A note on helicity conservation in steady fluid flows

Abstract

A theorem on helicity conservation proved by Moffatt (1969) for the flows of inviscid barotropic fluids is generalized, for steady flows, to any fluid in which vorticity field lines are material. To make this generalization, the helicity within a volume V enclosed by a material surface S must be defined by the volume integral,formula herewhere v is the fluid velocity, m is a unit vector tangent to a vorticity line, λ is the vorticity line stretch (Casey & Naghdi 1991), and J is the determinant of the deformation gradient tensor. For the case of an inviscid barotropic fluid, ([Hscr ]′S differs only by a constant factor from the helicity integral defined originally by Moffatt (1969). The condition under which ([Hscr ]′S is invariant under steady fluid motion is also the condition necessary and sufficient for the existence of a permanent system of surfaces on which both the stream lines and the vorticity lines lie (Sposito 1997). These surfaces and the helicity invariant ([Hscr ]′S figure importantly in the topological classification of integrable steady fluid flows, including flows with dissipation, in which vorticity lines are material.