Solar Equilibrium à la Grad-Shafranov

Abstract

This dissertation investigates the extent to which features of the solar interior can be described by magnetohydrodynamic equilibria. Essentially, we solve the generalized Grad-Shafranov equation, with observational constraints serving as (incomplete) boundary conditions, thereby offering a family of plausible internal solar profiles.

Numerical simulations can offer insight into the interior dynamics and help identify which ingredients are necessary to reproduce particular observations. However, they are computationally intensive; fully resolved simulations of the solar interior will likely remain hypothetical for several solar cycles. Fortunately, despite being rife with turbulence, many features of the Sun can be understood analytically from an equilibrium perspective (e.g., Parker's laminar model of the supersonic solar wind).

To help identify which features admit an equilibrium description, we analyze stationary axisymmetric ideal magnetohydrodynamic flows for solar-relevant parameters. Our numerical scheme for obtaining global solutions uses the Lagrangian formulation of the resulting generalized Grad-Shafranov equation, employing a novel method for incorporating unconstrained boundary conditions (Chapter 3).

Beginning with the outer layers of the Sun, we show that the hydrodynamic limit is sufficient to describe the observed deviation from the cylindrical rotation in the solar Convection Zone (Chapter 4). Moreover, the inclusion of a poloidal flow results in a slowing of rotation at the surface, qualitatively similar to the Near Surface Shear Layer (Chapter 5).

Turning inward, we then investigate the effects of including a magnetic field, and its relationship to the Tachocline and the Radiative Interior (Chapter 6). The presence of both a poloidal field and poloidal flow can result in the equilibrium equations transitioning to hyperbolic type, and could lead to discontinuities or steep gradients characteristic of the Tachocline. While observations at the solar surface indicate that these transitions occur suggestively close to the Tachocline, we find that there exist solutions that remain smooth throughout, though they might not be robust to perturbations.

Coupled with the increasing sensitivity of extraterrestrial seismic and magnetic measurements, our framework could offer plausible extrapolations into the hidden interiors of other astrophysical objects, helping to determine which features have an equilibrium description, and which are necessarily dynamical in origin.