Axisymmetric Interchange Calculations with NIMROD

Axisymmetric interchange instabilities in ideal magnetohydrodynamics are considered for a code verification exercise. Linearized equations are solved for cylindrical equilibria with uniform axial current density and uniform axial magnetic field. Analytical results are obtained from eigenmode analysis, and different NIMROD computations are performed with the plane of finite elements representing either the r-z plane or the r-θ plane of the periodic cylinder. Accuracy is confirmed for interchange without axial magnetic field with both representations and with the r-z mesh for the large-axial-field case. No-slip conditions, which reduce the growth rate by 5.5%, are required with large-axial field and the r-θ mesh to avoid a numerical error. Possible improvements to the NIMROD representation are discussed. 1. Introduction At nearly eleven years into the NIMROD project (http://nimrodteam.org), the code has been well exercised on many applications and test problems, only some of which have been documented. When the code converges, we have confidence that it provides accurate results for the specified parameters. Nonetheless, there are computations where we have to modify parameters, usually increasing diffusivities, in order to achieve convergence. The difficulties typically appear as high-wavenumber modes that grow rapidly. However, in linear computations of edge-localized modes (ELMs) with low values of electrical resistivity, we observe slow convergence properties and an unphysical dependence on the diffusivity for controlling magnetic divergence error. Convergence on ELMs is achieved at increased values of resistivity, but the value is not realistic for large tokamaks, and high-order polynomials (degree 5 and larger) are still required. It is also worth noting where the code works extremely well. At low pressure, even very slow magnetic reconnection processes are reproduced accurately. Tearing modes tend to be low wavenumber, but very localized features exist near the reconnection region. In addition, the project changed the standard for resolving anisotropy in MHD computations without mesh alignment. The challenges associated with interchange behavior have received significant attention in the numerical literature. The book by Gruber and Rappaz [1] provides an extensive treatment of ideal MHD eigenvalue computations with finite elements. It emphasizes the 'non-standard' or singular nature of ideal MHD computations—when viewed as Sturm-Liouville problems—that results from resonant surfaces. In addition, stiffness in the time-dependent MHD system manifests itself as spectral sensitivity to point-wise numerical errors associated with compression in some finite element representations, leading to spectral pollution where a new unresolved mode appears as each new element is added. Degtyarev and Medvedev review …