Phase vortex spirals

For a complex scalar field ψ(x, y) in the plane, the flow lines (integral curves of the current field Im ψ*∇ψ) typically spiral slowly in or out of a phase vortex (where ψ = 0), with the distance between successive windings decreasing as 2πKr3 near the vortex at r = 0. The coefficient K depends on the derivatives of ψ at the vortex. In three dimensions, the flow spiral migrates slowly along the vortex line, in a helix whose pitch is proportional to r2. For fields with well-defined orbital angular momentum, the flow lines can be determined explicitly not just near the vortex but also globally. The explicit forms of flow lines near phase extrema and saddles are also found.