Towards the interpretation of Uranus's eccentric magnetic field

Abstract

Coriolis forces stimulate dynamo action in a rapidly-rotating fluid by promoting complexities in the pattern of fluid motions, notably departures from symmetry about the axis of rotation. This pattern and its time variations determine the instantaneous form and temporal behaviour of the magnetic field so produced. Instantaneous magnetic fields will usually exhibit in their broad-scale features approximate alignment with the rotation axis. This is borne out by observations of the magnetic fields of the Earth, Jupiter and Saturn, and it is likely on general grounds that Neptune will be found to have an aligned magnetic field. But, as is shown by laboratory and theoretical studies of thermal convection in rapidly-rotating fluids, for some ranges of rotation speed, rate of heating, etc. certain patterns can occur which in electrically-conducting fluids would produce magnetic fields exhibiting departures from alignment with the rotation axis, which instantaneously could be quite pronounced but would average out to very small values over sufficiently long periods of time. These findings indicate obvious strategies for theoretical studies towards the interpretation of Uranus's eccentric magnetic field (which need not invoke departures from axial symmetry in the thermal, mechanical or electrical boundary conditions of the dynamo region within the planet) and for further observational studies.     

On the modified Taylor constraint

A dynamo driven by motions unaffected by viscous forces is termed magnetostrophic. Although such a model might describe magnetic field generation in Earth’s core well, a magnetostrophic dynamo has not yet been found even though Taylor [Proc. R. Soc. Lond. A 1963, 274, 274–283] devised an apparently viable method of finding one. His method for determining the fluid velocity from the magnetic field and the energy source involved only the evaluation of integrals along lines parallel to the Earth’s axis of rotation and the solution of a second-order ordinary differential equation. It is demonstrated below that an approximate solution of this equation for a broad family of magnetic fields is immediate. Furthermore inertia, which was neglected in Taylor’s theory, is restored here, so that the modified theory includes torsional waves, whose existence in the Earth’s core has been inferred from observations of the length of day. Their theory is reconsidered.     

Dynamos with weakly convecting outer layers: implications for core-mantle boundary interaction

Convection in the Earth's core is driven much harder at the bottom than the top. This is partly because the adiabatic gradient steepens towards the top, partly because the spherical geometry means the area involved increases towards the top, and partly because compositional convection is driven by light material released at the lower boundary and remixed uniformly throughout the outer core, providing a volumetric sink of buoyancy. We have therefore investigated dynamo action of thermal convection in a Boussinesq fluid contained within a rotating spherical shell driven by a combination of bottom and internal heating or cooling. We first apply a homogeneous temperature on the outer boundary in order to explore the effects of heat sinks on dynamo action; we then impose an inhomogeneous temperature proportional to a single spherical harmonic Y ₂² in order to explore core-mantle interactions. With homogeneous boundary conditions and moderate Rayleigh numbers, a heat sink reduces the generated magnetic field appreciably; the magnetic Reynolds number remains high because the dominant toroidal component of flow is not reduced significantly. The dipolar structure of the field becomes more pronounced as found by other authors. Increasing the Rayleigh number yields a regime in which convection inside the tangent cylinder is strongly affected by the magnetic field. With inhomogeneous boundary conditions, a heat sink promotes boundary effects and locking of the magnetic field to boundary anomalies. We show that boundary locking is inhibited by advection of heat in the outer regions. With uniform heating, the boundary effects are only significant at low Rayleigh numbers, when dynamo action is only possible for artificially low magnetic diffusivity. With heat sinks, the boundary effects remain significant at higher Rayleigh numbers provided the convection remains weak or the fluid is stably stratified at the top. Dynamo action is driven by vigorous convection at depth while boundary thermal anomalies dominate in the upper regions. This is a likely regime for the Earth's core.     

