An accurate implementation of the compressibility terms in the equation of state in a low order pressure gradient scheme for sigma coordinate ocean models

In a previous study, the authors studied a low order pressure gradient force (PGF) scheme referred to as the Primitive-Modified scheme, that appears to be equivalent to the Pressure-Jacobian PGF (Lin, 1997). The scheme was successfully tested on the seamount experiment using a simplified equation of state (EOS). Yet, a complete equation of state, including compressibility effect, can raise a serious problem of accuracy. A new implementation is thus proposed in the present paper. The scheme is rewritten using a (numerically equivalent) geopotential formulation. The PGF truncation errors are removed by computing the EOS compressibility terms with potential temperature and salinity interpolated on a suitable geopotential level. The so-called Equivalent Geopotential Formulation (EGF) method is compared to the Finite-Volume approach proposed by Adcroft et al. (2008).