(omega-user-eos)=

# 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.

Two choices of EOS are provided by Omega: a linear 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. However, this option is only recommended for simpler idealized test cases as its accuracy is not sufficient for real ocean simulations. The TEOS-10 EOS is a 75-term polynomial expression from [Roquet et al. 2015](https://www.sciencedirect.com/science/article/pii/S1463500315000566) that approximates the [Thermodynamic Equation of Seawater 2010](https://www.teos-10.org/pubs/TEOS-10_Manual.pdf), but in a less complex and more computationally efficient manner, and is the preferred EOS for real ocean simulations in Omega.

The user-configurable options are: `EosType` (choose either `Linear` or `Teos-10`), as well as the parameters needed for the linear EOS.

```yaml
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` and `BruntVaisalaFreq` are also calculated by the EOS.

## Displaced Specific Volume

The `Eos` class calculates the density of a parcel of fluid that is adiabatically displaced by a relative `k` levels, 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` EOS option, `SpecVolDisplaced` will be the same as `SpecVol` since the specific volume calculation is independent of pressure/depth.

## Brunt Vaisala Frequency

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