Numerical algorithm for Monte Carlo simulation of 4-D neoclassical diffusion

 

Anatolii Gurin¹ and Victor Goloborod’ko¹

 

¹Institute for Nuclear Research of National Academy of Science, Kiev, 03028, Ukraine

 

Abstract

     Based on the stochastic differential equations (SDE) theory [1] of diffusion Markov processes, the system of stochastic drift equations of motion of charged particles in a toroidal plasma corresponding to the drift theory with Coulomb collisions in general form is developed. For isotropic plasma in an axially symmetric magnetic field, a system of stochastic differential equations (SDEs) for kinetic energy, a pitch-parameter and a classical two-dimensional diffusion of particles in the poloidal cross-section of plasma is formulated. In order to describe the particle spatial diffusion, the equation for collisional evolution of the particle toroidal angular momentum is used. Such a system correctly considers the drift of trapped and circulating particles and the radial Brownian motion, which leads to the effects of neoclassical diffusion. The 1st-order Millstein-type algorithm for numerical integration of these equations is presented. This algorithm does not require calculation of derivatives for kinetic coefficients, i.e. belongs to the Runge-Kutta class. The algorithm also eliminates the need to calculate the repeated Ito continual integrals in the simulation of two-dimensional poloidal diffusion, and therefore can serve as a convenient basis for simulating enhanced neoclassical plasma diffusion by the Monte Carlo method, avoiding the analytical models of "banana" trajectories of particles in toroidal plasma. Thus, approach developed is accelerated in comparison with conventional Monte Carlo algorithms. Based on these theoretical considerations the numerical code was developed. Herein we present the results of numerical simulations for slowing down  and  2D spatial diffusive motion of fast ions in tokamak obtained with a proposed algorithm. Also we present the temporal evolution of fast ion trajectories in the tokamak poloidal cross-section. It is demonstrated that this algorithm may be effectively applied also for the calculation of fast ion loss poloidal distribution over the first wall in tokamaks.

1. A. Gurin, V. Goloborod’ko, Application of the Ito theory for Monte Carlo simulation of plasma diffusion J. Plasma Phys. (2018), vol. 84, 905840314, Cambridge University Press 2018, doi:10.1017/S0022377818000594