Stochastic Parker Field -- Effects of Footpoint motion on the Interplanetary magnetic field

Shortly after Parker developed the hydrodynamic solar wind model, Leighton considered how the footpoint motion of IMF can modify the original Parker spiral. He assumed a diffusion of the footpoint which contains an intrinsic difficulty in that the resulting field lines can have length unbounded in the limit that the correlation length in the diffusion model tends to zero. We extend the Leighton's diffusion model describing the turbulent mixing of magnetic footpoints on the solar wind source surface by a spherical Ornstein-Uhlenbeck process with a constant drift given by the rotation of the Sun. Our model contains two parameters: the Lagrangian integral timescale τ_L, and the root-mean-square footpoint velocity V_rms. The Lagrangian velocity and the positions of magnetic footpoints on the solar wind source surface are obtained from the solutions of a set of stochastic differential equations. The spherical diffusion model of Leighton is recovered in the singular Markov limit when the Lagrangian integral timescale tends to zero while keeping the footpoint diffusivity finite. In contrast to the magnetic field lines driven by standard Brownian processes on the solar wind source surface, the interplanetary magnetic field lines are smooth differentiable functions with finite path lengths in our model. The probability distributions of path length at 1 au are computed numerically and are shown to develop a significant skewness when the width of the distributions increases.