Reply to “Comments on ‘Diathermal Heat Transport in a Global Ocean Model’”

Abstract

AbstractHochet and Tailleux (2019), in a comment on Holmes et al. (2019), argue that under the incompressible Boussinesq approximation the “sum of the volume fluxes through any kind of control volume must integrate to zero at all times.” They hence argue that the expression in Holmes et al. (2019) for the change in the volume of seawater warmer than a given temperature is inaccurate. Here we clarify what is meant by the term “volume flux” as used in Holmes et al. (2019) and also more generally in the water-mass transformation literature. Specifically, a volume flux across a surface can occur either due to fluid moving through a fixed surface, or due to the surface moving through the fluid. Interpreted in this way, we show using several examples that the statement from Hochet and Tailleux (2019) quoted above does not apply to the control volume considered in Holmes et al. (2019). Hochet and Tailleux (2019) then derive a series of expressions for the water-mass transformation or volume flux across an isotherm in the general, compressible case. In the incompressible Boussinesq limit these expressions reduce to a form (similar to that provided in Holmes et al. 2019) that involves the temperature derivative of the diabatic heat fluxes. Due to this derivative, can be difficult to robustly estimate from ocean model output. This emphasizes one of the advantages of the approach of Holmes et al. (2019), namely, does not appear in the internal heat content budget and is not needed to describe the flow of internal heat content into and around the ocean.