Reduced model for approximating collisional losses in a magnetic mirror

Accurately predicting plasma density and temperature in magnetic mirrors requires precise confinement time and ambipolar potential estimations. This study focuses on a Pastukhov-type calculation of confinement time using a Dougherty model collision operator in magnetic mirrors. Gyrokinetic codes like Gkeyll, GENE, GENE-X, and GX can use the Dougherty operator due to its improved accuracy compared to the Krook operator while being simpler to implement and test [1]. However, the Dougherty collision operator assumes a constant collisionality, deviating from the realistic plateau and 1/v³ drop-off. In the context of Gkeyll's initial studies, this study aims to quantify the impact of the Dougherty operator's approximations on confinement time and determine necessary collision rate adjustments [2]. Pastukhov's seminal work described collisional losses in magnetic mirrors, with subsequent studies providing generalizations and higher-order corrections [3, 4, 5, 6]. Inspired by this methodology, we present a novel derivation to address this challenge. To assess our approach, we compare results with a FEniCS DOLFINx finite element solver [7]. Findings reveal that using the Dougherty collision operator alters the scaling behavior of the loss rate with the ambipolar potential from φ e^φ to φ^1/2 e^φ. We propose adjusting the collision frequency in codes utilizing this reduced collision operator to match the ambipolar potential.