Convection Driven Dynamos in Rotating Spheres

Of the objects in the solar system the Earth, Mercury, Jupiter, Saturn, Uranus, Neptune, Ganymede, and the Sun exhibit a magnetic field. These magnetic fields are believed to be generated by the magnetohydrodynamic dynamo process, in which current, generated as electrically conducting fluid crosses magnetic field lines, regenerates the magnetic field. Although most of the bodies listed above are believed to consist of a fluid outer core with a solid inner core, i.e. a spherical shell geometry, the full sphere dynamo problem is of physical interest as the dynamo of the early Earth, the ancient dynamo of Mars, and possibly Venus, the Moon and (currently) Mercury, are believed to have had no solid inner core. In this thesis we consider numerically the problem of magnetic field generation in a full sphere of rotating uniformly conducting fluid driven by a volumetric heat source. In order to numerically integrate the governing system of equations we combine the poloidal-toroidal field representation of Elsasser (1946) and Bullard&Gellman (1954) with an implicit/explicit multi-step Gear timestepping method and finite differences in radius. For the implicit radial differencing we develop a generalised compact finite-difference method which results in high order/low bandwidth timestepping systems, and we demonstrate that this method is competitive with other finite-difference methods: standard finite differences, Padé finite-differences, and the combined compact finitedifference schemes of Chu&Fan (1998). The numerical integrator is applied to three physical problems of interest. The first is kinematic dynamo action in a sphere. We investigate the possibility of dynamo action for flows with a missing component in spherical polar coordinates and find the growth rates are highly sensitive to changes in the truncation level. Nevertheless, we do find a working kinematic dynamo with axisymmetric velocity with no azimuthal component which demonstrates convincing convergence. The second problem we consider is that of thermal convection in the absence of a magnetic field in a rotating sphere. We fix the Ekman and Prandtl number (E; Pr) = (5 10¿4; 0:7) and obtain an estimate of the critical Rayleigh number Rac for the onset of convection, and describe the main characteristic of the flow for the convection solutions for Ra 1:4 Rac and Ra 5 Rac. These solutions are primarily for comparison for solutions computed in the third problem: dynamical dynamo action in a rotating sphere. The primary aim is to survey dynamo solutions for the fixed Ekman and Prandtl numbers (E; Pr) = (5 10¿4; 0:7), for magnetic Prandtl number varied from 1 to 40 and the modified Rayleigh number varied up to a few times the critical value for the onset of convection. We consider the solutions through the lens of dynamo scaling laws, but find no universally satisfactory theoretical or numerical scaling law. We also consider a weak/strong field classification of the solutions, finding highly localised force balances. We finish by considering three solutions in detail which represent three distinct classes of dynamo solution: an oscillating dipolar solution, an oscillating quadrupolar solution and a chaotic solution which oscillates between two different hemispherical states. Finally, we develop a first approach to the problem of dynamo action in a fluid sphere as it cools (with no internal heat source), and we present some first convective solutions which function exactly as we expect: the convection dieing down as the fluid cools

Taylor state dynamos found by optimal control: axisymmetric examples

Earth’s magnetic field is generated in its fluid metallic core through motional induction in a process termed the geodynamo. Fluid flow is heavily influenced by a combination of rapid rotation (Coriolis forces), Lorentz forces (from the interaction of electrical currents and magnetic fields) and buoyancy; it is believed that the inertial force and the viscous force are negligible. Direct approaches to this regime are far beyond the reach of modern high-performance computing power, hence an alternative ‘reduced’ approach may be beneficial. Taylor (Proc. R. Soc. Lond. A, vol. 274 (1357), 1963, pp. 274–283) studied an inertia-free and viscosity-free model as an asymptotic limit of such a rapidly rotating system. In this theoretical limit, the velocity and the magnetic field organize themselves in a special manner, such that the Lorentz torque acting on every geostrophic cylinder is zero, a property referred to as Taylor’s constraint. Moreover, the flow is instantaneously and uniquely determined by the buoyancy and the magnetic field. In order to find solutions to this mathematical system of equations in a full sphere, we use methods of optimal control to ensure that the required conditions on the geostrophic cylinders are satisfied at all times, through a conventional time-stepping procedure that implements the constraints at the end of each time step. A derivative-based approach is used to discover the correct geostrophic flow required so that the constraints are always satisfied. We report a new quantity, termed the Taylicity, that measures the adherence to Taylor’s constraint by analysing squared Lorentz torques, normalized by the squared energy in the magnetic field, over the entire core. Neglecting buoyancy, we solve the equations in a full sphere and seek axisymmetric solutions to the equations; we invoke - and -effects in order to sidestep Cowling’s anti-dynamo theorem so that the dynamo system possesses non-trivial solutions. Our methodology draws heavily on the use of fully spectral expansions for all divergenceless vector fields. We employ five special Galerkin polynomial bases in radius such that the boundary conditions are honoured by each member of the basis set, whilst satisfying an orthogonality relation defined in terms of energies. We demonstrate via numerous examples that there are stable solutions to the equations that possess a rapidly decreasing spectrum and are thus well-converged. Classic distributions for the - and -effects are invoked, as well as new distributions. One such new -effect model possesses oscillatory solutions for the magnetic field, rarely before seen. By comparing our Taylor state model with one that allows torsional oscillations to develop and decay, we show the equilibrium state of both configurations to be coincident. In all our models, the geostrophic flow dominates the ageostrophic flow. Our work corroborates some results previously reported by Wu & Roberts (Geophys. Astrophys. Fluid Dyn., vol. 109 (1), 2015, pp. 84–110), as well as presenting new results; it sets the stage for a three-dimensional implementation where the system is driven by, for example, thermal convection

