Abstract/Summary

Holmes et al. (2019) have proposed a new theoretical framework for studying ocean heat uptake in potential temperature coordinates. One important step in their derivations requires understanding the temporal changes of the volume of water V with temperature greater than some value, which they write as the sum of two terms. The first one is due to the surface freshwater fluxes and is well defined, but the second one— attributed to the volume fluxes through the lower boundary of the domain—is given no explicit expression. What the authors mean exactly is unclear, however, because in the incompressible Boussinesq approximation, the use of a divergenceless velocity field implies that the sum of the volume fluxes through any kind of control volume must integrate to zero at all times. In this comment, we provide two alternative explicit mathematical expressions linking the volume change of Holmes et al. (2019) to the diabatic sources and sinks of heat that clarify their result. By contrasting Holmes et al.’s (2019) approach with that for a fully compressible ocean, it is concluded that the volume considered by Holmes et al. (2019) is best interpreted as a proxy for the Boussinesq mass M0 5 r0V, where r0 is the reference Boussinesq density. If V were truly meant to represent volume rather than a proxy for the Boussinesq mass, the Boussinesq expression for dV/dt would have to be regarded as inaccurate because of its neglect of the volume changes resulting from mean density changes.