Effect of the Boussinesq approximation: Turbulence studies with GRILLIX in slab geometry

The drift‐reduced global Braginskii system is implemented in GRILLIX, a plasma turbulence code that is able to treat diverted geometries by using the flux‐coordinate independent approach (FCI). We solve a four‐field model, evolving the density, the electron temperature, the vorticity, and parallel ion momentum. The difference between the system with the Boussinesq approximation (BS) and the full system (FS) is investigated. The Boussinesq approximation is widely used as it reduces the numerical and computational complexity significantly. In order to understand the impact of the Boussinesq approximation, a periodic flux tube geometry is chosen as the computational domain. The conservation of energy and particles is derived and tested within the code, both in FS and BS. Moreover, we will discuss how the Boussinesq approximation has subtle effects on the energy theorem of the model, and a conserved energy‐like quantity is only obtained if the Boussinesq approximation is applied in a consistent way. We will also present numerical and computational techniques in order to relax the Boussinesq approximation. The code is verified by the method of manufactured solutions, which yields the expected order of accuracy. A fundamental result found in the paper is that the Boussinesq approximation only has a minor effect on non‐linear simulations, at least in the regime studied here.