Photospheric convection in strong magnetic fields

1996MNRAS.283.1153W Mon. Not. R. Astron. Soc. 283, 1153-1164 (1996) Photospheric convection in strong magnetic fields N. O. Weiss, D. P. Brownjohn, P. C. Matthews* and M. R. E. Proctor Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW Accepted 1996 July 17. Received 1996 July 11; in original form 1996 March 4 ABSTRACT Key words: convection - MHD - Sun: granulation - Sun: magnetic fields - sunspots - stars: magnetic fields. 1 INTRODUCTION The magnetic field of a late-type star is generated by dynamo action deep in its convection zone. The detailed structure of the fields that are observed depends, however, upon their interaction with convection near the surface of the star. Th.e most prominent features are starspots. They are dark because the magnetic field is so strong that convective transport is substantially inhibited. At the other extreme, weak fields are transported passively: magnetic flux is swept to the boundaries of convection cells and moves along them to accumulate at nodes in an evolving network. Fields of intermediate strength alter the pattern of convection in the photosphere, an effect that can be observed by studying small-scale structures on the Sun. High-resolution solar observations show that, outside sunspots and pores, nearly all of the magnetic flux is confined to isolated sheets or tubes, with fields that are locally intense. At the photospheric level these features are almost completely evacuated and the field strength approaches the value Bp .. 1500 G for which the magnetic pressure is equal to the ambient gas pressure. The mean flux density in a plage region or the magnetic network is Bo = f B p ' where f is a local filling factor. As Bo and f increase, the Lorentz force becomes dynamically more powerful and the pattern of granular convection changes (ritle et al. 1992). For Bo < 150 G (f < 0.1) normal granulation is scarcely affected. The bright cores of granules, where hot gas is rising, have a spacing of around * Present address: Department of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD. 1.8 Mm and are enclosed by dark intergranular lanes, with downward motion, where magnetic fields are located. For 150 < Bo < 600 G (0.1 < f < 0.4) Title et al. find that the granulation is abnormal, with a spacing of only 1.1 Mm, while magnetic fields form a perforated network along which flux moves like a 'magnetic fluid'. Strong fields are associated with bright points in line emission and downward velocities. Only for Bo > 600 G are magnetic features dark, with a diminished downward flow. If enough magnetic flux accumulates, a dark pore is formed. Pores have diameters of 1.5-7.0 Mm and fields of around 2000 G. Within them are bright features that correspond to umbral dots in sunspots (Muller 1992; Bonet, Sobotka & Vazquez 1995). If the total flux 4> exceeds a critical value 4>c'" 7 TWb (or 7 x 1020 mx), the pore develops a penumbra and becomes a sunspot (Thomas & Weiss 1992); indeed, spots can form with diameters of only 3.6 Mm and fluxes of 2 TWb (Rucklidge, Schmidt & Weiss 1995). In the umbra of a sunspot the field is roughly vertical, with a strength of up to 3000 G. Within the umbra there are small bright points called umbral dots, visible against a dark background with weak fluctuations (Muller 1992). Umbral dots are present in all sunspots but large spots contain dark nuclei that are free ofthem (Maltby 1992; Muller 1992; Sobotka, Bonet & Vazquez 1993). There is, however, a distinction between peripheral umbral dots, which are related to bright features moving in from the penumbra, and central umbral dots, which are convective features (Sobotka et al. 1993, 1995). To understand these different patterns of magnetoconvection we need to probe beneath the surface of a star. Since observations only penetrate to a continuum optical depth TO.5 .. 10 and experiments © 1996RAS © Royal Astronomical Society • Provided by the NASA Astrophysics Data System The effect of magnetic fields on convection at the surfaces of cool stars can be explored by comparing the results of detailed numerical experiments with high-resolution solar observations. We have investigated non-linear three-dimensional magnetoconvection in a fully compressible perfect gas. In this paper we study the effect of an imposed magnetic field on the pattern of convection in a deep stratified layer. When the field is strong enough to dominate the motion we find steady convection with rising plumes on a deformed hexagonal lattice, and a magnetic network at the upper boundary. This gives way to spatially modulated oscillations for weaker fields. As the field strength is further reduced the oscillations become more violent and irregular, and their horizontal scale increases. Magnetic flux moves rapidly along the network that encloses the ephemeral plumes; when the imposed field is relatively weak, intense fields appear at junctions in the network, where the magnetic pressure is comparable to the gas pressure and an order of magnitude greater than the dynamic pressure. This behaviour is related to convection in sunspots and plages and to the structure of intergranular magnetic fields on the Sun. 1996MNRAS.283.1153W 1154 N. O. Weiss et al. 2 SETTING UP THE MODEL PROBLEM The system that we shall investigate is a straightforward threedimensional extension of the fully compressible two-dimensional configuration that was studied earlier (Weiss et al. 1990, which will be referred to as Paper I). Thus we take a layer of depth d containing a perfect monatomic gas, with fixed temperatures To and To + !l.T at its upper and lower boundaries respectively. The gas is electrically conducting and there is an imposed magnetic field such that the mean flux density corresponds to a uniform vertical field Bo. We assume that the z-axis points downwards, in the direction of the gravitational acceleration g. The origin is chosen so that z = Zo at the upper boundary, where Zo = Tod/!l.T, and we restrict attention to the region {O:s:x:s: M; O:s:y:s: M;Zo :s:z :S:Zo + d}; that is to say, we choose a box with square cross-section and aspect ratio A. This geometry naturally imposes constraints on the solutions that we can find. 2.1 The background atmosphere In the absence of any motion there is a uniformly stratified equilibrium solution, corresponding to a polytrope of index m = (gd/UT) - 1, where 1{ is the gas constant. Then the temperature T(z) and the density p(z) are given by T = !l.Tz/d, p = po(zlzo)m, (2.1) where Po = p(zo), and the superadiabatic gradient 1 7- 1 (2.2) (V - V.d) = (m+ 1) - - 7 - ' with 7 = 5/3. We assume that the thermal conductivity K, the electrical conductivity (/1{)'I/)-1, the shear viscosity p., the magnetic permeability /1{) and the heat capacity cp are all constant. Then the magnetic diffusivity '1/ is likewise constant but the thermal diffusivity K = K/cpp and the viscous diffusivity ,,= p./p bo (...truncated)