  We present numerical simulations of hydrodynamic overshooting convection in
local Cartesian domains. We find that a substantial fraction of the lower part
of the convection zone (CZ) is stably stratified according to the Schwarzschild
criterion while the enthalpy flux is outward directed. This occurs when the
heat conduction profile at the bottom of the CZ is smoothly varying, based
either on a Kramers-like opacity prescription as a function of temperature and
density or a static profile of a similar shape. We show that the subadiabatic
layer arises due to nonlocal energy transport by buoyantly driven downflows in
the upper parts of the CZ. Analysis of the force balance of the upflows and
downflows confirms that convection is driven by cooling at the surface. We find
that the commonly used prescription for the convective enthalpy flux being
proportional to the negative entropy gradient does not hold in the stably
stratified layers where the flux is positive. We demonstrate the existence of a
non-gradient contribution to the enthalpy flux, which is estimated to be
important throughout the convective layer. A quantitative analysis of downflows
indicates a transition from a tree-like structure where smaller downdrafts
merge into larger ones in the upper parts to a structure in the deeper parts
where a height-independent number of strong downdrafts persist. This change of
flow topology occurs when a substantial subadiabatic layer is present in the
lower part of the CZ.
