Non-local transport based on the fractional Fokker–Planck Equation model

Other conference contribution, 2018

Recently a non-local (non-diffusive) heat flux model based on a fractional derivative of plasma pressure was proposed for the heat transport in the JET tokamak plasmas [1], where the degree $\alpha$ of the fractional derivative i.e. non-locality (non-diffusivity), of the heat flux was defined though a simple power balance analysis of the steady state. The findings showed that the fractional degree in all of the analysed plasmas was $\alpha < 2$ for both ion and electron channels, suggesting that the heat transport in these plasmas is likely to be of a non-local (non-diffusive) nature. Thus, a study of anomalous diffusion of heat transport using a Fokker-Planck description with fractional velocity derivatives while keeping the non-linear terms is strongly called for. The distribution functions are found using numerical means for varying degree of fractionality of the stable L\'{e}vy distribution. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy. We find that the ratio of the generalized entropy and expectation is increasing with decreasing fractionality towards the well known so-called sub-diffusive domain, indicating a self-organising behavior. Here it is pertinent to keep in mind that the success of a fractional or non-local diffusion model indicates that there is lack of physics in current transport models, namely the super-diffusive character of heat transport, as such it is not only a simplified transport model. When the experimentally found values of the fractional derivatives are used in the model, within a good agreement the experimental heat fluxes were reproduced.