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 $p$.

Convention on dynamic versus kinematic pressure

In Oceananigans, the pressure $p$ refers to "kinematic pressure" (with dimensions velocity squared), i.e., the dynamic pressure scaled with the reference fluid density $\rho_0$.

  • 1Named after Boussinesq (1903) although used earlier by Oberbeck (1879), the Boussinesq approximation neglects density differences in the momentum equation except when associated with the gravitational term. It is an accurate approximation for many flows, and especially so for oceanic flows where density differences are very small. See Vallis (2017, section 2.4) for an oceanographic introduction to the Boussinesq equations and Vallis (2017, Section 2.A) for an asymptotic derivation. See Kundu (2015, Section 4.9) for an engineering introduction.