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Changeset 10414 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_SBC.tex – NEMO

Ignore:
Timestamp:
2018-12-19T00:02:00+01:00 (5 years ago)
Author:
nicolasmartin
Message:
  • Comment \label commands on maths environments for unreferenced equations and adapt the unnumbered math container accordingly (mainly switch to shortanded LateX syntax with \[ ... \])
  • Add a code trick to build subfile with its own bibliography
  • Fix right path for main LaTeX document in first line of subfiles (\documentclass[...]{subfiles})
  • Rename abstract_foreword.tex to foreword.tex
  • Fix some non-ASCII codes inserted here or there in LaTeX (\[0-9]*)
  • Made a first iteration on the indentation and alignement within math, figure and table environments to improve source code readability
File:
1 edited

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

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    910 
    1011\newpage 
    11 $\ $\newline    % force a new ligne 
     12 
    1213%---------------------------------------namsbc-------------------------------------------------- 
    1314 
    1415\nlst{namsbc} 
    1516%-------------------------------------------------------------------------------------------------------------- 
    16 $\ $\newline    % force a new ligne 
    1717 
    1818The ocean needs six fields as surface boundary condition: 
     
    161161 
    162162%-------------------------------------------------TABLE--------------------------------------------------- 
    163 \begin{table}[tb]   \begin{center}   \begin{tabular}{|l|l|l|l|} 
    164 \hline 
    165 Variable description             & Model variable  & Units  & point \\  \hline 
    166 i-component of the surface current  & ssu\_m & $m.s^{-1}$   & U \\   \hline 
    167 j-component of the surface current  & ssv\_m & $m.s^{-1}$   & V \\   \hline 
    168 Sea surface temperature          & sst\_m & \r{}$K$      & T \\   \hline 
    169 Sea surface salinty              & sss\_m & $psu$        & T \\   \hline 
    170 \end{tabular} 
    171 \caption{  \protect\label{tab:ssm} 
    172   Ocean variables provided by the ocean to the surface module (SBC). 
    173   The variable are averaged over nn{\_}fsbc time step, 
    174   $i.e.$ the frequency of computation of surface fluxes.} 
    175 \end{center}   \end{table} 
     163\begin{table}[tb] 
     164  \begin{center} 
     165    \begin{tabular}{|l|l|l|l|} 
     166      \hline 
     167      Variable description             & Model variable  & Units  & point \\  \hline 
     168      i-component of the surface current  & ssu\_m & $m.s^{-1}$   & U \\   \hline 
     169      j-component of the surface current  & ssv\_m & $m.s^{-1}$   & V \\   \hline 
     170      Sea surface temperature          & sst\_m & \r{}$K$      & T \\   \hline 
     171      Sea surface salinty              & sss\_m & $psu$        & T \\   \hline 
     172    \end{tabular} 
     173    \caption{ 
     174      \protect\label{tab:ssm} 
     175      Ocean variables provided by the ocean to the surface module (SBC). 
     176      The variable are averaged over nn{\_}fsbc time step, 
     177      $i.e.$ the frequency of computation of surface fluxes. 
     178    } 
     179  \end{center} 
     180\end{table} 
    176181%-------------------------------------------------------------------------------------------------------------- 
    177182 
     
