Optimal dynamos in the cores of terrestrial exoplanets: Magnetic field generation and detectability

A potentially promising way to gain knowledge about the internal dynamics of extrasolar planets is by remote measurement of an intrinsic magnetic field. Strong planetary magnetic fields, maintained by internal dynamo action in an electrically conducting fluid layer, are helpful for shielding the upper atmosphere from stellar wind induced mass loss and retaining water over long (Gyr) time scales. Here we present a whole planet dynamo model that consists of three main components: an internal structure model with composition and layers similar to the Earth, an optimal mantle convection model that is designed to maximize the heat flow available to drive convective dynamo action in the core, and a scaling law to estimate the magnetic field intensity at the surface of a terrestrial exoplanet. We find that the magnetic field intensity at the core surface can be up to twice the present-day geomagnetic field intensity, while the magnetic moment varies by a factor of 20 over the models considered. Assuming electron cyclotron emission is produced from the interaction between the stellar wind and the exoplanet magnetic field we estimate the cyclotron frequencies around the ionospheric cutoff at 10 MHz with emission fluxes in the range 10⁻⁴-10⁻⁷ Jy, below the current detection threshold of radio telescopes. However, we propose that anomalous boosts and modulations to the magnetic field intensity and cyclotron emission may allow for their detection in the future.     

行星和太阳内部的磁流体力学

许多行星 (如木卫三 ,水星 ,地球 ,木星和土星 )和恒星 (如太阳 )具有内部磁场。对这些磁场的存在和变化的解释对行星科学家和天体物理学家是一个巨大的挑战。本文试图总结行星和恒星的导电流体内部磁流体力学研究的新近发展和困难。一般由热对流驱动的流动通过磁流体力学过程产生并维持在行星和恒星中的磁场。在行星中磁流体力学过程强烈地受到转动 ,磁场和球几何位型的综合影响。其动力学的关键方面涉及科里奥利力和洛伦兹力间的相互作用。在太阳中其流线 ,即处于对流层的薄的剪切流层在太阳的磁流体力学过程中扮演了一个基本的角色 ,并由之产生了 1 1年的太阳黑子周期。本文也给出了一个新的非线性三维太阳发电机模型。     

Numerical dynamo models of Uranus' and Neptune's magnetic fields

The non-axisymmetric, non-dipolar magnetic fields of Uranus and Neptune are markedly different from the axially-dipolar dominated fields of the other planets in our Solar System with active dynamos. Stanley and Bloxham [Stanley, S., Bloxham, J., 2004. Nature 428, 151-153] used numerical modeling to demonstrate that Uranus' and Neptune's unusual fields could be the result of a different convective region geometry in these planets. They found that a numerical dynamo operating in a thin shell surrounding a stably-stratified fluid interior produces magnetic field morphologies similar to those of Uranus and Neptune. This geometry for the convective region was initially proposed by Hubbard et al. [Hubbard, W.B., Podolak, M., Stevenson, D.J., 1995. In: Cruickshank, D. (Ed.), Neptune and Triton. Univ. of Arizona Press, Tucson, pp. 109-138] to explain both the magnetic field morphology as well as the low intrinsic heat flows from these planets. Here we examine the influence of varying the stable layer radius in numerical models and compare the results to thin shell models surrounding solid inner cores. We find that a limited range of stable-layer shell thicknesses exist in which Uranus/Neptune-like field morphologies result. This allows us to put constraints on the size of the convective layers in Uranus and Neptune.     

Axisymmetric field generation within an ambient axial field

The generation of magnetic field in a homogeneous, electrically conducting fluid – as required for the dynamo generation of the fields of many astrophysical bodies – is normally a threshold process; the dynamo mechanism, applicable to such bodies in unmagnetised environments, requires motions of sufficient strength to overcome the innate magnetic diffusion. In the presence of an ambient field, however, the critical nature of the field generation process is relaxed. Motions can distort and amplify the ambient field for all amplitudes of flow. For motions with appropriate geometries, an internal ‘dynamo‐like’ field of appreciable strength can be generated, even for relatively weak flows. At least a minority of planets, moons and other bodies exist within significant external astrophysical fields. For these bodies, the ambient field problem is more relevant than the classical dynamo problem, yet it remains relatively little studied. In this paper we consider the effect of an axial ambient field on a spherical mean‐field α ²ω dynamo model, through nonlinear calculations with α ‐quenching feedback. Ambient fields of varying strengths, and both stationary and oscillatory in time, are imposed. Particular focus is placed on the effects of these fields on the equatorial symmetry and the time dependence of the preferred solutions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The phase diagram of water and the magnetic fields of Uranus and Neptune

