On Galilean invariance of mean kinetic helicity

While kinetic helicity is not Galilean invariant locally, it is known [Moffatt, J. Fluid Mech. 35, 117 (1969)] 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. Here, we provide a direct mathematical proof 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.