3.5 Idealized magneto-convection simulations

Magneto-convection has been investigated in idealized setups for several decades and have been reviewed by Proctor and Weiss (1982Jump To The Next Citation Point), Hurlburt et al. (2000Jump To The Next Citation Point), Schüssler (2001) and Weiss (2002). Apart from the Rayleigh number characterizing the degree of convective stability, these studies typically focus on how magneto-convection patterns change with imposed field strength (typically expressed through the Chandrasekhar number), the ratio of magnetic to thermal diffusivity ζ = η∕κ as well as the field inclination angle. Early studies summarized by Proctor and Weiss (1982) were based on the Boussinesq approximation, followed by 2D compressible studies (Hurlburt and Toomre, 1988; Weiss et al., 1990Jump To The Next Citation Point). Here, it was found that for ζ < 1 convection in strong field regions is oscillatory, while steady overturning motions are present for ζ > 1. It has been conjectured by Weiss et al. (1990) that umbral dots can be explained through oscillatory magneto-convection in the sub-photospheric layers (ζ < 1 is realized in the upper most 2 Mm of a sunspot umbra). Matthews et al. (1995) and Weiss et al. (1996) expanded this work to 3D where convection takes place in a lattice of pulsating dots.UpdateJump To The Next Update Information The regime of moderately strong field was studied by Tao et al. (1998), here magnetic field separates from convective motions (flux-separation), which is realized in granulation and plage regions. Weiss et al. (2002) presented 3D studies of flux separation in photospheric convection, varying the field strength while using a fixed ζ profile that varies (bottom to top) between 2.2 to 0.2. Based on this work it was also suggested that in certain regions of the umbra an intermediate regime with flux separation on small scales is realized.

Hurlburt et al. (1996) investigated 2D magneto convection in inclined field. Here oscillatory convection transitions to traveling waves that can lead to both pattern motion and average horizontal flows near the top boundary. Hurlburt et al. (2000) presented the corresponding 3D traveling wave pattern for different inclination angles. They found convection cells with a pattern motion toward the umbra, while fluid is rapidly moving outward in the wake of the traveling convection cells. It has been speculated by the authors that several aspects of penumbral structure and flows are represented by traveling wave magneto-convection.

While idealized simulations point toward oscillatory and traveling wave like convection under the condition ζ < 1, which is realized about 2 Mm beneath the photosphere, MHD simulations with radiative transfer and a realistic equation of state (described in Section 3.6.2) show the immediate transition to overturning convection in umbra as well as penumbra. To our knowledge it has not been thoroughly studied which additional ingredient (radiative transfer, partial ionization, location of photospheric boundary away from domain boundary allowing for convective overshoot) is responsible for the change of behavior compared to the idealized models summarized above.

Recent magneto-convection studies by Thomas et al. (2002a,b), Weiss et al. (2004Jump To The Next Citation Point), and Brummell et al. (2008Jump To The Next Citation Point) focused on the role of turbulent magnetic pumping for the formation and maintenance of a sunspot penumbra. Overall, pumping was found to be very efficient in the idealized setups to hold down magnetic field lines near the outer edge of the penumbra and it was conjectured that this process together with a convective fluting instability is responsible for the formation of penumbrae as well as the Evershed flow in terms of a siphon flow in the overarching flux loops resulting from this process.

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