### Abstract:

A new density variable, empirically corrected for pressure, is constructed. This is done by first fitting compressibility (or sound speed) computed from global ocean datasets to an empirical function of pressure and in situ density (or specific volume). Then, by replacing true compressibility by this best-fit virtual compressibility in the thermodynamic density equation, an exact integral of a Pfaffian differential form can be found; this is called orthobaric density. The compressibility anomaly (true minus best-fit) is not neglected, but used to develop a gain factor ψ on the irreversible processes that contribute to the density equation and drive diapycnal motion. The complement of the gain factor, ψ − 1, multiplies the reversible motion of orthobaric density surfaces to make a second contribution to diapycnal motion. The gain factor is a diagnostic of the materiality of orthobaric density: gain factors of 1 would indicate that orthobaric density surfaces are as material as potential density surfaces. Calculations of the gain factor for extensive north–south ocean sections in the Atlantic and Pacific show that it generally lies between 0.8 and 1.2.
Orthobaric density in the ocean possesses advantages over potential density that commend its use as a vertical coordinate for both descriptive and modeling purposes. A geostrophic streamfunction exists for the momentum equations transformed to orthobaric density coordinates so that the gradients of orthobaric density surfaces give precisely the geostrophic shear. A form of Ertel's potential vorticity can be defined whose evolution equation contains no contribution from the baroclinicity vector. Orthobaric density surfaces are invariant to the choice of reference pressure. All of these are properties that potential density lacks.
In the continuous limit, patched potential density surfaces, which are formed by joining segments of locally referenced potential density surfaces in various pressure ranges and are extensively used in descriptive physical oceanography, become a particular form of orthobaric density surfaces. The method of selecting the segments is equivalent to choosing the virtual compressibility function. This correspondence is a useful aid in interpreting patched potential density surfaces. In particular, there is a material flow across such surfaces that is analogous to the reversible flow across orthobaric isopycnals.