A comparison of no-slip, stress-free and inviscid models of rapidly rotating fluid in a spherical shell.

Philip W Livermore,Rainer Hollerbach,Lewis M Bailey

doi:10.1038/srep22812

Philip W Livermore, Rainer Hollerbach + Show 1 more

Open Access

https://doi.org/10.1038/srep22812

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Journal: Scientific Reports	Publication Date: Mar 16, 2016
Citations: 13	License type: CC BY 4.0

Affiliation: University of Leeds

Abstract

We investigate how the choice of either no-slip or stress-free boundary conditions affects numerical models of rapidly rotating flow in Earth’s core by computing solutions of the weakly-viscous magnetostrophic equations within a spherical shell, driven by a prescribed body force. For non-axisymmetric solutions, we show that models with either choice of boundary condition have thin boundary layers of depth E1/2, where E is the Ekman number, and a free-stream flow that converges to the formally inviscid solution. At Earth-like values of viscosity, the boundary layer thickness is approximately 1 m, for either choice of condition. In contrast, the axisymmetric flows depend crucially on the choice of boundary condition, in both their structure and magnitude (either E−1/2 or E−1). These very large zonal flows arise from requiring viscosity to balance residual axisymmetric torques. We demonstrate that switching the mechanical boundary conditions can cause a distinct change of structure of the flow, including a sign-change close to the equator, even at asymptotically low viscosity. Thus implementation of stress-free boundary conditions, compared with no-slip conditions, may yield qualitatively different dynamics in weakly-viscous magnetostrophic models of Earth’s core. We further show that convergence of the free-stream flow to its asymptotic structure requires E ≤ 10−5.

Highlights

We investigate how the choice of either no-slip or stress-free boundary conditions affects numerical models of rapidly rotating flow in Earth’s core by computing solutions of the weakly-viscous magnetostrophic equations within a spherical shell, driven by a prescribed body force
In this paper we have investigated the role of the choice of mechanical boundary condition on the boundary layers, free-stream and zonal flows in weakly-viscous numerical solutions
Irrespective of boundary condition, non-axisymmetric solutions converge to the free-stream flow outside thin boundary layers

Summary

Introduction

We investigate how the choice of either no-slip or stress-free boundary conditions affects numerical models of rapidly rotating flow in Earth’s core by computing solutions of the weakly-viscous magnetostrophic equations within a spherical shell, driven by a prescribed body force. Two numerical parameters are of particular importance: the Rossby number, Ro, a measure of the magnitude of fluid inertia, and the Ekman number, E, a measure of the magnitude of viscosity Both of these non-dimensional numbers are believed small in Earth’s core, Ro ~ 10−6 − 10−9 (depending on the details of non-dimensionalisation) and E ~ 10−15, leading to a likely dominant magnetostrophic force balance between rotational, buoyancy, pressure and magnetic Lorentz forces in the bulk of the core. Away from the equator, the depth of these boundary layers scales as E1/2, giving an Earth-like value of approximately 1 m thick, about a million times smaller than the radial depth of the fluid core, well beyond the reach of state-of-the-art time-dependent 3D numerical models. Because the core is rapidly rotating, the boundary layers may well be active in the core, a well known example of which is the action of Ekman pumping, the movement of fluid into and out of the boundary layers and the formation of a secondary circulation of magnitude E1/2 5,6

Objectives

Conclusion