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 in the momentum equation.

Seawater buoyancy

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

\[ b = - \frac{g}{\rho_0} \rho' \left (\theta, S, d \right ) \, , \tag{eq:seawater-buoyancy}\]

where $g$ is gravitational acceleration, $\rho_0$ is the reference density. The function $\rho'(\theta, S, d)$ in the seawater buoyancy relationship that links conservative temperature, salinity, and depth to the density perturbation is called the equation of state. Both $\theta$ and $S$ obey the tracer conservation equation.

Linear equation of state

Buoyancy is determined from a linear equation of state via

\[ b = g \left ( \alpha_\theta \theta - \beta_S S \right ) \, ,\]

where $g$ is gravitational acceleration, $\alpha_\theta$ is the thermal expansion coefficient, and $\beta_S$ is the haline contraction coefficient.

Nonlinear equation of state

Buoyancy is determined by the simplified equations of state introduced by Roquet et al (2015).