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 pressure-gradient 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):

\[\rho = \rho_0 + \rho_1(T,S) + q_1(T,S) \ |z|\]

where \(\rho_1(T,S)\) is the sea-water 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 free-surface 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 Brunt-Väisäla frequency etc.

The Brunt-Väisäla frequency \(N\) (at horizontal \(\rho\) and vertical \(w\) points) is defined by:

\[N^2 = - \frac{g}{\rho_0} \frac{\partial \rho_{\theta}}{\partial z}\]

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 pressure-gradient force in an oceanic model with a non-aligned 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.