### 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 (1982), Hurlburt et al. (2000), 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 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., 1990). Here, it was found that for convection in strong
field regions is oscillatory, while steady overturning motions are present for . It has
been conjectured by Weiss et al. (1990) that umbral dots can be explained through oscillatory
magneto-convection in the sub-photospheric layers ( 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.Update 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 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 , 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.