1. Max-Planck-Institut für Plasmaphysik, EURATOM Association Different numerical approaches to 3D transport modelling of fusion devices Alexander Kalentyev Max-Planck Institut für Plasmaphysik, EURATOM Association Stellarator Theory Division

2. Max-Planck-Institut für Plasmaphysik, EURATOM Association Introduction 3D effects: In tokamaks near divertor plates stellarators are intrinsically 3D Ergodicity: Perturbation coils in tokamaks (TEXTOR-DED, DIII-D) In stellarators ergodic region always present

3. Max-Planck-Institut für Plasmaphysik, EURATOM Association Transport equations

4. Max-Planck-Institut für Plasmaphysik, EURATOM Association Finite volume approach (BoRiS) plasma core (non- ergodic) ergodic region island (non- ergodic) Divertors Generalized Newton solver Special application - W7-X using Boozer coordinates for 7 separate domains

5. Max-Planck-Institut für Plasmaphysik, EURATOM Association Flexibility of BoRiS Solution of the Navier-Stokes equations for a flow in a square cavity Predicted streamlines Influence of the under-relaxation parameters on convergence rate Convergence region Peric et al. 1988

6. Scrape Off Layer Plasma core Wall Parallel direction Radial direction Ergodic region Enhancement of radial transport due to contribution from parallel transport Rechester Rosenbluth, Physical Review Letters, 1978 Electron temperature r Max-Planck-Institut für Plasmaphysik, EURATOM Association Transport in an ergodic region

7. Kolmogorov length L K is a measure of field line ergodicity exponential divergence Typical value in W7-X : L K = 10 – 30 m Max-Planck-Institut für Plasmaphysik, EURATOM Association Kolmogorov length

8. central cut backward cut forward cut x1x1 x2x2 x3x3 One coordinate aligned with the magnetic field to minimize numerical diffusion Area is conserved Use a full metric tensor Local system shorter than Kolmogorov length to handle ergodicity Max-Planck-Institut für Plasmaphysik, EURATOM Association Local magnetic coordinates

9. Max-Planck-Institut für Plasmaphysik, EURATOM Association Interface problem 1) Optimized mesh (finite-difference scheme)  Problem: numerical diffusion induced by interpolation on the interface

10. Max-Planck-Institut für Plasmaphysik, EURATOM Association Monte-Carlo 1st Order Algorithm Random process random step Realization Diffusion Convection Monte-Carlo combined with Interpolated Cell Mapping High accuracy transformation of the perpendicular coordinates of a particle (mapping between cuts) needed!

11. Max-Planck-Institut für Plasmaphysik, EURATOM Association Finite Difference Approach Fieldline tracing Triangulation Metric coefficients Transport code Grid Neighborhoods Temperature solution Magnetic field Linearization matrix Mesh optimization

12. Max-Planck-Institut für Plasmaphysik, EURATOM Association “Semi-implicit” scheme Implicit scheme „Semi-implicit“ scheme Memory usage: 7 times less Solver: 50 times faster

13. Max-Planck-Institut für Plasmaphysik, EURATOM Association Results

14. Max-Planck-Institut für Plasmaphysik, EURATOM Association Conclusion and Future Work Conclusion Comparisons between three different codes for a W7-X geometry were done. Future Work To complete the physics (including all transport equations). To compare results in more realistic cases (including target plates, finite beta).

16. Max-Planck-Institut für Plasmaphysik, EURATOM Association Conduction-convection Convection-conduction equation for a „fluid quantity“ f: x 1 =const x 2 =const x3x3 reference cut „Magnetic“ coordinate system: - contribution from D || in D 33 only Metric tensor: determined by field line tracing

17. Max-Planck-Institut für Plasmaphysik, EURATOM Association Monte-Carlo 1 st Order Algorithm Fokker-Planck Eq. for pseudoscalar density of test particles, Random process Requirement Realization diffusion, convectionsink, source random step independent random numbers physics: diffusion and convection of the “fluid quantity” Higher order schemes in 3D get much too complex Interpretation as probabilistic approximation of Green functions possible