Changeset 10406


Ignore:
Timestamp:
2018-12-18T11:25:09+01:00 (23 months ago)
Author:
nicolasmartin
Message:

Edition of math environments

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

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

    r10354 r10406  
    259259Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and 
    260260using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side, 
    261 \begin{equation*} {\begin{array}{*{20}l}  
     261\[ {\begin{array}{*{20}l}  
    262262w_s  \;\frac{\partial u}{\partial s}  
    263263   = \frac{\partial s}{\partial t} \;  \frac{\partial u }{\partial s} 
    264264   = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \quad ,  
    265265\end{array} }      
    266 \end{equation*} 
     266\] 
    267267leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 
    268268$i.e.$ the total $s-$coordinate time derivative : 
     
    370370 
    371371The horizontal pressure gradient term can be transformed as follows: 
    372 \begin{equation*} 
     372\[ 
    373373\begin{split} 
    374  -\frac{1}{\rho _o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z 
    375  & =-\frac{1}{\rho _o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial p}{\partial s}} \right] \\ 
    376 & =-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma _1 }{\rho _o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ 
    377 &=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1 
     374 -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z 
     375 & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial p}{\partial s}} \right] \\ 
     376& =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma _1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ 
     377&=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma _1 
    378378\end{split} 
    379 \end{equation*} 
     379\] 
    380380Applying similar manipulation to the second component and 
    381381replacing $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: 
    382382\begin{equation} \label{apdx:A_grad_p_1} 
    383383\begin{split} 
    384  -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
    385 &=-\frac{1}{\rho _o \,e_1 } \left(     \left.              {\frac{\partial p}{\partial i}} \right|_s  
     384 -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
     385&=-\frac{1}{\rho_o \,e_1 } \left(     \left.              {\frac{\partial p}{\partial i}} \right|_s  
    386386                                                  + g\;\rho  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right) \\ 
    387387% 
    388  -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
    389 &=-\frac{1}{\rho _o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s  
     388 -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
     389&=-\frac{1}{\rho_o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s  
    390390                                                   + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) \\ 
    391391\end{split} 
     
    400400and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. 
    401401The pressure is then given by: 
    402 \begin{equation*}  
     402\[  
    403403\begin{split} 
    404404p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left(  \rho_o \, d + 1 \right) \; e_3 \; dk   \\ 
    405405   &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk     
    406406\end{split} 
    407 \end{equation*} 
     407\] 
    408408Therefore, $p$ and $p_h'$ are linked through: 
    409409\begin{equation} \label{apdx:A_pressure} 
     
    411411\end{equation} 
    412412and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: 
    413 \begin{equation*}  
     413\[  
    414414\frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
    415 \end{equation*} 
     415\] 
    416416 
    417417Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and 
     
    419419\begin{equation} \label{apdx:A_grad_p_2} 
    420420\begin{split} 
    421  -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
     421 -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
    422422&=-\frac{1}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s  
    423423                                       + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\ 
    424424% 
    425  -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
     425 -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
    426426&=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s  
    427427                                        + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\ 
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_B.tex

    r10354 r10406  
    2020\subsubsection*{In z-coordinates} 
    2121In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: 
    22 \begin{eqnarray} \label{apdx:B1} 
     22\begin{align} \label{apdx:B1} 
    2323 &D^T = \frac{1}{e_1 \, e_2}      \left[ 
    2424  \left. \frac{\partial}{\partial i} \left(  \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right. 
     
    2626+ \left. \frac{\partial}{\partial j} \left(  \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z   \right)   \right|_z      \right] 
    2727+ \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) 
    28 \end{eqnarray} 
     28\end{align} 
    2929 
    3030\subsubsection*{In generalized vertical coordinates} 
     
    156156\end{equation} 
    157157where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials: 
    158 \begin{equation*} 
     158\[ 
    159159a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
    160160\qquad , \qquad 
    161161a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
    162162\right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
    163 \end{equation*} 
     163\] 
    164164 
    165165In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, 
     
    204204The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes. 
    205205Let us demonstrate the second one: 
    206 \begin{equation*} 
     206\[ 
    207207\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv 
    208208          = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, 
    209 \end{equation*} 
     209\] 
    210210and since 
    211211\begin{subequations} 
     
    249249where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, 
    250250relative to $s$-coordinate surfaces: 
    251 \begin{equation*} 
     251\[ 
    252252r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} 
    253253\qquad , \qquad 
    254254r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
    255255\right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. 
    256 \end{equation*} 
     256\] 
    257257 
    258258To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious. 
     
    325325Using \autoref{eq:PE_div}, the definition of the horizontal divergence, 
    326326the third componant of the second vector is obviously zero and thus : 
    327 \begin{equation*} 
     327\[ 
    328328\Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) 
    329 \end{equation*} 
     329\] 
    330330 
    331331Note that this operator ensures a full separation between 
  • 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% ------------------------------------------------------------------------------------------------------------- 
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_E.tex

    r10354 r10406  
    2626For example, in the $i$-direction: 
    2727\begin{equation} \label{eq:tra_adv_ubs2} 
    28 \tau _u^{ubs} = \left\{  \begin{aligned} 
    29   & \tau _u^{cen4} + \frac{1}{12} \,\tau"_i     & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
    30   & \tau _u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2}       <       0 
     28\tau_u^{ubs} = \left\{   \begin{aligned} 
     29  & \tau_u^{cen4} + \frac{1}{12} \,\tau"_i      & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
     30  & \tau_u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2}       <       0 
    3131                   \end{aligned}    \right. 
    3232\end{equation} 
    3333or equivalently, the advective flux is 
    3434\begin{equation} \label{eq:tra_adv_ubs2} 
    35 U_{i+1/2} \ \tau _u^{ubs}  
     35U_{i+1/2} \ \tau_u^{ubs}  
    3636=U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} 
    3737- \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
    3838\end{equation} 
    3939where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and 
    40 $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$. 
     40$\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$. 
    4141By choosing this expression for $\tau "$ we consider a fourth order approximation of $\partial_i^2$ with 
    4242a constant i-grid spacing ($\Delta i=1$). 
    4343 
    4444Alternative choice: introduce the scale factors:   
    45 $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta _{i+1/2}[\tau] \right]$. 
     45$\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta_{i+1/2}[\tau] \right]$. 
    4646 
    4747 
     
