Determination of the instantaneous geostrophic flow within the three-dimensional magnetostrophic regime.

Colin M Hardy,Jitse Niesen,Jiawen Luo,Kuan Li,Philip W Livermore

doi:10.1098/rspa.2018.0412

Abstract

In his seminal work, Taylor (1963 Proc. R. Soc. Lond. A 274, 274–283. (doi:10.1098/rspa.1963.0130).) 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 generalization of Taylor's method that enables correct evaluation of the instantaneous geostrophic flow for any three-dimensional 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 three-dimensional case we show that this is absent and indeed the geostrophic flow appears to be everywhere regular.

Highlights

Earth’s magnetic field is generated by a self-excited dynamo process through the flow of electrically conducting fluid in the outer core
We have discussed in some detail how the geostrophic flow, a fundamental part of any magnetostrophic dynamo, might be determined
Of particular note is that we have shown why the method introduced by Taylor [9] fails in most cases, because of its intrinsic assumption that the initial magnetic field structure must satisfy a higher-order boundary condition

Summary

Introduction

Earth’s magnetic field is generated by a self-excited dynamo process through the flow of electrically conducting fluid in the outer core. Beginning with the work of [2,3], it was noted that by artificially increasing these two parameters by many orders of magnitude to typical values of Ro = 10−3, E = 10−7 [4], the numerically difficult rapid timescales and short length scales are smoothed, allowing larger time steps, and permitting a longer time period to be studied for a given finite computer resource Such ( mainstream) models can reproduce many characteristics of Earth’s geomagnetic field, several studies have cast doubt as to whether they obey the correct force balance within the core [1,5,6], some evidence points to models being on the cusp of faithfully representing Earth’s core [7,8]

Objectives

Methods

Findings

Conclusion