On the Topological Number in Kelvin’s Circulation of Inviscid Fluids

Abstract Kelvin’s circulation, as a loop integral of the velocity field, is first shown to be equal to ΓC = 2π∆nΓ1 + constant in an inviscid fluid, with Γ1 a unit circulation. The integer ∆n is interpreted as the number of net 2π-rotations of the velocity vector in one cyclicality along the circulation loop and hence it is a topological number. Different from the linkage number in the helicity invariant, which only exists 3D space, the topological number in the circulation invariant is independent of the spatial dimension. A further study reveals that this new topological phase 2π∆n is rooted at the gauge invariance of the circulation and it is a counterpart of the Berry’s phase in quantum system.