Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

The Godbillon-Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a three-dimensional manifold are satisfied. The magnetic Godbillon-Vey helicity invariant in magnetohydrodynamics (MHD) is a higher-order helicity invariant that occurs for flows in which the magnetic helicity density , where is the magnetic vector potential and is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon-Vey field and the Godbillon-Vey helicity density in general MHD flows in which either or . A conservation law for occurs in flows for which . For the evolution equation for contains a source term in which is coupled to via the shear tensor of the background flow. The transport equation for also depends on the electric field potential , which is related to the gauge for , which takes its simplest form for the advected gauge in which where is the fluid velocity. An application of the Godbillon-Vey magnetic helicity to nonlinear force-free magnetic fields used in solar physics is investigated. The possible uses of the Godbillon-Vey helicity in zero helicity flows in ideal fluid mechanics, and in zero helicity Lagrangian kinematics of three-dimensional advection, are discussed.