This paper describes a series of three-dimensional simulations of shallow inefficient convection in the outer layers of the Sun.The computational domain is a closed box containing the convection—radiation transition layer, located at the top of the solar convection zone.The most salient features of the simulations are that: (i) the position of the lower boundary can have a major effect on the characteristics of solar surface convection (thermal structure, kinetic energy and turbulent pressure); (ii) the width of the box has only a minor effect on the thermal structure, but a more significant effect on the dynamics (rms velocities); (iii) between the surface and a depth of 1 Mm, even though the density and pressure increase by an order of magnitude, the vertical correlation length of vertical velocity is always close to 600 km; (iv) in this region the vertical velocity cannot be scaled by the pressure or the density scaleheight; this casts doubt on the applicability of the mixing length theory, not only in the superadiabatic layer, but also in the adjacent underlying layers; (v) the final statistically steady state is not strictly dependent on the initial atmospheric stratification.