Hughes, C. W. 
ORCID: https://orcid.org/0000-0002-9355-0233.
2008
A form of potential vorticity equation for depth-integrated flow with a free surface.
Journal of Physical Oceanography, 38 (5).
1131-1136.
10.1175/2007JPO3809.1
A form of linear, barotropic potential vorticity equation is derived for an ocean with a free surface, in
which only one scalar variable appears (ocean bottom pressure, or subsurface pressure). Unlike quasigeostrophic
or rigid-lid derivations, the only approximation made (apart from linearization) is that changes in
the circulation must be slow compared with the inertial frequency. Effects of stratification are included, but
only parametrically in the sense that density is treated as a given quantity or forcing term rather than a
variable.
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