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.