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

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

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

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

Simulation setup  grid resolution: 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

+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

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

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

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

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

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

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

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

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

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” 

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

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

[G] [K] Upper photosphere: Temperature distribution vertical magnetic field Temperature Horizontal cuts 300 km above  =1

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

Flux cancellation event magnetic field temperature continuum brightness 300 km above 300 km above time [s]

Flux cancellation event magnetic field 300 km above 300 km above vertical velocity time [s] down up

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

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

3D view of field lines view from top