    7676 
    7777NB 2: In a forthcoming release four options will be proposed for the vertical component used in the UBS scheme. 
    78 $\tau _w^{ubs}$ will be evaluated using either \textit{(a)} a centered $2^{nd}$ order scheme, 
     78$\tau_w^{ubs}$ will be evaluated using either \textit{(a)} a centered $2^{nd}$ order scheme, 
    7979or \textit{(b)} a TVD scheme, or \textit{(c)} an interpolation based on conservative parabolic splines following 
    8080\citet{Shchepetkin_McWilliams_OM05} implementation of UBS in ROMS, or \textit{(d)} an UBS. 
     
    8383NB 3: It is straight forward to rewrite \autoref{eq:tra_adv_ubs} as follows: 
    8484\begin{equation} \label{eq:tra_adv_ubs2} 
    85 \tau _u^{ubs} = \left\{  \begin{aligned} 
    86    & \tau _u^{cen4} + \frac{1}{12} \tau"_i      & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
    87    & \tau _u^{cen4} - \frac{1}{12} \tau"_{i+1}  & \quad \text{if }\ u_{i+1/2}       <       0 
     85\tau_u^{ubs} = \left\{   \begin{aligned} 
     86   & \tau_u^{cen4} + \frac{1}{12} \tau"_i    & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
     87   & \tau_u^{cen4} - \frac{1}{12} \tau"_{i+1}   & \quad \text{if }\ u_{i+1/2}       <       0 
    8888                   \end{aligned}    \right. 
    8989\end{equation} 
     
    9191\begin{equation} \label{eq:tra_adv_ubs2} 
    9292\begin{split} 
    93 e_{2u} e_{3u}\,u_{i+1/2} \ \tau _u^{ubs}  
     93e_{2u} e_{3u}\,u_{i+1/2} \ \tau_u^{ubs}  
    9494&= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\ 
    9595& - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
     
    107107\begin{equation} \label{eq:tra_ldf_lap} 
    108108\begin{split} 
    109 D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\;  e_{3T} } &\left[ {\quad \delta _i  
    110 \left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2}  
     109D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\;  e_{3T} } &\left[ {\quad \delta_i  
     110\left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta_{i+1/2}  
    111111\left[ T \right]} \right]} \right. 
    112112\\ 
    113 &\ \left. {+\; \delta _j \left[  
    114 {A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta _{j+1/2} \left[ T  
     113&\ \left. {+\; \delta_j \left[  
     114{A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta_{j+1/2} \left[ T  
    115115\right]} \right)} \right]\quad } \right] 
    116116\end{split} 
     
    121121\begin{split} 
    122122D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\ 
    123 & \delta _i \left[  \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2}  
     123& \delta_i \left[  \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2}  
    124124        \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 
    125     \delta _i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2}  
     125    \delta_i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2}  
    126126           [T] \right] \right] \right] 
    127127\end{split} 
     
    133133\begin{split} 
    134134D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\ 
    135 & \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2}  
     135& \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2}  
    136136       \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}  
    137     \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2}  
     137    \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2}  
    138138         [T] \right] \right] \right] 
    139139\end{split} 
     
    143143\begin{split} 
    144144F_u^{lT} = - \frac{1}{12} 
    145  e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta _{i+1/2}  
     145 e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta_{i+1/2}  
    146146       \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}  
    147     \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta _{i+1/2}  
     147    \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta_{i+1/2}  
    148148         [T] \right] \right] 
    149149\end{split} 
     
    158158\end{equation} 
    159159if the velocity is uniform ($i.e.$ $|u|=cst$) and 
    160 choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right]$ 
     160choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right]$ 
    161161 
    162162sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): 
     
    164164\begin{split} 
    165165F_u^{lT} 
    166 &= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right] \right] 
     166&= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] 
    167167\end{split} 
    168168\end{equation} 
     
    173173\begin{split} 
    174174F_u^{lT} 
    175 &= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta _i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right] \right] \\ 
    176 &= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta _i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta _{i+1/2}[\tau] \right] \right] 
     175&= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta_i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] \\ 
     176&= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta_i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta_{i+1/2}[\tau] \right] \right] 
    177177\end{split} 
    178178\end{equation} 
     
    191191\begin{subequations} \label{eq:dt_mt} 
    192192\begin{align} 
    193  \delta _{t+\rdt/2} [q]     &=  \  \ \,   q^{t+\rdt}  - q^{t}     \\ 
     193 \delta_{t+\rdt/2} [q]     &=  \  \ \,   q^{t+\rdt}  - q^{t}      \\ 
    194194 \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2 
    195195\end{align} 
     
    202202\begin{equation} \label{eq:LF} 
    203203   \frac{\partial q}{\partial t}  
    204          \equiv \frac{1}{\rdt} \overline{ \delta _{t+\rdt/2}[q]}^{\,t}  
     204         \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t}  
    205205      =         \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} 
    206206\end{equation}  
     
