Description

All of the amazing structure of the Sun's atmosphere is created by its magnetic field. To understand this structure, we need to understand how different parts of the solar surface are connected by lines of magnetic force known as field lines.

To calculate a field line, one solves an initial value problem for an Ordinary Differential Equation describing the path the line of magnetic force takes through space. This project will explore how calculating many field lines can be used to map the connections of magnetic fields like the one in the solar atmosphere. Specifically, we will study what's known as the squashing factor (Q). Q is a measure of how much two field lines that start close to one another diverge, which makes it useful for mapping out boundaries between different magnetic domains.

Field lines in a model of the solar magnetic field (from Titov et al. 2011). The heat map on the surface shows Q, which picks out the boundaries between different magnetic domains.

We will start by exploring solving methods for initial value problems and testing them on simple analytical magnetic fields. You would then be encouraged to create your own Python code to calculate field lines and Q numerically for an analytical field where an analytical form for both exists to compare with.

Beyond this point there are many potential avenues for specialisation. The more numerically inclined could look into interpolation methods to extend their tracing code to data on a grid. Those more theoretically inclined could look at different methods of calculating Q. Those more physically motivated could investigate how regions of the surface swept out by moving lines of Q (for a time dependent magentic field) are analogous to solar flare ribbons.

Prerequisites

Analysis in Many Variables II (or Mathematical Methods in Physics) is essential. Special Relativity & Electromagnetism II would be useful. Basic Python knowledge from Programming I, along with familiarity with numerical approximations of derivatives from Numerical Analysis II are the main programing skills required.

Resources

For further details of magnetic fields and field lines see this Wiki. There is plenty of available literature on numerically solving initial value problems, see for instance this (in particular see the Runge-Kutta methods). To read more about Q see for instance chapter 5 of the Living Review by Longcope.