Magnetic flux separation in photospheric convection

N. O. Weiss 0 M. R. E. Proctor 0 D. P. Brownjohn 0 0 Department of Applied Mathematics and Theoretical Physics, University of Cambridge , Cambridge CB3 9EW A B S T R A C T Three-dimensional non-linear magnetoconvection in a strongly stratified compressible layer exhibits different patterns as the strength of the imposed magnetic field is reduced. There is a transition from a magnetically dominated regime, with small-scale convection in slender hexagonal cells, to a convectively dominated regime, with clusters of broad rising plumes that confine the magnetic flux to narrow lanes where fields are locally intense. Both patterns can coexist for intermediate field strengths, giving rise to flux separation: clumps of vigorously convecting plumes, from which magnetic flux has been excluded, are segregated from regions with strong fields and small-scale convection. A systematic numerical investigation of these different states shows that flux separation can occur over a significant parameter range and that there is also hysteresis. The results are related to the fine structure of magnetic fields in sunspots and in the quiet Sun. I N T R O D U C T I O N It is only recently that high-resolution observations, whether from space or from the ground, have been able to reveal the fine structure of magnetic fields at the solar surface (Title 2000). Within sunspot umbrae, where the strongest fields are found, there are small bright features, the diameters of which range down to the limit set by resolution (Sobotka, Brandt & Simon 1997a). Outside active regions, in the quiet Sun, magnetic fields are confined to the network of cooler sinking material that encloses the bright rising granules (Lin & Rimmele 1999). Where these fields are locally intense they correspond to bright points in CH G-band emission, which move rapidly within the narrow intergranular lanes (Berger & Title 1996). In the atmosphere above, the slender flux loops that have been revealed by the Transition Region and Coronal Explorer (TRACE) have comparable cross-sections. At the same time, the advent of powerful supercomputers has made it possible to model three-dimensional magnetoconvection in a compressible layer, and to compare the results of numerical experiments with structures that are actually observed in the Sun (and must be present in other magnetically active stars). The gas at the solar photosphere is highly conducting and so magnetic fields tend to move with the fluid. Where rising plumes impinge on the stably stratified atmosphere above, the fluid moves horizontally outwards, carrying magnetic flux into the network, where it accumulates preferentially at the nodes. In simple models of cellular convection, flux expulsion allows the field to be concentrated into isolated sheets or tubes from which the overturning motion is excluded (Proctor & Weiss 1982). By segregating magnetic fields (which impede convection) from the motion it becomes possible for convection to set in subcritically: in the most extreme cases, vigorous isolated eddies (or convectons) appear against a background of stationary or feebly convecting fluid (Blanchflower 1999; Blanchflower & Weiss 2002). Flux expulsion becomes less likely in fully turbulent convection, although a turbulent stellar convection zone can still pump magnetic flux downwards into the underlying radiative interior (Tobias et al. 1998). In a compressible layer that is strongly superadiabatic there is a sharp contrast between the broad convective plumes that appear when there is no magnetic field and the narrow cells that are formed when the field is strong enough for the Lorentz force to dominate the motion. Thus there is a competition between two different scales (or phases) of convection. Flux separation occurs when these two phases coexist and are separated by fronts. This is illustrated in Fig. 1, which shows a snapshot of the magnetic field strength and the vertical temperature gradient at the top of a convecting layer in the presence of an externally imposed magnetic field. There are several clusters of broad and vigorously convecting plumes from which the field has been excluded. Each cluster is surrounded by strong magnetic fields that only allow weaker, small-scale convection to occur. Similar patterns, with coherent structures, arise in other complex systems (for example, when the two phases correspond to magnetized and unmagnetized states). In this paper we present a systematic study of flux separation in a strongly stratified layer. It is apparent from Fig. 1 that this effect can only appear in a numerical experiment if the computational box is sufficiently wide. Flux separation was first recognized by Tao et al. (1998) in a box with aspect ratio (normalized width) = 8. Earlier work on three-dimensional compressible magnetoconvection had been restricted (for reasons of economy) to narrow boxes with 83 (Weiss et al. 1996). The changes produced as the aspect ratio is increased have since been investigated in both two dimensions (Blanchflower, Rucklidge & Weiss 1998) and three (Rucklidge et al. 2000). The aspect ratios we adopt here are large enough to ensure that the patterns found do not depend on the finite size of the computational box. Nordlund & Stein (1989) have simulated non-linear magnetoconvection in a realistic model of the solar atmosphere but most other three-dimensional calculations have been restricted to the Boussinesq approximation. The transition from small-scale dynamo action (Cattaneo 1999) to magnetoconvection has been systematically explored (Emonet, Cattaneo & Weiss 2001) but it seems that flux separation is a less robust feature of incompressible convection. In the next section we describe the model system that will be investigated, commenting on the choice of boundary conditions and on linear behaviour. The results of our numerical experiments are contained in Section 3, where we distinguish between small-scale convection (in a magnetically dominated regime) and flux separation (in a regime with moderate field strengths). There is also an intermediate regime with hysteresis. When the imposed flux density is weak, convection concentrates magnetic flux into lanes that enclose clusters of plumes, where the fields are locally intense. These different patterns of non-linear magnetoconvection are analysed in Section 4 and the qualitative features of our results are discussed in Section 5. In the final section, we relate these numerical experiments to the fine structure that is associated with magnetoconvection at the solar surface. Our results have been summarized in a brief preliminary survey (Weiss & Proctor 2001) and some colour images were reproduced by Weiss (2001). Readers are urged to view the time-dependent evolution of these patterns in the movies which are available on the website http://www.damtp.cam.ac.uk/user/afdnld/movies. T H E M O D E L P R O B L E M We shall investigate fully compressible three-dimensional convectio (...truncated)