Turbulent Diffusion of Magnetic Field Lines in the Heliosphere

Due to solar wind turbulence, Parker spirals are stochastic. The dispersion of magnetic field lines is described by a convection-diffusion equation for the field line density distribution which is a function of the two heliographic angles in addition to the radial distance. Taking into account the radial evolution of the turbulence, the three-dimensional convection-diffusion equation is transformed into a set {of} stochastic differential equations which is solved numerically using both a forward and backward formulation. By tracing a large number of stochastic Parker spirals, the field line density distribution is constructed at any point in the heliosphere. It is shown that the angular part of the distribution function can be well-fitted by a two-dimensional Gaussian with standard deviation close to $ 25^{\circ}$ at 1 AU. The simulations also confirm that the magnetic field lines are underwound, on average, for strong enough turbulence intensity. Applying the backward approach, magnetic field lines are traced from an observer at 1 AU back to the Sun, quantifying the probability of magnetic connection when interplanetary turbulence is accounted for. It is shown that the angular uncertainty of $\sim 25^{\circ}$ is sharply reduced to $\sim 4^{\circ}$ when the field lines are traced back to the solar wind source surface from 0.25 AU.