Localized plumes in three-dimensional compressible magnetoconvection

Mon. Not. R. Astron. Soc. 412, 555–560 (2011) doi:10.1111/j.1365-2966.2010.17926.x Localized plumes in three-dimensional compressible magnetoconvection S. M. Houghton1 and P. J. Bushby2 1 School 2 School of Mathematics, University of Leeds, Leeds LS2 9JT of Mathematics & Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU Accepted 2010 October 27. Received 2010 September 29; in original form 2010 July 13 ABSTRACT Key words: convection – magnetic fields – MHD – sunspots – Sun: surface magnetism. 1 I N T RO D U C T I O N Using modern instruments, such as the Solar Optical Telescope onboard Hinode and the Swedish 1-m Solar Telescope on La Palma, it is possible to make detailed observations of magnetic fields and convection at the surface of the Sun. Sunspots are the most prominent magnetic features on the solar surface. A typical sunspot consists of a central umbral region, surrounded by a complex filamentary penumbra. Umbral regions appear dark because their surface temperatures are (typically) only 70–85 per cent of the mean surface temperature of the non-magnetic photosphere (see, for example, Thomas & Weiss 2008). This reduction in temperature is due to the fact that the convective transport of heat is impeded within sunspot umbrae by the presence of strong, near-vertical magnetic fields (which can often exceed 3000 G). Detailed observations of sunspot umbrae have shown that they are not uniformly dark. In almost all sunspots, bright point-like structures can be observed – these are known as umbral dots (Danielson 1964). These bright features are warmer than their immediate surroundings, but are (generally) cooler than the surrounding photosphere (see, for example, Sobotka & Hanslmeier 2005; Kitai et al. 2007). It is difficult to determine the characteristic size of an umbral dot, although these features are always small compared to the umbral diameter. In a recent study, Kitai et al. (2007) found that the umbral dots in one particular sunspot had typical diameters of approximately 220–350 km, although a significant number appeared to be much smaller than this (possibly below the resolution limit for the Solar Optical Telescope on Hinode). Umbral dots are also  E-mail: (SMH); (PJB)  C 2010 The Authors C 2010 RAS Monthly Notices of the Royal Astronomical Society  short-lived features. Kitai et al. (2007) found that most of the umbral dots in their survey had lifetimes of between 5 and 20 min. In an earlier study, Sobotka, Brandt & Simon (1997) found a much broader range of lifetimes for umbral dots (with a small percentage lasting longer than 2 h), although, like Kitai et al. (2007), they found a mean lifetime of approximately 15 min. Most umbral dots exhibit no systematic proper motions. However, those that appear to form at the umbral/penumbral boundary (which are often associated with penumbral grains) tend to migrate radially inwards towards the centre of the umbra (Sobotka et al. 1995; Kitai et al. 2007). There is some observational evidence for weak upflows within umbral dots (Socas-Navarro et al. 2004; Bharti, Jain & Jaaffrey 2007) as well as downflows around their edges (Bharti, Jain & Jaaffrey 2007; Ortiz, Rubio & van der Voort 2010). Clearly, the observations indicate that umbral dots correspond to convective plumes within sunspot umbrae. Further theoretical support for this conclusion comes from the work of Deinzer (1965), who determined that convective motions must be present within the umbra, as radiative processes alone could not transport sufficient energy to the surface. Theoretical studies of umbral convection tend to be based upon local models of magnetoconvection in a Cartesian domain. It is well known that a strong vertical magnetic field tends to inhibit convective motions in an electrically conducting fluid (Chandrasekhar 1961). When the dynamics are dominated by magnetic fields, convection takes the form of weak, narrow plumes. In an idealized model of magnetoconvection, Weiss, Proctor & Brownjohn (2002) found a steady, almost hexagonal pattern of convection in the magnetically dominated regime. More recently, Schüssler & Vöogler (2006; see also Bharti, Beeck & Schüssler 2010) have carried out a more realistic set of calculations, including the effects of partial ionization and radiative transfer. These simulations produced a Within the umbrae of sunspots, convection is generally inhibited by the presence of strong vertical magnetic fields. However, convection is not completely suppressed in these regions: bright features, known as umbral dots, are probably associated with weak, isolated convective plumes. Motivated by observations of umbral dots, we carry out numerical simulations of three-dimensional, compressible magnetoconvection. By following solution branches into the subcritical parameter regime (a region of parameter space in which the static solution is linearly stable to convective perturbations), we find that it is possible to generate a solution which is characterized by a single, isolated convective plume. This solution is analogous to the steady magnetohydrodynamic convectons that have previously been found in two-dimensional calculations. These results can be related, in a qualitative sense, to observations of umbral dots. 556 S. M. Houghton and P. J. Bushby 2 P RO B L E M D E S C R I P T I O N A N D S E T- U P We consider the evolution of a layer of compressible, electrically conducting fluid, heated from below, in the presence of an imposed magnetic field. Various properties of the fluid, including the thermal conductivity, K, the shear viscosity, μ, the magnetic diffusivity, η, the magnetic permeability, μ0 , and the specific heat capacities at constant pressure and density (cP and cV , respectively) are assumed to be constant. At a position x and time t, we define ρ(x, t), T(x, t) and u(x, t) to be the fluid density, temperature and velocity field (respectively), whilst B(x, t) represents the magnetic field. This fluid occupies a three-dimensional Cartesian domain with 0 ≤ z ≤ d and 0 ≤ x, y ≤ 8d. The axes of this coordinate system are orientated so that the z-axis points vertically downwards, parallel to the constant gravitational acceleration, g = g ẑ. For this model problem, periodic boundary conditions are imposed in the x- and y-directions, whilst the upper and lower boundaries (at z = 0 and z = d) are assumed to be impermeable and stress free. Furthermore, fixed temperature boundary conditions are applied at the upper and lower boundaries with T = T 0 at z = 0 and T = T 0 + T at z = d (T > 0). It is also assumed that the horizontal components of any magnetic fields that are present vanish at z = 0 and z = d. When the layer is static, the imposed magnetic field is uniform and vertical, i.e. B = B0 ẑ. Before writing down the governing equations for this system, we can express these in non-dimensional form. More details of this procedure can be found in Matthews, Proctor & Weiss (199 (...truncated)