On Galilean Invariance of Mean Kinetic Helicity

While kinetic helicity is not Galilean invariant locally, it is known (K. Moffatt, Journal of Fluid Mechanics, 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, \textit{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.