4.5. Equation of State¶
Related CPP options:
SALINITY 
Activate salinity as an active tracer 
NONLIN_EOS 
Activate nonlinear equation of state 
SPLIT_EOS 
Activate the split of the nonlinear equation of state in adiabatic
and compressible parts for reduction of pressure gradient errors

Preselected options:
# define SALINITY
# define NONLIN_EOS
# define SPLIT_EOS
The density is obtained from temperature and salinity (if SALINITY defined) via a choice of linear \(\rho(T)\) or nonlinear \(\rho(T,S,P)\) equation of state (EOS) described in Shchepetkin and McWilliams (2003). The nonlinear EOS corresponds to the UNESCO formulation as derived by Jackett and McDougall (1995) that computes in situ density as a function of potential temperature, salinity and pressure.
To reduce errors of pressuregradient scheme associated with nonlinearity of compressibility effects, Shchepetkin and McWilliams (2003) introduced a Taylor expansion of this EOS that splits it into an adiabatic and a linearized compressible part (SPLIT_EOS):
where \(\rho_1(T,S)\) is the seawater density perturbation at the standard pressure of 1 Atm (sea surface), \(q_1\) is the compressibility coefficient, and \(z\) is absolute depth, i.e. the distance from freesurface to the point at which density is computed. This splitting of the EOS into two separate contributions allows for the representation of spatial derivatives of density as the sum of adiabatic derivatives and the compressible part. This makes it straightforward to remove pressure effects so as to reduce pressure gradient errors, compute neutral directions, enforce stable stratification, compute BruntVäisäla frequency etc.
The BruntVäisäla frequency \(N\) (at horizontal \(\rho\) and vertical \(w\) points) is defined by:
where \(\rho_{\theta}\) is potential density, .i.e., the density that a parcel would acquire if adiabatically brought to depth \(z_w\).
Shchepetkin, A.F., McWilliams, J.C., 2003: A method for computing horizontal pressuregradient force in an oceanic model with a nonaligned vertical coordinate. J. Geophys. Res. 108 (C3), 3090.
Shchepetkin, A.F., McWilliams, J.C., 2011. Accurate Boussinesq oceanic modeling with a practical, ‘‘stiffened’’ equation of state. Ocean Modell. 38, 41–70.