Calculations for the Practical Applications of Quadratic Helicity in MHD

TL;DR

This paper introduces the quadratic helicity in MHD, generalizes inequalities relating magnetic energy and helicity, and demonstrates its practical computation and invariance properties in magnetic dynamo models.

Contribution

It presents a new quadratic helicity invariant, extends the Arnol'd inequality, and provides methods for its numerical computation and application in MHD.

Findings

Quadratic helicity generalizes classical helicity.

Flow of quadratic helicity matches the square of classical helicity in $\

Numerical computations demonstrate practical applications of quadratic helicity.

Abstract

For the quadratic helicity $\chi^{(2)}$ we present a generalization of the Arnol'd inequality which relates the magnetic energy to the quadratic helicity, which poses a lower bound. We then introduce the quadratic helicity density using the classical magnetic helicity density and its derivatives along magnetic field lines. For practical purposes we also compute the flow of the quadratic helicity and show that for an $\alpha^2$-dynamo setting it coincides with the flow of the square of the classical helicity. We then show how the quadratic helicity can be extended to obtain an invariant even under compressible deformations. Finally, we conclude with the numerical computation of $\chi^{(2)}$ which show cases the practical usage of this higher order topological invariant.