Exploration of anomalous transport based on the use of general conformable fractional derivative in tokamak plasmas

Abstract

This study investigates anomalous transport in tokamak plasmas by employing general conformable fractional derivatives (GCFDs) and utilizing general conformable fractional diffusion equations (GCFDEs). GCFDs, which are local derivatives utilizing fractional conformable functions, exhibit properties similar to those of ordinary derivatives. The action can be defined by employing the definition of the inverse operation of GCFDs, and the general conformable fractional equation of motion (GCFEM) is derived through the calculus of variations. Introducing a damping term to the GCFEM results in the general conformable fractional Langevin equation (GCFLE). Solutions of the GCFLE indicate a scaling law for the mean squared displacement (MSD) ⟨x2⟩∝tα/Γ1+α, linking MSD scaling to the order α of the GCFD if the conformable fractional function ψt,α=Γαt1−α, where Γx is the gamma function. Therefore, the general conformable fractional diffusion coefficient (GCFDC) Dψ,α is defined as the ratio of the classical diffusion coefficient to ψt,α. From the definition of the running diffusion coefficient, it is found that when the Kubo number is much greater than unity, indicating that the system is in a turbulent state, both the classical and the GCFDC are inversely proportional to α—the power of the magnitude of the background magnetic field. After constructing a GCFDE based on the scaling law of MSD, it is applied to investigate the formation of hollow temperature profiles during off-axis heating in magnetically confined plasmas. Simulation results reveal the crucial role of the fractional conformable function in sustaining the long-term existence of these hollow temperature profiles as it can impede thermal conduction.