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Changeset 10406 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.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/chap_TRA.tex

    r10354 r10406  
    2727The two active tracers are potential temperature and salinity. 
    2828Their prognostic equations can be summarized as follows: 
    29 \begin{equation*} 
     29\[ 
    3030\text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC} 
    3131                   \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP}) 
    32 \end{equation*} 
     32\] 
    3333 
    3434NXT stands for next, referring to the time-stepping. 
    … …  
    7676\begin{equation} \label{eq:tra_adv} 
    7777ADV_\tau =-\frac{1}{b_t} \left(  
    78 \;\delta _i \left[ e_{2u}\,e_{3u} \;  u\; \tau _u  \right] 
    79 +\delta _j \left[ e_{1v}\,e_{3v}  \;  v\; \tau _v  \right] \; \right) 
    80 -\frac{1}{e_{3t}} \;\delta _k \left[ w\; \tau _w \right] 
     78\;\delta_i \left[ e_{2u}\,e_{3u} \;  u\; \tau_u  \right] 
     79+\delta_j \left[ e_{1v}\,e_{3v}  \;  v\; \tau_v  \right] \; \right) 
     80-\frac{1}{e_{3t}} \;\delta_k \left[ w\; \tau_w \right] 
    8181\end{equation} 
    8282where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. 
    … …  
    125125  the vertical boundary condition is applied at the fixed surface $z=0$ rather than on the moving surface $z=\eta$. 
    126126  There is a non-zero advective flux which is set for all advection schemes as 
    127   $\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $, 
     127  $\left. {\tau_w } \right|_{k=1/2} =T_{k=1} $, 
    128128  $i.e.$ the product of surface velocity (at $z=0$) by the first level tracer value. 
    129129\item[non-linear free surface:] 
    … …  
    194194For example, in the $i$-direction : 
    195195\begin{equation} \label{eq:tra_adv_cen2} 
    196 \tau _u^{cen2} =\overline T ^{i+1/2} 
     196\tau_u^{cen2} =\overline T ^{i+1/2} 
    197197\end{equation} 
    198198 
    … …  
    213213For example, in the $i$-direction: 
    214214\begin{equation} \label{eq:tra_adv_cen4} 
    215 \tau _u^{cen4}  
    216 =\overline{   T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2} 
     215\tau_u^{cen4}  
     216=\overline{   T - \frac{1}{6}\,\delta_i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2} 
    217217\end{equation} 
    218218In the vertical direction (\np{nn\_cen\_v}\forcode{ = 4}), 
    … …  
    238238 
    239239At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), 
    240 an additional hypothesis must be made to evaluate $\tau _u^{cen4}$. 
     240an additional hypothesis must be made to evaluate $\tau_u^{cen4}$. 
    241241This hypothesis usually reduces the order of the scheme. 
    242242Here we choose to set the gradient of $T$ across the boundary to zero. 
    … …  
    260260\begin{equation} \label{eq:tra_adv_fct} 
    261261\begin{split} 
    262 \tau _u^{ups}&= \begin{cases} 
     262\tau_u^{ups}&= \begin{cases} 
    263263               T_{i+1}  & \text{if $\ u_{i+1/2} <     0$} \hfill \\ 
    264264               T_i         & \text{if $\ u_{i+1/2} \geq 0$} \hfill \\ 
    265265              \end{cases}     \\ 
    266266\\ 
    267 \tau _u^{fct}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen} -\tau _u^{ups} } \right) 
     267\tau_u^{fct}&=\tau_u^{ups} +c_u \;\left( {\tau_u^{cen} -\tau_u^{ups} } \right) 
    268268\end{split} 
    269269\end{equation} 
    … …  
    287287 
    288288For stability reasons (see \autoref{chap:STP}), 
    289 $\tau _u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while 
    290 $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. 
     289$\tau_u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while 
     290$\tau_u^{ups}$ is evaluated using the \textit{before} tracer. 
    291291In other words, the advective part of the scheme is time stepped with a leap-frog scheme 
    292292while a forward scheme is used for the diffusive part.  
    … …  
    306306For example, in the $i$-direction : 
    307307\begin{equation} \label{eq:tra_adv_mus} 
    308    \tau _u^{mus} = \left\{      \begin{aligned} 
    309          &\tau _i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) 
     308   \tau_u^{mus} = \left\{      \begin{aligned} 
     309         &\tau_i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) 
    310310         &\ \widetilde{\partial _i \tau}  & \quad \text{if }\;u_{i+1/2} \geqslant 0      \\ 
    311          &\tau _{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right) 
     311         &\tau_{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right) 
    312312         &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0 
    313313   \end{aligned}    \right. 
