Construction And Evolution Of Knotted Vortex Tubes In Incompressible Schrodinger Flow

We propose a theoretical method for constructing an initial two-component wave function that can be transformed into a knotted velocity field with finite kinetic energy and enstrophy. The wave function is constructed using two complex-valued polynomials, with one determining the desired shape of the knotted central axis and the other encoding the twisting nature of vortex lines, which facilitates the study of helicity conversions. We construct six knotted vortex fields with various centerline and twist helicity as initial conditions for direct numerical simulation of incompressible Schrodinger flow (ISF) in a periodic box. Although the evolution of morphological structure is similar for ISF and classical viscous flow, with all the knots becoming untied after a short time to form one or more separate vortex rings, their statistics are quite different. During the critical period of vortex reconnection, the increase in enstrophy is much more moderate in ISF than in viscous flow, indicating that the Landau-Lifshitz term in ISF inhibits the energy cascade from large to small scales. We also find that the centerline helicity changes dramatically during reconnection, which is consistent with the evolution of the geometrical shape of vortex lines.