Horizontal/Vertical 2nd Order Tracer Diffusive Operators

In z-coordinates

In $ z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by:
    $\displaystyle D^T = \frac{1}{e_1   e_2} \left[
\left. \frac{\partial}{\partial...
...c{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)$ (B.1)

In generalized vertical coordinates

In $ s$-coordinates, we defined the slopes of $ s$-surfaces, $ \sigma_1$ and $ \sigma _2$ by (A.1) and the vertical/horizontal ratio of diffusion coefficient by $ \epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by:

$\displaystyle D^T = \left. \nabla \right\vert _s \cdot \left[ A^{lT} \;\Re \cdot \left. \nabla \right\vert _s T \right]  \;\;$where$\displaystyle \;\Re =\left( {{\begin{array}{*{20}c} 1 \hfill & 0 \hfill & {-\si...
...ill & {\varepsilon +\sigma _1 ^2+\sigma _2 ^2} \hfill  \end{array} }} \right)$ (B.2)

or in expanded form:
\begin{subequations}\begin{align*}{\begin{array}{*{20}l} D^T=& \frac{1}{e_1 e_2...
...}{\partial s}} \right)\;\;} \right] \end{array} } \end{align*}\end{subequations}

Equation (B.2) is obtained from (B.1) without any additional assumption. Indeed, for the special case $ k=z$ and thus $ e_3=1$, we introduce an arbitrary vertical coordinate $ s=s(i,j,z)$ as in Appendix A and use (A.1) and (A.2). Since no cross horizontal derivative $ \partial _i \partial _j $ appears in (B.1), the ($ i$,$ z$) and ($ j$,$ z$) planes are independent. The derivation can then be demonstrated for the ($ i$,$ z$$ \to$ ($ j$,$ s$) transformation without any loss of generality:

$\displaystyle \begin{align*}{\begin{array}{*{20}l} D^T&=\frac{1}{e_1 e_2} \lef...
...ial s}} \hfill  \end{array}}} \right) \left( T \right)} \right] \end{align*} $  

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