    239244 
    240245%--------------------------------------------------TABLE-------------------------------------------------- 
    241 \begin{table}[htbp]  
    242 \begin{center} 
    243 \begin{tabular}{|l|c|c|c|} 
    244 \hline 
    245                          & daily or weekLLL          & monthly                   &   yearly          \\   \hline 
    246 \np{clim}\forcode{ = .false.} & fn\_yYYYYmMMdDD.nc  &   fn\_yYYYYmMM.nc   &   fn\_yYYYY.nc  \\   \hline 
    247 \np{clim}\forcode{ = .true.}        & not possible                  &  fn\_m??.nc             &   fn                \\   \hline 
    248 \end{tabular} 
    249 \end{center} 
    250 \caption{ \protect\label{tab:fldread} 
    251   naming nomenclature for climatological or interannual input file, as a function of the Open/close frequency. 
    252   The stem name is assumed to be 'fn'. 
    253   For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week 
    254   ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon'). 
    255   The 'YYYY', 'MM' and 'DD' should be replaced by the actual year/month/day, always coded with 4 or 2 digits. 
    256   Note that (1) in mpp, if the file is split over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc', 
    257   where 'PPPP' is the process number coded with 4 digits; 
    258   (2) when using AGRIF, the prefix '\_N' is added to files, where 'N'  is the child grid number.} 
    259 \end{table} 
     246  \begin{table}[htbp] 
     247    \begin{center} 
     248      \begin{tabular}{|l|c|c|c|} 
     249        \hline 
     250        & daily or weekLLL         & monthly                   &   yearly          \\   \hline 
     251        \np{clim}\forcode{ = .false.}  & fn\_yYYYYmMMdDD.nc  &   fn\_yYYYYmMM.nc   &   fn\_yYYYY.nc  \\   \hline 
     252        \np{clim}\forcode{ = .true.}         & not possible                  &  fn\_m??.nc             &   fn                \\   \hline 
     253      \end{tabular} 
     254    \end{center} 
     255    \caption{ 
     256      \protect\label{tab:fldread} 
     257      naming nomenclature for climatological or interannual input file, as a function of the Open/close frequency. 
     258      The stem name is assumed to be 'fn'. 
     259      For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week 
     260      ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon'). 
     261      The 'YYYY', 'MM' and 'DD' should be replaced by the actual year/month/day, always coded with 4 or 2 digits. 
     262      Note that (1) in mpp, if the file is split over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc', 
     263      where 'PPPP' is the process number coded with 4 digits; 
     264      (2) when using AGRIF, the prefix '\_N' is added to files, where 'N'  is the child grid number. 
     265    } 
     266  \end{table} 
    260267%-------------------------------------------------------------------------------------------------------------- 
    261268   
     
    378385 
    379386Symbolically, the algorithm used is: 
    380 \begin{equation} 
    381 f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))} 
    382 \end{equation} 
     387\[ 
     388  f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))} 
     389\] 
    383390where function idx() transforms a one dimensional index src(k) into a two dimensional index, 
    384391and wgt(1) corresponds to variable "wgt01" for example. 
     
    391398The symbolic algorithm used to calculate values on the model grid is now: 
    392399 
    393 \[ \begin{split} 
    394 f_{m}(i,j) =  f_{m}(i,j) +& \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}      
    395               +   \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }    \\ 
    396               +& \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} }    
    397               +   \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} } 
    398 \end{split} 
     400\[ 
     401  \begin{split} 
     402    f_{m}(i,j) =  f_{m}(i,j) +& \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))} 
     403    +   \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }    \\ 
     404    +& \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} } 
     405    +   \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} } 
     406  \end{split} 
    399407\] 
    400408The gradients here are taken with respect to the horizontal indices and not distances since 
     
    638646 
    639647%--------------------------------------------------TABLE-------------------------------------------------- 
    640 \begin{table}[htbp]   \label{tab:CORE} 
    641 \begin{center} 
    642 \begin{tabular}{|l|c|c|c|} 
    643 \hline 
    644 Variable desciption              & Model variable  & Units   & point \\    \hline 
    645 i-component of the 10m air velocity & utau      & $m.s^{-1}$         & T  \\  \hline 
    646 j-component of the 10m air velocity & vtau      & $m.s^{-1}$         & T  \\  \hline 
    647 10m air temperature              & tair      & \r{}$K$            & T   \\ \hline 
    648 Specific humidity             & humi      & \%              & T \\      \hline 
    649 Incoming long wave radiation     & qlw    & $W.m^{-2}$         & T \\      \hline 
    650 Incoming short wave radiation    & qsr    & $W.m^{-2}$         & T \\      \hline 
    651 Total precipitation (liquid + solid)   & precip & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
    652 Solid precipitation              & snow      & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
    653 \end{tabular} 
    654 \end{center} 
     648\begin{table}[htbp] 
     649  \label{tab:CORE} 
     650  \begin{center} 
     651    \begin{tabular}{|l|c|c|c|} 
     652      \hline 
     653      Variable desciption              & Model variable  & Units   & point \\    \hline 
     654      i-component of the 10m air velocity & utau      & $m.s^{-1}$         & T  \\  \hline 
     655      j-component of the 10m air velocity & vtau      & $m.s^{-1}$         & T  \\  \hline 
     656      10m air temperature              & tair      & \r{}$K$            & T   \\ \hline 
     657      Specific humidity             & humi      & \%              & T \\      \hline 
     658      Incoming long wave radiation     & qlw    & $W.m^{-2}$         & T \\      \hline 
     659      Incoming short wave radiation    & qsr    & $W.m^{-2}$         & T \\      \hline 
     660      Total precipitation (liquid + solid)   & precip & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
     661      Solid precipitation              & snow      & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
     662    \end{tabular} 
     663  \end{center} 
    655664\end{table} 
    656665%-------------------------------------------------------------------------------------------------------------- 
     
