We discuss cross-field diffusion of energetic particles using compressional and noncompressional two-dimensional turbulence models by performing test particle simulations. For both models, the diffusion coefficient, defined in the classical way, exhibits a timescale dependence, suggesting that the underlining physical process should be described by Lévy statistics. The diffusion coefficient for long timescales is classified in terms of the Kubo number, K = bL ⊥ /ρ, where b is the standard

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... ion of magnetic field fluctuation in units of the background field, ρ is the particle Larmor raduis, and L ⊥ is the field turbulence scale length. While the well-known JKG theorem predicts that a particle cannot move more than about one gyroradius normal to the magnetic field in a system with two or less spatial dimensions, the cross-field diffusion does take place in our models since they are exceptions to the theorem: we argue that particle motion on the flux surface is not prohibited in general, and in particular, it is not bounded when the background magnetic field is exactly parallel to the ignorable coordinate.