1. Decay of a simulated bipolar field in the solar surface layers Alexander Vögler Robert H. Cameron Christoph U. Keller Manfred Schüssler Max-Planck-Institute for Solar System Research Katlenburg-Lindau, Germany Magnetohydrodynamics of Stellar Interiors MSI Workshop, Cambridge, September 6-15, 2004

2. Regimes of solar magneto-convection  horizontal scale of convection decreases  convective energy transport decreases G-band image: KIS/VTT, Obs. del Teide, Tenerife sunspot umbra plage ‘quiet’ Sun quiet Sun plage sunspot umbra average B

3. The MPS/UofC Radiation MHD (MURaM) Code  3D compressible MHD  full non-local, frequency-dependent radiative transfer (LTE)  partial ionisation (11 elements)  open lower boundary

4. Energy equation Momentum equation Continuity equation Induction equation Radiative Transfer Equation The MURaM Code

5. Simulation setup  grid resolution: 288 2 x 100 pts. (∆x=∆y=21 km, ∆z=14 km)  vertical field at upper/lower boundary  start with non-magnetic convection 6 Mm 1.4 Mm 600 km 800 km  =1 brightness vertical velocity temperature

6. +200 G -200 G A B  dominant horizontal wavenumber vectors: Simulation setup  introduce initially vertical, bipolar field  two cases: k = (1, 1) k 0 (1,-1) k 0 k = (1, 0) k 0 L = 6 Mm k 0 = 2  /L

7. Case A: time evolution vertical magnetic field +200 G weak fields -200 G 0 G horizontal cuts near visible surface +2 kG -2 kG Decay of magnetic surface flux

8.  Exponential decay of energy  Decay rates depend on initial condition  Early phase AB  2.2  Decay of magnetic energy min

9.  Similar decay rate for both runs  Memory of the initial conditions is lost after two hours Late phase Decay of magnetic energy

10. Decay rates: spectral energy distribution k=(1, 1), (1,-1) k=(1, 0), (0, 1)  Case A: inverse energy cascade  Late phase: - fundamental fourier mode dominates - similar spectral energy distribution in both cases A B

11. with U = 1 km s -1 = 1 Mm Ansatz:   U t 3/1  AB  1.95  2 B 2 A |k||k| AB  experimental rates consistent with compare Decay rates and “turbulent“ diffusion

12. Statistical properties at the surface Probability density of field strength Histograms of energy distribution time = 0... 180 min

13. Statistical properties at the surface Decay of superequipartition fields (|B| > 500 G)  strong fields contain ~70% of the total energy  contribution to total flux decreases with time

14.  weak fields carry significant fraction of surface flux  energetically unimportant Statistical properties at the surface Decay of weak fields (85% area coverage)

15. Field line topology  Four classes of field line topologies:  initially, all flux is in and state  flux removal requires reconnection “up” “down” “U-loop” “inverse U-loop” 

16. Field line topology: spatial distribution field strength topology near bottom near top

17. Field line topology unsigned flux vs. height  -flux increases with depth  Almost no -flux  dominant in subsurface layers t=15 min t=140 min

18. +600 0 -600 [G] [K] 7100 3800 Upper photosphere: Temperature distribution vertical magnetic field Temperature Horizontal cuts 300 km above  =1

19. Upper photosphere: Scatter-plots of Temperature versus... joule heating vertical velocity vertical Lorentz force dp/dz – rho*g div u

20. Flux cancellation event magnetic field temperature continuum brightness 300 km above 300 km above time [s] 0 30 60 90 120 150

21. Flux cancellation event magnetic field 300 km above 300 km above vertical velocity time [s] 0 30 60 90 120 150 down up

22. Vertical temperature structure at flux cancellation site x Temperature rise (  Emission lines ?) 1 Mm

23. 3D view of field lines view from top Field line topology

24. 3D view of field lines view from top