Finite difference code for 3D edge modelling Max-Planck-Institut für Plasmaphysik, EURATOM Association Finite difference code for 3D edge modelling Electron heat conduction equation To be solved in regions with : Non-linearity Extreme anisotropy Complex geometry Ergodicity

Transport in the Plasma Edge Max-Planck-Institut für Plasmaphysik, EURATOM Association Transport in the Plasma Edge What is the energy transport mechanism in the plasma edge region? Is it dominated by long open field lines or short open field lines?

Local Magnetic Coordinate System Max-Planck-Institut für Plasmaphysik, EURATOM Association Local Magnetic Coordinate System Local system shorter than Kolmogorov length to handle ergodicity forward cut One coordinate aligned with the magnetic field to minimize numerical diffusion x2 x3 central cut Area is conserved x1 Use a full metric tensor backward cut

Vector form of the transport equation Max-Planck-Institut für Plasmaphysik, EURATOM Association Vector form of the transport equation metric volume Generalized spatial coordinate Conduction tensor

Finite Difference Method Max-Planck-Institut für Plasmaphysik, EURATOM Association Finite Difference Method Points are generated along magnetic field lines Temperature at each point is a function of the temperature at the neighboring points Consistent local magnetic coordinates for the whole stencil  = -18°  = 0°  = 18° Field line segment

Max-Planck-Institut für Plasmaphysik, EURATOM Association Optimizing the Mesh  We use ‘closed’ field lines to minimize numerical diffusion :

Max-Planck-Institut für Plasmaphysik, EURATOM Association 3D mesh structure Standard finite beta mesh in W7-X 20 Poincaré plots ~25000 points

Summary of computational process Max-Planck-Institut für Plasmaphysik, EURATOM Association Summary of computational process Generate the Neighborhood arrays for each mesh point Magnetic equilibrium Finite difference transport code; ‚FINDIF‘ Ax=b Mesh W7-X Solution Generate the full metric tensor for each mesh point

Mesh structure in a vacuum field Max-Planck-Institut für Plasmaphysik, EURATOM Association Mesh structure in a vacuum field Poincaré plot at =36º from a W7-X vacuum mesh Outer boundary Open field lines on island flux surfaces Closed field lines in an island Open field lines elsewhere in the edge region Closed field lines forming core flux surfaces

Temperature solution on a W7-X vacuum mesh Max-Planck-Institut für Plasmaphysik, EURATOM Association Temperature solution on a W7-X vacuum mesh Innermost core flux surface Particle density: 1e20 m-3 Radial diffusivity: χ= 1 m2s-1 LCFS Boundary conditions: 200 eV in the core 10 eV at the wall Sheath condition at the ends of the open field lines Outer boundary

Solution on a Poincaré plot Max-Planck-Institut für Plasmaphysik, EURATOM Association Solution on a Poincaré plot  = 36°

Mesh structure in a finite beta field Max-Planck-Institut für Plasmaphysik, EURATOM Association Mesh structure in a finite beta field Poincaré plot at =36º from W7-X finite beta mesh Outer boundary Open field lines on island flux surfaces Open field lines elsewhere in the edge region Closed ergodic field lines Closed field lines forming core flux surfaces

Temperature solution on a W7-X finite beta mesh Max-Planck-Institut für Plasmaphysik, EURATOM Association Temperature solution on a W7-X finite beta mesh Innermost core flux surface LCFS Closed ergodic field lines surrounding the plasma core Outer boundary

Solution on a Poincaré plot Max-Planck-Institut für Plasmaphysik, EURATOM Association Solution on a Poincaré plot  = 36°

Comparison of vacuum and finite beta solutions Max-Planck-Institut für Plasmaphysik, EURATOM Association Comparison of vacuum and finite beta solutions Solution on normalized field lines in a vacuum field (Low ergodicity) Solution on normalized field lines in a finite beta field (High ergodicity) Particle density: 1e20 m-3 Radial diffusivity: χ= 1 m2s-1

Radial temperature profiles (1) Max-Planck-Institut für Plasmaphysik, EURATOM Association Radial temperature profiles (1) Poincaré plot from W7-X vacuum mesh Poincaré plot from W7-X finite beta mesh  = 0°  = 0°

Radial temperature profiles (2) Max-Planck-Institut für Plasmaphysik, EURATOM Association Radial temperature profiles (2) Vacuum case Finite beta case

Variation of κ χ = 1 m2s-1 κ = χne ne = 1e20 m-3 Max-Planck-Institut für Plasmaphysik, EURATOM Association Variation of κ χ = 1 m2s-1 κ = χne ne = 1e20 m-3 Solutions on W7-X finite beta mesh with different radial diffusivities Core boundary condition held constant at 200 eV χ = 0.1 m2s-1 χ = 0.01 m2s-1

Max-Planck-Institut für Plasmaphysik, EURATOM Association Variation of κ|| Solutions on W7-X finite beta mesh with different core boundary conditions Κ = χne χ = 1 m2s-1 ne = 1e20 m-3 300 eV (~13 MW/m2) 100 eV (~4 MW/m2)

Power loading on the divertor plates (1) Max-Planck-Institut für Plasmaphysik, EURATOM Association Power loading on the divertor plates (1) Feeding fluxes at the ends of the open field lines Divertor plate Green points are the ends of open field lines which form island flux surfaces. Red points are the ends of open field lines which lie elsewhere in the edge region.

