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 , 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. (2004), and Brummell et al. (2008) 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.
Living Rev. Solar Phys. 8, (2011), 3
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