Paul H. Roberts

A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle

We present the first three-dimensional (3D), time-dependent, self-consistent numerical solution of the magneto- hydrodynamic (MHD) equations that describe thermal convection and magnetic field generation in a rapidly rotating spherical fluid shell with a solid conducting inner core. This solution, which serves as a crude analog for the geodynamo, is a self-sustaining supercritical dynamo that has maintained a magnetic field for three magnetic diffusion times, roughly 40 000 years. Fluid velocity in the outer core reaches a maximum of 0.4 cm s-1, and at times the magnetic field can be as large as 560 gauss. Magnetic energy is usually about 4000 times greater than the kinetic energy of the convection that maintains it. Viscous and magnetic coupling to both the inner core below and the mantle above cause time-dependent variations in their respective rotation rates; the inner core usually rotates faster than the mantle and decadal variations in the length of the day of the mantle are similar to those observed for the Earth. The pattern and amplitude of the radial magnetic field at the core-mantle boundary (CMB) and its secular variation are also similar to the Earths. The maximum amplitudes of the longitudinally averaged temperature gradient, shear flow, helicity, and magnetic field oscillate between the northern and southern hemispheres on a time scale of a few thousand years. However, only once in many attempts does the field succeed in reversing its polarity because the field in the inner core, which has the opposite polarity to the field in most of the outer core, usually does not have enough time to reverse before the field in the outer core changes again. One successful magnetic field reversal occurs near the end of our simulation.

The role of the Earth's mantle in controlling the frequency of geomagnetic reversals

A series of computer simulations of the Earths dynamo illustrates how the thermal structure of the lowermost mantle might affect convection and magnetic-field generation in the fluid core. Eight different patterns of heat flux from the core to the mantle are imposed over the core–mantle boundary. Spontaneous magnetic dipole reversals and excursions occur in seven of these cases, although sometimes the field only reverses in the outer part of the core, and then quickly reverses back. The results suggest correlations among the frequency of reversals, the duration over which the reversals occur, the magnetic-field intensity and the secular variation. The case with uniform heat flux at the core–mantle boundary appears most ‘Earth-like’. This result suggests that variations in heat flux at the core–mantle boundary of the Earth are smaller than previously thought, possibly because seismic velocity anomalies in the lowermost mantle might have more of a compositional rather than thermal origin, or because of enhanced heat flux in the mantles zones of ultra-low seismic velocity.

EQUATIONS GOVERNING CONVECTION IN EARTH'S CORE AND THE GEODYNAMO

Abstract Convection in Earths fluid core is regarded as a small deviation from a well-mixed adiabatic state of uniform chemical composition. The core is modeled as a binary alloy of iron and some lighter constituent, whose precise chemical composition is unknown but which is here assumed to be FeAd, where Ad = Si, O or S. The turbulent transport of heat and light constituent is considered, and a simple ansatz is proposed in which this is modeled by anisotropic diffusion. On this basis, a closed system of equations and boundary conditions is derived that governs core convection and the geodynamo. The dual (thermal + compositional) nature of core convection is reconsidered. It is concluded that compositional convection may not dominate thermal convection, as had previously been argued by Braginsky (Soviet Phys. Dokl., v. 149, p. 8, 1963; Geomag, and Aeron., v. 4, p. 698, 1964), but that the two mechanisms are most probably comparable in importance. The key parameters leading to this conclusion are isolated...

On the thermal instability of a rotating-fluid sphere containing heat sources

The theory of marginal convection in a uniformly rotating, self-gravitating, fluid sphere, of uniform density and containing a uniform distribution of heat sources, is developed to embrace modes of convection which are asymmetric with respect to the axis of rotation. It is shown that these modes are the most unstable, except for the smallest Taylor numbers, T (a measure of the rotation rate); i.e. for any T and o) (Prandtl number), the lowest Rayleigh number (a measure of the temperature gradients in the sphere) is associated with an asymmetric motion. This is demonstrated both by an expansion method suitable for small T, and by asymptotic theory for T oo. For large T, the eigenmode most easily excited is small in amplitude everywhere except near a cylindrical surface, of radius about half that of the sphere, and coaxial with the diameter parallel to the angular velocity vector.

