Nondiffusive Transport in Plasma Turbulence: A Fractional Diffusion Approach

Numerical evidence of nondiffusive transport in three-dimensional, resistive pressure-gradient-driven plasma turbulence is presented. It is shown that the probability density function (pdf) of tracer particles' radial displacements is strongly non-Gaussian and exhibits algebraic decaying tails. To model these results we propose a macroscopic transport model for the pdf based on the use of fractional derivatives in space and time that incorporate in a unified way space-time nonlocality (non-Fickian transport), non-Gaussianity, and nondiffusive scaling. The fractional diffusion model reproduces the shape and space-time scaling of the non-Gaussian pdf of turbulent transport calculations. The model also reproduces the observed superdiffusive scaling.

Figure 1

Non-Gaussian probability density function of tracer particles in plasma turbulence. The triangles denote the results from the turbulence model in Eqs. (1, 2, 3, 4). The solid line is the analytical solution in Eq. (11) of the symmetric $(w^{+}=w^{-})$ fractional diffusion model in Eq. (5) with $\alpha =3/4$, $\beta =1/2$, and $\chi =0.09$. The inset on the left shows the parabolic dependence of the core of the pdf according to Eq. (13), and the log-log inset on the right shows the algebraic decay of the tails with exponent $1+\alpha =7/4$.

Figure 2

Time evolution of the probability density function of tracer particles in plasma turbulence. The circles and the triangles denote the results from the turbulence model in Eqs. (1, 2, 3, 4). The solid line is the analytical solution in Eq. (14) of the symmetric $(w^{+}=w^{-})$ fractional diffusion transport model in Eq. (5) with $\alpha =3/4$, $\beta =1/2$, and $\chi =0.09$. In agreement with the asymptotic result, the dashed lines in the inset show an algebraic dependence of the tails with exponent $|\beta |=1/2$.