[Paper Review] Three-dimensional solutions for the geostrophic flow in the Earth's core

Colin M. Hardy, Philip W. Livermore+3 more|arXiv (Cornell University)|Jun 18, 2018

Geomagnetism and Paleomagnetism Studies43 references3 citations

TL;DR

This paper presents a generalized method to compute the instantaneous geostrophic flow in Earth's core for arbitrary three-dimensional Taylor states, correcting an error in Taylor's original 1963 formulation that neglected proper boundary conditions. By rigorously enforcing boundary constraints, the authors derive a fully regular geostrophic flow without logarithmic singularities on the rotation axis, even in non-axisymmetric configurations, and present the first full-sphere 3D examples of such flows.

ABSTRACT

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.

Motivation & Objective

• To resolve the long-standing inaccuracy in Taylor's 1963 formulation of geostrophic flow, which incorrectly treated boundary conditions for non-axisymmetric magnetic fields.
• To develop a general method for computing the instantaneous geostrophic flow that is consistent with the full 3D Taylor state condition, including proper enforcement of boundary conditions.
• To demonstrate that the geostrophic flow is everywhere regular in 3D, even in the presence of non-axisymmetric magnetic fields, contrary to the logarithmic singularity predicted in axisymmetric cases.
• To provide a computationally efficient framework for modeling long-term geodynamo evolution by correctly slaving the geostrophic flow to the magnetic field under the magnetostrophic approximation.

Proposed method

• The method reformulates the geostrophic flow as a solution to a modified second-order ordinary differential equation that incorporates full boundary conditions, correcting Taylor’s original approach.
• It uses a Galerkin spectral method with spherical harmonic and radial polynomial basis functions (Jacobi polynomials) to represent the magnetic field and velocity fields in a divergence-free, orthonormal basis.
• The ageostrophic flow is computed by solving the magnetostrophic equation via modal expansion, with unknown coefficients determined by matching powers of radius and solving analytically using symbolic computation (e.g., Maple).
• The geostrophic flow is extracted by removing the cylindrically-averaged azimuthal component from the total velocity, ensuring uniqueness and consistency with the Taylor constraint.
• The method ensures $L^2$ orthonormality and regularity at the origin, and satisfies the impenetrability condition on the core-mantle boundary.
• The solution is validated by projecting the magnetic field onto the Galerkin basis and solving for coefficients via spherical harmonic transforms and radial integrals.

Experimental results

Research questions

• RQ1 Does Taylor’s original 1963 second-order ODE for geostrophic flow remain valid when non-axisymmetric magnetic fields and proper boundary conditions are considered?
• RQ2 Can a fully three-dimensional, non-axisymmetric Taylor state produce a geostrophic flow that is regular everywhere, including on the rotation axis?
• RQ3 What is the correct mathematical formulation for the geostrophic flow that ensures the Lorentz torque vanishes at all radii and times, as required by the Taylor constraint?
• RQ4 How can the geostrophic flow be uniquely determined in 3D when the magnetostrophic equation alone does not constrain it?
• RQ5 What are the implications of boundary condition misapplication in previous models for the long-term evolution of the geodynamo?

Key findings

• The original Taylor 1963 equation for geostrophic flow is invalid in general due to an incomplete treatment of boundary conditions, particularly in non-axisymmetric cases.
• The corrected method produces a geostrophic flow that is everywhere regular in 3D, eliminating the logarithmic singularity on the rotation axis that appears in axisymmetric solutions.
• The first full-sphere 3D examples of geostrophic flows driven by non-axisymmetric Taylor states are successfully computed and validated.
• The geostrophic flow is uniquely determined by enforcing the time-continuity of the Taylor constraint, ensuring consistency with the evolving magnetic field.
• The method enables accurate, computationally efficient modeling of long-term geodynamo evolution by correctly slaving the geostrophic flow to the magnetic field in the magnetostrophic regime.
• The use of Galerkin modes with Jacobi polynomials ensures $L^2$ orthonormality, regularity at the origin, and accurate projection of magnetic and velocity fields.