Dirichlet boundary conditions for arbitrary-shaped boundaries in stellarator-like magnetic fields for the Flux-Coordinate Independent method

Abstract

We present a technique for handling Dirichlet boundary conditions with the Flux Coordinate Independent (FCI) parallel derivative operator with arbitrary-shaped material geometry in general 3D magnetic fields. The FCI method constructs a finite difference scheme for ∇ by following the field lines between poloidal planes and interpolating within planes, rather than having a field-aligned mesh on flux surfaces.Doing so removes the need for field-aligned coordinate systems that suffer from singularities in the metric tensor at null points in the magnetic field (or equivalently, when q → ∞). One cost of this method is that as the field lines are not on the mesh, they may leave the domain at any point between neighbouring planes, complicating the application of boundary conditions. The Leg Value Fill (LVF) boundary condition scheme presented here involves an extrapolation/interpolation of the boundary value onto the field line end point. The usual finite difference scheme can then be used unmodified. We implement the LVF scheme in BOUT++ and use the Method of Manufactured Solutions to verify the implementation in a rectangular domain, and show that it doesn't modify the error scaling of the finite difference scheme. We outline the use of LVF for arbitrary wall geometry.We also demonstrate the feasibility of using the FCI approach in non-axisymmetric configurations for a simple diffusion model in a "straight stellarator" magnetic field.A Gaussian blob diffuses along the field lines, tracing out flux surfaces. Dirichlet boundary conditions impose a last closed flux surface (LCFS) that confines the density. Including a poloidal limiter moves the LCFS to a smaller radius.The expected scaling of the numerical perpendicular diffusion, which is a consequence of the FCI method, in stellarator-like geometry is recovered. A novel technique for increasing the parallel resolution during post-processing, in order to reduce artefacts in visualisations, is described. a) Electronic mail: Peter.Hill@york.ac.uk 2

Towards nonaxisymmetry; initial results using the Flux Coordinate Independent method in BOUT++

“…Inherent numerical perpendicular diffusion as a function of poloidal mesh spacing in the straight stellarator test case. The fit shows third order convergence, which is broadly in line with previous work [10]. From Figures 5 and 6 it appears that the optimal resolution for an FCI mesh is 256 by 256, as the perpendicular diffusion is at least 10 −8 smaller than the parallel diffusion.…”

“…Recently the Flux Coordinate Independent method for calculating parallel derivatives [10,11] has been implemented in BOUT++. This method for calculating parallel derivatives has been implemented in other codes [12,13], and is intuitively straightforward, as described in Figure 1.…”

“…The FCI method has been implemented into BOUT++ and tested via the method of manufactured solutions [10], which determined that the operators converge to second order, as is expected for the central difference schemes in use. One of the aims of the research presented here is to provide the components necessary to simulate stellarator turbulence cases and evaluate the computational and developmental work required.…”

“…A Poincaré plot of this configuration was created to ensure the existence of closed flux surfaces, as shown in Figure 3. This configuration has been implemented into the FCI grid generator [10], where it is possible to change several parameters including rotational transform, toroidal field and helical coil current/position. A typical grid with 16 Cartesian poloidal planes with a uniform 256 x 256 resolution can be generated in approximately 45 seconds.…”

The effects of non-uniform drive on plasma filaments

“…While BOUT++ is capable of simulating complex geometries [3][4][5], one can often explore complex phenomena by simplifying the problem. For this reason, isolated filament simulations in slab geometries are often employed [6][7][8].…”