The interior of giant planets can give valuable information on formation and evolution processes of planetary systems. However, the interior and evolution of Uranus and Neptune is still largely unknown. In this paper, we compare water-rich three-layer structure models of these planets with predictions of shell structures derived from magnetic field models. Uranus and Neptune have unusual non-dipolar magnetic fields contrary to that of the Earth. Extensive three-dimensional simulations of Stanley and Bloxham (Stanley, S., Bloxham, J. [2004]. Nature 428, 151-153) have indicated that such a magnetic field is generated in a rather thin shell of at most 0.3 planetary radii located below the H/He rich outer envelope and a conducting core that is fluid but stably stratified. Interior models rely on equation of state data for the planetary materials which have usually considerable uncertainties in the high-pressure domain. We present interior models for Uranus and Neptune that are based on ab initio equation of state data for hydrogen, helium, and water as the representative of all heavier elements or ices. Based on a detailed high-pressure phase diagram of water we can specify the region where superionic water should occur in the inner envelope. This superionic region correlates well with the location of the stably-stratified region as found in the dynamo models. Hence we suggest a significant impact of the phase diagram of water on the generation of the magnetic fields in Uranus and Neptune.     

On self–killing and self–creating dynamos

Numerical studies with a spherical dynamo model have shown two remarkable phenomena. The model consists of a spherical body of an electrically conducting incompressible uid surrounded by free space. In addition to a rotation of the body an inner motion due to a given forcing is considered which satisfies a no–slip condition at the boundary. The full interaction of magnetic field and motion is taken into account. Starting from a fluid motion capable of dynamo action and a very weak magnetic field it was observed that the growing magnetic field destroys the dynamo property of the motion and then decays, and that the system ends up in a state with another motion incapable of dynamo action and zero magnetic field. In another case with a motion unable to prevent small magnetic fields from decay it proved to be possible that stronger magnetic fields deform it so that a dynamo starts to work which enables the system to approach a steady state with a finite magnetic field.     

Spherical Oscillatory alpha2 Dynamo Induced by Magnetic Coupling between a Fluid Shell and an Inner Electrically Conducting Core: Relevance to the Solar Dynamo

A two-layer spherical alpha2 dynamo model consisting of an inner electrically conducting core (magnetic diffusivity lambdai and radius ri) with alpha=0 surrounded by an electrically conducting spherical shell (magnetic diffusivity lambdao and radius ro) with a constant alpha is shown to exhibit oscillatory behavior for values of beta=lambdai&solm0;lambdao and ri&solm0;ro relevant to the solar dynamo. Time-dependent dynamo solutions require ri&solm0;ro>/=0.55 and beta相似文献   

Invisible dynamo in mean-field models

The inverse problem in a spherical shell to find the two-dimensional spatial distributions of the α-effect and differential rotation in a mean-field dynamo model has been solved. The derived distributions lead to the generation of a magnetic field concentrated inside the convection zone. The magnetic field is shown to have no time to rise from the region of maximum generation located in the lower layers to the surface in the polarity reversal time due to magnetic diffusion. The ratio of the maximum magnetic energy in the convection zone to its value at the outer boundary reaches two orders of magnitude or more. This result is important in interpreting the observed stellar and planetary magnetic fields. The proposed method of solving the inverse nonlinear dynamo problem is easily adapted for a wide class of mathematical-physics problems.     

