Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

Abstract

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 = A·B = A·(∇ × A) = 0, where A is the magnetic vector potential and B is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon-Vey field η = A × B/|A| 2 and the Godbillon-Vey helicity density h gv = η·(∇ × η) in general MHD flows in which either h m = 0 or h m = 0. A conservation law for h gv occurs in flows for which h m = 0. For h m = 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 ψ, which is related to the gauge for A, which takes its simplest form for the advected A gauge in which ψ = A · u where 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.

The Godbillon-Vey invariant as topological vorticity compression and obstruction to steady flow in ideal fluids

“…Vorticity compression is the core nonlinearity of the Euler equations, and given the strong relationship between GV and dynamics, we speculate that flows with GV ≠ 0 will be particularly interesting from a dynamical perspective. The helical compression interpretation, essentially Thurston’s ‘helical wobble’ for foliations [28,43], can be seen as follows. Suppose F is the foliation induced by ω , and N is a unit vector normal to it.…”

“…In particular, Tur & Yanovsky [27], establish the local conservation of GV, where it arises in the case of a hydrodynamic system described by a 1-form S , evolving as (∂ t + L U ) S = 0. In Webb et al [28], the potential application of GV to ideal fluids is discussed. In general, the vector potential A for ω satisfying A · ω = 0 may be written as A = U + V , where U is the fluid velocity and V is curl-free.…”

A geometric look at MHD and the Braginsky dynamo

“…See Arnold & Khesin (1998), Moffatt & Dormy (2019) for more discussion, for example of gauge invariance. If the magnetic helicity density h M = α ∧ B vanishes everywhere, the resulting Pfaff integrability condition means that locally α defines a family of surfaces Φ = const., with α ∝ dΦ, and a further Godbillon-Vey helicity invariant may be defined, as discussed by Webb (2018), Webb et al (2019) and Machon (2020).…”

The Godbillon-Vey Invariant as Topological Vorticity Compression and Obstruction to Steady Flow in Ideal Fluids

“…Vorticity compression is the core nonlinearity of the Euler equations, and given the strong relationship between between GV and dynamics, we speculate that flows with GV = 0 will be particularly interesting from a dynamical perspective. The helical compression interpretation, essentially Thurston's 'helical wobble' for foliations [27,41], can be seen as follows. Suppose F is the foliation induced by ω, and N is a unit vector normal to it.…”

“…In particular, Tur and Yanovsky [26], establish the local conservation of GV , where it arises in the case of a hydrodynamic system described by a 1-form S, evolving as (∂ t +L U )S = 0. In Webb et al [27] the potential application of GV to ideal fluids is discussed. In general, the vector potential A for ω satisfying A•ω = 0 may be written as A = U + V , where U is the fluid velocity and V is curl-free.…”