Transport equation describing fractional Lévy motion of suprathermal ions in TORPEX

Suprathermal ions, created by fusion reactions or by additional heating, will play an important role in burning plasmas such as the ones in ITER or DEMO. Basic plasma experiments, with easy access for diagnostics and well-controlled plasma scenarios, are particularly suitable to investigate the transport of suprathermal ions in plasma waves and turbulence. Experimental measurements and numerical simulations have revealed that the transport of fast ions in the presence of electrostatic turbulence in the basic plasma toroidal experiment TORPEX is generally non-classical. Namely, the mean-squared radial displacement of the ions does not scale linearly with time, but 〈r2(t)〉 ~ tγ, with γ ≠ 1 generally, γ > 1 corresponding to superdiffusion and γ < 1 to subdiffusion. A generalization of the classical model of diffusion, the so-called fractional Lévy motion, which encompasses power-law (Lévy) statistics for the displacements and correlated temporal increments, leads to non-classical dynamics such as that observed in the experiments. On a macroscopic scale, this results in fractional differential operators, which are used to model non-Gaussian, non-local anomalous transport in a growing number of applied fields, including plasma physics. In this paper, we show that asymmetric fractional Lévy motion can be described by a diffusion equation using space-fractional differential operator with skewness. Numerical simulations of tracers in TORPEX turbulence are performed. The time evolution of the radial particle position distribution is shown to be described by solutions of the fractional diffusion equation corresponding to asymmetric fractional Lévy motion in sub- and superdiffusive cases.