A NOTE ON HELICITY CONSERVATION IN STEADY FLUID-FLOWS

A theorem on helicity conservation proved by Moffatt (1969) for the fl ows of inviscid barotropic fluids is generalized, for steady flows, to any fluid in which vorticity field lines are material. To make this g eneralization, the helicity within a volume V enclosed by a material s urface S must be defined by the volume integral, H-s'(t) = integral(v) (lambda/J) m . v dV, where v is the fluid velocity, m is a unit vector tangent to a vorticity line, lambda is the vorticity line stretch (Ca sey & Naghdi 1991), and J is the determinant of the deformation gradie nt tensor. For the case of an inviscid barotropic fluid, H-s' differs only by a constant factor from the helicity integral defined originall y by Moffatt (1969). The condition under which H-s' is invariant under steady fluid motion is also the condition necessary and sufficient fo r the existence of a permanent system of surfaces on which both the st ream lines and the vorticity lines lie (Sposito 1997). These surfaces and the helicity invariant H-s' figure importantly in the topological classification of integrable steady fluid hows, including flows with d issipation, in which vorticity lines are material.