Introduction / Notations

Notation used in this appendix in the demonstations :

fluxes at the faces of a $T$-box:

$$U = e_{2u} e_{3u}\; u \qquad V = e_{1v} e_{3v}\; v \qquad W = e_{1w} e_{2w}\; \omega$$

volume of cells at $u$-, $v$-, and $T$-points:

$$b_u = e_{1u} e_{2u} e_{3u} \qquad b_v = e_{1v} e_{2v} e_{3v} \qquad b_t = e_{1t} e_{2t} e_{3t}$$

partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$

$dv=e_1 e_2 e_3 di dj dk$ is the volume element, with only $e_3$ that depends on time. $D$ and $S$ are the ocean domain volume and surface, respectively. No wetting/drying is allow ($i.e.$ $\frac{\partial S}{\partial t} = 0$) Let $k_s$ and $k_b$ be the ocean surface and bottom, resp. ($i.e.$ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth).

$$z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s} e_3(\tilde{k}) \;d\tilde{k} = \eta - \int\limits_k^{k_s} e_3 \;d\tilde{k}$$

Continuity equation with the above notation:

$$\frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t} \biggl\{ \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0$$

A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when:

$$\partial_t \left( \int_D{ Q\;dv } \right) =0$$

Noting that the coordinate system used .... blah blah

$$\partial_t \left( \int_D {Q\;dv} \right) = \int_D { \partial_t \l... ...i dj dk } = \int_D { \frac{1}{e_3} \partial_t \left( e_3 Q \right) dv } =0$$

equation of evolution of $Q$ written as the time evolution of the vertical content of $Q$ like for tracers, or momentum in flux form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when :

$$\partial_t \left( \int_D{ \frac{1}{2} Q^2\;dv } \right) = \int_D{ \frac{1}{2} \partial_t \left( \frac{1}{e_3}\left( e_3 Q \right)^2 \right) e_1e_2\;di dj dk }$$

$$= \int_D { Q \;\partial_t\left( e_3 Q \right) e_1e_2\;di dj dk } - \int_D { \frac{1}{2} Q^2 \partial_t (e_3) \;e_1e_2\;di dj dk }$$

that is in a more compact form :

$$\partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) = \int_D { \frac{Q}{e_3} \partial_t \left( e_3 Q \right) dv } - \frac{1}{2} \int_D { \frac{Q^2}{e_3} \partial_t (e_3) \;dv }$$ (C.1)

equation of evolution of $Q$ written as the time evolution of $Q$ like for momentum in vector invariant form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when :

$$\partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) = \int_D { \frac{1}{2} \partial_t \left( e_3 Q^2 \right) \;e_1e_2\;di dj dk }$$

$$= \int_D { Q \partial_t Q \;e_1e_2e_3\;di dj dk } + \int_D { \frac{1}{2} Q^2 \partial_t e_3 \;e_1e_2\;di dj dk }$$

that is in a more compact form :

$$\partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) = \int_D { Q \partial_t Q \;dv } + \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv }$$ (C.2)