A positive solution of a problem by V.I.Arnol'd about higher analog of the
asymptotic Hopf invariant of divergence-free vector fields is presented. A
higher asymptotic invariant of magnetic fields, which is not expressed from the
asymptotic linking numbers of magnetic lines is constructed and a non-trivial
example of an asymptotic invariant is presented. A definition of an asymptotic
finite-order invariant for classical links is presented. Various problems,
which are investigated by means of the magnetic helicity invariant presented.
Solutions of the considered problems could be investigated more precisely by
higher analogs of the helicity invariant.