A novel approach to ion–ion Langevin self-collisions in particle-in-cell modules applied to hybrid MHD codes

In order to have a better closure for magnetohydrodynamic (MHD) equations, a common approach is to obtain the ion fluid pressure tensor by directly computing the moments of an ion distribution function, obtained by a particle-in-cell solver of the Vlasov or Boltzmann equation. This is the so-called hybrid approach. Long MHD simulations are required for problems such as investigating the properties of the sawtooth cycle. In such long hybrid simulations, collisions are required to relax the distribution function after violent MHD events, and to obtain the self-consistent neoclassical transport. In this paper, we present a new approach to ion self-collisions, based on temperature- and velocity-shifted Maxwellian distributions. It is shown that the approach emulates the effect of the background reaction, without the need to explicitly implement it. Arbitrariness in the choice of the closest Maxwellian is removed. The model compares very well with binary collision Monte-Carlo simulations. The practical implementation as a Fokker–Planck module in a hybrid kinetic/MHD simulation code is discussed. This requires an additional manipulation in order to conserve energy and momentum.