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Changeset 10406 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_C.tex – NEMO

Ignore:
Timestamp:
2018-12-18T11:25:09+01:00 (5 years ago)
Author:
nicolasmartin
Message:

Edition of math environments

Location:
NEMO/trunk/doc/latex
Files:
4 edited

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  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_C.tex

    r10354 r10406  
    2525 
    2626fluxes at the faces of a $T$-box: 
    27 \begin{equation*} 
     27\[ 
    2828U = e_{2u}\,e_{3u}\; u  \qquad  V = e_{1v}\,e_{3v}\; v  \qquad W = e_{1w}\,e_{2w}\; \omega     \\ 
    29 \end{equation*} 
     29\] 
    3030 
    3131volume of cells at $u$-, $v$-, and $T$-points: 
    32 \begin{equation*} 
     32\[ 
    3333b_u = e_{1u}\,e_{2u}\,e_{3u}  \qquad  b_v = e_{1v}\,e_{2v}\,e_{3v}  \qquad b_t = e_{1t}\,e_{2t}\,e_{3t}     \\ 
    34 \end{equation*} 
     34\] 
    3535 
    3636partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$ 
    … …  
    4747 
    4848Continuity equation with the above notation: 
    49 \begin{equation*} 
     49\[ 
    5050\frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t}  \biggl\{  \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0 
    51 \end{equation*} 
     51\] 
    5252 
    5353A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when: 
    54 \begin{equation*} 
     54\[ 
    5555\partial_t \left( \int_D{ Q\;dv } \right) =0 
    56 \end{equation*} 
     56\] 
    5757Noting that the coordinate system used ....  blah blah 
    58 \begin{equation*} 
     58\[ 
    5959\partial_t \left( \int_D {Q\;dv} \right) =  \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } 
    6060                                                       =  \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0 
    61 \end{equation*} 
     61\] 
    6262equation of evolution of $Q$ written as 
    6363the time evolution of the vertical content of $Q$ like for tracers, or momentum in flux form, 
    … …  
    162162$\ $\newline    % force a new ligne 
    163163The prognostic ocean dynamics equation can be summarized as follows: 
    164 \begin{equation*} 
     164\[ 
    165165\text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } 
    166166                  {\text{COR} + \text{ADV}                       } 
    167167         + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} 
    168 \end{equation*} 
     168\] 
    169169$\ $\newline    % force a new ligne 
    170170 
    … …  
    365365 
    366366For the ENE scheme, the two components of the vorticity term are given by: 
    367 \begin{equation*} 
     367\[ 
    368368- e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h    \equiv  
    369369   \left( {{  \begin{array} {*{20}c} 
    … …  
    373373      \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i}       \hfill \\ 
    374374   \end{array}} }    \right) 
    375 \end{equation*} 
     375\] 
    376376 
    377377This formulation does not conserve the enstrophy but it does conserve the total kinetic energy. 
    … …  
    471471 
    472472The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE~: 
    473 \begin{equation*} 
     473\[ 
    474474    \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv 
    475475=  -   \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv 
    476476   +   \frac{1}{2} \int_D {  \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv }  \\ 
    477 \end{equation*} 
     477\] 
    478478Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry property of the $\delta$ operator) 
    479479and the continuity equation, then \autoref{eq:DOM_di_adj} again, 
    … …  
    544544For example KE can also be discretized as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. 
    545545This leads to the following expression for the vertical advection: 
    546 \begin{equation*} 
     546\[ 
    547547\frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h 
    548548\equiv \left( {{\begin{array} {*{20}c} 
    … …  
    552552\left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\ 
    553553\end{array}} } \right) 
    554 \end{equation*} 
     554\] 
    555555a formulation that requires an additional horizontal mean in contrast with the one used in NEMO. 
    556556Nine velocity points have to be used instead of 3. 
    … …  
    588588the change of KE due to the work of pressure forces is balanced by 
    589589the change of potential energy due to buoyancy forces:  
    590 \begin{equation*} 
     590\[ 
    591591- \int_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv  
    592592= - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv 
    593593  + \int_D g\, \rho \; \partial_t (z)  \;dv 
    594 \end{equation*} 
     594\] 
    595595 
    596596This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. 
    … …  
    771771This altered Coriolis parameter is discretised at an f-point. 
    772772It is given by: 
    773 \begin{equation*} 
     773\[ 
    774774f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; 
    775775\equiv \; 
    776776f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right]  
    777777                                               -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u}  \right] \right) 
    778 \end{equation*} 
     778\] 
    779779 
    780780Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. 
    … …  
    842842\end{flalign*} 
    843843Applying similar manipulation applied to the second term of the scalar product leads to: 
    844 \begin{equation*} 
     844\[ 
    845845 -  \int_D \textbf{U}_h \cdot     \left(                 {{\begin{array} {*{20}c} 
    846846\nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 
    … …  
    848848\equiv + \sum\limits_{i,j,k}  \frac{1}{2}  \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) 
    849849   \biggl\{     \left(   \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t}   \right) \; b_t     \biggr\}     
    850 \end{equation*} 
     850\] 
    851851which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv $. 
    852852\autoref{eq:C_ADV_KE_flux} is thus satisfied. 
    … …  
    10321032 
    10331033conservation of a tracer, $T$: 
    1034 \begin{equation*} 
     1034\[ 
    10351035\frac{\partial }{\partial t} \left(   \int_D {T\;dv}   \right)  
    10361036=  \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0 
    1037 \end{equation*} 
     1037\] 
    10381038 
    10391039conservation of its variance: 
    … …  
    11561156The lateral momentum diffusion term dissipates the horizontal kinetic energy: 
    11571157%\begin{flalign*} 
    1158 \begin{equation*} 
     1158\[ 
    11591159\begin{split} 
    11601160\int_D \textbf{U}_h \cdot  
    … …  
    11981198\quad \leq 0       \\ 
    11991199\end{split} 
    1200 \end{equation*} 
     1200\] 
    12011201 
    12021202% ------------------------------------------------------------------------------------------------------------- 
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