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tokens: and Ultrascale The evaluation of the pathlength from the SDE steps in x and z directions, as given by Equation ( 7 ), proved unphysical. However, it does provide a possibility to solve the problem of determining the pathlength of a diffusing particle, given a suitable physical framework. Here, we employ field-line random walk as the framework for determining the pathlength of a diffusing particle. In the SDE method, the stepsize does not have a physical meaning, but in the physical world it does. Particle cross-field diffusion is believed to be dominated by their following the turbulent random-walk of the magnetic fieldlines (e.g., Fraschetti Jokipii 2011 ). The fieldlines do not, however, meander at infinitesimal scales, since physical processes damp small-scale fluctuations. Thus, a length scale that would describe the meandering of the fieldlines is a good candidate for evaluation of the pathlength of a particle in turbulent magnetic fields. Such a length scale can be derived from the definition of a fieldline diffusion coefficient and the concept of an ultrascale, . The fieldline diffusion coefficient for 2D-dominated turbulence is given bywhere B is the magnitude of the ambient background magnetic field, and is the turbulence variance (Matthaeus et al. 1995 ). For 2D turbulence spectrum S ( k ), the ultrascale is defined aswhere k is the wavenumber. Matthaeus et al. ( 1999 ) gives the ultrascale an interpretation as the representative scale size of turbulent closed magnetic 2D structures, “magnetic islands,” in the cross-field direction, x . The fieldlines in 2D-dominated turbulence can be thought to be either trapped in magnetic islands or meandering freely around these islands (Ruffolo et al. 2003 ; Chuychai et al. 2007 ). We can thus consider the ultrascale to be the relevant cross-field length scale for the meandering of the untrapped fieldlines around the islands that are of size . The distance Δ z FL along the mean field direction as the fieldline propagates a cross-field length can then be evaluated using the fieldline diffusion coefficient, Equation ( 13 ), using the general definition of a diffusion coefficient , where the distance along the z -axis takes the place of time in the denominator for fieldline diffusion. If we consider the mean square cross-field step given as the ultrascale, , we can writeEquation ( 15 ) gives a natural interpretation to the fieldline diffusion coefficient in Equation ( 13 ): the fieldline random walk across the mean field direction is described as random walk with stepsize , with the ratio between the steps along and across the field, , equal to B / δB ⊥ . Using the steplength as defined by the turbulent island size, given by Equation ( 15 ), we can solve the pathlength of the meandering fieldline as Gaussian random walk withThe pathlength of a particle following such a fieldline can then be estimated and integrated using equationwith the steps along and across the field given by Equations (