Turbulent diffusion of lines and circulations

We study material lines and passive vectors in a model of turbulent flow at infinite-Reynolds number, the Kraichnan–Kazantsev ensemble of velocities that are white-noise in time and rough (Hölder continuous) in space. It is argued that the phenomenon of “spontaneous stochasticity” generalizes to material lines and that conservation of circulations generalizes to a “martingale property” of the stochastic process of lines.