    219219\int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt}  
    220220   &\equiv \sum\limits_{0}^{N}  
    221          {\frac{1}{\rdt} q^t \ \overline{ \delta _{t+\rdt/2}[q]}^{\,t} \ \rdt}  
    222       \equiv \sum\limits_{0}^{N}  { q^t \ \overline{ \delta _{t+\rdt/2}[q]}^{\,t} } \\ 
    223    &\equiv \sum\limits_{0}^{N}  { \overline{q}^{\,t+\Delta/2}{ \delta _{t+\rdt/2}[q]}} 
    224       \equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta _{t+\rdt/2}[q^2] }\\ 
    225    &\equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta _{t+\rdt/2}[q^2] } 
     221         {\frac{1}{\rdt} q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} \ \rdt}  
     222      \equiv \sum\limits_{0}^{N}  { q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} } \\ 
     223   &\equiv \sum\limits_{0}^{N}  { \overline{q}^{\,t+\Delta/2}{ \delta_{t+\rdt/2}[q]}} 
     224      \equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta_{t+\rdt/2}[q^2] }\\ 
     225   &\equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta_{t+\rdt/2}[q^2] } 
    226226      \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) 
    227227\end{split} \end{equation} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_iso.tex

    r10354 r10406  
    155155noting that the $e_{{3}_{i+1/2}^k}$ in the area $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with 
    156156the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq:tra_ldf_iso} 
    157 \begin{equation*} 
     157\[ 
    158158  \left(F_u^{13} \right)_{i+\hhalf}^k = \Alts_{i+\hhalf}^k 
    159159  {e_{2}}_{i+1/2}^k \overline{\overline 
    160160    r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k}, 
    161 \end{equation*} 
     161\] 
    162162where 
    163 \begin{equation*} 
     163\[ 
    164164  \overline{\overline 
    165165   r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k} 
    166166  \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}}, 
    167 \end{equation*} 
     167\] 
    168168and here and in the following we drop the $^{lT}$ superscript from $\Alt$ for simplicity. 
    169169Unfortunately the resulting combination $\overline{\overline{\delta_k\bullet}}^{\,i,k}$ of a $k$ average and 
     
    200200  \label{eq:i13} 
    201201  \left( F_u^{13}  \right)_{i+\frac{1}{2}}^k = \Alts_{i+1}^k a_1 s_1 
    202   \delta _{k+\frac{1}{2}} \left[ T^{i+1} 
     202  \delta_{k+\frac{1}{2}} \left[ T^{i+1} 
    203203  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}  + \Alts _i^k a_2 s_2 \delta 
    204204  _{k+\frac{1}{2}} \left[ T^i 
    205205  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\ 
    206    +\Alts _{i+1}^k a_3 s_3 \delta _{k-\frac{1}{2}} \left[ T^{i+1} 
     206   +\Alts _{i+1}^k a_3 s_3 \delta_{k-\frac{1}{2}} \left[ T^{i+1} 
    207207  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}  +\Alts _i^k a_4 s_4 \delta 
    208208  _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}, 
     
    218218  \label{eq:i31} 
    219219  \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} =  \Alts_i^{k+1} a_{1}' 
    220   s_{1}' \delta _{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} 
    221    +\Alts_i^{k+1} a_{2}' s_{2}' \delta _{i+\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i+\frac{1}{2}}^{k+1}\\ 
    222   + \Alts_i^k a_{3}' s_{3}' \delta _{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k 
    223   +\Alts_i^k a_{4}' s_{4}' \delta _{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k. 
     220  s_{1}' \delta_{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} 
     221   +\Alts_i^{k+1} a_{2}' s_{2}' \delta_{i+\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i+\frac{1}{2}}^{k+1}\\ 
     222  + \Alts_i^k a_{3}' s_{3}' \delta_{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k 
     223  +\Alts_i^k a_{4}' s_{4}' \delta_{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k. 
    224224\end{multline} 
    225225 
     
    275275  - \left( \Alts_i^{k+1} a_{1} + \Alts_i^{k+1} a_{2} + \Alts_i^k 
    276276    a_{3} + \Alts_i^k a_{4} \right) 
    277   \frac{\delta _{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}}, 
     277  \frac{\delta_{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}}, 
    278278\end{equation} 
    279279where the areas $a_i$ are as in \autoref{eq:i13}. 
     
    647647(here the $\sigma_i$ are the slopes of the coordinate surfaces relative to geopotentials) 
    648648\autoref{eq:PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require 
    649 \begin{equation*} 
     649\[ 
    650650  |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01. 
    651 \end{equation*} 
     651\] 
    652652and then recalculate the slopes $r_i$ relative to coordinates. 
    653653Each individual triad slope 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_ASM.tex

    r10354 r10406  
    4545is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH as 
    4646additional tendency terms to the prognostic equations: 
    47 \begin{eqnarray}     \label{eq:wa_traj_iau} 
     47\begin{align}     \label{eq:wa_traj_iau} 
    4848{\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})]  
    4949\; + \; F_{i} \delta \tilde{\bf x}^{a}  
    50 \end{eqnarray} 
     50\end{align} 
    5151where $F_{i}$ is a weighting function for applying the increments $\delta\tilde{\bf x}^{a}$ defined such that 
    5252$\sum_{i=1}^{N} F_{i}=1$. 
     
    5858In addition, two different weighting functions have been implemented. 
    5959The first function employs constant weights,  
    60 \begin{eqnarray}    \label{eq:F1_i} 
     60\begin{align}    \label{eq:F1_i} 
    6161F^{(1)}_{i} 
    6262=\left\{ \begin{array}{ll} 
     
    6565   0     &    {\rm if} \; \; \; t_{i} > t_{n} 
    6666  \end{array} \right.  
    67 \end{eqnarray} 
     67\end{align} 
    6868where $M = m-n$. 
    6969The second function employs peaked hat-like weights in order to give maximum weight in the centre of the sub-window, 
    7070with the weighting reduced linearly to a small value at the window end-points: 
    71 \begin{eqnarray}     \label{eq:F2_i} 
     71\begin{align}     \label{eq:F2_i} 
    7272F^{(2)}_{i} 
    7373=\left\{ \begin{array}{ll} 
     
    7777   0                            &   {\rm if} \; \; \; t_{i}        >    t_{n} 
    7878  \end{array} \right. 
    79 \end{eqnarray} 
     79\end{align} 
    8080where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even.  
    8181The weights described by \autoref{eq:F2_i} provide a smoother transition of the analysis trajectory from 
     
