NoteA note on helicity conservation in Leray models of incompressible flow

We prove conservation of a regularized helicity HL:=∫Ωu⋅wdx for the Leray model (and its variants) of turbulent flow, where w is the solution of a Leray-regularized vorticity equation. The usual definition of helicity is H=∫Ωu⋅(∇×u)dx, which is considered by Navier-Stokes flows, but is not a conserved quantity of the Leray model. However, if u is a Leray solution, then the difference between H and HL is that HL uses a regularized vorticity and H uses the curl of a regularized velocity. The results are extended to show that the standard Crank-Nicolson finite element method for Leray models conserves both discrete energy and discrete regularized helicity.