Comments on the Recent Discussion Concerning the Article “Diathermal Heat Transport in a Global Ocean Model”

A 2019 comment by Hochet and Tailleux and the corresponding reply by Holmes et al. discuss the volume and mass balance on a control volume bounded by a given isotherm and the ocean free surface. This note partly reconciles the terms of discrepancy on volume or mass transport appearing in these publications by proving that the integral expressions in the comment of Hochet and Tailleux, using a particular parameterization of the moving surfaces in Cartesian coordinates, correspond to the mass transport across the moving surfaces, as long as the mass density is included, as given in direct vector notation by Holmes et al. in their reply.