When implementing friction operators in a numerical model, it is important to maintain as much of the continuum properties as feasible. For constant viscosity models, a direct discretization based on the formulas of Bryan (1969) (see Section 9.8) seem to be sufficient. These formulas consist of a Laplacian plus extra ``metric terms'' which arise from the spherical geometry. When using non-constant viscosities, Wajsowicz (1993) pointed out that there are extra metric terms proportional to the derivatives of the viscosity (Section 9.8). The metric terms are cumbersome to discretize, and they generally introduce computational modes to the discreted friction operator on a B-grid. Hence, they are often ignored as in Cox (1984), but at the cost of no longer maintaining angular momentum conservation, as discussed previously.
For general orthogonal curvilinear coordinates, the metric terms are tedious to compute. In this case, it becomes even more clear that it is preferable to directly discretize the expressions (9.126) and (9.127). However, some attempts to do so on the B-grid have led to the presence of computational modes. Indeed, when using the Smagorinsky viscosity, these approaches can lead to unacceptably noisy solutions, more-so than found using the Laplacian plus metric approach.
In conclusion, either approach is unsatisfying, and so another approach is required. The purpose of this section is to introduce a different approach based on the same functional formalism applied to the isoneutral diffusion operator (Appendix C and Griffies, Gnanadesikan, Pacanowski, Larichev, Dukowicz, and Smith, 1998). This method provides a general framework that leads to a suitable discretization of the friction operator.