  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.