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)     

Planetary dynamos

Planetary magnetic fields are thought to be generated by magnetohydrodynamic dynamos acting in the convecting, electrically conducting fluid cores of these cosmic bodies. Similar processes are believed to produce a wide variety of other cosmic magnetic fields, including the fields of the Sun and stars. At present, we understand the basic physical processes involved in dynamo magnetic field generation. However, a detailed understanding still eludes us both because of continuing uncertainty about planetary interior structures, properties, and fluid motions, and because of our still primitive capacity for dealing with the complex, nonlinear dynamical processes involved in the fully elaborated dynamo process.     

On the generation of the large-scale and turbulent magnetic fields in solar-type stars

It is thought that the large-scale solar-cycle magnetic field is generated in a thin region at the interface of the radiative core (RC) and solar convection zone (SCZ). We show that the bulk of the SCZ virogoursly generates a small-scale turbulent magnetic field. Rotation, while not essential, increases the generation rate of this field.Thus, fully convective stars should have significant turbulent magnetic fields generated in their lower convection zones. In these stars the absence of a radiative core, i.e., the absence of a region of weak buoyancy, precludes the generation of a large-scale magnetic field, and as a consequence the angular momentum loss is reduced. This is, in our opinion, the explanation for the rapid rotation of the M-dwarfs in the Hyades cluster.Adopting the Utrecht's group terminology, we argue that the residual chromospheric emission should have three distinctive components: the basal emission, the emission due to the large-scale field, and the emission due to the turbulent field, with the last component being particularly strong for low mass stars.In the conventional dynamo equations, the dynamo frequencies and the propagation of the dynamo wave towards the equator are based on the highly questionable assumption of a constant . Furthermore, meridional motions, a necessary consequence of the interaction of rotation with convection, are ignored. In this context we discuss Stenflo's results about the global wave pattern decomposition of the solar magnetic field and conclude that it cannot be interpreted in the framework of the conventional dynamo equations.We discuss solar dynamo theories and argue that the surface layers could be essential for the generation of the poloidal field. If this is the case an -effect would not be needed at the RC-SCZ interface (where the toroidal field is generated). The two central problems facing solar dynamo theories may the transport of the surface poloidal field to the RC-SCZ interface and the uncertainty about the contributions to the global magnetic field by the small-scale magnetic features.Visitor, National Solar Observatory, National Optical Astronomy Observatories.The National Optical Astronomy Observatories are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.     

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.     

Models of magnetic field generation in partly stable planetary cores: Applications to Mercury and Saturn

A substantial part of Mercury's iron core may be stably stratified because the temperature gradient is subadiabatic. A dynamo would operate only in a deep sublayer. We show that such a situation arises for a wide range of values for the heat flow and the sulfur content in the core. In Saturn the upper part of the metallic hydrogen core could be stably stratified because of helium depletion. The magnetic field is unusually weak in the case of Mercury and unusually axisymmetric at Saturn. We study numerical dynamo models in rotating spherical shells with a stable outer region. The control parameters are chosen such that the magnetic Reynolds number is in the range of expected Mercury values. Because of its slow rotation, Mercury may be in a regime where the dipole contribution to the internal magnetic field is weak. Most of our models are in this regime, where the dynamo field consists mainly of rapidly varying higher multipole components. They can hardly pass the stable conducting layer because of the skin effect. The weak low-degree components vary more slowly and control the structure of the field outside the core, whose strength matches the observed field strength at Mercury. In some models the axial dipole dominates at the planet's surface and in others the axial quadrupole is dominant. Differential rotation in the stable layer, representing a thermal wind, is important for attenuating non-axisymmetric components in the exterior field. In some models that we relate to Saturn the axial dipole is intrinsically strong inside the dynamo. The surface field strength is much larger than in the other cases, but the stable layer eliminates non-axisymmetric modes. The Messenger and Bepi Colombo space missions can test our predictions that Mercury's field is large-scaled, fairly axisymmetric, and shows no secular variations on the decadal time scale.     

A 2D study of the effects of the size of a solid core on the equatorial flow in giant planets

Zonal winds simulated in two-dimensional computer models of turbulent convection in the equatorial plane of giant planets have greater surface amplitudes for cases with smaller solid cores, and therefore larger buoyancy driving, all other properties being equal. This differential rotation in radius is maintained by the convergence of angular momentum flux, which occurs because of the convective flow that develops due to the effects of planetary rotation and density stratification. The superposition of the convective flow and the stronger zonal flow produces wave-like, instead of cellular convection.     