    9191\begin{equation} \label{eq:asm_dmp} 
    9292\left\{ \begin{aligned} 
    93  u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta _{i+1/2} \left( {A_D 
     93 u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta_{i+1/2} \left( {A_D 
    9494\;\chi^{n-1}_I } \right) \\ 
    9595\\ 
    96  v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta _{j+1/2} \left( {A_D 
     96 v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta_{j+1/2} \left( {A_D 
    9797\;\chi^{n-1}_I } \right) \\ 
    9898\end{aligned} \right., 
     
    101101\begin{equation} \label{eq:asm_div} 
    102102\chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
    103                 \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] 
    104                        +\delta _j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). 
     103                \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] 
     104                       +\delta_j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). 
    105105\end{equation} 
    106106By the application of \autoref{eq:asm_dmp} and \autoref{eq:asm_dmp} the divergence is filtered in each iteration, 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIU.tex

    r10354 r10406  
    6161The warm layer is calculated using the model of \citet{Takaya_al_JGR10} (TAKAYA10 model hereafter). 
    6262This is a simple flux based model that is defined by the equations 
    63 \begin{eqnarray} 
     63\begin{align} 
    6464\frac{\partial{\Delta T_{\rm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p 
    6565\nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,} 
    6666\label{eq:ecmwf1} \\ 
    6767L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:ecmwf2} 
    68 \end{eqnarray} 
     68\end{align} 
    6969where $\Delta T_{\rm{wl}}$ is the temperature difference between the top of the warm layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal. 
    7070In equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water, 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DOM.tex

    r10354 r10406  
    113113\begin{subequations} \label{eq:di_mi} 
    114114\begin{align} 
    115  \delta _i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)    \\ 
     115 \delta_i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)     \\ 
    116116 \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2 
    117117\end{align} 
     
    124124These operators have the following discrete forms in the curvilinear $s$-coordinate system: 
    125125\begin{equation} \label{eq:DOM_grad} 
    126 \nabla q\equiv    \frac{1}{e_{1u} } \delta _{i+1/2 } [q] \;\,\mathbf{i} 
    127       +  \frac{1}{e_{2v} } \delta _{j+1/2 } [q] \;\,\mathbf{j} 
    128       +  \frac{1}{e_{3w}} \delta _{k+1/2} [q] \;\,\mathbf{k} 
     126\nabla q\equiv    \frac{1}{e_{1u} } \delta_{i+1/2 } [q] \;\,\mathbf{i} 
     127      +  \frac{1}{e_{2v} } \delta_{j+1/2 } [q] \;\,\mathbf{j} 
     128      +  \frac{1}{e_{3w}} \delta_{k+1/2} [q] \;\,\mathbf{k} 
    129129\end{equation} 
    130130\begin{multline} \label{eq:DOM_lap} 
     
    138138defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, and $f$-points, 
    139139and its divergence defined at $t$-points: 
    140 \begin{eqnarray}  \label{eq:DOM_curl} 
     140\begin{align}  \label{eq:DOM_curl} 
    141141 \nabla \times {\rm{\bf A}}\equiv & 
    142142      \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right)  &\ \mathbf{i} \\  
    143143 +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1  \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right)  &\ \mathbf{j} \\ 
    144144 +& \frac{1}{e_{1f} \,e_{2f}    } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2  \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right)  &\ \mathbf{k} 
    145  \end{eqnarray} 
    146 \begin{eqnarray} \label{eq:DOM_div} 
     145 \end{align} 
     146\begin{align} \label{eq:DOM_div} 
    147147\nabla \cdot \rm{\bf A} \equiv  
    148148    \frac{1}{e_{1t}\,e_{2t}\,e_{3t}} \left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] 
    149149                                           +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right] 
    150 \end{eqnarray} 
     150\end{align} 
    151151 
    152152The vertical average over the whole water column denoted by an overbar becomes for a quantity $q$ which 
     
    181181\begin{align}  
    182182\label{eq:DOM_di_adj} 
    183 \sum\limits_i { a_i \;\delta _i \left[ b \right]}  
    184    &\equiv -\sum\limits_i {\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\ 
     183\sum\limits_i { a_i \;\delta_i \left[ b \right]}  
     184   &\equiv -\sum\limits_i {\delta_{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\ 
    185185\label{eq:DOM_mi_adj} 
    186186\sum\limits_i { a_i \;\overline b^{\,i}}  
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DYN.tex

    r10354 r10406  
    1919 
    2020The prognostic ocean dynamics equation can be summarized as follows: 
    21 \begin{equation*} 
     21\[ 
    2222\text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } 
    2323                  {\text{COR} + \text{ADV}                       } 
    2424         + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} 
    25 \end{equation*} 
     25\] 
    2626NXT stands for next, referring to the time-stepping. 
    2727The first group of terms on the rhs of this equation corresponds to the Coriolis and advection terms that 
     
    7373The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows: 
    7474\begin{equation} \label{eq:divcur_cur} 
    75 \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta _{i+1/2} \left[ {e_{2v}\;v} \right] 
    76                           -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) 
     75\zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right] 
     76                          -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) 
    7777\end{equation}  
    7878 
     
    8181\begin{equation} \label{eq:divcur_div} 
    8282\chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
    83       \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u} \right] 
    84              +\delta _j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right) 
     83      \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right] 
     84             +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right) 
    8585\end{equation}  
    8686 
     
    109109\begin{aligned} 
    110110\frac{\partial \eta }{\partial t} 
    111 &\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta _i \left[ {e_{2u}\,e_{3u}\;u} \right] 
    112                                                                                   +\delta _j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} }  
    113            -    \frac{\textit{emp}}{\rho _w }   \\ 
    114 &\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho _w } 
     111&\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta_i \left[ {e_{2u}\,e_{3u}\;u} \right] 
     112                                                                                  +\delta_j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} }  
     113           -    \frac{\textit{emp}}{\rho_w }   \\ 
     114&\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho_w } 
    115115\end{aligned} 
    116116\end{equation} 
    117117where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),  
    118118expressed in Kg/m$^2$/s (which is equal to mm/s), 
    119 and $\rho _w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation). 
     119and $\rho_w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation). 
    120120If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then 
    121121\textit{emp} can be written as the evaporation minus precipitation, minus the river runoff.  
     
