Abstract:
The helicity integral (in the form of the Hopf integral) plays an important role for equations of magnetic hydrodynamics (MHD). Its topological meaning was revealed by V.I. Arnold (1974) as the mean value of the asymptotic limit of Gauss integrals over pairs of magnetic lines.
In 2012 the author repeated Arnold's construction and defined the quadratic helicity integral. There was proposed a conjecture that solutions of standard MHD problems could be refined by using this integral. The conjecture was criticized by some authors and the main reasonings were the following: the quadratic helicity is not stable and it is reduced to the square of helicity under a $C^1$-small deformation of the magnetic field. The purpose of this talk is to respond to objections and confirm the conjecture. At the end, I will tell about the cubic magnetic helicities: they turned out to be eight, and five of them are not expressed via the helicity and quadratic helicity.
