Are planetary dynamos driven by gravitational settling?

The basic features of gravitationally driven convection are summarized, and an expression for the power available to drive a hydromagnetic dynamo is given. Some thermal consequences of gravitationally driven convection are presented.     

Control volume method for hydromagnetic dynamos in rotating spherical shells: Testing the code against the numerical dynamo benchmark

Hydromagnetic dynamos in rotating spherical shells are investigated using the control volume method. We present a validation of our code against the numerical dynamo benchmark. It is successfully benchmarked and we are able to conclude that the control volume method is another numerical method available for numerical modelling of self-consistent dynamos. In addition, the efficiency of our numerical code is tested. Computations provide conclusions that dynamo codes based on the spectral methods are much more efficient than our code based on the control volume method at the study of global fields on small and medium size parallel computers. However, our code could be much more efficient than codes based on the spectral methods on very large parallel computers, especially at the study of turbulence.     

Nonlinear galactic dynamos

Abstract

We present trial calculations for a simple nonlinear ‘‘thick disk'’ galaxy dynamo. The nonlinearity is a simple α-quenching. Our strictly axisymmetric solution demonstrates the possibility of a nonlinear interaction between modes of opposite parity. We suggest that a three dimensional model might exhibit a similar persistent interaction between axisymmetric and non-axisymmetric modes.     

Dipole moment scaling for convection-driven planetary dynamos

Scaling laws are derived for the time-average magnetic dipole moment in rotating convection-driven numerical dynamo models. Results from 145 dynamo models with a variety of boundary conditions and heating modes, covering a wide section of parameter space, show that the time-average dipole moment depends on the convective buoyancy flux F. Two distinct regimes are found above the critical magnetic Reynolds number for onset of dynamo action. In the first regime the external magnetic field is dipole-dominant, whereas for larger buoyancy flux or slower rotation the external field is dominated by higher multipoles and the dipole moment is reduced by a factor of 10 or more relative to the dipolar regime. For dynamos driven by basal heating, the dipole moment M increases like M  F^1/3 in the dipolar regime. Reversing dipolar dynamos tend to cluster near the multipolar transition, which is shown to depend on a local Rossby number parameter. The geodynamo lies close to this transition, suggesting an explanation for polarity reversals and the possibility of a weaker dipole earlier in Earth history. Internally heated dynamos generate smaller dipole moments overall and show a gradual transition from dipolar to multipolar states. Our scaling yields order of magnitude agreement with the dipole moments of Earth, Jupiter, Saturn, Uranus, Neptune, and Ganymede, and predicts a multipolar-type dynamo for Mercury.     

An explicit solution for static unbounded helical dynamos

Abstract

The Lortz dynamo with helical symmetry is re-examined. It is shown that by imposing appropriate boundary conditions the set of possible solutions can be broken down into various classes characterized by the behavior of the mean magnetic field. It is found that, as the cylindrical radius, s, tends to zero, <B_Φ> ～ 0(s^j), <B_z> ～ const + 0(s^j?i), where j>5. It is proved that the azimuthal wavenumber associated with the j=5 class is necessarily equal to 2. The existence of at least one cylindrical surface inside which the dynamo is self-sustained is demonstrated. A new simple explicit solution is obtained. The topology the magnetic field is studied and three-dimensional pictures of the magnetic field lines are exhibited. Finally, a criterion for reversal of the magnetic field as a function of radius is ohtained and is applied to our solution.     

Residual fields from extinct dynamos

Abstract

The paper explores some of the many facets of the problem of the generation of magnetic fields in convective zones of declining vigor and/or thickness. The ultimate goal of such work is the explanation of the magnetic fields observed in A-stars. The present inquiry is restricted to kinematical dynamos, to show some of the many possibilities, depending on the assumed conditions of decline of the convection. The examples serve to illustrate in what quantitative detail it will be necessary to describe the convection in order to extract any firm conclusions concerning specific stars.

The first illustrative example treats the basic problem of diffusion from a layer of declining thickness. The second adds a buoyant rise to the field in the layer. The third treats plane dynamo waves in a region with declining eddy diffusivity, dynamo coefficient, and large-scale shear. The dynamo number may increase or decrease with declining convection, with an increase expected if the large-scale shear does not decline as rapidly as the eddy diffusivity. It is shown that one of the components of the field may increase without bound even in the case that the dynamo number declines to zero.     

