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Solar convection plays an important role in transporting energy, chemical elements, and magnetic fields, in maintaining differential rotation, and in the overall operation of the solar dynamo. Helioseismic techniques like ring-diagram and time-distance used to infer the subsurface structure and amplitudes of deep convection give conflicting results leading to the so-called convective conundrum. Hence, understanding the onset and driving mechanisms of deep solar convection is of crucial importance. In this work, we present 2D and 3D Cartesian simulations that explore the onset and setup of a deep solar convection zone self-consistently atop a radiative interior. We treat the radiative transfer using a Kramers-like opacity, along with a highly unstable near-surface layer, allowing for a self-consistent coupling between the solar convection zone and the radiative interior. We investigate the depth of the formed convection zone, the extent of overshooting convective motions, and the convective spectrum at various depths for a range of non-dimensional parameters. The findings of this study help us better understand the driving processes behind solar convection at the multitude of spatial and temporal scales in which it manifests itself.