Description

This project is based on the surface flux-transport model for the evolution of magnetic flux patterns on the Sun's surface. First proposed by R.B. Leighton in the 1960s, this model has been very successful in explaining the intriguing evolution of these large-scale patterns. New magnetic flux emerges rapidly in concentrated "active regions" (sunspots), before undergoing a gradual spreading out and decay. In addition, the decaying patterns are carried around by large-scale surface motions. In essence, the model is an advection-diffusion equation for a passive scalar field, and is therefore similar to advection-diffusion processes occuring in many other contexts.

A surface flux-transport simulation of the Sun's surface, looking down from the north pole. Red represents positive magnetic flux (i.e., out of the Sun), and blue negative.

Although it is possible to find some simplified solutions, it is usually necessary to solve the underlying advection-diffusion equation numerically. Hence, the project will involve developing numerical solutions by computer for the evolution of the magnetic flux.

Depending on your interests, you could focus either on applying the model to physical problems, or on the methods of numerical solution in their own right. For the former, you might investigate recent suggested improvements to the model, in the form of additional nonlinearities. For the latter, the basic question is: how can we design numerical methods to solve the underlying differential equation both efficiently and accurately on the sphere?

Prerequisites

There are no essential prerequisites, although Numerical Analysis II and Analysis in Many Variables II would be helpful. Continuum Mechanics III would be a good co-requisite. The project will require some computer programming, and previous experience will be an advantage but not a necessity (I can help you get started). I recommend using either MATLAB (available on the CIS network) or Python (available freely).

Resources

To get a flavour of the physical model, you could try Neil Sheeley's article Surface Evolution of the Sun's Magnetic Field: A Historical Review of the Flux-Transport Mechanism. Or Chapter 2 of this review. Leighton's original paper is also an accessible read. For the numerical solution of partial differential equations, you will find introductory material in many textbooks on Numerical Analysis. For example, A first course in the numerical analysis of differential equations / A. Iserles. Start by looking at finite-difference methods.