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Changeset 10406 – NEMO

Changeset 10406


Ignore:
Timestamp:
2018-12-18T11:25:09+01:00 (4 years ago)
Author:
nicolasmartin
Message:

Edition of math environments

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

Legend:

Unmodified
Added
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  • NEMO/trunk/doc/latex

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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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