Mean Cross-Field Displacement of Magnetic Field Lines: Full Nonlinear Calculation and Comparison with Generalized Quasi-linear Results in the Solar Wind

Generalized quasi-linear (GQL) calculations of magnetic field-line transport can now be performed on all length scales and for any three-dimensional turbulence spectrum. In those GQL calculations, the transverse turbulent field is evaluated along a line that follows the average field direction instead of a real field line. However, the range of validity for this approximation is still undetermined. To establish this range of validity, a full nonlinear calculation of the mean cross-field displacement X ≡ ⟨(Δx)²⟩^1/2 is made, in which the turbulent field is evaluated along real field lines. As Δx increases, nonlinearities appear that progressively average out wavevectors with decreasing transverse components, but as long as the power in those wavevectors remains small, the GQL results remain accurate. The nonlinear calculation reveals a new nonlinear diffusion regime. It occurs beyond the point where X reaches 2^1/2ξk = 2^1/2L, k₀ being the parallel wavenumber below which the projected turbulence spectrum is flat and ξ ≡ L/L the ratio of the perpendicular to parallel correlation lengths. At the transition to this nonlinear diffusion regime, the nonlinear and GQL predictions start departing from each other if the GQL diffusion regime has not yet been reached; that is, if X exceeds ξ times the distance Δz elapsed along the main field. In quiet solar wind within 1 AU from the Sun, X < Δz, and the GQL prediction remains accurate at least up to Δz ~ a few L. For isotropic turbulence, it would remain accurate at all scales even for turbulence levels enhanced tenfold.