Stochastic Parker Spirals in the Solar Wind

An analytic model for the angular dispersion of magnetic field lines resulting from the turbulence in the solar wind and at the solar source surface is presented. The heliospheric magnetic field lines in our model are derived from a Hamiltonian with the pair of canonically conjugated variables the cosine of the heliographic colatitude and the longitude φ. In the diffusion approximation, the Parker spirals are modeled by a set of stochastic differential equations for θ and φ as functions of r. These stochastic Parker spirals are realizations of a standard random walk on a sphere of increasing radius, superimposed on an angular drift due to solar rotation. The Green function solution of the Fokker-Planck equation describing the angular diffusion of the field line density is obtained in terms of spherical harmonics. Magnetic field lines traced from an observer back to the Sun are realizations of a Brownian bridge. Our model incorporates the effect of the random footpoint motions at the source surface, which is associated with the zero-frequency component of the solar wind turbulence. Assuming that the footpoint motion is diffusive, its contribution to the angular diffusivity of the stochastic Parker spirals is then given by the angular diffusivity of the footpoints divided by the solar wind speed and is controlled by a unique parameter, which is the Kubo number.