    355355\begin{equation} \label{eq:dynkeg} 
    356356\left\{ \begin{aligned} 
    357  -\frac{1}{2 \; e_{1u} }  & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\ 
    358  -\frac{1}{2 \; e_{2v} }  & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]     
     357 -\frac{1}{2 \; e_{1u} }  & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\ 
     358 -\frac{1}{2 \; e_{2v} }  & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]     
    359359\end{aligned} \right. 
    360360\end{equation}  
     
    373373\begin{equation} \label{eq:dynzad} 
    374374\left\{     \begin{aligned} 
    375 -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2}  \;\delta _{k+1/2} \left[ u \right]\  }^{\,k}  \\ 
    376 -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2}  \;\delta _{k+1/2} \left[ u \right]\  }^{\,k}  
     375-\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2}  \;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  \\ 
     376-\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2}  \;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  
    377377\end{aligned}         \right. 
    378378\end{equation}  
     
    414414\begin{multline} \label{eq:dyncor_metric} 
    415415f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right)  \\ 
    416    \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right]   
    417                                                                  -  \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right]  }  \ \right) 
     416   \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right]   
     417                                                                 -  \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right]  }  \ \right) 
    418418\end{multline}  
    419419 
     
    434434\begin{aligned} 
    435435\frac{1}{e_{1u}\,e_{2u}\,e_{3u}}  
    436 \left(      \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]     
    437           + \delta _{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right.  \ \;   \\ 
    438 \left.   + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\ 
     436\left(      \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]     
     437          + \delta_{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right.  \ \;   \\ 
     438\left.   + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\ 
    439439\\ 
    440440\frac{1}{e_{1v}\,e_{2v}\,e_{3v}}  
    441 \left(     \delta _{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right]  
    442          + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right.  \ \, \, \\ 
    443 \left.  + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\ 
     441\left(     \delta_{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right]  
     442         + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right.  \ \, \, \\ 
     443\left.  + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\ 
    444444\end{aligned} 
    445445\right. 
     
    490490\end{cases} 
    491491\end{equation} 
    492 where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. 
     492where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$. 
    493493This results in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error 
    494494\citep{Shchepetkin_McWilliams_OM05}. 
     
    562562\begin{equation} \label{eq:dynhpg_zco_surf} 
    563563\left\{ \begin{aligned} 
    564                \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k=km}  
    565 &= \frac{1}{2} g \   \left. \delta _{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\ 
    566                   \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k=km}  
    567 &= \frac{1}{2} g \   \left. \delta _{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\ 
     564               \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k=km}  
     565&= \frac{1}{2} g \   \left. \delta_{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\ 
     566                  \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k=km}  
     567&= \frac{1}{2} g \   \left. \delta_{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\ 
    568568\end{aligned} \right. 
    569569\end{equation}  
     
    572572\begin{equation} \label{eq:dynhpg_zco} 
    573573\left\{ \begin{aligned} 
    574                \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k}  
    575 &=             \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k-1}  
    576 +    \frac{1}{2}\;g\;   \left. \delta _{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\ 
    577                   \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k}  
    578 &=                \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k-1}  
    579 +    \frac{1}{2}\;g\;   \left. \delta _{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\ 
     574               \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k}  
     575&=             \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k-1}  
     576+    \frac{1}{2}\;g\;   \left. \delta_{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\ 
     577                  \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k}  
     578&=                \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k-1}  
     579+    \frac{1}{2}\;g\;   \left. \delta_{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\ 
    580580\end{aligned} \right. 
    581581\end{equation}  
     
    622622\begin{equation} \label{eq:dynhpg_sco} 
    623623\left\{ \begin{aligned} 
    624  - \frac{1}                   {\rho_o \, e_{1u}} \;   \delta _{i+1/2} \left[  p^h  \right]  
    625 + \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta _{i+1/2} \left[  z_t   \right]    \\ 
    626  - \frac{1}                   {\rho_o \, e_{2v}} \;   \delta _{j+1/2} \left[  p^h  \right]   
    627 + \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta _{j+1/2} \left[  z_t   \right]    \\ 
     624 - \frac{1}                   {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  p^h  \right]  
     625+ \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  z_t   \right]    \\ 
     626 - \frac{1}                   {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  p^h  \right]   
     627+ \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  z_t   \right]    \\ 
    628628\end{aligned} \right. 
    629629\end{equation}  
     
    695695\begin{equation} \label{eq:dynhpg_lf} 
    696696\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 
    697    -\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right] 
     697   -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right] 
    698698\end{equation} 
    699699 
     
    701701\begin{equation} \label{eq:dynhpg_imp} 
    702702\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 
    703    -\frac{1}{4\,\rho _o \,e_{1u} } \delta_{i+1/2} \left[ p_h^{t+\rdt} +2\,p_h^t +p_h^{t-\rdt}  \right] 
     703   -\frac{1}{4\,\rho_o \,e_{1u} } \delta_{i+1/2} \left[ p_h^{t+\rdt} +2\,p_h^t +p_h^{t-\rdt}  \right] 
    704704\end{equation} 
    705705 
     
