Conservation Properties on Vertical Momentum Physics

As for the lateral momentum physics, the continuous form of the vertical diffusion of momentum satisfies several integral constraints. The first two are associated with the conservation of momentum and the dissipation of horizontal kinetic energy:

$$\int\limits_D \frac{1} {e_3 }\; \frac{\partial } {\partial k} \le... ...} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv \qquad \quad = \vec{\textbf{0}}$$
and

$$\int\limits_D \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial... ...^{ vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv \quad \leq 0$$

The first property is obvious. The second results from:

$$\int\limits_D \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial... ...\frac{A^{ vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\;dv$$

$$\equiv \sum\limits_{i,j,k} \left( u\; \delta_k \left[ \frac{A_u^{... ...vm}} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right]\; e_{1v} e_{2v} \right)$$

$$k$$

$$\equiv - \sum\limits_{i,j,k} \left( \frac{A_u^{ vm}} {e_{3uw}} \... ...\delta_{k+1/2} \left[ v \right] \right)^2\; e_{1v} e_{2v} \right) \quad \leq 0$$

The vorticity is also conserved. Indeed:

$$\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times \le... ...{ vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv$$

$$\equiv \sum\limits_{i,j,k} \frac{1} {e_{3f}}\; \frac{1} {e_{1f} e_{2f}} \bigg\{ \biggr. \quad \delta_{i+1/2} \left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right] \right)$$

$$\biggl. - \delta_{j+1/2} \left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{1} {e_{3uw}}\... ...\left[ u \right] \right] \right) \biggr\} \; e_{1f} e_{2f} e_{3f} \; \equiv 0$$

If the vertical diffusion coefficient is uniform over the whole domain, the enstrophy is dissipated, $i.e.$

$$\int\limits_D \zeta \textbf{k} \cdot \nabla \times \left( \fra... ...}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0$$

This property is only satisfied in $z$-coordinates:

$$\int\limits_D \zeta \textbf{k} \cdot \nabla \times \left( \fra... ... vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv$$

$$\equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \; \biggl\{ \biggr. \quad \delta_{i+1/2} \left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{A_v^{ vm}} {e_{3vw}} \delta_{k+1/2}[v] \right] \right)$$

$$- \delta_{j+1/2} \biggl. \left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{ vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) \biggr\}$$

$$\equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \biggl\{ \biggr. \quad \frac{1} {e_{3v}} \delta_k \left[ \frac{A_v^{ vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v} v \right] \right] \right]$$

$$\biggl. - \frac{1} {e_{3u}} \delta_k \left[ \frac{A_u^{ vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u} u \right] \right] \right] \biggr\}$$

Using the fact that the vertical diffusion coefficients are uniform, and that in $z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so that: $e_{3f} =e_{3u} =e_{3v} =e_{3t}$ and $e_{3w} =e_{3uw} =e_{3vw}$, it follows:

$$\equiv A^{ vm} \sum\limits_{i,j,k} \zeta \;\delta_k \left[ \frac... ...ft[ e_{2v} v \right] - \delta_{j+1/ 2} \left[ e_{1u} u \right] \Bigr] \right]$$

$$\equiv - A^{ vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}} \left( \delta_{k+1/2} \left[ \zeta \right] \right)^2 \; e_{1f} e_{2f} \; \leq 0$$

Similarly, the horizontal divergence is obviously conserved:

$$\int\limits_D \nabla \cdot \left( \frac{1} {e_3 }\; \frac{\partia... ...}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0$$

and the square of the horizontal divergence decreases ($i.e.$ the horizontal divergence is dissipated) if the vertical diffusion coefficient is uniform over the whole domain:

$$\int\limits_D \chi \;\nabla \cdot \left( \frac{1} {e_3 }\; \frac{... ...}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0$$

This property is only satisfied in the $z$-coordinate:

$$\int\limits_D \chi \;\nabla \cdot \left( \frac{1} {e_3 }\; \frac{... ... vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv$$

$$\equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t} e_{2t}} \biggl\{ \Biggr. \quad \delta_{i+1/2} \left( \frac{e_{2u}} {e_{3u}} \delta_k \left[ \frac{A_u^{ vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right)$$

$$\Biggl. + \delta_{j+1/2} \left( \frac{e_{1v}} {e_{3v}} \delta_k \left[ \frac{A_v^{ vm}} {e_{3vw}} \delta_{k+1/2} [v] \right] \right) \Biggr\} \; e_{1t} e_{2t} e_{3t}$$

$$\equiv A^{ vm} \sum\limits_{i,j,k} \chi \biggl\{ \biggr. \quad \delta_{i+1/2} \left( \delta_k \left[ \frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u} u \right] \right] \right)$$

$$\biggl. + \delta_{j+1/2} \left( \delta_k \left[ \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v} v \right] \right] \right) \biggr\}$$

$$\equiv -A^{ vm} \sum\limits_{i,j,k} \frac{\delta_{k+1/2} \left[ ... ...ft[ e_{2u} u \right] + \delta_{j+1/2} \left[ e_{1v} v \right] \Bigr] \biggr\}$$

$$\equiv -A^{ vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t} e_{2t} \;\chi \right]$$

$$\equiv -A^{ vm} \sum\limits_{i,j,k} \frac{e_{1t} e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right] \right)^2 \quad \equiv 0$$