Bounds on the infimum decay rate for axisymmetric incompressible dynamos

Abstract

The variational lower bound v > 0.39π² determined by Ivers (1984) for the infimum decay rate v of axisymmetric poloidal magnetic fields is corroborated numerically and an upper bound v < 0.66°² established. This is achieved by correcting and extending results for certain flows considered by Chandrasekhar (1956).     

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.     

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.     

Ponderomotive effects in magnetospheric hydromagnetic waves

The ponderomotive force (PMF) is a ubiquitous nonlinear wave effect arising in plasma physics when applied wave fields or plasma parameters have significant spatial gradients. Some basic properties of the PMF are described as they relate to ULF hydromagnetic waves in the magnetosphere. Examples are given of recent results obtained using both analytical and numerical techniques for waves from the lowest frequencies (determined by the dimensions of the magnetosphere) up to the vicinity of the ion cyclotron frequency. These results include the possibilities that the PMF may transport plasma over large distances in the magnetosphere, and that the PMF may energize magnetospheric ions significantly. In particular the PMF may play a role in transporting and energizing O⁺ ions from the ionosphere into the body of the magnetosphere. The PMF can also generate nonlinear coupling between the slow magnetosonic mode and the other hydromagnetic modes. This should lead to limitation of density enhancements and, notably in the case of standing Alfvén waves, to spatial harmonic generation, secularly growing frequency shifts, and saturation of driven wave fields. Some implications of these results for the magnetosphere are discussed.Based on an invited review given in IAGA Symposium 3.07, Nonlinear and Kinetic Effects in ULF Waves, at the IUGG XXI General Assembly, Boulder, Colorado, July 2-14, 1995.     

Structure of magnetic fields generated by galactic dynamos

Abstract

We investigated global axisymmetric (m = 0) and non-axisymmetric (m = 1) modes of magnetic fields generated by the galactic dynamo including the α²-dynamo. The α²-dynamo is responsible for the field generation in the central region of galaxies where the shear of galactic rotation is weak (e.g. M51). The highest growth rate of m = 1 modes is always smaller than that of m = 0 modes; thus m = 1 modes of the standard galactic dynamo cannot explain the dominance of the bisymmetric fields in spiral galaxies. Radial extent of each m = 1 mode is too narrow to reproduce the observed bisymmetric structure extending over a disk.     

A unified approach to a class of slow dynamos

Abstract

Dynamo action in a highly conducting fluid with small magnetic diffusivity η is particularly sensitive to the topology of the flow. The sites of rapid magnetic field regeneration, when they occur, appear to be located at the stagnation points or in regions where the particle paths are chaotic. Elsewhere only slow dynamo action is to be expected. Two such examples are the nearly axially symmetric dynamo of Braginsky and the generalisation to smooth velocity fields of the Ponomarenko dynamo. Here a method of solution is developed, which applies to both these examples and is applicable to other situations, where magnetic field lines are close to either closed or spatially periodic contours. Particular attention is given to field generation in the neighbourhood of resonant surfaces where growth rates may be intermediate between the slow diffusive and fast convective time scales. The method is applied to the case of the two-dimensional ABC-flows, where it is shown that such intermediate dynamo action can occur on resonant surfaces.     

Detecting thermal boundary control in surface flows from numerical dynamos

The geomagnetic field and secular variation exhibit asymmetrical spatial features which are possibly originating from an heterogeneous thermal control of the Earth's lower mantle on the core. The identification of this control in magnetic data is subject to several difficulties, some of which can be alleviated by the use of core surface flow models. Using numerical dynamos driven by heterogeneous boundary heat flux, we confirm that within the parameter space accessible to simulations, time average surface flows obey a simple thermal wind equilibrium between the Coriolis and buoyancy forces, the Lorentz, inertial and viscous forces playing only a secondary role, even for Elsasser numbers significantly larger than 1. Furthermore, we average the models over the duration of three vortex turnovers, and correlate them with a longer time average which fully reveals the signature of boundary heterogeneity. This allows us to quantify the possibility of observing mantle control in core surface flows averaged over a short time period. A scaling analysis is performed in order to apply the results to the Earth's core. We find that three vortex turnovers could represent between 100 and 360 years of Earth time, and that the heat flux heterogeneity at the core-mantle boundary could be large enough to yield an observable signature of thermal mantle control in a time average core surface flow within reach of the available geomagnetic data.     