Can Core Flows inferred from Geomagnetic Field Models explain the Earth's Dynamo?

We test the ability of large scale velocity fields inferred from geomagnetic secular variation data to produce the global magnetic field of the Earth.Our kinematic dynamo calculations use quasi-geostrophic (QG) flows inverted from geomagnetic field models which, as such, incorporate flow structures that are Earth-like and may be important for the geodynamo.Furthermore, the QG hypothesis allows straightforward prolongation of the flow from the core surface to the bulk.As expected from previous studies, we check that a simple quasi-geostrophic flow is not able to sustain the magnetic field against ohmic decay.Additional complexity is then introduced in the flow, inspired by the action of the Lorentz force.Indeed, on centenial time-scales, the Lorentz force can balance the Coriolis force and strict quasi-geostrophy may not be the best ansatz.When the columnar flow is modified to account for the action of the Lorentz force, magnetic field is generated for Elsasser numbers larger than 0.25 and magnetic Reynolds numbers larger than 100.This suggests that our large scale flow captures the relevant features for the generation of the Earth's magnetic field and that the invisible small scale flow may not be directly involved in this process.Near the threshold, the resulting magnetic field is dominated by an axial dipole, with some reversed flux patches.Time-dependence is also considered, derived from principal component analysis applied to the inverted flows.We find that time periods from 120 to 50 years do not affect the mean growth rate of the kinematic dynamos.Finally we notice the footprint of the inner-core in the magnetic field generated deep in the bulk of the shell, although we did not include one in our computations

Three-dimensional solutions for the geostrophic flow in the Earth's core

In his seminal work, Taylor (1963) argued that the geophysically relevant limit for dynamo action within the outer core is one of negligibly small inertia and viscosity in the magnetohydrodynamic equations. Within this limit, he showed the existence of a necessary condition, now well known as Taylor's constraint, which requires that the cylindrically-averaged Lorentz torque must everywhere vanish; magnetic fields that satisfy this condition are termed Taylor states. Taylor further showed that the requirement of this constraint being continuously satisfied through time prescribes the evolution of the geostrophic flow, the cylindrically-averaged azimuthal flow. We show that Taylor's original prescription for the geostrophic flow, as satisfying a given second order ordinary differential equation, is only valid for a small subset of Taylor states. An incomplete treatment of the boundary conditions renders his equation generally incorrect. Here, by taking proper account of the boundaries, we describe a generalisation of Taylor's method that enables correct evaluation of the instantaneous geostrophic flow for any 3D Taylor state. We present the first full-sphere examples of geostrophic flows driven by non-axisymmetric Taylor states. Although in axisymmetry the geostrophic flow admits a mild logarithmic singularity on the rotation axis, in the fully 3D case we show that this is absent and indeed the geostrophic flow appears to be everywhere regular.Comment: 29 Pages, 8 figure

Numerical simulations of current generation and dynamo excitation in a mechanically-forced, turbulent flow