Reducing the non-axisymmetry of a planetary dynamo and an application to saturn

Abstract

If a conducting fluid shell is undergoing spin-axisymmetric differential rotation and overlies the dynamo generating region of a planet then it is capable of greatly reducing the non-spin-axisymmetric components of the generated field, provided the appropriate magnetic Reynolds number is large. The model, closely related to the electromagnetic skin effect, is quantified and applied to Saturn. The observed small dipole tilt (～ 1°) of Saturn's magnetic field can be explained because of the presence of a stably stratified conducting layer overlying the dynamo region. This layer is a predicted consequence of the thermal evolution, arises because of the limited solubility of helium in metallic hydrogen (Stevenson, 1980), and appears to be required by the Voyager infrared observations indicating depletion of helium from Saturn's atmosphere. The much larger dipole tilt angles of Jupiter and the Earth indicate the absence of any such stable, differentially rotating layer with a large magnetic Reynolds number.     

Necessary conditions for the magnetohydrodynamic dynamo

Abstract

A magnetohydrodynamic, dynamo driven by convection in a rotating spherical shell is supposed to have averages that are independent of time. Two cases are considered, one driven by a fixed temperature difference R and the other by a given internal heating rate Q. It is found that when q, the ratio of thermal conductivity to magnetic diffusivity, is small, R must be of order q ^?4/3 and Q of order q ^?2 for dynamo action to be possible; q is small in the Earth's core, so it is hoped that the criteria will prove useful in practical as well as theoretical studies of dynamic dynamos. The criteria can be further strengthened when the ohmic dissipation of the field is significant in the energy balance. The development includes the derivation of two necessary conditions for dynamo action, both based on the viscous dissipation rate of the velocity field that drives the dynamo.     

A dynamo model with double diffusive convection for Mercury's core

A recent dynamo model for Mercury assumes that the upper part of the planet's fluid core is thermally stably stratified because the temperature gradient at the core–mantle boundary is subadiabatic. Vigorous convection driven by a superadiabatic temperature gradient at the boundary of a growing solid inner core and by the associated release of light constituents takes place in a deep sub-layer and powers a dynamo. These models have been successful at explaining the observed weak global magnetic field at Mercury's surface. They have been based on the concept of codensity, which combines thermal and compositional sources of buoyancy into a single variable by assuming the same diffusivity for both components. Actual diffusivities in planetary cores differ by a large factor. To overcome the limitation of the codensity model, we solve two separate transport equations with different diffusivities in a double diffusive dynamo model for Mercury. When temperature and composition contribute comparable amounts to the buoyancy force, we find significant differences to the codensity model. In the double diffusive case convection penetrates the upper layer with a net stable density stratification in the form of finger convection. Compared to the codensity model, this enhances the poloidal magnetic field in the nominally stable layer and outside the core, where it becomes too strong compared to observation. Intense azimuthal flow in the stable layer generates a strong axisymmetric toroidal field. We find in double diffusive models a surface magnetic field of the observed strength when compositional buoyancy plays an inferior role for driving the dynamo, which is the case when the sulphur concentration in Mercury's core is only a fraction of a percent.     

Properties of nonlinear dynamo waves

Abstract

Dynamo theory offers the most promising explanation of the generation of the sun's magnetic cycle. Mean field electrodynamics has provided the platform for linear and nonlinear models of solar dynamos. However the nonlinearities included arc (necessarily) arbitrarily imposed in these models. This paper conducts a systematic survey of the role of nonlinearities in the dynamo process, by cousidering the behaviour of dynamo waves in the nonlinear regime. It is demonstrated that only by considering realistic nonlinearities that are non-local in space and time can modulation of the basic dynamo wave be achieved. Moreover this modulation is greatest when there is a large separation of timescales provided by including a low magnetic Prandtl number in the equation for the velocity perturbations.     

Convective dynamos with intermediate and strong fields

Abstract

The weak-field Benard-type dynamo treated by Soward is considered here at higher levels of the induced magnetic field. Two sources of instability are found to occur in the intermediate field regime M ～ T ^1/12, where M and T are the Hartmann and Taylor numbers. On the time scale of magnetic diffusion, solutions may blow up in finite time owing to destabilization of the convection by the magnetic field. On a faster time scale a dynamic instability related to MAC-wave instability can also occur. It is therefore concluded that the asymptotic structure of this dynamo is unstable to virtual increases in the magnetic field energy.

