## Abstract

Traditionally, the conservation equations in oceanography include the Boussinesq approximation, and the velocity variable is interpreted as the Eulerian mean velocity averaged over turbulent scales. If such a view is adopted, then the conservation equations for tracers contain errors that are often as large as the diapycnal mixing term. This result has been known for about a decade and, at face value, implies that all Boussinesq ocean models contain leading-order errors in their conservation equations. To date there has not yet been a solution proposed to avoid this conundrum. Here it is shown that the conundrum can be solved by interpreting the horizontal velocity vector carried by Boussinesq ocean models as the average horizontal mass flux per unit area divided by the constant reference density that appears in the horizontal momentum equation. The authors argue that the vector labeled the “velocity” in present ocean models is not, and never was, the Eulerian mean velocity. If it were, then the conservation equations for salinity anomaly and potential temperature would contain systematic errors whose magnitude would be as large as the diapycnal mixing terms. By interpreting the model's horizontal “velocity” as being proportional to the horizontal mass flux per unit area, the conservation equations in the present generation of Boussinesq models are actually much more accurate even than previously thought. In particular, when these Boussinesq models achieve a steady state, they are actually almost fully non-Boussinesq, and in a nonsteady state there is no systematic error in the diapycnal advective/diffusive balance due to the Boussinesq approximation. With the above interpretation of the model's “velocity,” it is also relatively simple to change the model code to make it fully non-Boussinesq even when the flow is unsteady.

A conclusion of the authors' work is that the Boussinesq approximation actually consists of three parts, not two, as has been assumed in the past. Traditionally, the Boussinesq approximation consists of replacing (i) the equation for conservation of mass by the equation for conservation of volume and (ii) the density that appears in the temporal and advection operators by a constant reference density. Here the authors show that it is also important to (iii) ensure that using a divergence free velocity to advect tracer does not lead to significant error, an aspect of the Boussinesq approximation that has previously been overlooked.

*Corresponding author address:* Dr. Trevor J. McDougall, CSIRO Division of Marine Research, GPO Box 1538, Hobart, TAS 7001, Australia. Email: Trevor.McDougall@marine.csiro.au