1 Introduction

The solar convection zone is the ultimate driver of activity in the solar chromosphere and corona. It is the only available source of mechanical energy. Upper atmosphere activity and heating is empirically intimately connected with the presence of magnetic fields. Hence the need to understand the behavior of magnetic fields at the solar surface. The solar magnetic field is produced by dynamo action within the convection zone. Thus, to understand the energy source of the chromosphere and corona we need to understand the solar dynamo, magneto-convection, and the transport of magnetic flux through the convection zone. For recent reviews see Fan (2009Jump To The Next Citation Point) and Charbonneau (2010Jump To The Next Citation Point). For probing the subsurface layers of the Sun, our best tools are the various techniques of local helioseismology (Gizon and Birch, 2005). Accurate modeling of the rise through the convection zone and emergence of magnetic flux, of sunspots and active regions is needed for improving helioseismic probing of solar subsurface structure.

Convection is the transport of energy by bulk mass motions. In a convection zone, energy is transported as thermal energy, except in layers where hydrogen is only partially ionized, in which case most of the energy is transported as ionization energy. Typically, the motions are slow compared to the sound speed so that approximate horizontal pressure balance is maintained. As a result, warmer fluid is less dense and buoyant while cooler fluid is denser and gets pulled down by gravity. For a detailed review of solar surface convection see Nordlund et al. (2009Jump To The Next Citation Point).

The topology of convection is controlled by mass conservation (Stein and Nordlund, 1989). Convection has a horizontal cellular pattern, with the warm fluid ascending in separate fountain-like cells surrounded by lanes of cool descending fluid. In a stratified atmosphere, with density decreasing outward, most of the ascending fluid must turn over and be entrained in the downflows within a density scale height (ignoring gradients in velocity and filling factor). Fluid moving a distance Δr in an atmosphere with a density gradient dln ρ∕dr would, if its density remained constant, be overdense compared to its surroundings by a factor Δ ρ∕ρ = − (d ln ρ∕dr)Δr. This is unstable and produces a pressure excess in the upflow cell interiors that pushes the fluid to turn over into the surrounding downflow lanes. Since the fluid velocity decreases inward from the top of the convection zone, its derivative has opposite sign to that of the density, so the length scale for entrainment is increased. Warm upflows diverge and tend to be laminar, while cool downflows converge and tend to be turbulent. Temperature in stellar convection zones increases inward, so the scale height and, as a result, the size of the horizontal convective cellular pattern also increase inward. Think of the rising fluid as a cylinder. As described above, most of the fluid entering at the bottom of the cylinder must leave through its sides within a scale height. If the ratio of vertical to horizontal velocities does not change much with depth, then the radius of the cylinder can increase in proportion to the scale height and still maintain mass conservation (Stein and Nordlund, 1998Jump To The Next Citation Point).

The solar surface is covered with magnetic features with spatial sizes ranging from unobservably small to hundreds of megameters. Their distribution is featureless (Parnell et al., 2009Jump To The Next Citation Point; Thornton and Parnell, 2011Jump To The Next Citation Point). Large-scale magnetic structures, sunspots and active regions, possess some well defined global properties (Hathaway, 2010Jump To The Next Citation Point). The main observed properties of small scale magnetic structures are (de Wijn et al., 2009Jump To The Next Citation Point): Strong fields tend to be vertical and weaker fields horizontal. The strongest vertical fields are in pressure equilibrium with their surroundings and tend to occur in the magnetic network and the intergranular lanes. Horizontal fields are found predominantly inside granules and near the edges of granules. Horizontal field properties are similar in the quiet Sun, plage, and polar regions (Ishikawa and Tsuneta, 2009Jump To The Next Citation Point). Three orders of magnitude more magnetic flux emerges in the quiet Sun than emerges in active regions (Thornton and Parnell, 2011Jump To The Next Citation Point). This new flux is first seen as horizontal field inside granules followed by the appearance of vertical field at the ends of the horizontal field (Centeno et al., 2007Jump To The Next Citation Point; Martínez González and Bellot Rubio, 2009Jump To The Next Citation Point; Ishikawa et al., 2010Jump To The Next Citation Point; Guglielmino et al., 2012Jump To The Next Citation Point).

In the presence of magnetic fields, convection is altered by the Lorentz force, while convection influences the magnetic field via the curl(v × B ) term in the induction equation.

