We discuss here the currently most complete model implemented in GRILLIX and available as braginskii. It is based on the drift-reduced Braginskii equations, but with corrections of different complexity for low collisionality. The model is fully electromagnetic, with induction and flutter. It is also fully global, in the sense that no approximations are made on the amplitude of fluctuations and all parametric non-linearities are kept.
Literature on the derivation and implications of drift reduced Braginskii models can be found e.g. in Zeiler 1997, Scott 1997 and Simakov 2003. We follow in particular the Habilitation theses of Andreas Zeiler and Bruce Scott, but they cannot be found online. A good alternative to the latter are the two new books Scott 2021, Vol.1 and Scott 2021, Vol.2. These are generally amazing references for the understanding of turbulence in magnetised plasmas, particularly in the plasma edge.
The Braginskii equations for the plasma require boundary conditions for closure, both along and across the magnetic field. In parallel direction we apply sheath boundary conditions using the penalisation technique (TODO: link!). In perpendicular direction, somewhat ad-hoc no-flux boundary conditions are applied.
Additionally, the plasma is strongly coupled to neutral gas. For the plasma model, this is realized via source terms, provided from an external neutral gas model. In the other direction, the neutrals dynamics and source terms depend on plasma parameters, in particular the plasma density and temperatures. This means that the coupling is weak and both the plasma and the neutral gas models should be exchangeable. Currently, a diffusive fluid model for neutral gas is implemented in GRILLIX.
Limitations of the model should be pointed out. Firstly, being a fluid model, it does not naturally capture kinetic effects. Most importantly, these are trapped particles and Landau damping. Approximations for these effects are being tested, though. Secondly, only low-order finite Larmor radius (FLR) effects can be considered in drift-reduced models, equivalent to the long-wavelength limit of gyro-averaging.
And lastly, the Braginkii equations are only valid for a simple plasma, i.e. a plasma with only one electron and one ion species. Burning plasmas will necessarily consist of at least 3 ion species (deuterium, tritium and helium) in comparable concentrations. Additionally, plasmas always contain some level of various impurity ions, sometimes in significant concentrations, also on todays experiments. Therefore, the model is being extended to the multi-ion Zhdanov closure.