On Galilean Invariance of Mean Kinetic Helicity

While kinetic helicity is not Galilean invariant locally, it is known [1] that its spatial integral quantifies the degree of knottedness of vorticity field lines. Being a topological property of the flow, mean kinetic helicity is Galilean invariant. We provide a direct mathematical proof and discuss that kinetic helicity is Galilean invariant when spatially integrated over regions enclosed by vorticity surfaces, i.e., surfaces of zero vorticity flux. We also discuss so-called ``relative'' kinetic helicity, which is Galilean invariant when integrated over any region in the flow.

[1] Moffatt K. (1969), J. of Fluid Mech., doi: 10.1017/S0022112069000991.

[2] Soltani Tehrani, D. & Aluie, H. (2023), Phys. of Fluids, doi: 10.1063/5.0178926.