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

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.

Spherical convective dynamos in the rapidly rotating asymptotic regime

“…Our interpretation is that the configuration used by these authors mostly applies to the tangent cylinder of a spherical shell, while most of the energy transfer in our simulations occurs elsewhere in low-latitude regions (Yadav et al 2016a). We also obtain zonal flows of similar amplitude regardless of mechanical boundary conditions, another confirmation that they are thermal-wind limited in dipole-dominated spherical convective dynamos (Aubert 2005;Yadav et al 2013a), and that possible residual effects of the boundary viscous drag (Livermore et al 2016) are not present in our calculations.…”

“…Types CE and ST have zonal flows of similar amplitude despite the change in boundary conditions from stress-free to rigid (not shown, previously already documented by Yadav et al 2013a). Livermore et al (2016) have suggested that the amplitude of zonal flows may asymptotically scale differently depending on mechanical boundary conditions if these are limited by the residual viscosity. This does not apply here because zonal flows in spherical, convective, dipole-dominated dynamos such as those discussed here are thermal-wind limited (Aubert 2005).…”

Geomagnetic polar minima do not arise from steady meridional circulation

“…It can be seen from Fig. 2 that the relative amplitude of polar magnetic minima in the kinematic surveys follows the classical E 1/2 scaling in the no-slip cases and the E 1 scaling in the free-slip cases (38). In the no-slip kinematic cases,…”

“…Anticyclonic (polar) vortices exist below the CMB in both cases, however a cyclonic vortex exists above the inner core boundary (ICB) only in the kinematic case. In the dynamic case, the violation of the Taylor's constraint (36) generates a strong z-invariant zonal flow (37,38), which swamps the cyclonic vortical flow near the ICB (SI Appendix, Figs. S2 and S3).…”

“…S4). The inclusion of Lorentz forces drives much stronger polar vortical flows in the dynamic calculations, with strong z -invariant component of the zonal flows (36)(37)(38). For example, at E = 3 × 10 −6 and AT W = 600, the amplitude of the z -invariant zonal flow inside the TC exceeds that of the z -varying zonal flow (thermal wind) in both the free-slip case and the no-slip case (SI Appendix, Figs.…”