Polar confinement of the Sun's interior magnetic field by laminar magnetostrophic flow

Abstract

The global-scale interior magnetic field Bi needed to account for the Sun's observed differential rotation can be effective only if confined below the convection zone in all latitudes including, most critically, the polar caps. Axisymmetric solutions are obtained to the nonlinear magnetohydrodynamic equations showing that such polar confinement can be brought about by a very weak downwelling flow U ~ 10−5cms−1 over each pole. Such downwelling is consistent with the helioseismic evidence. All three components of the magnetic field B decay exponentially with altitude across a thin, laminar ‘magnetic confinement layer’ located at the bottom of the tachocline and permeated by spiralling field lines. With realistic parameter values, the thickness of the confinement layer ~10−3 of the Sun's radius, the thickness scale being the magnetic advection–diffusion scale δ = η/U where the magnetic (ohmic) diffusivity η ≈ 4.1 × 102cm2s−1. Alongside baroclinic effects and stable thermal stratification, the solutions take into account the stable compositional stratification of the helium settling layer, if present as in today's Sun, and the small diffusivity of helium through hydrogen, χ ≈ 0.9 × 101cm2s−1. The small value of χ relative to η produces a double boundary-layer structure in which a ‘helium sublayer‘ of smaller vertical scale (χ/η)1/2δ is sandwiched between the top of the helium settling layer and the rest of the confinement layer. Solutions are obtained using both semi-analytical and purely numerical, finite-difference techniques. The confinement-layer flows are magnetostrophic to excellent approximation. More precisely, the principal force balances are between Lorentz, Coriolis, pressure-gradient and buoyancy forces, with relative accelerations negligible to excellent approximation. Viscous forces are also negligible, even in the helium sublayer where shears are greatest. This is despite the kinematic viscosity being somewhat greater than χ. We discuss how the confinement layers s at each pole might fit into a global dynamical picture of the solar tachocline. That picture, in turn, suggests a new insight into the early Sun and into the longstanding enigma of solar lithium depletion.

A Self-consistent Model of the Solar Tachocline

“…The polar field cases presented in Section 3.2 closely resemble the axisymmetric polar model of Wood & McIntyre (2011). The magnetic field is confined by the downwelling meridional flow in the tachocline, whose penetration depth into the radiation zone depends on the strength of the magnetic field.…”

“…On this basis, Gough & McIntyre argued that the polar magnetic field can only be confined by meridional flows that downwell in the tachocline and hold the field in an essentially laminar advection-diffusion balance. Such downwelling meridional flows are, in fact, expected in the high-latitude tachocline, as a result of "gyroscopic pumping" by the retrograde rotation of the overlying convection zone (Spiegel & Zahn 1992;McIntyre 2000;Wood & McIntyre 2011;Wood & Brummell 2012;Miesch et al 2012). The characteristic velocity of this downwelling, U , say, can be estimated from the observed differential rotation of the tachocline, if we assume that the tachocline is in thermal-wind balance and local thermal equilibrium (Gough & McIntyre 1998;McIntyre 2007, p. 194).…”

Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit

“…The turbulence within this layer is believed to play a crucial role in redistributing angular momentum and mixing and yet remains poorly understood (see e.g. Tobias 2005;Wood & McIntyre 2011). Recent investigations using both direct numerical simulations and direct statistical simulations have demonstrated that magnetic fields can strongly affect the dynamics of stably stratified turbulent flows (Tobias, Diamond & Hughes 2007;Tobias, Dagon & Marston 2011).…”

Pattern formation in rapidly oscillating peculiar A stars

“…However, being axisymmetric, difficulty was experienced in preventing magnetic field close to the axis of symmetry from penetrating through the tachocline and adding a direct Lorentz coupling between the convection zone and the radiative interior, almost eliminating the polar tachocline shear. This might be regarded as raising a serious problem, notwithstanding the recent demonstration by Wood and McIntyre (2011) of the existence of a steady state with a field wholly confined to the radiative interior. However, gyroscopic pumping inevitably causes an almost horizontal flow in the tachocline away from the poles (and, in the case of the sun, from the equator too) towards the latitudes AE 0 (e.g.…”