changed by the action of external sources. Relatively weak external background fields (as low as 2% of the averaged undisturbed field at the core-mantle boundary) may change the energy balance and alter the regime over which natural isolated dynamos operate.     

On non-axisymmetric magnetic equilibria in stars

In previous work, stable approximately axisymmetric equilibrium configurations for magnetic stars were found by numerical simulation. Here, I investigate the conditions under which more complex, non-axisymmetric configurations can form. I present numerical simulations of the formation of stable equilibria from turbulent initial conditions and demonstrate the existence of non-axisymmetric equilibria consisting of twisted flux tubes lying horizontally below the surface of the star, meandering around the star in random patterns. Whether such a non-axisymmetric equilibrium or a simple axisymmetric equilibrium forms depends on the radial profile of the strength of the initial magnetic field. The results could explain observations of non-dipolar fields on stars such as the B0.2 main-sequence star τ Sco or the pulsar 1E 1207.4-5209. The secular evolution of these equilibria due to Ohmic and buoyancy processes is also examined.     

Planetary dynamos: effects of electrically conducting flows overlying turbulent regions of magnetic field generation

A fully three-dimensional, nonlinear, time-dependent, multi-layered spherical kinematic dynamo model is used to study the effect on the observable external magnetic field of flow in an electrically conducting layer above a spherical turbulent dynamo region in which the α effect generates the magnetic field. It is shown that the amplitude and structure of an observable planetary magnetic field are largely determined by the magnitude and structure of the flow in the overlying layer. It is also shown that a strong-field planetary dynamo can be readily produced by the effect of an electrically conducting flow layer at the top of a convective core. The overlying layer and the underlying convective region constitute a magnetically strongly coupled system. Such overlying layers might exist at the top of the Earth's core due to chemical or thermal causes, in the cores of other terrestrial planets for similar reasons, and in Saturn due to the differentiation of helium from hydrogen. An electrically conducting and differentially rotating layer could exist above the metallic hydrogen region in Jupiter and affect the jovian magnetic field similar to the overlying layers in other planets. Lateral temperature gradients resulting in thermal winds drive the flow in the overlying layers. All planetary magnetic fields could be maintained by similar turbulent convective dynamos in the field-generation regions of planets with the differences among observable magnetic fields due to different circulations in the overlying electrically conducting layers.     

Dynamo action in simulations of penetrative solar convection with an imposed tachocline

We summarize new and continuing three-dimensional spherical shell simulations of dynamo action by convection allowed to penetrate downward into a tachocline of rotational shear. The inclusion of an imposed tachocline allows us to examine several processes believed to be essential in the operation of the global solar dynamo, including differential rotation, magnetic pumping, and the stretching and organization of fields within the tachocline. In the stably stratified core, our simulations reveal that strong axisymmetric magnetic fields (of ∼ 3000 G strength) can be built, and that those fields generally exhibit a striking antisymmetric parity, with fields in the northern hemisphere largely of opposite polarity to those in the southern hemisphere. In the convection zone above, fluctuating fields dominate over weaker mean fields. New calculations indicate that the tendency toward toroidal fields of antisymmetric parity is relatively insensitive to initial magnetic field configurations; they also reveal that on decade-long timescales, the magnetic fields can briefly enter (and subsequently emerge from) states of symmetric parity.We have not yet observed any overall reversals of the field polarity, nor systematic latitudinal propagation. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Speculations on the origin of the magnetic field of Mercury

The intrinsic magnetic field of Mercury may be generated by a dynamo process in a liquid core, despite the planet's slow rotation rate. It is argued that, provided the core of Mercury is partially liquid, it will be dynamically very similar to that of the Earth.     

Observations of Mercury's magnetic field

This paper presents a study of magnetic field data obtained by Mariner 10 during the third and final encounter with the planet Mercury on 16 March 1975. A well developed bow shock and modest magnetosphere, previously observed at first encounter on 29 March 1974, were again observed. In addition, a much stronger magnetic field near closest approach, 400γ versus 98γ, was observed at an altitude of 327 km and approximately 68° north Mercurian latitude. Spherical harmonic analysis of the data provides an estimate of the centered planetary magnetic dipole of 5.0 × 10²² gauss-cm³ with the axis tilted 12° to the rotation axis and in the same sense as Earth's. The interplanetary field was sufficiently different between first and third encounters that in addition to the very large field magnitude observed it argues strongly against a complex induction process generating the observed planetary field. While a possibility exists that Mercury possesses a remanent field due to magnetization early in its formation, a present day active dynamo seems to be a more likely candidate for its origin. The existence of such a dynamo argues for a mature planetary interior with a well-developed core.     

Transport Effects in the Evolution of the Global Solar Magnetic Field

The axisymmetric component of the large-scale solar magnetic fields has a pronounced poleward branch at higher latitudes. In order to clarify the origin of this branch we construct an axisymmetric model of the passive transport of the mean poloidal magnetic field in the convective zone, including meridional circulation, anisotropic diffusivity, turbulent pumping and density pumping. For realistic values of the transport coefficients we find that diffusivity is prevalent, and the latitudinal distribution of the field at the surface simply reflects the conditions at the bottom of the convective zone. Pumping effects concentrate the field to the bottom of the convective zone; a significant part of this pumping occurs in a shallow subsurface layer, normally not resolved in dynamo models. The phase delay of the surface poloidal field relative to the bottom poloidal field is found to be small. These results support the double dynamo wave models, may be compatible with some form of a mixed transport scenario, and exclude the passive transport theory for the origin of the polar branch.     

