Equation of State (EOS)

The equation of state (EOS) for the ocean describes the relationship between specific volume of seawater (in \(\textrm{m}^3/\textrm{kg}\); the reciprocal of density) and temperature (in \(^{\circ}\textrm{C}\)), salinity (in \(\textrm{g/kg}\)), and pressure (in \(\textrm{dbar}\)). Through the hydrostatic balance (which relates density/specific volume gradients to pressure gradients), the equation of state provides a connection between active tracers (temperature and salinity) and the fluid dynamics.

Three choices of EOS are provided by Omega: a linear EOS, a constant EOS, and a TEOS-10 EOS. The linear EOS simplifies the relationship by excluding the influence of pressure and using constant expansion/contraction coefficients, making the specific volume a simple linear function of temperature and salinity. The constant EOS is intended for idealized/debug-style configurations and sets the specific volume to 1/RhoSw for all active cells/layers (with RhoSw set to 1026.0 in GlobalConstants.h). However, both simplified options should only be used for simpler idealized test cases and are not appropriate for realistic ocean simulations. The TEOS-10 EOS is a 75-term polynomial expression from Roquet et al. 2015 that approximates the Thermodynamic Equation of Seawater 2010, but in a less complex and more computationally efficient manner, and is the preferred EOS for real ocean simulations in Omega.

The user-configurable option EosType can be set to linear, teos10, or constant. Parameters in the Linear subsection are used only when EosType is set to linear.

Eos:
   EosType: teos10
   Linear:
      DRhoDT: -0.2
      DRhoDS: 0.8
      RhoT0S0: 1000.0

where DRhoDT is the thermal expansion coefficient (\(\textrm{kg}/(\textrm{m}^3 \cdot ^{\circ}\textrm{C})\)), DRhoDS is the saline contraction coefficient (\(\textrm{kg}/\textrm{m}^3\)), and RhoT0S0 is the reference density at (T,S)=(0,0) (in \(\textrm{kg}/\textrm{m}^3\)).

In addition to SpecVol, the displaced specific volume SpecVolDisplaced, the squared Brunt-Vaisala frequency BruntVaisalaFreqSq and the first derivatives of specific volume SpecVolDCt, SpecVolDSa and SpecVolDP are also calculated by the EOS.

TEOS-10 Helper Conversions

When using the teos10 EOS option, helper methods are available for common temperature conversions used by ocean thermodynamics workflows:

calcCtFreezing(Sa, P, SaturationFract) computes the Conservative Temperature freezing point from absolute salinity and pressure.
calcPtFromCt(Sa, Ct) converts Conservative Temperature to potential temperature.
calcCtFromPt(Sa, Pt) converts potential temperature to Conservative Temperature.

These helper methods are available through the EOS implementation but do not replace the standard computeSpecVol, computeSpecVolDisp, or computeBruntVaisalaFreqSq calculations.

First Derivatives of Specific Volume

The Eos class can also compute the first derivatives of the specific volume with respect to conservative temperature (in \(\textrm{m}^3\textrm{kg}^{-1}\,^{\circ}\textrm{C}^{-1}\)), absolute salinity (in \(\textrm{m}^3\textrm{g}^{-1}\)), and pressure (in \(\textrm{m}^3\textrm{kg}^{-1}\textrm{Pa}^{-1}\)). These are needed by the higher-order horizontal pressure gradient, which expands the specific volume about a reference state within each layer instead of evaluating the full equation of state at every quadrature point.

There is no user-configurable option associated with the derivatives. They are computed on request by the parts of the model that need them, in the same pass that computes the specific volume, so selecting teos10 does not make the model slower unless a scheme that uses them is enabled. They are available for all three EosType choices: for teos10 they are the analytic derivatives of the same 75-term polynomial, for linear they follow from the configured DRhoDT and DRhoDS and have no pressure dependence, and for constant they are zero.

The derivatives are stored in the SpecVolDCt, SpecVolDSa and SpecVolDP fields of the Eos field group and can be requested in a stream’s contents just like SpecVol.

The thermal expansion and haline contraction coefficients that enter the squared Brunt-Vaisala frequency are computed from these same derivatives.

Displaced Specific Volume

The Eos class calculates the density of a parcel of fluid that is adiabatically displaced by a relative k levels (k counted positive downward), capturing the effects of pressure/depth changes. This is primarily used to calculate quantities for determining the water column stability (i.e. the stratification) and the vertical mixing coefficients (viscosity and diffusivity). Note: when using the Linear or constant EOS option, SpecVolDisplaced will be the same as SpecVol since the specific volume calculation is independent of pressure/depth.

Squared Brunt Vaisala Frequency

The Eos class also calculates the squared Brunt Vaisala Frequency, which is used by the VertMix class to determine water column stability for both the convective adjustment and shear-instability driven mixing. When using the Linear EOS option, the BruntVaisalaFreqSq is calculated using linear coefficients. When using the Teos-10 EOS option, the BruntVaisalaFreqSq is calculated with non-linear coefficients according to the TOES-10. When using the constant EOS option, BruntVaisalaFreqSq is zero in this implementation. For all options, SpecVol must be calculated prior to calculating BruntVaisalaFreqSq, as it is an input to the BruntVaisalaFreqSq calculation.