Influences of $\delta$B Contribution and Parallel Inertial Term of Energetic Particles on MHD-Kinetic Hybrid Simulations: A Case Study of the 1/1 Internal Kink Mode

TL;DR

This study examines how the $ abla imes extbf{P}_h$ term and magnetic field perturbations affect hybrid MHD-kinetic simulations of the 1/1 internal kink mode, revealing limitations of simplified models especially with anisotropic energetic particles.

Contribution

The paper derives analytical formulations for pressure- and current-coupling schemes and assesses their accuracy in simulating internal kink mode stability, highlighting the importance of $ abla imes extbf{P}_h$ and magnetic perturbations.

Findings

Simplified models are adequate for isotropic EP distributions.

Models fail with anisotropic EP distributions.

EP orbit width influences internal kink mode stability.

Abstract

The magnetohydrodynamic-kinetic (MHD-kinetic) hybrid model [Park et. al., 1992] has been widely applied in studying energetic particles (EPs) problems in fusion plasmas for past decades. The pressure-coupling scheme or the current-coupling scheme is adopted in this model. However, two noteworthy issues arise in the model application: firstly, the coupled term introduced in the pressure-coupling scheme, $\left( \nabla \cdot \mathbf{P}_{\mathrm{h}} \right)_{\bot}$, is often simplified by $\nabla \cdot \mathbf{P}_{\mathrm{h}}$, which is equivalent to neglecting the parallel inertial term of EPs; secondly, besides the $\delta f $ contribution caused by changing in the EP distribution function, the magnetic field perturbation (the $\delta \mathbf{B} $ contribution) generated during development of the instabilities should also be considered, but it is often ignored in existing hybrid simulations. In this paper, we derive the analytical formulations under these two coupling schemes and then numerically study the representative case of the linear stability of the m/n=1/1 internal kink mode (IKM) [Fu et. al., 2006] by using the CLT-K code. It is found that the approximated models can still yield reasonable results when EPs are isotopically distributed. But it fails completely in cases with anisotropic EP distributions. In addition, we further investigate the influence of EP's orbit width on the stability of IKM and verify the equivalence between pressure-coupling scheme and the current-coupling scheme.