Quadratic Helicity in MHD

Arnold's asymptotic ergodic Hopf invariant of magnetic lines is an invariant for the ideal magnetohydrodynamics (MHD) and is called the helicity. V. I. Arnold conjectured (1986) that several higher invariants of links are also presentable in an ergodic asymptotic form. The conjecture is proved for the simplest invariant of this type, called the quadratic helicity. The quadratic helicity is the dispersion of the helicity density and is an invariant for the ideal MHD.
In my talk I will remind definitions and properties of the helicity and of the quadratic helicity. I introduce the quadratic helicity density (this is a joint result with Simon Candelaresi). As a new application of the quadratic helicity I explain why examples of geodesic flows by P. Dehornoy determine magnetic fields of minimal magnetic energy with a constant helicity density.