# Lateral/Vertical Momentum Diffusive Operators

The second order momentum diffusion operator (Laplacian) in the -coordinate is found by applying (2.7e), the expression for the Laplacian of a vector, to the horizontal velocity vector :

Using (2.7b), the definition of the horizontal divergence, the third componant of the second vector is obviously zero and thus :

Note that this operator ensures a full separation between the vorticity and horizontal divergence fields (see Appendix C). It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere.

The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in the -coordinate therefore takes the following form :

 (B.7)

that is, in expanded form:

Note Bene: introducing a rotation in (B.8) does not lead to a useful expression for the iso/diapycnal Laplacian operator in the -coordinate. Similarly, we did not found an expression of practical use for the geopotential horizontal/vertical Laplacian operator in the -coordinate. Generally, (B.8) is used in both - and -coordinate systems, that is a Laplacian diffusion is applied on momentum along the coordinate directions.

Gurvan Madec and the NEMO Team
NEMO European Consortium2017-02-17