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The vertical upwelling/diffusion model (VUDM) has historically played a keyrole 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 theestimation of transfer coefficients to the parameterisation of ocean heatuptake in Simple Climate Models (SCMs). Its conceptual value as a realistictheoretical model of the ocean heat balance has become increasingly unclearover the years however, because: 1) the different ways in which upwelling hasbeen linked to high-latitude deep water formation and downgradient diffusionlinked to vertical/diapycnal mixing have remained imprecise and somewhat ad-hocso far; 2) other effects such as isopycnal mixing, density-compensatedtemperature anomalies, meso-scale eddy-induced advection and the depth-varyingocean area have all be demonstrated to affect actual ocean heat uptake as well,but their incorporation into existing VUDM frameworks has been problematic. Inthis paper, a new process-based vertical advection/diffusion theoretical modelof ocean heat uptake is constructed that resolve all above difficulties. Thisnew model is obtained by coarse-graining the full three-dimensionaladvection/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 theisopycnally-averaged thickness-weighted potential temperature in terms of aneffective velocity that depends uniquely on the surface heating conditionallyintegrated in density classes, an effective diapycnal diffusivity controlled byisoneutral and dianeutral mixing, and an additional term linked to themeridional transport of density-compensated temperature anomalies by thediabatic residual overturning circulation.

A new process-based vertical advection/diffusion theoretical model of ocean heat uptake