    695704 
    696705%--------------------------------------------------TABLE-------------------------------------------------- 
    697 \begin{table}[htbp]   \label{tab:CLIO} 
    698 \begin{center} 
    699 \begin{tabular}{|l|l|l|l|} 
    700 \hline 
    701 Variable desciption           & Model variable  & Units           & point \\  \hline 
    702 i-component of the ocean stress     & utau         & $N.m^{-2}$         & U \\   \hline 
    703 j-component of the ocean stress     & vtau         & $N.m^{-2}$         & V \\   \hline 
    704 Wind speed module             & vatm         & $m.s^{-1}$         & T \\   \hline 
    705 10m air temperature              & tair         & \r{}$K$            & T \\   \hline 
    706 Specific humidity                & humi         & \%              & T \\   \hline 
    707 Cloud cover                   &           & \%              & T \\   \hline 
    708 Total precipitation (liquid + solid)   & precip    & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
    709 Solid precipitation              & snow         & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
    710 \end{tabular} 
    711 \end{center} 
     706\begin{table}[htbp] 
     707  \label{tab:CLIO} 
     708  \begin{center} 
     709    \begin{tabular}{|l|l|l|l|} 
     710      \hline 
     711      Variable desciption           & Model variable  & Units           & point \\  \hline 
     712      i-component of the ocean stress     & utau         & $N.m^{-2}$         & U \\   \hline 
     713      j-component of the ocean stress     & vtau         & $N.m^{-2}$         & V \\   \hline 
     714      Wind speed module             & vatm         & $m.s^{-1}$         & T \\   \hline 
     715      10m air temperature              & tair         & \r{}$K$            & T \\   \hline 
     716      Specific humidity                & humi         & \%              & T \\   \hline 
     717      Cloud cover                   &           & \%              & T \\   \hline 
     718      Total precipitation (liquid + solid)   & precip    & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
     719      Solid precipitation              & snow         & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
     720    \end{tabular} 
     721  \end{center} 
    712722\end{table} 
    713723%-------------------------------------------------------------------------------------------------------------- 
     
    773783When used to force the dynamics, the atmospheric pressure is further transformed into 
    774784an equivalent inverse barometer sea surface height, $\eta_{ib}$, using: 
    775 \begin{equation} \label{eq:SBC_ssh_ib} 
    776    \eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right)  
    777 \end{equation} 
     785\[ 
     786  % \label{eq:SBC_ssh_ib} 
     787  \eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right) 
     788\] 
    778789where $P_{atm}$ is the atmospheric pressure and $P_o$ a reference atmospheric pressure. 
    779790A value of $101,000~N/m^2$ is used unless \np{ln\_ref\_apr} is set to true. 
     
