Non-local Modelling of Radial Transport in the Tokamak Edge and SOL

The Scrape-Off Layer (SOL) is a fascinating region which links the core plasma of the
tokamak to the material components of the tokamak device. Understanding transport in
this region is essential to the design of fusion devices. The SOL is influenced strongly by
conditions at the turbulent edge, and characterised by the presence of non-local phenomena.

While the solution of full transport equations of the SOL is possible, these tend to be
intensive in both labour and computing time - Simplest possible analytical modelling is then worth pursuing. Advection-Diffusion models have been attempted with limited success - but models which incorporate the statistical behaviour of the non-local phenomena present have been promising.

On this basis, we develop an ‘observational’ random walk. The key element of this approach is a jump function which determines the probability of a random walker making a particular jump during an observation interval of duration τ . We show that non-Gaussian jump functions result in behaviour that cannot be characterised with an advection-diffusion model, and find evidence that such jump functions may be observed in SOL simulations dominated by coherent structures.

We construct a stochastic flux equation which relates the jump function to the flux. Through use of the general central limit theorem, we prove that in the case that the jump function converges to the Levy-α stable (LαS) distribution, the transport behaviour may be described by a non-local ‘fractional-advection, fractional-diffusion’ equation (FADE) - of which the classical advection-diffusion equation is just one specific case.

We apply the work to simulations of the 2D SOL. The measured jump functions appear to be inhomogeneous, anisotropic, and non-Gaussian - and indeed, belong to the LαS distribution. Numerical solutions of FADEs obtained from the jump functions match well to the average radial density profiles generated by the 2D SOL simulations.