Modelling of lower hybrid current drive in the presence of spatial radial diffusion

Abstract

A code to solve the full three dimensional Fokker-Planck kinetic equation in the presence of lower hybrid current drive (LHCD) is applied to study numerically the problem of modelling LHCD, including the combined effects of spatial radial diffusion and an electric field. The code uses the Beliaev-Budker relativistic collision operator, and applies a method of fractional steps to the numerical solution of the time evolution equation for the three dimensional Fokker-Planck equation. The importance of the numerical approach associated with an exact relativistic treatment for the solution is emphasized in order to study accurately all the physics associated with this problem, especially the shape of the distribution function and the physics associated with its hot tail as, for instance, the fast electron transport and lower hybrid absorbed power profiles. The aim of the simulation is to match hard X ray imaging profiles and current density profiles. An accurate relativistic treatment of the hot tail helps make the study of these problems less phenomenological and more physical. Some preliminary results from the Tokamak de Varennes (TdeV) are analysed in order to illustrate the power of the method

Numerical characterization of bump formation in the runaway electron tail

“…where K ab [f b ] is the collisional friction vector and D ab [f b ] is the collisional diffusion tensor. The relativistic Braams-Karney collision operator is used in this paper [16,17]. So-called knock-on collisions represent a 1/ ln Λ correction to the collision operators.…”

Weak turbulence theory for collisional plasmas

“…For numerical analysis of the relatively dense plasmas discussed in fusion research, collisional effects have been applied for decades in the context of quasilinear theory, by simply adding the collision operator to the right-hand of the equation for the time evolution of the particle distribution function [43][44][45][46][47]. It is also a common approximation in the literature to heuristically replace the collision operator by an effective collision frequency,…”

Advanced 3-D Electron Fokker-Planck Transport Calculations

“…Collisions are described by the Belaiev-Budker operator that is valid from classical to relativistic kinetic energies [5], while RF wave-particle interactions are taken into account within the usual quasi-linear diffusion approximation. All quantities are determined at the minimum magnetic field value B min .…”

“…During the last twenty years, progress in this domain followed regularly those obtained in the field of computers, enabling the description of physical processes with ever increasing realism [1][2][3][4][5]. During the last twenty years, progress in this domain followed regularly those obtained in the field of computers, enabling the description of physical processes with ever increasing realism [1][2][3][4][5].…”

“…In that case, plasma equilibrium, transport properties, RF wave propagation, current drive and current resistive diffusion must all be determined self-consistently [8]. Therefore, existing 3-D bounce-averaged relativistic Fokker-Planck codes rely on an operator splitting technique which strongly reduces memory requirements since at each time step Δt, momentum and spatial dynamics are alternatively described implicitly or explicitly (normal time advancing) [2][3][4][5]. For this purpose, the implicit or inverse time advancing technique is the most appropriate method, since large time step may be used without onset of numerical instabilities, which considerably speed-up convergence [9].…”