    783783\begin{equation} \label{eq:dynspg_exp} 
    784784\left\{ \begin{aligned} 
    785  - \frac{1}{e_{1u}\,\rho_o} \;   \delta _{i+1/2} \left[  \,\rho \,\eta\,  \right]   \\ 
    786  - \frac{1}{e_{2v}\,\rho_o} \;   \delta _{j+1/2} \left[  \,\rho \,\eta\,  \right]   
     785 - \frac{1}{e_{1u}\,\rho_o} \;   \delta_{i+1/2} \left[  \,\rho \,\eta\,  \right]    \\ 
     786 - \frac{1}{e_{2v}\,\rho_o} \;   \delta_{j+1/2} \left[  \,\rho \,\eta\,  \right]   
    787787\end{aligned} \right. 
    788788\end{equation}  
     
    10931093\begin{equation} \label{eq:dynldf_lap} 
    10941094\left\{ \begin{aligned} 
    1095  D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm}  
    1096 \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[  
     1095 D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm}  
     1096\;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[  
    10971097{A_f^{lm} \;e_{3f} \zeta } \right] \\  
    10981098\\ 
    1099  D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm}  
    1100 \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[  
     1099 D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm}  
     1100\;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[  
    11011101{A_f^{lm} \;e_{3f} \zeta } \right] \\  
    11021102\end{aligned} \right. 
     
    11271127\begin{split} 
    11281128 D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ 
    1129 &  \left\{\quad  {\delta _{i+1/2} \left[ {A_T^{lm}  \left(  
    1130     {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta _{i}[u] 
    1131    -e_{2t} \; r_{1t} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}} 
     1129&  \left\{\quad  {\delta_{i+1/2} \left[ {A_T^{lm}  \left(  
     1130    {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u] 
     1131   -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}} 
    11321132 \right)} \right]}   \right. 
    11331133\\  
    11341134& \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f}  
    1135 }\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f}  
    1136 \,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}}  
     1135}\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f}  
     1136\,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}}  
    11371137\right)} \right]  
    11381138\\  
     
    11501150\\ 
    11511151 D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} }    \\ 
    1152 &  \left\{\quad  {\delta _{i+1/2} \left[ {A_f^{lm}  \left(  
    1153     {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v] 
    1154    -e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}} 
     1152&  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left(  
     1153    {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v] 
     1154   -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}} 
    11551155 \right)} \right]}   \right. 
    11561156\\  
    11571157& \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t}  
    1158 }\,\delta _{j} [v] - e_{1t}\, r_{2t}  
    1159 \,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}}  
     1158}\,\delta_{j} [v] - e_{1t}\, r_{2t}  
     1159\,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}}  
    11601160\right)} \right]  
    11611161\\  
     
    12131213\begin{equation} \label{eq:dynzdf} 
    12141214\left\{   \begin{aligned} 
    1215 D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} } 
    1216                               \ \delta _{k+1/2} [\,u\,]         \right]     \\ 
     1215D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} } 
     1216                              \ \delta_{k+1/2} [\,u\,]         \right]     \\ 
    12171217\\ 
    1218 D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} } 
    1219                               \ \delta _{k+1/2} [\,v\,]         \right] 
     1218D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} } 
     1219                              \ \delta_{k+1/2} [\,v\,]         \right] 
    12201220\end{aligned}   \right. 
    12211221\end{equation}  
     
    12281228\begin{equation} \label{eq:dynzdf_sbc} 
    12291229\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} 
    1230     = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v } 
    1231 \end{equation} 
    1232 where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress vector in 
     1230    = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v } 
     1231\end{equation} 
     1232where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in 
    12331233the (\textbf{i},\textbf{j}) coordinate system. 
    12341234The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in  
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_LBC.tex

    r10354 r10406  
    4646\begin{equation} \label{eq:lbc_aaaa} 
    4747\frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT}  
    48 }{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u  
     48}{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u  
    4949\end{equation} 
    5050(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is zero inside land and 
     
    106106  Therefore, the vorticity along the coastlines is given by:  
    107107 
    108 \begin{equation*} 
     108\[ 
    109109\zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ , 
    110 \end{equation*} 
     110\] 
    111111where $u$ and $v$ are masked fields. 
    112112Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with 
    113113the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ : 
    114114\begin{equation} \label{eq:lbc_bbbb} 
    115 \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2}  
    116 \left[ {e_{2v} \,v} \right]-\delta _{j+1/2} \left[ {e_{1u} \,u} \right]}  
     115\zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2}  
     116\left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]}  
    117117\right)\;\mbox{mask}_f  
    118118\end{equation} 
     
    279279The whole domain dimensions are named \np{jpiglo}, \np{jpjglo} and \jp{jpk}. 
    280280The relationship between the whole domain and a sub-domain is: 
    281 \begin{eqnarray}  
     281\begin{align}  
    282282      jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\ 
    283283      jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj  \label{eq:lbc_jpi} 
    284 \end{eqnarray} 
     284\end{align} 
    285285where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis. 
    286286 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_OBS.tex

    r10354 r10406  
    578578All horizontal interpolation methods implemented in NEMO estimate the value of a model variable $x$ at point $P$ as 
    579579a weighted linear combination of the values of the model variables at the grid points ${\rm A}$, ${\rm B}$ etc.: 
    580 \begin{eqnarray} 
     580\begin{align} 
    581581{x_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} &  
    582582\frac{1}{w} \left( {w_{}}_{\rm A} {x_{}}_{\rm A} +  
     
    584584                   {w_{}}_{\rm C} {x_{}}_{\rm C} +  
    585585                   {w_{}}_{\rm D} {x_{}}_{\rm D} \right) 
    586 \end{eqnarray} 
     586\end{align} 
    587587where ${w_{}}_{\rm A}$, ${w_{}}_{\rm B}$ etc. are the respective weights for the model field at 
    588588points ${\rm A}$, ${\rm B}$ etc., and $w = {w_{}}_{\rm A} + {w_{}}_{\rm B} + {w_{}}_{\rm C} + {w_{}}_{\rm D}$. 
     
