34.6.1 Discretization of the new metric terms

Once the viscosity coefficients have been computed, either from some a priori profile or using a scheme such as that of Smagorinsky (Section 33.7), the new metric terms (Section 9.8) arising from the non-constant viscosity coefficient must be computed. The discretization of the new metric terms, which are located at the center of the U-cell, is given schematically by

$$new\_metric^{(1)} -{ \overline{ \delta_{\lambda} A }^{\phi} \over \cos\phi} ( \over... ...os\phi} ( \overline{ \delta_{\lambda} v }^{\lambda} + a^{-1} \, u \, \sin\phi )$$ = (34.71)

$$new\_metric^{(2)} { \overline{ \delta_{\lambda} A }^{\phi} \over \cos\phi } ( \over... ...\phi } (- \overline{\delta_{\lambda} u }^{\lambda} + a^{-1} \, v \, \sin\phi ).$$ = (34.72)

Recall that the finite difference operators $\delta_\lambda$ and $\delta_\phi$ absorb one factor of the earth's radius a. The averaging performed on the viscosity coefficients assumes they are defined on the corners of the U-cell. This is the natural positioning of these coefficients when employing the Smagorinsky scheme (see Section 33.7), and so the same positioning is employed in the general case described here. Note the presence of computational modes in the discretization of the velocity gradients. Again, they are of no consequence since the discretization of the corresponding Laplacian acts on all waves, thus eliminating the potential for this splitting in the metric terms to be harmful.