A further twist to helicity

In fluid dynamics, helicity measures the correlation between velocity and its curl, vorticity, over a spatial volume. Under ‘ideal’ conditions (vanishing viscosity and either homogeneneous density or when pressure may be regarded as a function of density alone), helicity is a topological invariant closely related to the knottedness of vortex lines (Moffatt 1969 J. Fluid Mech. 35 (1), 117–129). Helicity is conserved following a material volume for compact vorticity distributions, i.e. when the vorticity field is tangent to the surface of the volume. There is a related helicity invariant in ideal magnetohydrodynamics involving the correlation between the magnetic potential and its curl, the magnetic field. Helicity is a fragile invariant in the sense that relaxing any one of the ideal conditions results in non-conservation. Unlike energy and enstrophy (mean-square vorticity), helicity is not positive (or sign) definite. Viscous diffusion can create both positive and negative helicity when vortex lines reconnect, something which is topologically forbidden in an ideal fluid where vortex lines move as material curves. Moreover, variable density or more generally compressibility destroys conservation and weakens the association between helicity and vortex-line topology. Furthermore, in compressible flows, the velocity field is not entirely determined from the vorticity field. A recent paper by Boutros & Gibbon (2025) J. Fluid Mech. in this journal explains how one can extend the definition of helicity to control and limit the non-conservation of helicity. This offers a promising way forward in using helicity to characterise flow properties in computational studies of high Reynolds number flows.