4.2 Flux emergence in upper convection zone

Models for the flux emergence process in the upper most layers of the convection zone require fully compressible MHD since flows are approaching and exceeding the speed of sound in the photosphere. In addition, fully realistic simulations have to take into account partial ionization in the equation of state and radiative energy transport in the photospheric layers. Over the past few decades, realistic MHD simulations on granular scale evolved to a degree that allows a detailed comparison with high resolution observations (see review by Nordlund et al., 2009, and further references therein). Realistic MHD simulations of sunspot formation are much more demanding due to the substantially longer times and length scales inherent to this problem. While there has been in recent years a large body of idealized 3D MHD simulations addressing the emergence of flux from the photosphere into the corona, only very recently MHD simulations with radiative transfer and a realistic equation of state are utilized to address the last stages of flux emergence in the sub-photospheric layers. Cheung et al. (2007) considered the emergence of small flux concentrations (1018 – 1019 Mx) in the upper most 2 Mm of the convection zone, and Cheung et al. (2008) studied the flux emergence of (1020 Mx) tubes in the uppermost 5 Mm of the convection zone.Update While all these cases show signatures in the photosphere that are in agreement with observations, such as highly distorted granulation, formation of supersonic downflows, separation of mixed polarities, they were not yet able to produce “pore-like” flux concentrations. In part this was due to the amount of flux considered, in part it was due to time-step and stability constraints imposed once the flux reached the photosphere and layers above, which did not allow running these simulations sufficiently long. The formation of a strong flux concentration was finally achieved by Cheung et al. (2010Jump To The Next Citation Point) building on code improvements introduced by Rempel et al. (2009aJump To The Next Citation Point,bJump To The Next Citation Point) and utilizing a new bottom boundary condition that accounts for a half-torus shaped flux tube entering the domain from beneath following Fan and Gibson (2003). In this case the rise of a twisted flux tube with 7 × 1021 Mx flux in the upper most 7.5 Mm of the convection zone lead to the formation of 2 pores, showing fine structure in form of umbral dots as well as light bridges (see Figure 14Watch/download Movie). All these simulations clearly show a strong influence of the near surface convection on the flux emergence process, leading to the formation of small scale loop like structures that emerge on a granular scale.

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Figure 14: mov-Movie (46891 KB) Results from a flux emergence simulation in a 92 × 49 × 8 Mm3 sized simulation domain. Top: Magnetogram at τRoss = 0.1. Bottom: Magnetic field strength on vertical cut along the indicated red dashed line. The presented snapshot at 15.3 hours after initialization of the simulation run shows the formation of a pair of opposite polarity spots with about 3 × 1021 Mx flux. This figure is reproduced from Cheung et al. (2010Jump To The Next Citation Point) by permission of the AAS.

A complementary approach was recently introduced by Stein et al. (2011a). Instead of prescribing initially a concentrated flux tube they emerged horizontal magnetic field across the bottom boundary in inflow regions in a 20 Mm deep domain. Experiments with different initial field strength revealed that only runs with less than 20 kG field strength at 20 Mm depth lead to flux emergence signatures in agreement with observational constraints. In this setup the overall amount of flux reaching the photosphere is too small to form an active region. Recently, Stein et al. (2011bJump To The Next Citation Point) reported on an experiment in which the magnetic field was in addition scaled up everywhere in the domain proportional to B on a time scale of 30 min to produce larger flux concentrations leading to a complex active region.

Expanding the simulation domains to also include chromospheric layers and the lower corona is computationally very expensive. As a consequence many emergence simulations do not include convection and focus entirely on the buoyant rise. Recently, Abbett (2007) developed a compressible MHD code in which the radiative energy transport is approximated through a combination of radiative diffusion beneath and cooling functions above the photosphere. This simplification allows to cover the full range from the upper convection zone into the corona and allows to simulate flux emergence and interaction with convective motions. A flux emergence simulation with realistic photospheric convection but simplified thermodynamics in the chromosphere was presented by Tortosa-Andreu and Moreno-Insertis (2009). In contrast to this Martínez-Sykora et al. (2008, 2009) focus on detailed physics of time dependent ionization and non-LTE radiative transfer. While the former approaches are more efficient to address large active region scale problems, the latter is required for in-depth comparison with observations.

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