1. Ergodic heat transport analysis in non-aligned coordinate systems S. Günter, K. Lackner, Q. Yu IPP Garching Problems with non-aligned coordinates? Description of new scheme First results

2. Problems with non-aligned coordinates Radial heat transport in multiple helicity magnetic fields enhanced by stochasticity For large  || /   (realistic values for hot plasmas ~ 10 10 ) careful treatment of parallel heat flux required

3. Problems with non-aligned coordinates Usually coordinate system aligned with magnetic field lines used, see e.g. Runov et al. (for static magnetic field) But for non-linear MHD calculations dynamically evolving magnetic fields need to be considered! For large  || /   (realistic values for hot plasmas ~ 10 10 ): small gradient along magnetic field lines cause large errors in temperature profile

4. An example: Interaction of NTMs with different helicities No simultaneous large NTMs of different helicities observed in experiments j BS   p

5. Analytic theory: for NTMs stabilising effect of additional helical field can be proven for small values of  || /   (incomplete temperature flattening) effect vanishes for  || /     Is there an effect remaining for realistic values of  || /   ? If so: new stabilisation method for NTMs can be propsed: stabilisation by external helical perturbation fields An example: Interaction of NTMs with perturbation fields Many other problems, but: so far no non-linear MHD code can deal with realistic  || /  

6. Proposal for a solution in non-aligned coordinate system In the following, for simplicity (not in the code): Cartesian coordinates in radial direction, Fourier decomposition within the unperturbed flux surface, only one perturbation field component Heat conduction equation for different Fourier components of temperature: … … To close the equations one should not truncate the Fourier series in T, but in q  heat flux along perturbed magnetic field line remains finite (nearly vanishing temperature gradients)

7. Fourier decomposition for perturbation To lowest order (for explanation): include only terms up to first order in q  T 2 adjusts itself such that q ||1 becomes small In the following, for simplicity (not in the code): Cartesian coordinates in radial direction, Fourier decomposition within the unperturbed flux surface, only one perturbation field component Heat conduction equation for different Fourier components of temperature:

8. Fourier decomposition for perturbation If one truncates in T (just as example, holds for any order): Cut after lowest order in temperature Enhancement of radial transport (T 1 would adjust to cancel the first term) To close the equations one should not truncate the Fourier series in T, but in q  heat flux along perturbed magnetic field line remains finite (nearly vanishing temperature gradients)

9. What about the radial derivatives? perturbation field: Introduces an additional error or order (  r) 2, but equations for each grid point ensure vanishing temperature gradients along perturbed field lines simplest discretisation at i’s grid point new scheme

10. Convergence properties: single magnetic island  || /   = 10 8 Still convergence only  (  r) -2 But: relative error reduced by factor of 10 Improvement increases for larger  || /   (~ (  || /   ) 1/2 ) -2

11. Example: Magnetic islands of two helicities

12.  || /   = 10 10 Magnetic islands seen in temperature contours, but still strong gradient in ergodic region Example: Magnetic islands of two helicities

13.  || /   = 10 12 Temperature gradient vanishes in ergodic region due to increased radial transport along magnetic field lines Example: Magnetic islands of two helicities

14. Coming back to: Interaction of NTMs with different helicities Is there an effect remaining for realistic values of  || /   ? If so: new stabilisation method for NTMs can be propsed: stabilisation by external helical perturbation fields (next talk) YES!

15. Summary and conclusion New scheme for solving heat conduction equation in non-aligned coordinates developed Successful test for realistic (and even higher) values of  || /   Method can be used in general (toroidal) non-linear MHD codes Generalisation to 3d Cartesian grid straightforward