The Godbillon-Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a 3D 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 $h_m={ \bf A}{ \bf\cdot}{ \bf B}={ \bf A}{ \bf\cdot}(\nabla\times{ \bf A})=0$, where ${ \bf A}$ is the magnetic vector potential and ${ \bf B}$ is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon-Vey field $\boldsymbol{ \eta}={ \bf A}\times{ \bf B}/|{ \bf A}|^2$ and the Godbillon-Vey helicity density $h_{ gv}=\boldsymbol{ \eta}{ \bf\cdot}(\nabla\times{ \boldsymbol\eta})$ in general MHD flows in which either $h_m=0$ or $h_m\neq 0$. A conservation law for $h_{ gv}$ occurs in flows for which $h_m=0$. For $h_m\neq 0$ the evolution equation for $h_{ gv}$ contains a source term in which $h_m$ is coupled to $h_{ gv}$ via the shear tensor of the background flow. The transport equation for $h_{ gv}$ also depends on the electric field potential $\psi$, which is related to the gauge for ${ \bf A}$, which takes its simplest form for the advected ${ \bf A}$ gauge in which $\psi={ \bf A\cdot u}$ where ${ \bf u}$ 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.
Categories: astro-ph.SR math-ph math.MP