Helicity in Fluids and MHD

Abstract

In this chapter we provide an overview of helicity and vorticity conservation laws in ideal fluid dynamics and MHD. For ideal barotropic fluids, in fluid mechanics, we derive the helicity conservation law for the helicity density h _f  = u ⋅ω, where ω = ∇×u is the fluid vorticity. The integral $H_f=\int _{V_m} h_f \ d^3x$ over a volume V _m moving with the fluid, is the fluid helicity. It is important in the description of the linkage of the vorticity streamlines (e.g. Moffatt (1969), Arnold and Khesin (1998)). In MHD, the integral $H_M=\int _{V_m} \mathbf {A}\cdot \mathbf {B}\ d^3x$ is the magnetic helicity, where B = ∇×A is the magnetic induction and A is the magnetic vector potential. It is referred to as the Chern Simons term in field theory (the Chern Simons term in Yang-Mills theory has a totally different form). It describes the linkage and self linkage of the magnetic field lines (Woltjer (1958), Berger and Field (1984)). The cross helicity $H_C=\int _{V_m} \mathbf {u}\cdot \mathbf {B}\ d^3x$ describes the linkage of the magnetic field flux tubes and the vorticity flux tubes. For the case of a barotropic gas with p = p(ρ), H _C is conserved following the flow, i.e. dH _C /dt = 0. For non-barotopic flows, a modifled form of the cross helicity, H _CNB is conserved following the flow. We derive topological invariants (topological charges) by determining invariants which are Lie dragged with the flow in Chapter 6 (e.g. Moiseev et al. (1982), Tur and Yanovsky (1993), Webb et al. (2014a)).