Modelling photospheric magnetoconvection

S. M. Blanchflower 0 A. M. Rucklidge 0 N. O. Weiss 0 0 Department of Applied Mathematics & Theoretical Physics, University of Cambridge , Cambridge CB3 9EW A B S T R A C T The increasing power of computers makes it possible to model the non-linear interaction between magnetic fields and convection at the surfaces of solar-type stars in ever greater detail. We present the results of idealized numerical experiments on two-dimensional magnetoconvection in a fully compressible perfect gas. We first vary the aspect ratio l of the computational box and show that the system runs through a sequence of convective patterns, and that it is only for a sufficiently wide box (l $ 6) that the flow becomes insensitive to further increases in l. Next, setting l 6, we decrease the field strength from a value strong enough to halt convection and find transitions to small-scale steady convection, next to spatially modulated oscillations (first periodic, then chaotic) and then to a new regime of flux separation, with regions of strong field (where convection is almost completely suppressed) separated by broad convective plumes. We also explore the effects of altering the boundary conditions and show that this sequence of transitions is robust. Finally, we relate these model calculations to recent high-resolution observations of solar magnetoconvection, in plage regions as well as in light bridges and the umbrae of sunspots. I N T R O D U C T I O N Magnetic fields interfere with convective transport in the photospheres of late-type stars. This interaction can be observed in detail at the surface of the Sun, where features that are only a few hundred kilometres across can now be resolved, revealing a variety of fine structure that depends on the local strength of the magnetic field. At the same time, rapid advances in computing power have made it possible to model non-linear magnetoconvection in regimes where numerical experiments can be contrasted with solar observations. In this paper we study the effects of varying the geometry and boundary conditions in idealized models, and identify different patterns of behaviour when the fields are weak or strong. These regimes are then related to convective structures on the Sun. In the solar photosphere, the strongest vertical fields are found in pores and sunspot umbrae, where convective plumes show up as umbral dots (Danielson 1964). These small bright features are present in all sunspots, though large spots contain isolated regions (dark nuclei) that are free of them (Muller 1992; Sobotka, Bonet & Vazquez 1993; Sobotka 1997). Until very recently it was thought that umbral dots had diameters of 180300 km and a filling factor of 310 per cent. With improved resolution (Sobotka 1997; Sobotka, Brandt & Simon 1997a,b) it is now clear that there is no typical diameter; rather, the number density of umbral dots increases with decreasing size, down to the limit of resolution at 0.28 arcsec (200 km). An average specimen has a diameter of 300 km and a lifetime of 14 min but the lifetimes range from 2 h for the largest bright dots to a few minutes for the smallest. Light bridges are bright linear features that cut across sunspots and exhibit a fine granular structure (Muller 1992; Sobotka et al. 1993; Sobotka 1997). The field within a light bridge is weaker than in the surrounding umbra, so the bridge resembles a slot, contained between magnetic walls that resist distortion, within which more vigorous convection can occur. Rimmele (1997) has followed the evolution of convective plumes within such a slot for a full hour and confirmed that bright granules are associated with upward motion. He also found that the velocity and intensity varied in a manner consistent with oscillatory convection. Outside sunspots, there is a distinction between plage regions (with average field strengths greater than 150 G) and quiet Sun. Plages are characterized by abnormal granulation, corresponding to convection with a smaller horizontal scale. The fields form a perforated network, including fine magnetic structures that give rise to isolated bright points (Title et al. 1992; Muller 1994; Sobotka, Bonet & Vazquez 1994). These appear most strikingly in the CH G-band and their dynamic behaviour indicates that magnetic flux moves rapidly through the intergranular network, forming ephemeral concentrations rather than isolated flux tubes (Berger et al. 1995; Berger & Title 1996; Berger et al. 1998). Within the photospheric network, magnetic structures are smaller and more nearly isolated, with diameters less than 1000 km and fields of 12 kG (Muller 1994). There are two approaches to modelling the non-linear interaction between convection and magnetic fields at the surface of a star like the Sun (Weiss 1997). The first attempts to simulate photospheric magnetoconvection in as much detail as possible; this approach was pioneered by Nordlund (1984), using the anelastic approximation, and later carried through to a fully compressible calculation that demonstrates the formation of the intergranular magnetic network (Nordlund & Stein 1989, 1990). The dynamical evolution of an isolated flux element has also been simulated in the same style, though so far only in a more restricted two-dimensional (2D) geometry (Steiner, Kno lker & Schu ssler 1994; Steiner et al. 1996, 1998). The second approach is less ambitious but more systematic: it relies on idealized models, where different physical processes can be isolated and the key parameters can be varied. The combined effects of stratification and compressibility were first studied in a 2D model which showed that convection in a slot should indeed exhibit spatially modulated oscillations (Weiss et al. 1990). Subsequently, three-dimensional (3D) computations revealed the changes that occur in both the scale and the pattern of convection as the overall field strength is varied (Weiss et al. 1996). At any stage such calculations are limited by the computing power that is available; in practice this means that turbulent motion cannot be faithfully represented, and that the normalized width of the computational box (its aspect ratio) is limited. Experience shows that the pattern of convection may be drastically altered if the aspect ratio is too small (Weiss et al. 1996). Unfortunately, these idealized 2D and 3D models have not so far reproduced the range of scales that is now known to exist for umbral dots. This discrepancy might be ascribed to various causes. First of all, the aspect ratio may be too narrow to allow a sufficient range of variation. Secondly, the choice of boundary conditions for the idealized models may be inappropriate. In these models the lower boundary is closed and kept at a fixed temperature, with a magnetic field that may vary but is constrained to be vertical. It is known that the return flow has a significant effect (Nordlund, Galsgaard & Stein 1994). Furthermore, the same conditions were applied at the upper bounda (...truncated)