1. 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

2. 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?

3. 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

4. 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

5. 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

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

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

8. 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

9. 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

10. 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

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

12. 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

13. 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

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

15. 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

16. 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°

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

18. 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

19. 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)

20. 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.

21. 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.

22. 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.

23. 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

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

25. 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

26. 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

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

28. 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

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

30. 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

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

32. 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

33. Max-Planck-Institut für Plasmaphysik, EURATOM Association
Triangulation

34. 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

35. 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

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