    806817is activated if \np{ln\_tide} and \np{ln\_tide\_pot} are both set to \np{.true.} in \ngn{nam\_tide}. 
    807818This translates as an additional barotropic force in the momentum equations \ref{eq:PE_dyn} such that: 
    808 \begin{equation}     \label{eq:PE_dyn_tides} 
    809 \frac{\partial {\rm {\bf U}}_h }{\partial t}= ... 
    810 +g\nabla (\Pi_{eq} + \Pi_{sal})  
    811 \end{equation}  
     819\[ 
     820  % \label{eq:PE_dyn_tides} 
     821  \frac{\partial {\rm {\bf U}}_h }{\partial t}= ... 
     822  +g\nabla (\Pi_{eq} + \Pi_{sal}) 
     823\] 
    812824where $\Pi_{eq}$ stands for the equilibrium tidal forcing and $\Pi_{sal}$ a self-attraction and loading term (SAL).  
    813825  
     
    816828For the three types of tidal frequencies it reads: \\ 
    817829Long period tides : 
    818 \begin{equation} 
    819 \Pi_{eq}(l)=A_{l}(1+k-h)(\frac{1}{2}-\frac{3}{2}sin^{2}\phi)cos(\omega_{l}t+V_{l}) 
    820 \end{equation} 
     830\[ 
     831  \Pi_{eq}(l)=A_{l}(1+k-h)(\frac{1}{2}-\frac{3}{2}sin^{2}\phi)cos(\omega_{l}t+V_{l}) 
     832\] 
    821833diurnal tides : 
    822 \begin{equation} 
    823 \Pi_{eq}(l)=A_{l}(1+k-h)(sin 2\phi)cos(\omega_{l}t+\lambda+V_{l}) 
    824 \end{equation} 
     834\[ 
     835  \Pi_{eq}(l)=A_{l}(1+k-h)(sin 2\phi)cos(\omega_{l}t+\lambda+V_{l}) 
     836\] 
    825837Semi-diurnal tides: 
    826 \begin{equation} 
    827 \Pi_{eq}(l)=A_{l}(1+k-h)(cos^{2}\phi)cos(\omega_{l}t+2\lambda+V_{l}) 
    828 \end{equation} 
     838\[ 
     839  \Pi_{eq}(l)=A_{l}(1+k-h)(cos^{2}\phi)cos(\omega_{l}t+2\lambda+V_{l}) 
     840\] 
    829841Here $A_{l}$ is the amplitude, $\omega_{l}$ is the frequency, $\phi$ the latitude, $\lambda$ the longitude, 
    830842$V_{0l}$ a phase shift with respect to Greenwich meridian and $t$ the time. 
     
    837849(\np{ln\_read\_load=.true.}) or use a ``scalar approximation'' (\np{ln\_scal\_load=.true.}). 
    838850In the latter case, it reads:\\ 
    839 \begin{equation} 
    840 \Pi_{sal} = \beta \eta 
    841 \end{equation} 
     851\[ 
     852  \Pi_{sal} = \beta \eta 
     853\] 
    842854where $\beta$ (\np{rn\_scal\_load}, $\approx0.09$) is a spatially constant scalar, 
    843855often chosen to minimize tidal prediction errors. 
     
    12301242The Stokes drift velocity $\mathbf{U}_{st}$ in deep water can be computed from the wave spectrum and may be written as:  
    12311243 
    1232 \begin{equation} \label{eq:sbc_wave_sdw} 
    1233 \mathbf{U}_{st} = \frac{16{\pi^3}} {g}  
    1234                 \int_0^\infty \int_{-\pi}^{\pi} (cos{\theta},sin{\theta}) {f^3} 
    1235                 \mathrm{S}(f,\theta) \mathrm{e}^{2kz}\,\mathrm{d}\theta {d}f 
    1236 \end{equation} 
     1244\[ 
     1245  % \label{eq:sbc_wave_sdw} 
     1246  \mathbf{U}_{st} = \frac{16{\pi^3}} {g} 
     1247  \int_0^\infty \int_{-\pi}^{\pi} (cos{\theta},sin{\theta}) {f^3} 
     1248  \mathrm{S}(f,\theta) \mathrm{e}^{2kz}\,\mathrm{d}\theta {d}f 
     1249\] 
    12371250 
    12381251where: ${\theta}$ is the wave direction, $f$ is the wave intrinsic frequency,  
     
