D. J. Ivers

Spectral interactions of rapidly-rotating anisotropic turbulent viscous and thermal diffusion in the Earth’s core

Abstract Rapidly-rotating anisotropic turbulence in the Earth’s core is modelled by the viscous and thermal diffusion tensors, D ν ≔2ρν 0 I +ρν ΩΩ ΩΩ and D κ ≔κ 0 I +κ ΩΩ ΩΩ , where I is the unit tensor, Ω is the angular velocity and the coefficients ν ΩΩ and κ ΩΩ are spherically-symmetric. Toroidal–poloidal spectral interactions are derived for the anisotropic parts of the body forces associated with the mean anisotropic viscous stress tensor, D ν ·(∇ v ) S and with its symmetric part. The stress tensors are linear in the trace-free symmetric part of the velocity gradient. Techniques of vector and tensor spherical harmonic analysis are used to find the vector spherical harmonic components of the body forces. From these components the toroidal–poloidal field interactions, (ΩΩtt n ) , (ΩΩst n ) , (ΩΩts n ) , (ΩΩss n ) , (ΩtΩt n ) , (ΩsΩt n ) , (ΩtΩs n ) and (ΩsΩs n ) , of the toroidal ( t n ) and poloidal ( s n ) momentum equations are derived using computer algebra. The temperature spectral interactions are also derived for the mean heat flux given by the turbulent thermal diffusion tensor. These spectral interactions represent a computationally practical first step in incorporating anisotropic turbulence models into existing dynamically-consistent angular–spectral geodynamo codes.

Spherical anisotropic diffusion models for the Earth's core

Abstract Estimates of the molecular values of magnetic, viscous and thermal diffusion in the Earths core suggest turbulent small-scale velocity, magnetic and temperature fluctuations. Instability arguments (Braginsky, S.I., Meytlis, V.P., 1990. Local turbulence in the Earths core. Geophys. Astrophys. Fluid Dynam. 55 (1990) 71–87.) indicate that the viscous and thermal diffusions of the resulting turbulence are strongly anistropic, the directions of anisotropy being determined by the mean magnetic field, the rotation of the core, the mean temperature gradient and gravity. Physical principles and invariance arguments are used to constrain the forms of the turbulent viscous stress tensor to eight types and the turbulent heat flux vector to a single type. The models are interpreted in terms of angular momentum, kinetic energy and entropy exchange between the mean and fluctuating fields. For the development of the turbulence models in spherical geometries, general non-linear spectral expansions are derived for one of the turbulent viscous stress tensors, the turbulent heat flux vector and their divergences using vector and tensor spherical harmonic methods and poloidal–toroidal representations. The spectral expansions, which are complicated and would be difficult to derive using other methods, include all invariance terms and can be programmed directly from the explicit forms given, thus laying the foundation for future computational studies of core turbulence. They are particularly suitable for linearised problems, such as anisotropic convection and magnetoconvection. The spectral techniques employed are applicable to all eight turbulent viscous stress tensors.

Planar velocity dynamos in a sphere

The Earths main magnetic field is generally believed to be due to a self-exciting dynamo process in the Earths fluid outer core. A variety of antidynamo theorems exist that set conditions under which a magnetic field cannot be indefinitely maintained by dynamo action against ohmic decay. One such theorem, the Planar Velocity Antidynamo Theorem, precludes field maintenance when the flow is everywhere parallel to some plane, e.g. the equatorial plane. This paper shows that the proof of the Planar Velocity Theorem fails when the flow is confined to a sphere, due to diffusive coupling at the boundary. Then, the theorem reverts to a conjecture. There is a need to either prove the conjecture, or find a functioning planar velocity dynamo. To the latter end, this paper formulates the toroidal–poloidal spectral form of the magnetic induction equation for planar flows, as a basis for a numerical investigation. We have thereby determined magnetic field growth rates associated with various planar flows in spheres. For most flows, the induced magnetic field decays with time, supporting a planar velocity antidynamo conjecture for a spherical conducting fluid. However, one flow is exceptional, indicating that magnetic field growth can occur. We also re-examine some classical kinematic dynamo models, converting the flows where possible to planar flows. For the flow of Pekeris et al. (Pekeris, C. L., Accad, Y. & Shkoller, B. 1973 Kinematic dynamos and the Earths magnetic field. Phil. Trans. R. Soc. A 275, 425–461), this conversion dramatically reduces the critical magnetic Reynolds number.

Ørsted and Magsat scalar anomaly fields

The scalar anomaly field determined from available Ørsted data is compared with the upward continued scalar anomaly field derived from Magsat data. Two techniques were used to remove the core field from the Ørsted satellite data. In the first method, monthly spherical harmonic core field models of degree and order 13 derived from scalar and vector data were subtracted, and in the second method, along-track high-pass filtering of scalar data only was used. In both methods, the binned residuals were interpolated to a sphere, and subsequently filtered. Monthly degree and order 13 spherical harmonic core field models were removed from Magsat vector data. The binned Magsat vector residuals were interpolated to a sphere, filtered, and upward continued by high degree spherical harmonic analysis. The corresponding Magsat scalar anomaly field at Ørsted altitude was then determined. For latitudes below 50 degrees, removal of the core field by signal processing techniques from presently available Ørsted data led to a scalar anomaly field in better agreement with that determined from Magsat data, than removal by spherical harmonic analysis.