    … …  
    317317 
    318318The time stepping is performed using a forward scheme, 
    319 that is the \textit{before} tracer field is used to evaluate $\tau _u^{mus}$. 
     319that is the \textit{before} tracer field is used to evaluate $\tau_u^{mus}$. 
    320320 
    321321For an ocean grid point adjacent to land and where the ocean velocity is directed toward land, 
    … …  
    339339For example, in the $i$-direction: 
    340340\begin{equation} \label{eq:tra_adv_ubs} 
    341    \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{       
     341   \tau_u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{       
    342342   \begin{aligned} 
    343343         &\tau"_i          & \quad \text{if }\ u_{i+1/2} \geqslant 0      \\ 
    … …  
    345345   \end{aligned}    \right. 
    346346\end{equation} 
    347 where $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$. 
     347where $\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$. 
    348348 
    349349This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error 
    … …  
    374374Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows: 
    375375\begin{equation} \label{eq:traadv_ubs2} 
    376 \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{   
     376\tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} \left\{     
    377377   \begin{aligned} 
    378378   & + \tau"_i       & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
    … …  
    382382or equivalently  
    383383\begin{equation} \label{eq:traadv_ubs2b} 
    384 u_{i+1/2} \ \tau _u^{ubs}  
    385 =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} 
     384u_{i+1/2} \ \tau_u^{ubs}  
     385=u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta_i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} 
    386386- \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
    387387\end{equation} 
    … …  
    521521\begin{equation} \label{eq:tra_ldf_lap} 
    522522D_t^{lT} =\frac{1}{b_t} \left( \; 
    523    \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right]  
    524 + \delta _{j}\left[ A_v^{lT} \;  \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right]  \;\right) 
     523   \delta_{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta_{i+1/2} [T] \right]  
     524+ \delta_{j}\left[ A_v^{lT} \;  \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta_{j+1/2} [T] \right]  \;\right) 
    525525\end{equation} 
    526526where  $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells and 
    … …  
    728728\begin{equation} \label{eq:tra_sbc} 
    729729\begin{aligned} 
    730  &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} }  &\overline{ Q_{ns}       }^t  & \\  
    731 & F^S =\frac{ 1 }{\rho _o  \,      \left. e_{3t} \right|_{k=1} }  &\overline{ \textit{sfx} }^t   & \\    
     730 &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} }  &\overline{ Q_{ns}       }^t  & \\  
     731& F^S =\frac{ 1 }{\rho_o  \,      \left. e_{3t} \right|_{k=1} }  &\overline{ \textit{sfx} }^t   & \\    
    732732 \end{aligned} 
    733733\end{equation}  
    … …  
    743743\begin{equation} \label{eq:tra_sbc_lin} 
    744744\begin{aligned} 
    745  &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} }    
     745 &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} }    
    746746           &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t  & \\  
    747747% 
    748 & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} }  
     748& F^S =\frac{ 1 }{\rho_o \,\left. e_{3t} \right|_{k=1} }  
    749749           &\overline{ \left( \;\textit{sfx} - \textit{emp} \;\left. S \right|_{k=1}  \right) }^t   & \\    
    750750 \end{aligned} 
    … …  
    14101410      A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, 
    14111411      the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. 
    1412       The horizontal difference is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and 
     1412      The horizontal difference is then given by: $\delta_{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and 
    14131413      the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$. 
    14141414    } 
    … …  
    14161416\end{figure} 
    14171417%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1418 \begin{equation*} 
     1418\[ 
    14191419\widetilde{T}= \left\{  \begin{aligned}   
    1420 &T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta _k T^{i+1}   
     1420&T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta_k T^{i+1}    
    14211421                        && \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$   }  \\ 
    14221422                              \\ 
    1423 &T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta _k T^{i+1} 
     1423&T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta_k T^{i+1} 
    14241424                        && \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   }  
    14251425            \end{aligned}   \right. 
    1426 \end{equation*} 
     1426\] 
    14271427and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are:  
    14281428\begin{equation} \label{eq:zps_hde} 
    14291429\begin{aligned} 
    1430  \delta _{i+1/2} T=  \begin{cases} 
     1430 \delta_{i+1/2} T=   \begin{cases} 
    14311431\ \ \ \widetilde {T}\quad\ -T^i     & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ 
    14321432                              \\ 
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