    12541267\citet{Breivik_al_JPO2014}: 
    12551268 
    1256 \begin{equation} \label{eq:sbc_wave_sdw_0a} 
    1257 \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \frac{\mathrm{e}^{-2k_ez}} {1-8k_ez}  
    1258 \end{equation} 
     1269\[ 
     1270  % \label{eq:sbc_wave_sdw_0a} 
     1271  \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \frac{\mathrm{e}^{-2k_ez}} {1-8k_ez}  
     1272\] 
    12591273 
    12601274where $k_e$ is the effective wave number which depends on the Stokes transport $T_{st}$ defined as follows: 
    12611275 
    1262 \begin{equation} \label{eq:sbc_wave_sdw_0b} 
    1263 k_e = \frac{|\mathbf{U}_{\left.st\right|_{z=0}}|} {|T_{st}|}  
    1264 \quad \text{and }\ 
    1265 T_{st} = \frac{1}{16} \bar{\omega} H_s^2  
    1266 \end{equation} 
     1276\[ 
     1277  % \label{eq:sbc_wave_sdw_0b} 
     1278  k_e = \frac{|\mathbf{U}_{\left.st\right|_{z=0}}|} {|T_{st}|} 
     1279  \quad \text{and }\ 
     1280  T_{st} = \frac{1}{16} \bar{\omega} H_s^2  
     1281\] 
    12671282 
    12681283where $H_s$ is the significant wave height and $\omega$ is the wave frequency. 
     
    12721287\citep{Breivik_al_OM2016}: 
    12731288 
    1274 \begin{equation} \label{eq:sbc_wave_sdw_1} 
    1275 \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \Big[exp(2k_pz)-\beta \sqrt{-2 \pi k_pz}  
    1276 \textit{ erf } \Big(\sqrt{-2 k_pz}\Big)\Big] 
    1277 \end{equation} 
     1289\[ 
     1290  % \label{eq:sbc_wave_sdw_1} 
     1291  \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \Big[exp(2k_pz)-\beta \sqrt{-2 \pi k_pz} 
     1292  \textit{ erf } \Big(\sqrt{-2 k_pz}\Big)\Big] 
     1293\] 
    12781294 
    12791295where $erf$ is the complementary error function and $k_p$ is the peak wavenumber. 
     
    12871303and its effect on the evolution of the sea-surface height ${\eta}$ is considered as follows:  
    12881304 
    1289 \begin{equation} \label{eq:sbc_wave_eta_sdw} 
    1290 \frac{\partial{\eta}}{\partial{t}} =  
    1291 -\nabla_h \int_{-H}^{\eta} (\mathbf{U} + \mathbf{U}_{st}) dz  
    1292 \end{equation} 
     1305\[ 
     1306  % \label{eq:sbc_wave_eta_sdw} 
     1307  \frac{\partial{\eta}}{\partial{t}} = 
     1308  -\nabla_h \int_{-H}^{\eta} (\mathbf{U} + \mathbf{U}_{st}) dz 
     1309\] 
    12931310 
    12941311The tracer advection equation is also modified in order for Eulerian ocean models to properly account  
     
    12981315can be formulated as follows:  
    12991316 
    1300 \begin{equation} \label{eq:sbc_wave_tra_sdw} 
    1301 \frac{\partial{c}}{\partial{t}} =  
    1302 - (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c} 
    1303 \end{equation} 
     1317\[ 
     1318  % \label{eq:sbc_wave_tra_sdw} 
     1319  \frac{\partial{c}}{\partial{t}} = 
     1320  - (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c} 
     1321\] 
    13041322 
    13051323 
     
