During my participation in the AMSI Summer Research over the last 6 weeks, I have had the opportunity to delve into the world of chaos! Specifically, the chaos of the Sun’s surface! Although dramatic sounding, the fields of mathematics that investigate the chaotic nature of processes are rich with physically valuable insight. In my research this summer, the motivation was the coronal heating problem. This problem posits the question: why does the corona of the Sun have temperatures on the order of ~1million K whereas the Sun’s surface has temperatures only on the order of ~5000K? This temperature difference is on similar order to that of liquid helium (~4K) and a blast furnace (~1200K), which is quite dramatic and certainly worth investigating!
One partial answer is a process called ‘magnetic field braiding’. The Sun possesses a strong magnetic field with magnetic field lines emanating from the surface in large loops that stretch out to the atmosphere and corona. In this process of magnetic field braiding, it is hypothesized that the chaotic and turbulent movements of the surface move and tangle these magnetic field lines and transport the energy from the surface to the corona.
To imagine this process occurring, one can imagine many boats in a bay with anchors on the sea floor such that the ships are free to move on the surface of the water, but the anchor is pinned to the floor. If the bay suddenly encountered a storm that buffeted the ships in all directions, then one would expect that the anchor chains would get tangled. This is similar to the braiding that physicists consider on magnetic field lines on the surface of the Sun!
Following that analogy, the amount that the anchors will get tangled will depend on how ‘strong’ the storm is. In a mathematical sense, we talk about a 2-dimensional fluid flow that has a measure of how chaotic the flow is. We call this quantity ‘topological entropy’. During my study I investigated many different avenues that mathematicians and physicists estimate this value for real fluid flows, such as those on the Sun’s surface, with the aim of determining which technique should be used in future studies requiring the estimation of topological entropy.
This investigation involved many fields of mathematics. These included, group theory, in which I investigated the Artin braid groups and their Burau representations that provide a lower bound of topological entropy, and topological fluid dynamics where this idea of topological entropy originates. A primary result of my work was the comparison of three estimation techniques, Direct Line Segment Growth, Finite-Time Braiding Exponent, and Ensemble-based Topological Entropy Calculation, to measure the topological entropy of an Aref Blinking Flow. Each technique had their strong and weak points, however, Ensemble-based Topological Entropy Calculation appeared to be the strongest candidate due to the low number of trajectories and shorter observation time required. This intersection of pure mathematics and physical application is exactly where I thought I would love to do research, and the AMSI Summer Research Scholarship has only increased my enthusiasm for mathematics research at this intersection!
Zachariah Jones
The University of Newcastle
