1. Modelling photospheric magnetoconvection in the weak field regime Paul Bushby & Steve Houghton (University of Cambridge) Acknowledgements: Mike Proctor, Nigel Weiss

2. Observational Motivation: (Ribbon-like magnetic structures) Taken from Berger et al. (2004) Fe I magnetogram showing magnetic elements in a remnant active region plage Magnetic field seems to be organised into “ribbon-like” structures

3. Taken from Domínguez Cerdena et al. (2003) Point-like magnetic structures The quiet Sun: Fe I magnetogram superimposed upon the observed granulation Almost point-like structures collect in the inter-granular lanes

4. Modelling photospheric magnetoconvection Two complementary approaches: “Realistic” models: (e.g. Stein & Nordlund, 2002; Vögler & Schϋssler, 2003) “Illustrative” models: (e.g. Weiss, Proctor & Brownjohn, 2002) Simpler models of magnetoconvection can still highlight important aspects of the interaction between convection and magnetic fields Faster calculations enable a broader survey of parameter space Successfully reproduce many observed features (at least in a qualitative sense) Fully compressible, realistic equation of state, radiative transfer Possible to make detailed comparisons with observations, including spectral features

5. Governing Equations (Matthews et al. 1995, JFM, 305, 281)

6. Model Set Up: Initial condition: Equations have an equilibrium state – a static polytrope with a uniform vertical magnetic field (imposed net flux through the layer): 8x8x1 Cartesian box z = 0 z = 1

7. Model Parameters: (Polytropic index) (Thermal stratification) Parameters imply that temperature and density vary by a factor of 11 across the layer The Rayleigh number does not appear in the equations, but it can be derived in terms of the other parameters: The Chandrasekhar number, Q, is varied (Ratio of magnetic to thermal diffusion) (Prandtl Number) (Rayleigh Number)

8. Computational Techniques: Typical computational grid utilises 256 points in each horizontal direction, 120 points vertically Horizontal derivatives evaluated in Fourier space (de-aliasing using the 2/3 rule) Vertical derivatives calculated in configuration space using a 4 th order finite difference scheme Explicit 3 rd order Adams Bashforth time-stepping (variable time-step) Periodic boundary conditions horizontally Vertical conditions: Impermeable, stress-free Vertical field Fixed temperature Idealised Boundary Conditions:

9. Larger values of Q: ( Weiss et. al, 2002) Horizontal cuts taken from just below the top of the layer, showing the temperature variation (darker = cooler) Q=2500 Q=1000 Steady“Flux-separated”

10. Results for Q=100 (1): Vertical Magnetic FieldTemperature Horizontal Cuts taken from just below the top of the layer: Dark regions on the temperature plot correspond to cooler fluid – the magnetic regions are relatively cool due to the local suppression of convective heat transport

11. Vertical Magnetic FieldTemperature Results for Q=100 (1): Horizontal Cuts taken from just below the top of the layer: Dark regions on the temperature plot correspond to cooler fluid – the magnetic regions are relatively cool due to the local suppression of convective heat transport

12. Results for Q=100 (1): Vertical Magnetic FieldTemperature Horizontal Cuts taken from just below the top of the layer: Dark regions on the temperature plot correspond to cooler fluid – the magnetic regions are relatively cool due to the local suppression of convective heat transport

13. Results for Q=100 (2): Fractal Dimension Self-similar field structures  fractal dimension? More realistic simulations successfully match fractal dimensions calculated from magnetograms of weak field regions (e.g. Janßen et al. 2003) What about more idealised simulations?

14. Fractal dimension can then be calculated from the formula: The same technique, when applied to the magnetic field found in solar plages, gives a value for the fractal dimension which lies in the range 1.45 – 1.60 (Schrijver et al. 1992, A&A, 253, L1) Method: Identify regions where the field exceeds some given threshold value Within each region, count the number (N) of “magnetic” pixels and work out the size, in terms of pixels, (l) of the smallest square that encloses the region

15. Each cross corresponds to a patch of magnetic field – this sample of 58 regions was generated from 6 different (well separated in time) snapshots of the surface of the computational domain Best fit line gives a fractal dimension of 1.58 – surprisingly good agreement with observations

16. Results for Q=100 (3): Partial Evacuation High magnetic pressure in regions of intense field  partial evacuation Dark regions in density plot (top) are areas of reduced density – closely mirrors the magnetic field distribution (bottom) Numerical point: Critical time-step is proportional to the minimum density, so evacuation implies smaller critical time-steps and longer computations

17. A segment of the minimum density time- series: Order of magnitude decrease over 1 time unit  order of magnitude decrease in the required time-step Obvious numerical implications Comment: Reduction in gas pressure is achieved by a reduction in both the temperature and the density

18. Variations in the imposed magnetic field strength: Right: Q=100 Bottom: Q=10 “Ribbon-like” magnetic structures are found for Q=100, whilst more localised (almost point- like) structures are favoured for Q=10

19. Q=100 Q=10 Probability density functions for the distribution of magnetic field near the upper boundary Sharp peaks at B=0 correspond to the extensive field-free regions The second clear peak on the Q=100 plot is absent for Q=10 – instead there is a broad “shoulder”

20. Comparing temperature plots: Right: Q=100 Below: Q=10 Without prior knowledge, it is very difficult to pick out the magnetic field locations in the Q=10 plot – the granulation pattern is relatively unaffected by the magnetic field

21. Conclusions Relatively idealised models successfully reproduce (in a qualitative sense) many observational features of photospheric magnetoconvection – even quantitative measures like the fractal dimension are in good agreement with observations When the imposed field is large enough, ribbon-like magnetic structures are found – analogy to solar plage A weaker imposed field tends to lead to the formation of point- like magnetic structures, with a granulation pattern that mimics quiet Sun regions Future Work: Small-scale dynamos