Introduction / Notations

Notation used in this appendix in the demonstations :

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

$\displaystyle 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:

$\displaystyle 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).

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

Continuity equation with the above notation:

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

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

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

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

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

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 :

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

that is in a more compact form :

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

$\displaystyle \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) =$ $\displaystyle \int_D { \frac{1}{2} \partial_t \left( e_3   Q^2 \right) \;e_1e_2\;di dj dk }$    
$\displaystyle =$ $\displaystyle \int_D { Q \partial_t Q \;e_1e_2e_3\;di dj dk } + \int_D { \frac{1}{2} Q^2   \partial_t e_3 \;e_1e_2\;di dj dk }$    

that is in a more compact form :

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

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