Description

In the 19th Century, Helmholtz and Lord Kelvin recognised that vortex lines are 'frozen' in ideal fluid flows, so that their linkage and knottedness must be conserved as the fluid evolves. The most familiar examples are (unknotted) smoke rings, which maintain their form as they move through the air, but in principle the Euler equations for an ideal fluid admit solutions of any knot or link type.

Vortex ring in water, in the form of a trefoil knot, by Kleckner and Irvine (Nature Physics).

However, it was not until the second half of the 20th Century that the invariant quantity known as the helicity of a fluid flow was discovered. Helicity is a topological quantity that describes the average linking of all pairs of vortex lines in a fluid. Mathematically similar is the magnetic helicity that plays an important role in electrically-conducting fluids, except that this describes the linking of magnetic field lines rather than vortex lines.

The conservation (or near-conservation) of helicity, along with the topological view of fluid motion more generally, has since proved to be an invaluable toolkit for understanding the fundamental nature of fluids.

This project will begin by studying the basic fluid equations from a topological viewpoint - in particular the transport of vector fields, and the concepts of vortex lines and tubes. We will learn about the important helicity invariant. Thereafter, interesting applications to explore might be (i) the calculation of minimum-energy states of a given helicity, (ii) the development of singularities in ideal flows, or (iii) the analogous magnetic helicity invariant in electrically-conducting fluids.

Prerequisites

Analysis in Many Variables II is essential, while Electromagnetism III (or Physics equivalent), Geometric Topology II and Differential Geometry III would be useful. You must take Continuum Mechanics IV as a co-requisite. You will not, however, need more advanced ideas from Topology.

Resources

The lecture notes on Topological fluid dynamics for fluid dynamicists by Stephen Childress are a good resource, as are the notes by Andrew Gilbert and Mitchell Berger. To get a flavour for research in the subject, you could try this article by Keith Moffatt. Or for a non-technical application of magnetic helicity, see my article here. See also this short piece in Quanta Magazine on fluid knots (related to the picture above).