  The vertical upwelling/diffusion model (VUDM) has historically played a key
role in shaping our ideas about how the heat balance is achieved in the ocean.
Its has been and is still widely used in many applications ranging from the
estimation of transfer coefficients to the parameterisation of ocean heat
uptake in Simple Climate Models (SCMs). Its conceptual value as a realistic
theoretical model of the ocean heat balance has become increasingly unclear
over the years however, because: 1) the different ways in which upwelling has
been linked to high-latitude deep water formation and downgradient diffusion
linked to vertical/diapycnal mixing have remained imprecise and somewhat ad-hoc
so far; 2) other effects such as isopycnal mixing, density-compensated
temperature anomalies, meso-scale eddy-induced advection and the depth-varying
ocean area have all be demonstrated to affect actual ocean heat uptake as well,
but their incorporation into existing VUDM frameworks has been problematic. In
this paper, a new process-based vertical advection/diffusion theoretical model
of ocean heat uptake is constructed that resolve all above difficulties. This
new model is obtained by coarse-graining the full three-dimensional
advection/diffusion for potential temperature carried by ocean climate models,
by using the same isopycnal analysis as in the theory of ocean water masses.
The resulting model describes the temporal evolution of the
isopycnally-averaged thickness-weighted potential temperature in terms of an
effective velocity that depends uniquely on the surface heating conditionally
integrated in density classes, an effective diapycnal diffusivity controlled by
isoneutral and dianeutral mixing, and an additional term linked to the
meridional transport of density-compensated temperature anomalies by the
diabatic residual overturning circulation.