Power loading on the divertor plates (2) Max-Planck-Institut für Plasmaphysik, EURATOM Association Power loading on the divertor plates (2) Angled fluxes at the ends of the open field lines Divertor plate Green points are the ends of open field lines which form island flux surfaces. Red points are the ends of open field lines which lie elsewhere in the edge region.

Max-Planck-Institut für Plasmaphysik, EURATOM Association Conclusions Ergodicity causes cascading of energy into regions which previously had lower energy flow. At the target plates this produces a reduction of the peak power. (indirect effect of ergodicity) Reducing the radial transport caused a decoupling of the open field lines in the edge region from the closed field lines in the core region. Varying the parallel transport caused only a scaling of the temperature solution. The longer open field lines had more effect in removing heat from the plasma core.

3D solution on a W7-X finite beta mesh Max-Planck-Institut für Plasmaphysik, EURATOM Association 3D solution on a W7-X finite beta mesh Particle density: 1e20 m-3 Radial diffusivity: 1 m2s-1 Boundary conditions: 200 eV - core 10 eV - wall Sheath condition at the ends of open field lines

Max-Planck-Institut für Plasmaphysik, EURATOM Association NCSX 24 Poincaré plots ~17000 mesh points  = 60°  = 0°

Summary and coming attractions Max-Planck-Institut für Plasmaphysik, EURATOM Association Summary and coming attractions The FINDIF code solves the electron energy equation in systems which have high anisotropy, complex geometry and ergodic effects. Ergodicity is handled by using local magnetic coordinates. The mesh is optimized by selecting ‘properly’ closed field lines (Δ, L||) in the mesh construction Inclusion of constant flux boundary condition in the plasma core Inclusion of convection term in the transport equation Inclusion of more transport equations

Solution on a W7-X vacuum mesh Max-Planck-Institut für Plasmaphysik, EURATOM Association Solution on a W7-X vacuum mesh Particle density: 1e20 m-3 Radial diffusivity: 1 m2s-1 LCFS Open field lines in the edge Boundary conditions: 200 eV in the core 10 eV at the wall Sheath condition at the ends of the open field lines

Solution on a Poincaré plot Max-Planck-Institut für Plasmaphysik, EURATOM Association Solution on a Poincaré plot

Solution on a W7-X finite beta mesh Max-Planck-Institut für Plasmaphysik, EURATOM Association Solution on a W7-X finite beta mesh LCFS Closed ergodic field lines surrounding the plasma core

Solution on a Poincaré plot Max-Planck-Institut für Plasmaphysik, EURATOM Association Solution on a Poincaré plot

Te r Ergodic Effects rLCFS Last closed flux surface, LCFS Max-Planck-Institut für Plasmaphysik, EURATOM Association Ergodic Effects Last closed flux surface, LCFS Plasma core (non-ergodic) ergodic region Enhances radial transport, due to contributions from parallel transport Te Flattening of temperature profile Scrape Off Layer r rLCFS

Max-Planck-Institut für Plasmaphysik, EURATOM Association Kolmogorov length Kolmogorov length, LK, is a measure of field line ergodicity 2 field lines diverging exponentially

Resolution of metric coefficients Max-Planck-Institut für Plasmaphysik, EURATOM Association Resolution of metric coefficients Resolution of g along a field line for 360º around W7-X Mesh with 360 cuts Mesh with 40 cuts Mesh with 20 cuts Mesh with 10 cuts

Max-Planck-Institut für Plasmaphysik, EURATOM Association Triangulation

Max-Planck-Institut für Plasmaphysik, EURATOM Association Matrix structure Flux surfaces in the plasma core Closed ergodic field lines surrounding the plasma core Islands Outer boundary

Location of source in a W7-X core mesh Max-Planck-Institut für Plasmaphysik, EURATOM Association Location of source in a W7-X core mesh W7-X core mesh containing: 20 Poincaré plots, ~13000 points, 9 field lines forming 9 closed flux surfaces in the plasma core, No field lines in the plasma edge  = -36° LCFS Source 100 eV

Max-Planck-Institut für Plasmaphysik, EURATOM Association Effect of source With NO source With source LCFS Held constant at 10 eV