Antidynamo theorems for non-radial flows

Abstract Magnetic fields induced by non-radial compressible flows (vr ≡ 0, ∇·v ≢ 0) in spherical conductors are shown to decay, or at least be incapable of indefinite amplification. The results herein supplement previously established antidynamo results for purely toroidal flows (vr ≡ ∇·v ≡ 0), with the allowance of compressibility necessitating new proofs with different senses of field decay. The poloidal field variable r·B decays in a global sense with an undetermined decay rate. A pointwise bound is established that limits the ultimate strength of the toroidal field scalar T, and shows that if the poloidal field is negligible and the conductivity uniform then the toroidal field decays to zero at no slower than the poloidal free-decay rate π2.

On the maintenance of magnetic fields by compressible flows and the Nernst–Ettingshausen effect

Abstract The antidynamo theorems of Namikawa and Matsushita (1970) preclude maintenance of a steady magnetic field, or non-steady poloidal magnetic field, by a spherically symmetric radial velocity. A simple reinterpretation of these theorems also precludes such field maintenance by a spherically symmetric Nernst-Ettingshausen thermomagnetic effect. For any non-steady magnetic field supported by the above mechanisms, we show (a) that the poloidal component has a decay time less than five free-decay times, and (b) that any harmonic of the toroidal component with angular length scale less than a critical value, will decay. The models of Namikawa and Matsushita for toroidal fields in radially contracting stars do not decay simply by (b) above, but nevertheless are shown to decay with lives shorter than a free-decay time. These models are seen to be special cases of a larger class of decaying models. A brief discussion of the maintenance of axisymmetric fields by a compressible flow or thermomagnetic effect i...

Off-centre Steiner points for Delaunay-refinement on curved surfaces

An extension of the restricted Delaunay-refinement algorithm for surface mesh generation is described, where a new point-placement scheme is introduced to improve element quality in the presence of mesh size constraints. Specifically, it is shown that the use of off-centre Steiner points, positioned on the faces of the associated Voronoi diagram, typically leads to significant improvements in the shape- and size-quality of the resulting surface tessellations. The new algorithm can be viewed as a Frontal-Delaunay approach - a hybridisation of conventional Delaunay-refinement and advancing-front techniques in which new vertices are positioned to satisfy both element size and shape constraints. The performance of the new scheme is investigated experimentally via a series of comparative studies that contrast its performance with that of a typical Delaunay-refinement technique. It is shown that the new method inherits many of the best features of classical Delaunay-refinement and advancing-front type methods, leading to the construction of smooth, high quality surface triangulations with bounded radius-edge ratios and convergence guarantees. Experiments are conducted using a range of complex benchmarks, verifying the robustness and practical performance of the proposed scheme. Development of a new unstructured triangulation algorithm for smooth surfaces.Hybridisation of existing Delaunay-refinement and advancing-front techniques.Combines desirable aspects of both methods: high element quality, robustness.Provably good behaviour: guaranteed termination, bounded element quality.

Extension of the namikawa-matsushita antidynamo theorem to toroidal fields

Abstract Poloidal magnetic fields induced by radial flows under spherically symmetric conditions are known to decay pointwise to zero on a diffusion time-scale. We here prove decay to zero on the diffusion time-scale, in an integral sense, of toroidal fields in the same conditions.

Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a tri-axial ellipsoid

Abstract Inertial waves often occur in geophysics and astrophysics since fluids dominated by rotation are common. A simple model to study inertial waves consists of a uniform incompressible fluid filling a rigid tri-axial ellipsoid, which rotates about an arbitrary axis fixed in an inertial frame. The waves are due to the Coriolis force, which can be treated mathematically as a skew-symmetric bounded linear operator acting on smooth inviscid flows in the ellipsoid. It is shown that the space of incompressible polynomial flows in the ellipsoid of degree N or less is invariant under . The symmetry of thus implies the Coriolis operator is non-defective with an orthogonal set of eigenmodes – Coriolis modes – in the finite-dimensional space of inviscid polynomial flows in the ellipsoid. The modes with non-zero eigenvalues are the inertial modes; the zero-eigenvalue modes are geostrophic. This shows the Coriolis modes are polynomials, enables their enumeration and leads to proof of their completeness by using the Weierstrass polynomial approximation theorem. The modes are tilted if the rotation axis is not aligned with a principal axis of the ellipsoid. A basic tool is that the solution of the polynomial Poisson-Neumann problem, i.e. Poisson’s equation with Neumann boundary condition and polynomial data, in an ellipsoid is a polynomial. The tilted Coriolis modes of degree one are explicitly constructed and shown to be the only modes with non-zero angular momentum in the boundary frame. All tilted geostrophic modes are also explicitly constructed.

Spherical single-roll dynamos at large magnetic Reynolds numbers

This paper concerns kinematic helical dynamos in a spherical fluid body surrounded by an insulator. In particular, we examine their behavior in the regime of large magnetic Reynolds number Rm, for which dynamo action is usually concentrated on a simple resonant stream surface. The dynamo eigensolutions are computed numerically for two representative single-roll flows using a compact spherical harmonic decomposition and fourth-order finite differences in radius. These solutions are then compared with the growth rates and eigenfunctions of the Gilbert and Ponty large Rm asymptotic theory [Geophys. Astrophys. Fluid Dyn. 93, 55 (2000)]. We find good agreement between the growth rates when Rm>104 and between the eigenfunctions when Rm>105.