Force-free pulsar magnetospheres

The equations to the steady, axisymmetric, charged pulsar magnetosphere given recently by Michel (1973a, b) and Scharlemann and Wagoner (1972, preprint) are generalized to non-axisymmetric systems that are steady in the frame rotating with the pulsar. It is shown that in all nonsingular models with cylindrical (but not axial) symmetry, and with a non-zero magnetic field component parallel to the axis, the magnetic field lines emanating from the pulsar are all trapped within the light-cylinder, so that there is no net electromagnetic energy flow across the light cylinder.     

Turbulent dynamo in asymptotic giant branch stars

Using recent results on the operation of turbulent dynamos, we show that a turbulent dynamo may amplify a large-scale magnetic field in the envelopes of asymptotic giant branch (AGB) stars. We propose that a slow rotation of the AGB envelope can fix the symmetry axis, leading to the formation of an axisymmetric magnetic field structure. Unlike solar-type αω dynamos, the rotation has only a small role in amplifying the toroidal component of the magnetic field; instead of an αω dynamo we propose an α ² ω . The magnetic field may reach a value of     , where B _e is the equipartition (between the turbulent and magnetic energy densities) magnetic field. The large-scale magnetic field is strong enough for the formation of magnetic cool spots on the AGB stellar surface. The spots may regulate dust formation, and hence the mass-loss rate, leading to axisymmetric mass loss and the formation of elliptical planetary nebulae (PNe). Despite its role in forming cool spots, the large-scale magnetic field is too weak to play a dynamic role and directly influence the wind from the AGB star, as required by some models. We discuss other possible problems in models where the magnetic field plays a dynamic role in shaping the AGB winds, and argue that they cannot explain the formation of non-spherical PNe.     

The MESSENGER mission to Mercury: scientific objectives and implementation

Mercury holds answers to several critical questions regarding the formation and evolution of the terrestrial planets. These questions include the origin of Mercury's anomalously high ratio of metal to silicate and its implications for planetary accretion processes, the nature of Mercury's geological evolution and interior cooling history, the mechanism of global magnetic field generation, the state of Mercury's core, and the processes controlling volatile species in Mercury's polar deposits, exosphere, and magnetosphere. The MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) mission has been designed to fly by and orbit Mercury to address all of these key questions. After launch by a Delta 2925H-9.5, two flybys of Venus, and two flybys of Mercury, orbit insertion is accomplished at the third Mercury encounter. The instrument payload includes a dual imaging system for wide and narrow fields-of-view, monochrome and color imaging, and stereo; X-ray and combined gamma-ray and neutron spectrometers for surface chemical mapping; a magnetometer; a laser altimeter; a combined ultraviolet–visible and visible-near-infrared spectrometer to survey both exospheric species and surface mineralogy; and an energetic particle and plasma spectrometer to sample charged species in the magnetosphere. During the flybys of Mercury, regions unexplored by Mariner 10 will be seen for the first time, and new data will be gathered on Mercury's exosphere, magnetosphere, and surface composition. During the orbital phase of the mission, one Earth year in duration, MESSENGER will complete global mapping and the detailed characterization of the exosphere, magnetosphere, surface, and interior.     

Solar cycle according to mean magnetic field data

To investigate the shape of the solar cycle, we have performed a wavelet analysis of the large–scale magnetic field data for 1960–2000 for several latitudinal belts and have isolated the following quasi-periodic components: ∼22, 7 and 2 yr. The main 22-yr oscillation dominates all latitudinal belts except the latitudes of ±30° from the equator. The butterfly diagram for the nominal 22-yr oscillation shows a standing dipole wave in the low-latitude domain  (∣θ∣≤ 30°)  and another wave in the sub-polar domain  (∣θ∣≥ 35°)  , which migrates slowly polewards. The phase shift between these waves is about π. The nominal 7-yr oscillation yields a butterfly diagram with two domains. In the low-latitude domain  (∣θ∣≤ 35°)  , the dipole wave propagates equatorwards and in the sub-polar region, polewards. The nominal 2-yr oscillation is much more chaotic than the other two modes; however the waves propagate polewards whenever they can be isolated.
We conclude that the shape of the solar cycle inferred from the large-scale magnetic field data differs significantly from that inferred from sunspot data. Obviously, the dynamo models for a solar cycle must be generalized to include large-scale magnetic field data. We believe that sunspot data give adequate information concerning the magnetic field configuration deep inside the convection zone (say, in overshoot later), while the large-scale magnetic field is strongly affected by meridional circulation in its upper layer. This interpretation suggests that the poloidal magnetic field is affected by the polewards meridional circulation, whose velocity is comparable with that of the dynamo wave in the overshoot layer. The 7- and 2-yr oscillations could be explained as a contribution of two sub-critical dynamo modes with the corresponding frequencies.     

Recent advances in planetary magnetism

During the past decade, significant advances in thein situ measurements of planetary magnetic fields have been made. The U.S.A. and U.S.S.R. have conducted spacecraft investigations of all the planets, from innermost Mercury out to Jupiter. Unexpectedly, Mercury was found to possess a global magnetic field but neither the Moon nor Venus do. The results at Mars are incomplete butif a global field exists, it is clearly quite weak. The main magnetic field of Jupiter has been measured directly for the first time and confirms, as well as augments appreciably, the past 2 decades of groundbased radio astronomical studies which provided indirect evidence of the field. Progress in developing analytically complete models of the dynamo process suggests a possible common origin for Mercury, Earth and Jupiter.Paper dedicated to Professor Hannes Alfvén on the occasion of his 70th birthday, 30 May, 1978.