Implementing general moment equations for parallel closures in NIMROD

Implementing an advanced closure module significantly extends the capability of MHD fluid codes such as NIMROD [1]. Non-Maxwellian parallel moment equations are implemented in NIMROD to obtain parallel closures. To derive the moment equations, we take moments of the first-order drift kinetic equation using orthogonal velocity polynomials. The system of parallel moment equations is then solved using 2D finite elements in the poloidal plane, considering an axisymmetric magnetic field. To overcome the memory limit associated with large problem sizes, the GMRES algorithm is used without the need for explicit matrix construction. The solution is shown to converge as the system size increases. The steady-state results are compared to analytic results using the moment-Fourier approach, which involves expanding moments and quantities related to the magnetic field in terms of Fourier series [2].

[1] C. R. Sovinec, A. H. Glasser, T. A. Gianakon, D. C. Barnes, R. A. Nebel, S. E. Kruger, D. D. Schnack, S. J. Plimpton, A. Tarditi, M. S. Chu, and NIMROD Team, “Nonlinear magnetohydrodynamics simulation using high-order finite elements”, Journal of Computational Physics, vol. 195, 355 (2004).

[2] J.-Y. Ji, E. D. Held, J. A. Spencer and Y.-S. Na, “Moment-Fourier approach to ion parallel fluid closures and transport for a toroidally confined plasma ”, Plasma Physics and Controlled Fusion, vol. 65, 035018 (2023).