A comprehensive two-fluid extended-MHD model is implemented in M3D-C1. This model includes sources for particles, momentum, current, and heat; anisotropic thermal diffusivity and viscosity (including gyroviscosity); and separate equations for the electron and ion temperatures.
More detailed descriptions of the implementation of the extended-MHD model in M3D-C1 can be found in SC Jardin et al. J. Phys.: Conf. Ser. 125 012044 (2008), J Breslau, N Ferraro, and S Jardin, Phys. Plasmas 16 092503 (2009), N Ferraro and S Jardin, J. Comp. Phys. 228 7742 (2009) (describing the two-fluid model), and NM Ferraro et al Nucl. Fusion 59 016001 (2019) (describing the two-temperature model). The implementations of additional models, including the resistive wall model, the model for impurities, runaway electrons, and kinetic species, are described below.
M3D-C1 optionally includes a resistive wall model, which is treated as a spatially resolved region of arbitrary thickness. This capability enables the study of so-called "resistive wall modes" (RWMs) in tokamaks, which are external kink modes that would be stable in the presence of a perfectly conducting wall, but which are unstable when the wall has finite resistivity.
In calculations that include a resistive wall, the computational domain is divided into three parts. The innermost part is the extended-MHD (XMHD) region, in which the full extended-MHD model is solved. Enclosing this region is the resistive wall (RW) region, in which only Maxwell's equations are solved, with E = eta_W J. Enclosing this region is the vacuum region, where only J = 0 is solved. The equations in these regions are solved self-consistently and simultaneously, using implicit time-stepping methods.
More information about the implementation and application of the resistive wall capability in M3D-C1 can be found in NM Ferraro, et al. Phys. Plasmas 23 056114 (2016).
M3D-C1 implements the KPRAD model for the ionization, radiation, and recombination of impurities. Coronal equilibrium is not assumed; each charge state of the impurity species is evolved. The primary application of this model has been in simulations of disruption mitigation. More information about the implementation of this model can be found in NM Ferraro et al. Nucl. Fusion 59 016001 (2019).
The kinetic particle model implemented in M3D-C1 is described in C Liu et al. Comput. Phys. Comm. 275 108313 (2022).
Runaway electrons (REs) are electrons that have been accelerated to high energy (> 1 MeV) by powerful electric fields that can be created during transient phases of tokamak operation. RE beams pose serious risks for reactor-scale tokamaks and must be avoided or mitigated. A description of the fluid model for runaway electrons implemented in M3D-C1 can be found in C Zhao et al. Nucl. Fusion 60 126017 (2021). The PIC kinetic model, described above, can also be used to model REs in M3D-C1; see C Liu et al. Phys. Rev. Lett. 131 085102 (2023).