    13321350So the atmospheric stress felt by the ocean circulation $\tau_{oc,a}$ can be expressed as:  
    13331351 
    1334 \begin{equation} \label{eq:sbc_wave_tauoc} 
    1335 \tau_{oc,a} = \tau_a - \tau_w 
    1336 \end{equation} 
     1352\[ 
     1353  % \label{eq:sbc_wave_tauoc} 
     1354  \tau_{oc,a} = \tau_a - \tau_w 
     1355\] 
    13371356 
    13381357where $\tau_a$ is the atmospheric surface stress; 
    13391358$\tau_w$ is the atmospheric stress going into the waves defined as: 
    13401359 
    1341 \begin{equation} \label{eq:sbc_wave_tauw} 
    1342 \tau_w = \rho g \int {\frac{dk}{c_p} (S_{in}+S_{nl}+S_{diss})} 
    1343 \end{equation} 
     1360\[ 
     1361  % \label{eq:sbc_wave_tauw} 
     1362  \tau_w = \rho g \int {\frac{dk}{c_p} (S_{in}+S_{nl}+S_{diss})} 
     1363\] 
    13441364 
    13451365where: $c_p$ is the phase speed of the gravity waves, 
     
    13741394 
    13751395%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1376 \begin{figure}[!t]    \begin{center} 
    1377 \includegraphics[width=0.8\textwidth]{Fig_SBC_diurnal} 
    1378 \caption{ \protect\label{fig:SBC_diurnal} 
    1379   Example of recontruction of the diurnal cycle variation of short wave flux from daily mean values. 
    1380   The reconstructed diurnal cycle (black line) is chosen as 
    1381   the mean value of the analytical cycle (blue line) over a time step, 
    1382   not as the mid time step value of the analytically cycle (red square). 
    1383   From \citet{Bernie_al_CD07}.} 
    1384 \end{center}   \end{figure} 
     1396\begin{figure}[!t] 
     1397  \begin{center} 
     1398    \includegraphics[width=0.8\textwidth]{Fig_SBC_diurnal} 
     1399    \caption{ 
     1400      \protect\label{fig:SBC_diurnal} 
     1401      Example of recontruction of the diurnal cycle variation of short wave flux from daily mean values. 
     1402      The reconstructed diurnal cycle (black line) is chosen as 
     1403      the mean value of the analytical cycle (blue line) over a time step, 
     1404      not as the mid time step value of the analytically cycle (red square). 
     1405      From \citet{Bernie_al_CD07}. 
     1406    } 
     1407  \end{center} 
     1408\end{figure} 
    13851409%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    13861410 
     
    14081432 
    14091433%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1410 \begin{figure}[!t]  \begin{center} 
    1411 \includegraphics[width=0.7\textwidth]{Fig_SBC_dcy} 
    1412 \caption{ \protect\label{fig:SBC_dcy} 
    1413   Example of recontruction of the diurnal cycle variation of short wave flux from 
    1414   daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm). 
    1415   The display is on (i,j) plane. } 
    1416 \end{center}   \end{figure} 
     1434\begin{figure}[!t] 
     1435  \begin{center} 
     1436    \includegraphics[width=0.7\textwidth]{Fig_SBC_dcy} 
     1437    \caption{ 
     1438      \protect\label{fig:SBC_dcy} 
     1439      Example of recontruction of the diurnal cycle variation of short wave flux from 
     1440      daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm). 
     1441      The display is on (i,j) plane. 
     1442    } 
     1443  \end{center} 
     1444\end{figure} 
    14171445%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    14181446 
     
    14541482On forced mode using a flux formulation (\np{ln\_flx}\forcode{ = .true.}), 
    14551483a feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$: 
    1456 \begin{equation} \label{eq:sbc_dmp_q} 
    1457 Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right) 
    1458 \end{equation} 
     1484\[ 
     1485  % \label{eq:sbc_dmp_q} 
     1486  Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right) 
     1487\] 
    14591488where SST is a sea surface temperature field (observed or climatological), 
    14601489$T$ is the model surface layer temperature and 
     
    14661495Converted into an equivalent freshwater flux, it takes the following expression : 
    14671496 
    1468 \begin{equation} \label{eq:sbc_dmp_emp} 
    1469 \textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)} 
    1470                                              {\left.S\right|_{k=1}} 
     1497\begin{equation} 
     1498  \label{eq:sbc_dmp_emp} 
     1499  \textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)} 
     1500  {\left.S\right|_{k=1}} 
    14711501\end{equation} 
    14721502 
     
    15981628% in ocean-ice models.  
    15991629 
     1630\biblio 
    16001631 
    16011632\end{document} 
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