Kinematic dynamo models

Spherical kinematic dynamo models with axisymmetric magnetic fields are examined, which arise from the mean field electrodynamics of Steenbeck and Krause, and also from the nearly axisymmetric limit of Braginskii. Four main cases are considered: (i) there is no mean flow, but the dynamo is maintained by microscale motions which create a mean electromotive force, (E), proportional to the mean magnetic field, B (the α effect); (ii) in addition to an α effect which creates poloidal mean field from toroidal, a mean toroidal shearing flow (angular velocity w) is present which creates toroidal mean field from poloidal more efficiently than by the α effect; (iii) in addition to the processes operative in (ii), a mean meridional circulation, m, is present; (iv) (E) is produced by a second order inductive process first isolated by Radler. When these processes are sufficiently strong, they can maintain magnetic fields. The resulting situations are known as (i) α2 dynamos, (ii) αω dynamos, (iii) αω dynamos with meridional circulations, and (iv) Rädler dynamos. Models of each type are considered below, but cases (ii) and (iii) give rise to particularly interesting results. If |m| is sufficiently small, or zero [case (ii)], the most easily excited dynamo is oscillatory and is of dipole type if αω′<0 in the northern hemisphere (and negative in the southern); here ω′ denotes the outward gradient of ω. The oscillation resembles a Parker dynamo wave, generated at the poles, absorbed at the equator and always moving towards lower latitudes, as for the butterfly diagrams of sunspots. If αω′>0 in the northern hemisphere, the direction of wave motion is reversed, and also the quadrupolar solution is more readily excited than the dipolar. If |m| is sufficiently large, and of the right magnitude and sense (which is model dependent), it is found that the dynamo which regenerates most easily is steady. It is of dipolar form if αω′>0 but quadrupolar if αω′<0. These models appear to be relevant to the Earth, where meridional circulations might be provided by, for example, Ekman pumping. Evidence for a remarkable symmetry property is adduced. If m and αω′ are reversed everywhere in the state in which the dipole (say) is most readily excited, it is found that the state in which a quadrupole is most easily regenerated is recovered, almost precisely. Moreover, the critical magnetic Reynolds number for each is closely similar. As a corollary, the critical Reynolds numbers for dipolar and quadrupolar solutions of opposite αω′ are nearly identical for the αω dynamo (m = 0).

Rotation and Magnetism of Earth's Inner Core

Three-dimensional numerical simulations of the geodynamo suggest that a superrotation of Earths solid inner core relative to the mantle is maintained by magnetic coupling between the inner core and an eastward thermal wind in the fluid outer core. This mechanism, which is analogous to a synchronous motor, also plays a fundamental role in the generation of Earths magnetic field.

Convection in horizontal layers with internal heat generation. Theory

A theoretical study has been made of an experiment by Tritton & Zarraga (1967) in which eonvective motions were generated in a horizontal layer of water (cooled from above) by the application of uniform heating. The marginal stability problem for such a layer is solved, and a critical Rayleigh number of 2772 is obtained, at which patterns of wave-number 2·63 times the reciprocal depth of the layer are marginally stable. The remainder of the paper is devoted to the finite amplitude convection which ensues when the Rayleigh number, R , exceeds 2772. The theory is approximate, the basic simplification being that, to an adequate approximation, Fourier decompositions of the convective motions in the horizontal ( x, y ) directions can be represented by their dominant (planform) terms alone. A discussion is given of this hypothesis, with illustrations drawn from the (better studied) Benard situation of convection in a layer heated below, cooled from above, and containing no heat sources. The hypothesis is then used to obtain ‘mean-field equations’ for the convection. These admit solutions of at least three distinct forms: rolls, hexagons with upward flow at their centres, and hexagons with downward flow at their centres. Using the hypothesis again, the stability of these three solutions is examined. It is shown that, for all R , a (neutrally) stable form of convection exists in the form of rolls. The wave-number of this pattern increases gradually with R. This solution is, in all respects, independent of Prandtl number. It is found, numerically, that the hexagons with upward motions in their centres are unstable, but that the hexagons with downward motions at their centres are completely stable, provided R exceeds a critical value (which depends on Prandtl number, P , and which for water is about 3 R c ), and provided the wave-number of the pattern lies within certain limits dependent on R and P .

