Drift-kinetic PIC model for simulations of longitudinal plasma confinement in mirror traps

TL;DR

This work extends a 1D2V drift-kinetic PIC code to simulate longitudinal plasma confinement in open magnetic traps by adding energy-conserving Coulomb collisions and absorbing conducting-wall boundaries. The ADEPT framework uses Ampère’s law to compute the electric field and employs a predictor–corrector scheme with current corrections to ensure global energy and local charge conservation, enabling large grid steps while accurately modeling near-wall and ambipolar effects. Validation shows Coulomb collisions agree with analytical relaxation theories when sufficient macroparticles per cell are used, and boundary conditions conserve energy while reproducing Debye-sheath–like structures. Comparisons with hybrid simulations reveal 15%–20% differences in electron temperature, potential, and density due to electron kinetics in expanders, highlighting the necessity of fully kinetic electrons for accurate confinement predictions in open traps.

Abstract

The paper presents a 1D2V electrostatic PIC model with a drift-kinetic description of all particle types aiming at simulating classical longitudinal plasma transport in axially symmetric open traps. The model generalizes the semi-implicit particle-in-cell method with exact conservation of energy and charge to the case of collisional plasma and adapts it to boundary conditions on perfectly conducting walls with a floating potential. Implementation of Coulomb collisions is tested on the problem of temperature relaxation in a two-component plasma and demonstrates good agreement with the analytical theory. Since quasi-neutrality of plasma is not strictly determined, the model is able to correctly reproduce the ambipolar electric potential profile up to the walls. At the same time, the main advantage of implicit PIC simulations - the ability to use large grid steps, many times larger than the Debye radius - does not prevent the correct modeling of the near-wall electric potential jump. The model satisfactorily reproduces the known results of the Debye sheath theory and the Bohm criterion. A comparison of stationary plasma profiles formed in a mirror trap in the presence of a constant particle source with the results of simulations using the hybrid code MIDAS showed that self-consistent consideration of electron kinetics in expanders leads to noticeable (at the level of 15 %) differences in the electron temperature, potential, and density of the confined plasma.

Figures (6)

• Figure 1: Time dependence of the (a) electron and (b) ion temperatures averaged over the entire system for different numbers of particles in the cell. In (b), the solid curves show the longitudinal ion temperature $T_{i\parallel}$, while the dotted curves show the transverse $T_{i\perp}$. $T_{i\parallel}$ is the average doubled longitudinal kinetic energy $<mv^2_\parallel>$ of ions, and $T_{i\perp}$ is their average transverse energy $<mv^2_\perp/2>$. The red curve is the theoretical dependence of $T_e$ on time \ref{['eq_time']}.
• Figure 2: Implementation of open boundary conditions.
• Figure 3: The energy of the system $E_{system} = E_p + E_{field} + E_{deleted\; p} - E_{added\;p}$ in the simulation with h=0.01 cm shown in Fig. 4 in units of its initial value $E_{system\;0}$ ($E_p$ is the energy of particles located inside the calculation region, $E_{field}$ is the energy of the electric field, $E_{deleted\; p}$ and $E_{added\;p}$ are the energies of the removed and injected particles).
• Figure 4: Results of PIC simulations with different spatial steps $h = 0.000145$ cm (blue curve) and $h = 0.01$ cm (orange curve): (a) dependence of the electric potential jump between the system center and the wall on time in units of the central electron temperature $T_{center}/e$; (b) time behavior of the electric potential jump in electron volts, (c) spatial distribution of the potential $e\phi/T_{center}$ in the steady state ($t=100$ ns); (d) electron temperature profiles. All graphs are time averaged.
• Figure 5: Near-wall plasma parameters for PIC simulations with $h = 0.000145$ cm (solid curve) and $h = 0.01$ cm (dotted curve) in the steady state ($t=100$ ns): (a) potential in units of the electron temperature at the center of the system $T_{center}$; (b) longitudinal ion velocity in units of the speed of sound $c_s = \sqrt{T_{center} / m_i}$. Red curves on both plots correspond to the relative density difference $(n_e-n_i)/n_i$ and show where the quasi-neutrality is violated (orange stripes). All graphs are time averaged.