# Buoyancy model and equations of state

The buoyancy model determines the relationship between tracers and the buoyancy $b$ in the momentum equation.

## Buoyancy tracer

The simplest buoyancy model uses buoyancy $b$ itself as a tracer: $b$ obeys the tracer conservation equation and is used directly in the momentum equations.

## Seawater buoyancy

For seawater buoyancy is, in general, modeled as a function of conservative temperature $T$, absolute salinity $S$, and depth below the ocean surface $d$ via

\[ \begin{equation} b = - \frac{g}{\rho_0} \rho' \left (T, S, d \right ) \, , \label{eq:seawater-buoyancy} \end{equation}\]

where $g$ is gravitational acceleration, $\rho_0$ is the reference density. The function $\rho'(T, S, d)$ in the seawater buoyancy relationship that links conservative temperature, salinity, and depth to the density perturbation is called the *equation of state*. Both $T$ and $S$ obey the tracer conservation equation.

### Linear equation of state

Buoyancy is determined from a linear equation of state via

\[ b = g \left ( \alpha T - \beta S \right ) \, ,\]

where $g$ is gravitational acceleration, $\alpha$ is the thermal expansion coefficient, and $\beta$ is the haline contraction coefficient.

### Nonlinear equation of state

Buoyancy is determined by the simplified equations of state introduced by F. Roquet, G. Madec, Trevor J. McDougall, Paul M. Barker (2015).