    597597  For example, the weight given to the field ${x_{}}_{\rm A}$ is specified as the product of the distances 
    598598  from ${\rm P}$ to the other points: 
    599   \begin{eqnarray} 
     599  \begin{align} 
    600600  {w_{}}_{\rm A} = s({\rm P}, {\rm B}) \, s({\rm P}, {\rm C}) \, s({\rm P}, {\rm D}) 
    601601  \nonumber 
    602   \end{eqnarray} 
     602  \end{align} 
    603603  where  
    604   \begin{eqnarray} 
     604  \begin{align} 
    605605   s\left ({\rm P}, {\rm M} \right )  
    606606     & \hspace{-2mm} = \hspace{-2mm} &  
     
    610610               \cos ({\lambda_{}}_{\rm M} - {\lambda_{}}_{\rm P})  
    611611                   \right\} 
    612    \end{eqnarray} 
     612   \end{align} 
    613613   and $M$ corresponds to $B$, $C$ or $D$. 
    614614   A more stable form of the great-circle distance formula for small distances ($x$ near 1) 
    615615   involves the arcsine function ($e.g.$ see p.~101 of \citet{Daley_Barker_Bk01}: 
    616    \begin{eqnarray} 
     616   \begin{align} 
    617617   s\left( {\rm P}, {\rm M} \right)  
    618618     & \hspace{-2mm} = \hspace{-2mm} &  
    619619      \sin^{-1} \! \left\{ \sqrt{ 1 - x^2 } \right\} 
    620620   \nonumber 
    621    \end{eqnarray} 
     621   \end{align} 
    622622   where 
    623    \begin{eqnarray} 
     623   \begin{align} 
    624624    x & \hspace{-2mm} = \hspace{-2mm} &  
    625625      {a_{}}_{\rm M} {a_{}}_{\rm P} + {b_{}}_{\rm M} {b_{}}_{\rm P} + {c_{}}_{\rm M} {c_{}}_{\rm P} 
    626626   \nonumber 
    627    \end{eqnarray} 
     627   \end{align} 
    628628   and  
    629    \begin{eqnarray} 
     629   \begin{align} 
    630630      {a_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm M},  
    631631      \nonumber \\ 
     
    641641      \nonumber 
    642642   \nonumber 
    643   \end{eqnarray} 
     643  \end{align} 
    644644 
    645645\item[2.] {\bf Great-Circle distance-weighted interpolation with small angle approximation.} 
    646646  Similar to the previous interpolation but with the distance $s$ computed as 
    647   \begin{eqnarray} 
     647  \begin{align} 
    648648    s\left( {\rm P}, {\rm M} \right)  
    649649     & \hspace{-2mm} = \hspace{-2mm} &  
     
    651651      + \left( {\lambda_{}}_{\rm M} - {\lambda_{}}_{\rm P} \right)^{2} 
    652652        \cos^{2} {\phi_{}}_{\rm M} } 
    653   \end{eqnarray} 
     653  \end{align} 
    654654  where $M$ corresponds to $A$, $B$, $C$ or $D$. 
    655655 
     
    719719denote the bottom left, bottom right, top left and top right corner points of the cell, respectively.  
    720720To determine if P is inside the cell, we verify that the cross-products  
    721 \begin{eqnarray} 
     721\begin{align} 
    722722\begin{array}{lllll} 
    723723{{\bf r}_{}}_{\rm PA} \times {{\bf r}_{}}_{\rm PC} 
     
    743743\end{array} 
    744744\label{eq:cross} 
    745 \end{eqnarray} 
     745\end{align} 
    746746point in the opposite direction to the unit normal $\widehat{\bf k}$ 
    747747(i.e., that the coefficients of $\widehat{\bf k}$ are negative), 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_SBC.tex

    r10405 r10406  
    1919\begin{itemize} 
    2020\item 
    21   the two components of the surface ocean stress $\left( {\tau _u \;,\;\tau _v} \right)$ 
     21  the two components of the surface ocean stress $\left( {\tau_u \;,\;\tau_v} \right)$ 
    2222\item 
    2323  the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$ 
     
    391391The symbolic algorithm used to calculate values on the model grid is now: 
    392392 
    393 \begin{equation*} \begin{split} 
     393\[ \begin{split} 
    394394f_{m}(i,j) =  f_{m}(i,j) +& \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}      
    395395              +   \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }    \\ 
     
    397397              +   \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} } 
    398398\end{split} 
    399 \end{equation*} 
     399\] 
    400400The gradients here are taken with respect to the horizontal indices and not distances since 
    401401the spatial dependency has been absorbed into the weights. 
  • 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                              \\ 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex

    r10354 r10406  
    13291329In the above formula, $h_{ab}$ denotes the height above bottom, 
    13301330$h_{wkb}$ denotes the WKB-stretched height above bottom, defined by 
    1331 \begin{equation*} 
     1331\[ 
    13321332h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; , 
    1333 \end{equation*} 
     1333\] 
    13341334The $n_p$ parameter (given by \np{nn\_zpyc} in \ngn{namzdf\_tmx\_new} namelist) 
    13351335controls the stratification-dependence of the pycnocline-intensified dissipation. 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_conservation.tex

    r10354 r10406  
    123123 
    124124\begin{equation} \label{eq:hpg_pe} 
    125 \int_D {-\frac{1}{\rho _o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv} 
     125\int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv} 
    126126\end{equation} 
    127127 
     
    143143surface pressure forces is exactly zero: 
    144144\begin{equation} \label{eq:spg} 
    145 \int_D {-\frac{1}{\rho _o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 
     145\int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 
    146146\end{equation} 
    147147 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex

    r10354 r10406  
    6868            +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)}    \right]_h 
    6969 -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h  
    70 -\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}} 
     70-\frac{1}{\rho_o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}} 
    7171  \end{equation} 
    7272  \begin{equation}     \label{eq:PE_hydrostatic} 
     