The Kinematic Theory of Solar Dynamo

Generation of the Sun‘s magnetic fields by self-inductive processes in the solar electrically conducting interior, the solar dynamo theory, is a fundamentally important subject in astrophysics. The kinematic dynamo theory concerns how the magnetic fields are produced by kinematically possible flows without being constrained by the dynamic equation. We review a number of basic aspects of the kinematic dynamo theory, including the magnetohydrodynamic approximation for the dynamo equation, the impossibility of dynamo action with the solar differential rotation, the Cowling‘s anti-dynamo theorem in the solar context, the turbulent alpha effect and recently constructed three-dimensional interface dynamos controlled by the solar tachocline at the base of the convection zone.     

The interaction of the stellar wind with an extrasolar planet—3D hybrid and drift-kinetic simulations

To be able to simulate the interaction of extrasolar planets with the stellar wind, a number of planetary parameters are required. Some of these (like planetary mass and radius) can be obtained directly from observational data. Other properties are not known very precisely. For example, up to now, there is no observation providing information on the strength of planetary magnetic moments. However, there is good reason to expect only very small magnetic moments for planets in very close orbits around their stars (like HD 209458 b and OGLE-TR-56 b). Thus, as a first step towards a more complete treatment, it seems reasonable to treat the interaction of the stellar wind with an unmagnetized planet. Calculations were performed for a nonconducting as well as for a weakly conducting planet. The interaction with the stellar wind and the resulting induced magnetosphere was simulated using a three dimensional hybrid code as well as in the drift-kinetic approximation. The effect of a interplanetary magnetic field oriented perpendicular to the incoming stellar wind was included. In the case of a weakly conducting body an asymmetrical Mach cone is formed, whereas for a nonconducting body no Mach cone is observed. These investigations will serve as the first step in the search for particular effects occurring at extrasolar planets, which could possibly lead to observable effects, e.g. radio emission. The results are also relevant for plasma structures near weakly conducting, unmagnetized bodies like the Earth's moon.     

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.     

A planetary dynamo model

Using the Lagrangian approach, the author considered the temporal evolution of an ensemble of interacting magnetohydrodynamic cyclones, obeying equations of the Langevin type, in a rotating medium. The problem is topical for fast-rotating convective objects: cores of planets and a number of stars, where the Rossby numbers are much less than unity and the geostrophic balance of forces is observed. In this work, results of simulation are given both for the two-dimensional case, when axes of cyclones can rotate only in the vertical plane, and for the three-dimensional case when the axes are rotating by two angles. It is shown that a change in the heat flux on the shell boundary impacts the frequency of reversals of the mean dipole magnetic field, which agrees with results of simulation in three-dimensional models of a planetary dynamo. Applications of the model for the giant planets are considered, and an explanation of certain episodes of the geomagnetic field in the past is offered.     

Zur Dynamotheorie stellarer und planetarer Magnetfelder I. Berechnung sonnenähnlicher Wechselfeldgeneratoren

Non-reflectionsymmetric turbulent motions in an electrically conducting fluid provide for an average electromotoric force α parallel to the average magnetic field . This effect – furthermore called α-effect – can lead to a self-excited built-up of magnetic fields by dynamo action. Since by actions of Coriolis forces turbulent motions in rotating systems generally are non-reflectionsymmetric, the explanation of stellar and planetary magnetic fields becomes possible. – In this first part spherical alternating field dynamos are calculated on conditions similar to those at the sun; α-effect and differential rotation provide for dynamo action. – The basic equations, which form a system of partial differential equations, are treated by a separation method; in this way our problem is transformed into a two-parametric eigenvalue problem for an infinite system of ordinary differential equations. The two eigenvalues determine the frequency and the amount of the induction actions necessary for the field maintenance. The solution of this problem was made by an electronic computer. – The results give evidence for the possibility of magnetohydrodynamic alternating field dynamos. Furthermore the magnitude of the calculated eigenvalues and the similarities between the calculated field and the observed solar magnetic field are showing that the sun realizes a dynamo of this kind.