  We analyse a sunspot simulation in an effort to understand the origin of the
convective instabilities giving rise to the penumbral and umbral distinct
regimes. We applied the criterion from Gough \& Tayler (1966), accounting for
the stabilising effect of the vertical magnetic field to investigate the
convective instabilities in a MURaM sunspot simulation. We find: (1) a highly
unstable shallow layer right beneath the surface extending all over the
simulation box in which convection is triggered by radiative cooling in the
photosphere; (2) a deep umbral core (beneath -5 Mm) stabilised against
overturning convection that underlies a region with stable background values
permeated by slender instabilities coupled to umbral dots; (3) filamentary
instabilities below the penumbra nearly parallel to the surface and undulating
instabilities coupled to the penumbra which originate in the deep layers. These
deep-rooted instabilities result in the vigorous magneto-convection regime
characteristic of the penumbra; (4) convective downdrafts in the granulation,
penumbra, and umbra develop at about 2 km/s, 1 km/s, and 0.1 km/s,
respectively, indicating that the granular regime of convection is more
vigorous than the penumbra convection regime which, in turn, is more vigorous
than the close-to-steady umbra; (5) the GT criterion outlines both the sunspot
magnetopause and peripatopause, highlighting the tripartite nature of the
sub-photospheric layers of magnetohydrodynamic (MHD) sunspot models; and,
finally, (6) the Jur\v{c}\'ak criterion is the photospheric counterpart of the
GT criterion in deep layers. The GT criterion as a diagnostic tool reveals the
tripartite nature of sunspot structure with distinct regimes of
magneto-convection in the umbra, penumbra, and granulation operating in
realistic MHD simulations.
