9.6 Comments on frictional and advective metric terms

This section benefited from discussions with Bob Hallberg and Gavin Schmidt.

When discretizing physical processes in MOM, it is desirable to formulate these processes in terms of the finite difference of a flux across the faces of a grid cell. Hence, the forcing terms in the continuous equations should be in the form of a divergence of a flux. However, it is not possible to do so for the friction acting on the zonal and meridional momentum on a sphere. Likewise, it is not possible to do so for advection of zonal and meridional momentum, as mentioned in Section 4.2. The purpose of this section is to provide some discussion of these ideas.

As seen in Section 9.3.7, the hydrostatic approximation brings the vertical friction to the flux-form $(\kappa_{m} \, {\bf u}_{,z})_{,z}$, regardless of the details of the horizontal coordinates. For brevity, the following discussion will therefore omit the vertical portion of the friction and just focus on the lateral components.