In an attempt to model stabilization of the dynamo in a strong-field regime we consider two approximations. In the first, a truncated expansion in three-dimensional plane waves is studied numerically. A second approach utilizes an ad hoc set of ordinary differential equations which contains many of the features of convection dynamos at all field energies. Both of these models exhibit temporal intermittency of the dynamo effect.     

On the role of magnetostrophic waves in geodynamo

It is shown that magnetostrophic waves which are generated in the equatorial plane of the Earth’s core due to the instability of the equatorial jet and which propagate almost transversely to the rotational axis off the tangent cylinder, have a negative helicity in the northern hemisphere and positive helicity in the southern hemisphere. When the wave trains propagate through the regions with a constant azimuthal magnetic field caused by the Ω-effect, this helicity distribution induces an electromotive force (emf) (due to the α-effect), which may lead to the maintenance of the initial dipole field by the scenario of the α-Ω dynamo.     

Origin of the magnetic fields in the giant planets

Recent observations and progress in the understanding of various requirements for the generation of magnetic fields permit much more definite conclusions to be drawn about the fields of the giant planets than was possible until quite recently. The Jovian magnetic field of about 4 gauss could be either of primordial origin or generated by a thermally driven dynamo. The expected Saturnian field of about 1 gauss can be similarly accounted for either by a thermally or by a precessionally driven dynamo. The presence of a field on Uranus of perhaps 0.1 gauss presents a problem because although it could be accounted for by a thermally driven dynamo operating in a highly conductive shell of hydrogen, the so far unobserved thermal flux and convection may be too low. If such a dynamo were to operate then one would expect the field to show seasonal variations. A precessional dynamo driven by Miranda seems to be marginally possible. On Neptune a conductive shell similar to that on Uranus appears to be much thinner, which perhaps explains the absence of an active dynamo driven either thermally or precessionally by Triton. It is, however, very likely that Neptune does have a magnetic field but that it is too weak to lead to observable electromagnetic radiations.     

Tuning of the mean-field geodynamo model

Parker’s two-dimensional (2D) dynamo model with an algebraic form of nonlinearity for the α-effect is considered. The model uses geostrophic distributions for the α-effect and differential rotation, which are derived from the three-dimensional (3D) convection models. The resulting configurations of the magnetic field in the liquid core are close to the solutions in Braginsky’s Z-model. The implications of the degree of geostrophy observed in the 3D dynamo models for the behavior of the mean magnetic field are explored. It is shown that the reduction in geostrophy leads to magnetic field reversals accompanied by the relative growth of the nondipole component of the field on the surface of the liquid core. The simulations with a random α-effect which causes turbulent pulsations are carried out. The approach is capable of producing realistic sequences of magnetic reversals.     

Periodic and aperiodic dynamo waves

Abstract

In order to show that aperiodic magnetic cycles, with Maunder minima, can occur naturally in nonlinear hydromagnetic dynamos, we have investigated a simple nonlinear model of an oscillatory stellar dynamo. The parametrized mean field equations in plane geometry have a Hopf bifurcation when the dynamo number D=1, leading to Parker's dynamo waves. Including the nonlinear interaction between the magnetic field and the velocity shear results in a system of seven coupled nonlinear differential equations. For D>1 there is an exact nonlinear solution, corresponding to periodic dynamo waves. In the regime described by a fifth order system of equations this solution remains stable for all D and the velocity shear is progressively reduced by the Lorentz force. In a regime described by a sixth order system, the solution becomes unstable and successive transitions lead to chaotic behaviour. Oscillations are aperiodic and modulated to give episodes of reduced activity.     

