Geometry of field lines in elasticity and MHD
Prof. Mitchell Berger (University of Exeter)
Thursday 26th May, 2016 14:00-15:00 Maths 204
I will explore two topics concerning the geometry of field lines. In elasticity theory, a solid in equilibrium has a divergence-free stress tensor. We can regard rows of the stress tensor as vector fields; mapping where their field lines go provides a picture of how forces flow through a solid. This leads to a method for finding average stress and strain in a volume in a way which automatically preserves the elastic energy.
The second part of the talk concerns the geometry of magnetic fields. The linking of magnetic field lines provides a conserved quantity known as magnetic helicity. I will discuss recent proposals for defining an absolute measure of helicity when the field lines are not closed, but instead have endpoints on a surface (e.g. on the surface of the sun). This will involve both a generalization of the poloidal-toroidal field decomposition and a simple application of the Gauss-Bonnet theorem.