RECENT GEODYNAMO SIMULATIONS AND OBSERVATIONS OF THE GEOMAGNETIC FIELD

In 1995, two groups [Kageyama et al., 1995; Glatzmaier and Roberts, 1995a, 1995b] reported results of numerical integrations of fully three-dimensional, fully nonlinear dynamos. Their papers were precursors of a stream ofsuch models that have focused particularly on the geodynamo. They provide us, in unprecedented detail, with spectacular realizations of interesting geomagnetic field behaviors, such as secular variation and even polarity reversals. The proliferation of models has, however, created some confusion and apparently conflicting results. This can be partly attributed to the different ways in which different groups have modeled the core, normalized their equations, defined their dimensionless parameters, chosen their boundary conditions, and selected their energy sources. This has made it difficult to compare the results of different simulations directly. In this paper, we first try, as far as possible, to overcome this difficulty, so that all reported results can be compared on common ground. We then review the results, emphasizing three major topics: (1) onset and evolution of convection, (2) character of the magnetic field generated, and (3) comparison with the observed geomagnetic field. Although there are large differences in the way that the simulations are defined, the magnetic fields that they generate have some surprising similarities. The fields are dominated by the axial dipole. In some models they are most strongly generated in shear layers near the upper and lower boundaries and near the tangent cylinder, an imaginary surface touching the inner core on its equator. Convection rolls occur within which a type of the a effect distorts the toroidal field lines to create poloidal magnetic field. Some features of the models are found to strongly affect the fields that they produce. In particular, the boundary conditions defining the energy flow (e.g., an inhomogeneous heat flux or distribution of buoyancy sources) are very influential and have been extensively studied. They change the frequency and the mode of magnetic polarity reversals as well as the ratio in strengths of the dipole and nondipole moments. As the ultimate goal of geodynamo simulations is to explain the features of the real geomagnetic field, it is essential that proper comparisons be made between simulation results and observations. It is remarkable that polarity reversals reminiscent of the paleomagnetically observed field reversals have already been simulated by some of the models. Other features such as drift of the field, its secular variation, and statistical properties of Gauss coefficients are discussed in this paper and are compared with observations. These comparisons are rather primitive, not only because self-consistent dynamo models are still too new and too few but also because many of the observations (and especially the paleomagnetic data) are themselves not yet reliable or decisive enough. The aim of the third part of this paper is therefore more to demonstrate the potential use of simulations than to elucidate the nature of geomagnetic field generation.

An anelastic evolutionary geodynamo simulation driven by compositional and thermal convection

Abstract We have extended our geodynamo simulation 40 000 years using a more realistic representation of the thermodynamics and convection. The anelastic approximation replaces the Boussinesq approximation; and compositional buoyancy, in addition to thermal buoyancy, drives convection in the fluid outer core. Boundary conditions at the inner core boundary model the freezing of the heavy constituent onto the solid inner core and the release of the light constituent into the fluid outer core. The resulting simulated magnetic field has a strongly dipole dominated structure outside the core, similar to the Earths field, so far displaying no tendency to reverse its dipole polarity. The non-dipolar structure of the field at the core-mantle boundary is also quite similar to the Earths,including the rate of its general westward drift. As in our original simulation, the solid inner core typically rotates about 1°/yr faster than the mantle, in agreement with recent seismic studies of the Earth. This three-dimensional self-consistent solution seems to be simulating the stable regime of the geodynamo between reversals.

Simulating the geodynamo

Three-dimensional numerical simulations of convection and magnetic field generation in the Earths core now span several hundred thousand years; the magnetic field created during most of this time has an intensity, structure and time dependence similar to the present geomagnetic field. Five models are described here. The first is a homogeneous Boussinesq model, driven steadily by heat sources on the inner core boundary. At about 36 000 years into the simulation, a reversal of the dipole moment occurs that resembles those seen in the paleomagnetic reversal record. The four subsequent models are inhomogeneous, that is they allow for the varying properties of the Earth with depth. They are also evolutionary, in that they are powered by the secular cooling of the Earth over geological time. This cooling causes the inner core to grow through freezing, with the concomitant release at the inner core boundary of not only latent heat of crystallization but also light constituents of core fluid that provide respect...