From stable dipolar towards reversing numerical dynamos

We are using a three-dimensional convection-driven numerical dynamo model without hyperdiffusivity to study the characteristic structure and time variability of the magnetic field in dependence of the Rayleigh number (Ra) for values up to 40 times supercritical. We also compare a variety of ways to drive the convection and basically find two dynamo regimes. At low Ra, the magnetic field at the surface of the model is dominated by the non-reversing axial dipole component. At high Ra, the dipole part becomes small in comparison to higher multipole components. At transitional values of Ra, the dynamo vacillates between the dipole-dominated and the multipolar regime, which includes excursions and reversals of the dipole axis. We discuss, in particular, one model of chemically driven convection, where for a suitable value of Ra, the mean dipole moment and the temporal evolution of the magnetic field resemble the known properties of the Earth’s field from paleomagnetic data.     

Dynamos on stream surfaces of a highly conducting fluid

Abstract

This paper discusses dynamo action in generalisations of the Ponomarenko dynamo at large magnetic Reynolds number. The original Ponomarenko dynamo consists of a spiralling flow in which the stream surfaces are concentric cylinders of circular cross section, and the flow depends only on distance from the axis in cylindrical polar coordinates.

In this study, the stream surfaces are allowed to be cylinders of arbitrary cross section, and the flow is only required to be independent of the coordinate along the cylinder axes. For smooth flows alpha and eddy diffusion effects are identified, in terms of the geometry of the stream surfaces, and asymptotic formulae for growth rates in the limit of large magnetic Reynolds number are obtained. Numerical support for these results is presented using direct simulation of dynamo action in selected flows at high conductivity. Finally the case is considered when in spherical polar coordinates the flow is independent of the azimuthal coordinate and the stream surfaces, which are tori, have arbitrary cross sections.     

Thermal histories of the core and mantle

Recognition that the cooling of the core is accomplished by conduction of heat into a thermal boundary layer (D″) at the base of the mantle, partly decouples calculations of the thermal histories of the core and mantle. Both are controlled by the temperature-dependent rheology of the mantle, but in different ways. Thermal parameters of the Earth are more tightly constrained than hitherto by demanding that they satisfy both core and mantle histories. We require evolution from an early state, in which the temperatures of the top of the core and the base of the mantle were both very close to the mantle solidus, to the present state in which a temperature increment, estimated to be ～ 800 K, has developed across D″. The thermal history is not very dependent upon the assumption of Newtonian or non-Newtonian mantle rheology. The thermal boundary layer at the base of the mantle (i.e., D″) developed within the first few hundred million years and the temperature increment across it is still increasing slowly. In our preferred model the present temperature at the top of the core is 3800 K and the mantle temperature, extrapolated to the core boundary without the thermal boundary layer, is 3000 K. The mantle solidus is 3860 K. These temperatures could be varied within quite wide limits without seriously affecting our conclusions. Core gravitational energy release is found to have been remarkably constant at ～ 3 × 10¹¹ W. nearly 20% of the core heat flux, for the past 3 × 10⁹ y, although the total terrestrial heat flux has decreased by a factor of 2 or 3 in that time. This gravitational energy can power the “chemical” dynamo in spite of a core heat flux that is less than that required by conduction down an adiabatic gradient in the outer core; part of the gravitational energy is used to redistribute the excess heat back into the core, leaving 1.8 × 10¹¹ W to drive the dynamo. At no time was the dynamo thermally driven and the present radioactive heating in the core is negligibly small. The dynamo can persist indefinitely into the future; available power 10¹⁰ y from now is estimated to be 0.3 × 10¹¹ W if linear mantle rheology is assumed or more if mantle rheology is non-linear. The assumption that the gravitational constant decreases with time imposes an implausible rate of decrease in dynamo energy. With conventional thermodynamics it also requires radiogenic heating of the mantle considerably in excess of the likely content of radioactive elements.     

Kinematic dynamo induced by helical waves

We investigate numerically the kinematic dynamo induced by the superposition of two helical waves in a periodic box as a simplified model to understand the dynamo action in astronomical bodies. The effects of magnetic Reynolds number, wavenumber and wave frequency on the dynamo action are studied. It is found that this helical-wave dynamo is a slow dynamo. There exists an optimal wavenumber for the dynamo growth rate. A lower wave frequency facilitates the dynamo action and the oscillations of magnetic energy emerge at some particular wave frequencies.     