Where the conductivity high, the magnetic field is frozen into the ionized plasma. Where the magnetic field is weak (magnetic energy small compared to kinetic energy), convective motions drag it around. To maintain force balance, locations of higher field strength (higher magnetic pressure) tend to have smaller plasma density and lower gas pressure. Diverging, overturning motions quickly sweep the field (on granular times of minutes) from the granules into the intergranular lanes (Tao et al., 1998aJump To The Next Citation Point; Emonet and Cattaneo, 2001Jump To The Next Citation Point; Weiss et al., 2002Jump To The Next Citation Point; Stein and Nordlund, 2004Jump To The Next Citation Point; Vögler et al., 2005Jump To The Next Citation Point; Stein and Nordlund, 2006Jump To The Next Citation Point). In hours (mesogranular times), the field tends to collect on a mesogranule scale. In days (supergranule times), the slower, large scale supergranule motions collect the field in the magnetic network at the supergranule boundaries. Convective flows produce a hierarchy of loop structures in rising magnetic flux. Slow upflows and buoyancy raise the flux, while fast downflows pin it down, which produces Ω- and U-loops (Cheung et al., 2007Jump To The Next Citation Point). The different scales of convective motion produce loops on these different scales, with smaller loops riding piggy-back in a serpentine fashion on the larger loops (Cheung et al., 2007Jump To The Next Citation Point; Stein et al., 2010bJump To The Next Citation Point). Dynamo action occurs in the turbulent downflows where the magnetic field lines are stretched, twisted, and reconnected, increasing the field strength (Nordlund et al., 1992Jump To The Next Citation Point; Cattaneo, 1999Jump To The Next Citation Point; Vögler and Schüssler, 2007Jump To The Next Citation Point; Schüssler and Vögler, 2008Jump To The Next Citation Point; Pietarila Graham et al., 2010Jump To The Next Citation Point).

Magnetic fields influence convection via the Lorentz force, which inhibits motion perpendicular to the field. As a result, the overturning motions that are essential for convection are suppressed and convective energy transport from the interior to the surface is reduced. Radiative energy loss to space continues, so regions of strong field cool relative to their surroundings. Being cooler, these locations have a smaller scale height. Plasma drains out of the magnetic field concentrations in a process called “convective intensification” or “convective collapse” (Parker, 1978Jump To The Next Citation Point; Spruit, 1979Jump To The Next Citation Point; Unno and Ando, 1979Jump To The Next Citation Point; Nordlund, 1986Jump To The Next Citation Point; Bercik et al., 1998Jump To The Next Citation Point; Grossmann-Doerth et al., 1998Jump To The Next Citation Point; Bushby et al., 2008Jump To The Next Citation Point). This process can continue until the magnetic pressure (plus a small gas pressure) inside the flux concentration equals the gas pressure outside, giving rise to a field strength much greater than the equipartition value with the dynamic pressure of the convective motions. These magnetic flux concentrations are cooler than their surroundings at the same geometric layer. However, because they are evacuated, their opacity is reduced so photons escape from deeper in the atmosphere (Wilson depression, Maltby, 2000Jump To The Next Citation Point). Where the magnetic concentrations are narrow, there is heating from their hotter side walls and they appear as bright points (Spruit, 1976Jump To The Next Citation Point). Where the concentrations are wide, the side wall heating is not significant and the flux concentrations appear darker than the surroundings as pores or sunspots.

Magnetic fields alter granules’ properties – producing smaller, elongated, lower intensity contrast, “abnormal” granules (Muller et al., 1989; Title et al., 1992; Bercik et al., 1998Jump To The Next Citation Point; Vögler, 2005Jump To The Next Citation Point; Cheung et al., 2007Jump To The Next Citation Point). Strong magnetic flux concentrations typically form in convective downflow lanes, especially at the vertices of several such lanes, due to the sweeping of flux by the diverging convective upflows (Vögler et al., 2005Jump To The Next Citation Point; Stein and Nordlund, 2006Jump To The Next Citation Point). They are surrounded by downflows which sometimes become supersonic.

Magneto-convection simulations have been very useful in understanding and interpreting observations. Sánchez Almeida et al. (2003), Khomenko et al. (2005), Shelyag et al. (2007), and Bello González et al. (2009) have used simulations to calibrate the procedures for analyzing and interpreting Stokes spectra in order to determine the solar vector magnetic field. Fabbian et al. (2010) has shown that magnetic fields alter line equivalent widths by altering the temperature stratification and by Zeeman broadening. These two effects act in opposite directions, but still leave a net result and hence alter abundance determinations. Zhao et al. (2007), Braun et al. (2007), Kitiashvili et al. (2009), Birch et al. (2010), and Braun et al. (2012) have used convection and magneto-convection simulation results to analyze local helioseismic inversion methods.

Magneto-convection is highly non-linear and non-local, so it needs to be modeled using numerical simulations. Two complementary approaches are being used to study magneto-convection, which we will call “idealized” and “realistic”. “Idealized” studies ignore complex physics by assuming a fully ionized, ideal plasma and energy transport by thermal conduction. Magneto-convection in the deep, slow moving, adiabatic portion of the convection zone satisfies these idealized assumptions and, in addition, can use the anelastic approximation whereby acoustic waves are eliminated from the calculation, which permits larger time steps. “Idealized” calculations are important for isolating and studying fundamental physical phenomena as well as for exploring parameter space (because they run fast). “Realistic” studies include complex physics – an equation of state for partially ionized gas, non-grey radiation transport and, in some cases, even some non-equilibrium effects. “Realistic” calculations are necessary to make quantitative comparisons with observations in order to understand the observations and to provide artificial data for evaluating data analysis procedures. In this review we focus on the “realistic” numerical modeling of solar surface magneto-convection. It updates and extends the section on magneto-convection from the review by Nordlund et al. (2009Jump To The Next Citation Point) of solar surface convection.

It is organized as follows: Section 2 states the equations that need to be solved. Section 3 describes the solar observations. Section 4 describes the simulation results for: dynamo action (4.1), flux emergence (4.2), flux concentrations (4.3), and pores and sunspots (4.4).

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