Description

All of the amazing structure of the Sun's atmosphere is created by its magnetic field. However, the tenuous nature of the outer solar atmosphere means we cannot observe this magnetic field directly. Thankfully for us what we see as the visible surface of the Sun is much denser which allows reliable measurements of the magentic field on this surface to be made routinely by satellites. These measurements can be used as a lower boundary condition to extrapolate a representation of the magnetic field above.

This project will explore such equilibrium solutions. The simplest of physical assumptions on the field leads to looking for solutions to Laplace's equation. More sophisticated methods such as force-free solutions and magnetohydrostatic solutions can give more accurate representations of the solar atmospheric magnetic field, but are more complex to calculate.

Magnetic field lines in a magnetic field solution extrapolated from the observed magentic field on the Sun's surface (black and white). Image from NASA).

We will begin by reviewing the equations that describe the magnetic field and exploring basic solutions (including force-free fields). Beyond this there is ample scope for further investigation. The more theoretically inclined could generalise these solutions further to include finite plasma pressure and gravity effects and/or explore solutions in spherical geometry. The more numerically minded could apply these techniques to observed data by writing their own python routine.

Prerequisites

Analysis in Many Variables II is essential. Fluid Mechanics III and/or Partial Differential Equations III would be useful to take alongside this project.

Resources

For reviews on magnetic equilibria see for instance the Living Review by Wiegelmann and Sakura or chapter 3 of Mackay and Yeates. A nice set of lecture notes on the subject can also be found here. An accessible chapter on the subject is chapter 3 of Plasma Physics by Sturrock.