## 1. Introduction

*ρ*are the reference Boussinesq and fluid densities, respectively. As a result, one of its peculiar properties is that the volume flux integrated over the boundary

*V*vanishes identically at all times, raising the question of how the control volume

*V*varies with time. Mathematically,

*V*represents the volume of all water warmer than a given temperature

*θ*, Holmes et al. (2019) suggest that the time variation of

*V*should be given by the formula

*G*as “the volume flux across the Θ isotherm, or the

*water-mass transformation*” and to

^{−1})

*into*the ocean associated with precipitation, evaporation, river runoff, and ice melt.” However, they did not provide any explicit mathematical expression for

*G*.

*G*as being due to the volume flux across the isotherm, Holmes et al. (2019) may have originally assumed (as happened to us and a few other colleagues) that the time variation of

*V*should be governed by the following equation:

*V*should, for instance, increase in an ocean experiencing net warming and decrease in an ocean experiencing net cooling.

A survey of the literature reveals that the above ambiguity can actually be traced back to Walin [1982, Eq. (2.2)] where, as above, the time variation of the volume is linked to the volume fluxes through its boundaries. In this comment, we seek to clarify the physics of the time variations of *V* by deriving an explicit mathematical expression for the term *G*. In particular, we aim to show that while *G*. Indeed, *G* is found to be related to the water mass transformation due to the diabatic sources and sinks of heat and salt, as expected from physical intuition, and also stated by Holmes et al. (2019). The derivation of such an expression is nontrivial, which might explain why it does not appear to have been published in the water-mass conversion literature before.

## 2. Theory

### a. Linking G to water mass transformations

*G*to water mass transformations, and to evaluate the validity and accuracy of the Boussinesq form of the results obtained, we seek expressions valid for a fully compressible ocean first. Thus, the conservation equations for mass and heat that we take as our starting point are

*θ*as proposed by Tailleux (2015). Note that Eq. (6) implies for the Lagrangian derivative of

*θ*:

*ρ*by the reference Boussinesq density

*θ*in addition to their volume

*θ*can be written as

*θ*along the isothermal lower surface

*V*can change either as the result of an addition/subtraction of mass or due to a change in the mean density

*O*(1) mm yr

^{−1}to the globally averaged sea level. This means that the two terms in Eq. (16) are often of comparable importance, and hence that it is in general not possible to justify neglecting the second term.

*ρ*by

*G*in either Eq. (17) or (18) represents the desired expression explicitly linking

*G*to water mass transformations, as postulated but not demonstrated by Holmes et al. (2019).

### b. Link to volume-integrated diabatic processes

*G*to water mass transformations, it is arguably impractical for diagnosing

*G*from ocean model outputs owing to its dependence on the boundary values of

*G*to volume-integrated diabatic effects instead. For simplicity, we restrict our discussion to the case of an incompressible Boussinesq ocean, as in Holmes et al. (2019). To that end, we use a pdf approach that physically amounts to sorting the ocean according to potential temperature. The underlying idea is to define a reference potential temperature profile

*z*. By construction,

*θ*and

*θ*and time

*t*alternatively as a function of

*t*through the identity

*θ*. As stated in the introduction, the use of divergenceless velocity field

*G*,

*F*in Eq. (33) is linked to the total surface flux, which in our expression is linked to

*M*and

*I*are related to the parameterized and numerical mixing, which in our expression appears as an effective diffusive flux. The main advantage of Eq. (32) is that all of its terms are arguably more familiar and easier to diagnose from numerical model outputs than local values of

## 3. Conclusions

In this comment, we derived two mathematically equivalent expressions [Eqs. (18) and (32)] for the term *G* entering Eq. (3) of Holmes et al. (2019) governing the time variations of *V* is best interpreted as pertaining to the Boussinesq mass *V* itself, since the boundary conditions that enter the problem belong to the mass budget. If the expression for

This work was supported by the OUTCROP NERC Grant NE/R010536/1. The comments of a reviewer helped clarify and simplify the derivations of this paper.

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