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We analyse a sunspot simulation in an effort to understand the origin of theconvective instabilities giving rise to the penumbral and umbral distinctregimes. We applied the criterion from Gough \& Tayler (1966), accounting forthe stabilising effect of the vertical magnetic field to investigate theconvective instabilities in a MURaM sunspot simulation. We find: (1) a highlyunstable shallow layer right beneath the surface extending all over thesimulation box in which convection is triggered by radiative cooling in thephotosphere; (2) a deep umbral core (beneath -5 Mm) stabilised againstoverturning convection that underlies a region with stable background valuespermeated by slender instabilities coupled to umbral dots; (3) filamentaryinstabilities below the penumbra nearly parallel to the surface and undulatinginstabilities coupled to the penumbra which originate in the deep layers. Thesedeep-rooted instabilities result in the vigorous magneto-convection regimecharacteristic 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 morevigorous than the penumbra convection regime which, in turn, is more vigorousthan the close-to-steady umbra; (5) the GT criterion outlines both the sunspotmagnetopause and peripatopause, highlighting the tripartite nature of thesub-photospheric layers of magnetohydrodynamic (MHD) sunspot models; and,finally, (6) the Jur\v{c}\'ak criterion is the photospheric counterpart of theGT criterion in deep layers. The GT criterion as a diagnostic tool reveals thetripartite nature of sunspot structure with distinct regimes ofmagneto-convection in the umbra, penumbra, and granulation operating inrealistic MHD simulations.

Characterization of magneto-convection in sunspots -- The Gough & Tayler stability criterion in MURaM sunspot simulations