The construction of exact Taylor states. II: The influence of an inner core

Abstract

The geodynamo mechanism, responsible for sustaining Earth’s magnetic field, is believed to be strongly influenced by the solid inner core through its influence on the structure of convection within the tangent cylinder. In the rapidly rotating low-viscosity regime of the geodynamo equations relevant to the Earth’s core, the magnetic field must satisfy a continuum of conditions known as Taylor’s constraint. Magnetic fields that satisfy this constraint, termed Taylor states, have the property that their axial magnetic torque vanishes when averaged over any geostrophic contour, cylinders of fluid coaxial with the rotational axis. In recent theoretical developments, we proved that when adopting a truncated spherical harmonic expansion, the continuous constraint in space reduced to a finite spectral set of conditions. Furthermore, an expedient choice of regular radial basis presents an under-determined problem when constructing Taylor states in a full-sphere showing the ubiquity of such solutions. A spherical-shell geometry, with a conducting inner core, complicates the formulation of Taylor’s constraint due to the partitioning of the geostrophic contours into three distinct regions, ostensibly trebling the stringency of the constraint. This raises questions as to the admissible structures of such Taylor states, and their relation to those in a full-sphere. In this paper, we address these issues in two stages. First, we present a mathematical characterisation of the structure of Taylor’s constraint in a spherical-shell. We then enumerate the effective number of conditions that must be satisfied by any magnetic field that is everywhere C ∞ inside the core, and show that, assuming an equal truncation in radial and solid angle representation, the number of conditions is approximately 5/3 times that for a full-sphere Taylor state. Second, we investigate the influence of the inner core on the structure of admissible Taylor states by constructing a low-degree family of optimally smooth observationally consistent examples in both a spherical-shell and a full-sphere. We show that the introduction of an inner core into a full-sphere increases the minimum magnetic field complexity, simply by virtue of the increased potency of Taylor’s constraint, a trait more pronounced in our quasi-axisymmetric models. We speculate that axisymmetric dynamo-generated exact Taylor states, particularly those generated in a spherical-shell, in general have small radial length scales that may be difficult to resolve.