The Boussinesq approximation

In Oceananigans.jl the fluid density $\rho$ is, in general, decomposed into three components:

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

where $\rho_0$ is a constant 'reference' density, $\rho_*(z)$ is a background density profile typically associated with the hydrostatic compression of seawater in the deep ocean, and $\rho'(\bm{x}, t)$ 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^[2] In this case, the mass conservation equation reduces to the continuity equation

\[ \bm{\nabla} \bm{\cdot} \bm{u} = \partial_x u + \partial_y v + \partial_z w = 0 \, . \tag{eq:continuity}\]

The momentum conservation equation

The equations governing the conservation of momentum in a rotating fluid, including buoyancy via the Boussinesq approximation and including the averaged effects of surface gravity waves at the top of the domain via the Craik-Leibovich approximation are

\[ \partial_t \bm{u} + \left ( \bm{u} \bm{\cdot} \bm{\nabla} \right ) \bm{u} + \left ( \bm{f} - \bm{\nabla} \times \bm{u}^S \right ) \times \bm{u} = - \bm{\nabla} \phi + b \bm{\hat z} - \bm{\nabla} \bm{\cdot} \bm{\tau} - \partial_t \bm{u}^S + \bm{F_u} \, , \tag{eq:momentum}\]

where $b$ is buoyancy, $\bm{\tau}$ is the kinematic stress tensor, $\bm{F_u}$ denotes an internal forcing of the velocity field $\bm{u}$, $\phi$ is the potential associated with kinematic and constant hydrostatic contributions to pressure, $\bm{u}^S$ is the 'Stokes drift' velocity field associated with surface gravity waves, and $\bm{f}$ is Coriolis parameter, or the background vorticity associated with the specified rate of rotation of the frame of reference.

The tracer conservation equation

The conservation law for tracers in Oceananigans.jl is

\[ \partial_t c + \bm{u} \bm{\cdot} \bm{\nabla} c = - \bm{\nabla} \bm{\cdot} \bm{q}_c + F_c \, , \tag{eq:tracer}\]

where $\bm{q}_c$ is the diffusive flux of $c$ and $F_c$ is an arbitrary source term. Oceananigans.jl permits arbitrary tracers and thus an arbitrary number of tracer equations to be solved simultaneously with the momentum equations.

2Incompressible fluids do not support acoustic waves.