    570570   -   \frac{1}{2\,e_1}           \frac{\partial}{\partial i} \left(  u^2+v^2   \right)  
    571571   -   \frac{1}{e_3    }  w     \frac{\partial u}{\partial k}      &      \\ 
    572    -   \frac{1}{e_1    }            \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho _o}    \right)     
     572   -   \frac{1}{e_1    }            \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho_o}    \right)     
    573573   &+   D_u^{\vect{U}}  +   F_u^{\vect{U}}      \\ 
    574574\\ 
     
    577577       -   \frac{1}{2\,e_2 }        \frac{\partial }{\partial j}\left(  u^2+v^2  \right)    
    578578       -   \frac{1}{e_3     }   w  \frac{\partial v}{\partial k}     &      \\ 
    579        -   \frac{1}{e_2     }        \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)     
     579       -   \frac{1}{e_2     }        \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)     
    580580    &+  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
    581581\end{split} \end{equation} 
     
    595595      +        \frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j}  \right) 
    596596                 - \frac{1}{e_3 }\frac{\partial \left( {         w\,u} \right)}{\partial k}    \\ 
    597 -   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho _o}   \right) 
     597-   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho_o}   \right) 
    598598+   D_u^{\vect{U}} +   F_u^{\vect{U}} 
    599599\end{multline} 
     
    607607      +        \frac{\partial \left( {e_1 \,v\,v} \right)}{\partial j}  \right) 
    608608                 - \frac{1}{e_3 } \frac{\partial \left( {        w\,v} \right)}{\partial k}    \\ 
    609 -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}    \right) 
     609-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}    \right) 
    610610+  D_v^{\vect{U}} +  F_v^{\vect{U}}  
    611611\end{multline} 
     
    771771   -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)  
    772772   -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\ 
    773    -   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)     
    774    +  g\frac{\rho }{\rho _o}\sigma _1  
     773   -   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o}    \right)     
     774   +  g\frac{\rho }{\rho_o}\sigma _1  
    775775   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad 
    776776\end{multline} 
     
    780780   -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)         
    781781   -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\ 
    782    -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)  
    783     +  g\frac{\rho }{\rho _o }\sigma _2    
     782   -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)  
     783    +  g\frac{\rho }{\rho_o }\sigma _2    
    784784   +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad 
    785785\end{multline} 
     
    796796      +        \frac{\partial \left( {e_1 \, e_3 \, v\,u} \right)}{\partial j}   \right) 
    797797   - \frac{1}{e_3 }\frac{\partial \left( { \omega\,u} \right)}{\partial k}    \\ 
    798    - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)     
    799    +  g\frac{\rho }{\rho _o}\sigma _1  
     798   - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o}    \right)     
     799   +  g\frac{\rho }{\rho_o}\sigma _1  
    800800   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad 
    801801\end{multline} 
     
    809809      +        \frac{\partial \left( {e_1 \; e_3  \,v\,v} \right)}{\partial j}   \right) 
    810810                 - \frac{1}{e_3 } \frac{\partial \left( { \omega\,v} \right)}{\partial k}    \\ 
    811    -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)  
    812     +  g\frac{\rho }{\rho _o }\sigma _2    
     811   -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)  
     812    +  g\frac{\rho }{\rho_o }\sigma _2    
    813813   +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad 
    814814\end{multline} 
     
    896896$\textit{z*} = 0$ and  $\textit{z*} = -H$ respectively. 
    897897Also the divergence of the flow field is no longer zero as shown by the continuity equation: 
    898 \begin{equation*}  
     898\[  
    899899\frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right) 
    900900      \left( r \; w\textit{*} \right) = 0  
    901 \end{equation*}  
     901\]  
    902902%} 
    903903 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

    r10354 r10406  
    106106The sea surface height is given by: 
    107107\begin{equation} \label{eq:dynspg_ssh} 
    108 \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T}  
    109 e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u}  
    110 \right]+\delta _j \left[ {e_{1v} e_{3v} v} \right]} \right)}  
     108\frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T}  
     109e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u}  
     110\right]+\delta_j \left[ {e_{1v} e_{3v} v} \right]} \right)}  
    111111\end{equation} 
    112112 
    113113where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs 
    114114(if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$, 
    115 and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. 
     115and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. 
    116116The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, 
    117117i.e. the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity).  
     
    120120\begin{equation} \label{eq:dynspg_exp} 
    121121\left\{ \begin{aligned} 
    122  - \frac{1}                      {e_{1u}} \; \delta _{i+1/2} \left[  \,\eta\,  \right]    \\ 
     122 - \frac{1}                      {e_{1u}} \; \delta_{i+1/2} \left[  \,\eta\,  \right]  \\ 
    123123 \\ 
    124  - \frac{1}                      {e_{2v}} \; \delta _{j+1/2} \left[  \,\eta\,  \right]   
     124 - \frac{1}                      {e_{2v}} \; \delta_{j+1/2} \left[  \,\eta\,  \right]   
    125125\end{aligned} \right. 
    126126\end{equation}  
    127127 
    128 Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in 
     128Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in 
    129129(\autoref{eq:dynspg_exp}).  
    130130 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_time_domain.tex

    r10354 r10406  
    163163\begin{equation} \label{eq:STP_imp_zdf} 
    164164\frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\rdt}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta  
    165 _k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta _{k+1/2} \left[ {T^{t+1}} \right]}  
     165_k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta_{k+1/2} \left[ {T^{t+1}} \right]}  
    166166\right] 
    167167\end{equation} 
     
    352352\begin{flalign*} 
    353353&\frac{\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}}{2\rdt} 
    354 \equiv \text{RHS}+ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]}  
     354\equiv \text{RHS}+ \delta_k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k+1/2} \left[ {T^{t+1}} \right]}  
    355355\right]      \\ 
    356356&\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} 
    357 \equiv {2\rdt} \ \text{RHS}+ {2\rdt} \ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]}  
     357\equiv {2\rdt} \ \text{RHS}+ {2\rdt} \ \delta_k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k+1/2} \left[ {T^{t+1}} \right]}  
    358358\right]      \\ 
    359359&\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} 
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