4.5 Flux form and finite volumes

In general, MOM implments tracer and momentum advection as the divergence of a flux, rather than the advective form ${\bf u} \cdot \nabla \Psi$. In the continuum with an incompressible fluid, the advective form ${\bf u} \cdot \nabla \Psi$ and flux form $\nabla \cdot (\Psi \, {\bf u})$ are equivalent. In a numerical model, the flux formulation provides a straightforward way to ensure conservation properties of scalar quantities, and it allows a clear finite volume interpretation of the discrete equations.

As discussed by Adcroft et al (1996), a finite volume approach aims to formulate the discrete equations as self-consistent approximations of the volume integrated continuum equations, where the volume integration is taken over the a grid cell control volume. Such an approach is natural on a C-grid. With the B-grid in MOM, there are difficulties. Most notably, the bottom for a tracer cell does admit a finite volume interpretation. However, the bottom velocity cells do not rest on the ocean bottom (see Section 22.3.3). This is a notable instance where MOM does not respect the traditional finite volume approach.