  Anomaly-diffusing energy balance models (AD-EBM) are routinely employed to
analyze and emulate the warming response of both observed and simulated Earth
systems. We demonstrate a deficiency in common multi-layer as well as
continuous-diffusion AD-EBM variants: They are unable to, simultaneously,
properly represent surface warming and the vertical distribution of heat
uptake. We show that this inability is due to the diffusion approximation. On
the other hand, it is well understood that transport of water from the surface
mixed layer into the ocean interior is achieved, in large part, by the process
of ventilation---a process associated with outcropping isopycnals. We,
therefore, start from a configuration of outcropping isopycnals and demonstrate
how an AD-EBM can be modified to include the effect of ventilation on ocean
uptake of anomalous radiative forcing. The resulting EBM is able to
successfully represent both surface warming and the vertical distribution of
heat uptake, and indeed, a simple four layer model suffices. The simplicity of
the models notwithstanding, the analysis presented and the necessity of the
modification highlight the role played by processes related to the down-welling
branch of global ocean circulation in shaping the vertical distribution of
ocean heat uptake.