Shear flow instability of rotating hydromagnetic fluids

Abstract

The effect of an axial magnetic field on the linear stability of shear flows in rotating systems is examined by extending Busse's analysis of the nonmagnetic case to fluids of high magnetic diffusivity in the presence of a magnetic field. The shear is caused by differential rotation which creates slight deviations from a state of rigid rotation, corresponding to a small Rossby number. It is found that the Rossby number for the onset of instability is larger when a magnetic field is present than when it is absent.     

Effects of an outer thin stably stratified layer on planetary dynamos

The presence of outer stably stratified layers in planetary cores has been suggested for Earth, Saturn and Mercury. In this study, we use a 3-D numerical dynamo model to investigate the effects of a thin stable layer surrounding a convecting interior on the produced magnetic field. We find that a stable layer can destabilize the field morphology through a thermal wind that produces unfavorable zonal flows throughout the core. The direction of these zonal flows is prograde in equatorial regions, unlike a model with no stable layer that has retrograde equatorial flows. Our models therefore suggest that the Earth does not have a stable layer since we observe a westward drift as opposed to an eastward drift. For Saturn, we find that due to coupling of the flows in the stable and unstable layers, the layer does not act to shear out the non-axisymmetry in the observed magnetic field, and therefore cannot explain Saturn’s axisymmetric magnetic field. For Mercury, we find that if the stable layer is thin, it can actively produce strong or weak surface fields and not necessarily attenuate smaller scale features through the skin effect.     

Meteorite paleomagnetism - From magnetic domains to planetary fields and core dynamos

Meteorites represent the earliest records of the evolution of the solar system, providing information on the conditions, processes and chronology for formation of first solids, planetesimals and differentiated bodies. Evidence on the nature of magnetic fields in the early solar system has been derived from chondritic meteorites. Chondrules, which are millimeter sized silicate spherules formed by rapid melting and cooling, have been shown to retain remanent magnetization records dating from the time of chondrule formation and accretion of planetesimals. Studies on different meteorite classes, including ordinary and carbonaceous chondrites, have however provided contrasting results with wide ranges for protoplanetary disk magnetic fields. Developments on instrumentation and techniques for rock magnetic and paleointensity analyses are allowing increased precision. Micromagnetic and an array of geochemical, petrographic and electronic microscopy analyses provide unprecedented resolution, characterizing rock magnetic properties at magnetic domain scales. We review studies on chondrules from the Allende meteorite that reveal relationships among hysteresis parameters and physical properties. Coercivity, remanent and saturation remanence parameters correlate with chondrule size and density; in turn related to internal chondrule structure, mineralogy and morphology. Compound, fragmented and rimmed chondrules show distinct hysteresis properties, related to mineral composition and microstructures. The remanent magnetization record and paleointensity estimates derived from the Allende and other chondrites support remanent acquisition under influence of internal magnetic fields within parent planetesimals. Results support that rapid differentiation following formation of calcium-aluminum inclusions and chondrules gave rise to differentiated planetesimals with iron cores, capable of generating and sustaining dynamo action for million year periods. The Allende chondrite may have derived from a partly differentiated planetesimal which sustained an internal magnetic field.     

A nonlinear,resistive boundary layer in rotating hydromagnetic flow

A new nonlinear boundary layer in rotating hydromagnetic flows is presented. The purpose of this layer is to provide a smooth transition between the magnetic field at a rigid, electrically insulating boundary and that far from the boundary. The boundary layer problem is solved in a cylindrical geometry, assuming that the variables obey the hydromagnetic analog of the von Karman similarity. The primary momentum balance within this boundary layer is between pressure, Coriolis and hydromagnetic forces; the viscous and nonlinear inertia forces are unimportant. Diffusion of the magnetic field is balanced by advection where the advecting fluid flow is induced by the action of the hydromagnetic body forces within the boundary layer. The structure of the layer depends upon a single parameter which measures the strength of the normal magnetic field. Steady solutions are presented and their uniqueness and temporal stability are analyzed. The relevance of this layer to the core of the Earth is discussed. It is estimated that this boundary layer would be at least as thick as one tenth of the core radius if it exists within the core.     

Steady hydromagnetic viscous flow in an annulus with porous walls

Summary An exact solution of electrically conducting viscous incompressible flow in an annulus with porous walls under an external radial magnetic field is obtained when the motion is due to both longitudinal motion of the inner boundary and a constant axial pressure gradient, and the fluid injection rate at one wall is equal to the fluid withdrawal rate at the other. The fluid may be injected at the outer wall and sucked at the inner or vice versa. The solution for the hydromagnetic flow between two flat plates has also been obtained as a limiting case of the annulus problem.