Horizontal/Vertical 2nd Order Tracer Diffusive Operators

In z-coordinates

In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by:

$$D^T = \frac{1}{e_1 \, e_2} \left[ \left. \frac{\partial}{\partial... ...c{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)$$ (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:

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

or in expanded form:

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:

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