The talk presents the current status of our understanding of the
geometrical structure of compressible convection in the Sun and stars.
In the simplest case of nonmagnetic convection in a chemically
homogeneous layer this problem was basically solved 20 years ago and
recent work only focused on a better understanding of intermittency and
fine structures. In the understanding of semiconvection a breakthrough
has recently been made by H.C. Spruit et al. in Garching. The study of
magnetoconvection is the subject of ongoing research, realistic
simulations being developed in the MPI Lindau while idealized
experimental setups have been studied by Weiss et al. in Cambridge and
recently by our group in Budapest. A common feature of semiconvection
and magnetoconvection is the importance of subcritical onset of
convective transport, making considerations based on linear stability
theory irrelevant for the nonlinear problem. Our most recent
simulations, however, indicate that in the strong field case this
subcritical mode has a structure corresponding more closely to the
so-called "convectons" of nonlinear analytic theory than to the
structure previously found by Weiss et al. The relation of small scale
structure observed in sunspots, such as umbral dots and light bridges,
to the structures seen in the simulations is discussed.
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