The role of turbulence in current generation and self-excitation of magnetic fields has been studied in the geometry of a mechanically driven, spherical dynamo experiment, using a three dimensional numerical computation. A simple impeller model drives a flow which can generate a growing magnetic field, depending upon the magnetic Reynolds number, Rm, and the fluid Reynolds number. When the flow is laminar, the dynamo transition is governed by a simple threshold in Rm, above which a growing magnetic eigenmode is observed. The eigenmode is primarily a dipole field tranverse to axis of symmetry of the flow. In saturation the Lorentz force slows the flow such that the magnetic eigenmode becomes marginally stable. For turbulent flow, the dynamo eigenmode is suppressed. The mechanism of suppression is due to a combination of a time varying large-scale field and the presence of fluctuation driven currents which effectively enhance the magnetic diffusivity. For higher Rm a dynamo reappears, however the structure of the magnetic field is often different from the laminar dynamo; it is dominated by a dipolar magnetic field which is aligned with the axis of symmetry of the mean-flow, apparently generated by fluctuation-driven currents. The fluctuation-driven currents have been studied by applying a weak magnetic field to laminar and turbulent flows. The magnetic fields generated by the fluctuations are significant: a dipole moment aligned with the symmetry axis of the mean-flow is generated similar to those observed in the experiment, and both toroidal and poloidal flux expulsion are observed.Comment: 14 pages, 14 figure

Generation of bursting magnetic fields by nonperiodic torsional flows

A mechanism for the cyclic generation of bursts of magnetic fields by nonlinear torsional flows of complex time dependence but very simple spatial structure is described. These flows were obtained numerically as axisymmetric solutions of convection in internally heated rotating fluid spheres in the Boussinesq approximation. They behave as repeated transients, which start with nearly periodic oscillations of the velocity field of slowly increasing amplitude. This regime is followed by a chaotic fast increase and a final decrease of the amplitude of, at least, one order of magnitude. The magnetic field decays due to the magnetic diffusion during the regular oscillations, but it grows in the form of bursts during the intervals of irregular time dependence of the velocity. The magnetic field is strongly localized in spirals, with spatial- and temporal-dependent intensity.Postprint (published version

The evolution of a magnetic field subject to Taylor′s constraint using a projection operator

In the rapidly rotating, low-viscosity limit of the magnetohydrodynamic equations as relevant to the conditions in planetary cores, any generated magnetic field likely evolves while simultaneously satisfying a particular continuous family of invariants, termed Taylor′s constraint. It is known that, analytically, any magnetic field will evolve subject to these constraints through the action of a time-dependent coaxially cylindrical geostrophic flow. However, severe numerical problems limit the accuracy of this procedure, leading to rapid violation of the constraints. By judicious choice of a certain truncated Galerkin representation of the magnetic field, Taylor′s constraint reduces to a finite set of conditions of size O(N), significantly less than the O(N3) degrees of freedom, where N denotes the spectral truncation in both solid angle and radius. Each constraint is homogeneous and quadratic in the magnetic field and, taken together, the constraints define the finite-dimensional Taylor manifolδ whose tangent plane can be evaluated. The key result of this paper is a description of a stable numerical method in which the evolution of a magnetic field in a spherical geometry is constrained to the manifold by projecting its rate of change onto the local tangent hyperplane. The tangent plane is evaluated by contracting the vector of spectral coefficients with the Taylor tensor, a large but very sparse 3-D array that we define. We demonstrate by example the numerical difficulties in finding the geostrophic flow numerically and how the projection method can correct for inaccuracies. Further, we show that, in a simplified system using projection, the normalized measure of Taylorization, t, may be maintained smaller than O(10-10) (where t= 0 is an exact Taylor state) over 1/10 of a dipole decay time, eight orders of magnitude smaller than analogous measures applied to recent low Ekman-number geodynamo model

Hybrid Radial hp/Angular Galerkin Methods in Linearised Rotating Magnetohydrodynamics in Spheres and Related Geometries.

This thesis investigates hp-methods, involving refinement of mesh (h) and increase of polynomial order (p) used in several applications in linearised rotating magnetohydrodynamics in spherical geometries. The two hp-methods, one based on the Galerkin method, the other on a Chebyshev collocation method, are applied to eigenproblems with boundary layers for solving for viscous inertial modes and the onset of thermal convection at low Ekman number in a strong magnetic field, Elsasser number = 1. Results in one-dimension indicate the Chebyshev method is more efficient for forced problems, whereas the Galerkin method is more efficient for eigenproblems. Both hp-methods on a suitably defined radial mesh outperform the p-version for low Ekman number (10-6), and indicate increased efficiency as the Ekman number decreases. The Chebyshev method may find its use in nonlinear timestepping problems, which is beyond the scope of this thesis. A full sphere element is considered where it is important to build the analytic behaviour of the solutions at the origin into the approximation method. A method based on the weak Poincaré equation in a rotating tilted triaxial ellipsoid is presented. Extensions to ellipsoidal geometries are indicated