The Boussinesq approximation

Oceananigans.jl often employs the Boussinesq approximation^[1]. In the Boussinesq approximation the fluid density $\rho$ is, in general, decomposed into three components:

\[ \rho(\boldsymbol{x}, t) = \rho_0 + \rho_*(z) + \rho'(\boldsymbol{x}, t) \, ,\]

where $\rho_0$ is a constant 'reference' density, $\rho_*(z)$ is a background density profile which, when non-zero, is typically associated with the hydrostatic compression of seawater in the deep ocean, and $\rho'(\boldsymbol{x}, t)$ is the dynamic component of density corresponding to inhomogeneous distributions of a buoyant tracer such as temperature or salinity.

The fluid buoyancy, associated with the buoyant acceleration of fluid, is defined in terms of $\rho'$ as

\[ b = - \frac{g \rho'}{\rho_0} \, ,\]

where $g$ is gravitational acceleration.

The Boussinesq approximation is valid when $\rho_* + \rho' \ll \rho_0$, which implies the fluid is approximately incompressible, and thus does not support acoustic waves. In this case, the mass conservation equation reduces to the continuity equation

\[ \begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{v} = \partial_x u + \partial_y v + \partial_z w = 0 \, . \label{eq:continuity} \end{equation}\]

Similarly, in the the momentum equations we can divide through with $\rho_0$ and use that $\rho_* + \rho' \ll \rho_0$ to get:

\[ \begin{equation} \partial_t \boldsymbol{v} + \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{v} + \dotsb = - \frac1{\rho_0} \boldsymbol{\nabla} p - g \frac{\rho}{\rho_0} \hat{\boldsymbol{z}} + \dotsb \, . \label{eq:momentum} \end{equation}\]

We refer to $p / \rho_0$ as the "kinematic pressure" with dimensions of velocity squared. Hereafter, we abuse notation a bit and denote the kinematic pressure simply as $p$.