Inner-core conductivity in numerical dynamo simulations

Speculation about its possible super-rotation has drawn the attention of many geophysical researchers to the Earth’s inner core. An issue of special interest for geodynamo modelling is the influence of the inner-core conductivity. It has been suggested that the finite magnetic diffusivity of the inner core prevents more frequent reversals of the Earth’s magnetic field. We explore the possible influence of the inner-core conductivity by comparing convection-driven 3D dynamo simulations with insulating or conducting inner cores (CIC) at various parameters. The influence on the field structure in the outer core is only marginal. The time behaviour of dipole-dominated non-reversing dynamos is also little affected. Concerning reversing dynamos, the inner-core conductivity reduces the number of short dipole-polarity intervals with a typical length of a few thousand years. Reversals are always correlated with low dipole strength and these short intervals are found in periods where the dipole moment stays low. Polarity intervals longer than about 10,000 years, where the dipole moment has time recover in strength, are equally likely in insulating and CIC models. Since these latter intervals are of more geophysical relevance, we conclude that the influence of the inner-core conductivity on Earth-like reversal sequences is insignificant for the dynamo model employed here.     

Past and present magnetism of the Moon

The characteristics of the remanent magnetism of lunar samples suggests that it was acquired in a magnetic field on the Moon. The most likely origin of the field is a dynamo process in a molten, electricallyconducting core, but generation of a transient magnetic field during large meteorite impacts cannot be entirely ruled out. The magnetizing process may be thermoremanence, acquired when the rocks cooled through, the Curie point of the constituent iron grains which carry the remanent magnetization, or it may involve shock at the time of a meteorite impact, with or without a partial thermoremanence arising from heating.Evidence from absolute and relative determinations of the ancient field strength from the sample magnetizations strongly favours a global lunar field. This is implied by a trend which shows the field rising to a maximum value of 100 T between about 3.9–3.7 by ago and then decaying to 5–10 T until3.1 by. Such a systematic variation of field with time is not expected to be derived from magnetizations acquired in transient, impact-generated fields varying randomly in intensity.Contributory evidence for a dynamo field is provided by measurements of present lunar surface fields, the present very small dipole moment of the Moon and accumulating evidence of variation of the axis of the lunar field with time. Although there is no direct evidence for the existence of a lunar core the relevant observations are consistent with the presence of a core of up to 400 km, in radius. There are some difficulties associated with the lunar dynamo mechanism and its energy source but the evidence for a lunar dynamo is accumulating, with important implications for the structure and thermal history of the Moon.     

古地磁学在地磁场起源研究中的应用

地磁场起源及其倒转是地球科学的难题之一。究其原因一方面是由于无法直接观测地球内部发生的物理过程,另一方面是由于缺乏理论与实验相结合的综合研究。本文以磁流体力学为基础,将古地磁学与αω发电机理论结合在一起进行分析和研究。得出了如下新观点:(1)洛仑兹力在地核发电过程起负反馈作用;(2)较差旋转控制着地磁场西向漂移,(3)α作用使地磁极偏离地球自转轴     

Mantle regulation of core cooling: A geodynamo without core radioactivity?

The case for radioactivity in the core based on the power requirements of the geodynamo is re-evaluated. Previous calculations of mantle regulation of core thermal evolution have used an inappropriate formula. New calculations with a more appropriate formula yield lower core heat loss in the past, thus mitigating the implication of unreasonably high past core and mantle temperatures. Multiple thermal evolutions leading to present heat flows are also demonstrated, depending on the efficiency of mantle removal of core heat, some with moderately high past core heat loss and some with low and steady core heat loss. The latter would permit a low- or moderate-power dynamo without core radioactivity. Key uncertainties are the efficiency of core cooling by the mantle, the thermal conductivity of the core and the energy or entropy flow required to maintain the dynamo. The present rate of heat loss from the core is argued to be still rather uncertain, and a commonly used estimate of the thermal conductivity of the core is shown plausibly to be too high and in any case to be uncertain by perhaps a factor of 2. The geochemical difficulties associated with postulating radioactive heat sources in the core are stressed.