  Structures seen in idealized numerical experiments on compressible
magnetoconvection in an imposed strong vertical magnetic field show important
differences from those detected in observations or realistic numerical
simulations of sunspot umbrae. To elucidate the origin of these discrepancies,
we present a series of idealized 3D compressible magnetoconvection experiments
that differ from previous such experiments in several details, bringing them
closer to realistic solar conditions. An initially vertical magnetic field B 0
is imposed on a time snapshot of fully developed solar-like turbulent
convection in a layer bounded by a stable layer from above. Upon relaxation to
a statistically steady state, the structure of the flow field and magnetic
field is examined. Instead of the vigorous granular convection (GRC) well known
to take place in magnetized or weakly magnetized convection, for high values of
B0 heat is transported by small-scale convection (SSC) in the form of narrow,
persistent convective columns consisting of slender upflows accompanied by
adjacent downflow patches, which are reminiscent of the 'convectons' identified
in earlier semianalytic models. For moderate field strengths, flux separation
(FXS) is observed: isolated field-free inclusions of GRC are embedded in a
strongly magnetized plasma with SSC. Between the SSC and FXS regimes, a
transitional regime (F/S) is identified where convectons dynamically evolve
into multiply segmented granular inclusions and back. Our results agree in some
aspects more closely with observed umbral structures than earlier idealized
models, because they do reproduce the strong localized, patchy downflows
immediately adjacent to the narrow convective columns. Based on recent
observations of umbral dots, we suggest that in some cases the conditions in
sunspot umbrae correspond to the newly identified F/S transitional regime.
