1. Turbulent transport coefficients from numerical experiments Axel Brandenburg & Matthias Rheinhardt (Nordita, Stockholm) Extracting concepts from grand challenge DNS –Alpha effect –Test-field method –Integral kernel Test-flow method: Reynolds stress Results for passive scalar

2. 2 Steve’s contributions to dynamo theory PEPI 1979

3. 3 Last chapter: interest in DNS results 1995

4. 4 … at a time of grand challenge … Need tools to interprete results Origin of large-scale magnetic field? an  effect after all? Glatzmaier & Roberts (1995)

5. 5  -effect dynamos (large scale) Differential rotation (surface layers: faster inside) Cyclonic convection; Buoyant flux tubes Equatorward migration New loop    - effect

6. 6 Mean-field theory Define suitable averages Fully analogous to passive scalar! still exact!

7. 7 Test-field method:  ij and  ij tensors Original equation (uncurled) Mean-field equation fluctuations Response to arbitrary mean fields

8. 8 Test fields Example:

9. 9 Validation: Roberts flow SOCA normalize SOCA result Brandenburg, Rädler, Schrinner (2009, A&A)

10. 10 Kinematic  and  t independent of Rm (2…200) Sur et al. (2008, MNRAS)

11. 11 Scale-dependence: nonlocality

12. 12 Time-dependent case Hubbard & Brandenburg (2009, ApJ)

13. 13 Importance of time-dependence

14. 14 Momentum equation: nonlinear cf. Rheinhardt & Brandenburg (2010, A&A) linear in u U

15. 15 Results for Roberts flow Rheinhardt & Brandenburg (2010, in prep) ansatz

16. 16 Results for passive scalar transport Rotation Magnetic fields Linear shear flow

17. 17 Effects of rotation Brandenburg, Svedin, Vasil (2009, MNRAS)

18. 18 Effects of magnetic field Brandenburg, Svedin, Vasil (2009, MNRAS)

19. 19 Effects of linear shear Madarassy & Brandenburg,, (2009, PRE) U=(0, Sx, 0) Sh=S/uk=-0.13 Sh=-0.2

20. 20 Conclusions Test fields provide detailed insight into grand challenge DNS Non-locality in space and time – effects of boundaries –strong stratification? Reynolds and Maxwell stresses –Differential rotation of the Sun –Active regions? Applications to LES? –Averages still 3-D