The parallel boundary condition for turbulence simulations in low magnetic shear devices

Abstract

Flux tube simulations of plasma turbulence in stellarators and tokamaks typically employ coordinates which are aligned with the magnetic field lines. Anisotropic turbulent fluctuations can be represented in such field-aligned coordinates very efficiently, but the resulting non-trivial boundary conditions involve all three spatial directions, and must be handled with care. The standard "twistand-shift" formulation of the boundary conditions [Beer, Cowley, Hammett Phys. Plasmas 2, 2687(1995] was derived assuming axisymmetry and is widely used because it is efficient, as long as the global magnetic shear is not too small. A generalization of this formulation is presented, appropriate for studies of non-axisymmetric, stellarator-symmetric configurations, as well as for axisymmetric configurations with small global shear. The key idea is to replace the "twist" of the standard approach (which accounts only for global shear) with the integrated local shear. This generalization allows one significantly more freedom when choosing the extent of the simulation domain in each direction, without losing the natural efficiency of field-line-following coordinates. It also corrects errors associated with naive application of axisymmetric boundary conditions to non-axisymmetric configurations. Simulations of stellarator turbulence that employ the generalized boundary conditions require much less resolution than simulations that use the (incorrect, axisymmetric) boundary conditions. We also demonstrate the surprising result that (at least in some cases) an easily implemented but manifestly incorrect formulation of the boundary conditions does not change important predicted quantities, such as the turbulent heat flux. * mfmartin@umd.edu

Stellarator microinstabilities and turbulence at low magnetic shear

“…Recently, a more accurate parallel boundary condition for stellarator symmetric flux tubes has been derived in Martin et al. (2018). This approach removes the discontinuities present at the ends of the simulation domain in the current conventional treatment, which can be seen at of the curve in figure 1.…”

“…In such circumstances, accurate flux-tube simulations can be performed using the geometry for only one poloidal turn, as was done in , . Recently, a more accurate parallel boundary condition for stellarator symmetric flux tubes has been derived in Martin et al (2018). This approach removes the discontinuities present at the ends of the simulation domain in the current conventional treatment, which can be seen at θ = ±4π of the k 2 ⊥ curve in figure 1.…”

Eliminating turbulent self-interaction through the parallel boundary condition in local gyrokinetic simulations

“…More recent work considered the self-interaction of linear eigenmodes and revealed that their parallel self-interaction can even interfere with the numerical convergence of nonlinear simulations with respect to the domain size [11,12] §. Additionally, it has been explored in gyrokinetic simulations of low magnetic shear stellarators [13,14]. This paper will show that parallel self-interaction can be eliminated by lengthening the simulation domain in the parallel direction to multiple poloidal turns or making the binormal width of the domain large.…”

Gyrokinetic simulations in stellarators using different computational domains

“…In nonlinear simulations, a box in the radial and binormal directions is defined around the central field line defining the FT, and a set of radial and binormal wavenumbers k x and k y are considered. The fulfilment of the periodicity condition in the parallel direction for all the modes considered in the simulation is more restrictive in this case, with a strong influence of the magnetic shear [1,2,22].…”

“…The lack of axisymmetry in stellarators introduces complexity at several levels. First, the twist-and-shift boundary condition in flux tube simulations is questionable [2] due to the three-dimensional dependence of the equilibrium quantities affecting micro-instabilities, such as magnetic field line curvature and magnetic shear. As a consequence of this dependence, different flux tubes over a given flux surface are in general not equivalent to each other [3].…”