Changeset 10419 for NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_SBC.tex

Timestamp:

2018-12-19T20:46:30+01:00 (5 years ago)

Author:

smasson

Message:

dev_r10164_HPC09_ESIWACE_PREP_MERGE: merge with trunk@10418, see #2133

Location:

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex

Files:

• . (modified) (1 prop)
• NEMO (modified) (1 prop)
• NEMO/subfiles (modified) (1 prop)
• NEMO/subfiles/chap_SBC.tex (modified) (24 diffs)

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_SBC.tex

@@ -1 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -9 +10 @@
 
 \newpage
-$\ $\newline    % force a new ligne
+
 %---------------------------------------namsbc--------------------------------------------------
 
 \nlst{namsbc}
 %--------------------------------------------------------------------------------------------------------------
-$\ $\newline    % force a new ligne
 
 The ocean needs six fields as surface boundary condition:
 \begin{itemize}
 \item
-  the two components of the surface ocean stress $\left( {\tau _u \;,\;\tau _v} \right)$
+  the two components of the surface ocean stress $\left( {\tau_u \;,\;\tau_v} \right)$
 \item
   the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$
@@ -161 +161 @@
 
 %-------------------------------------------------TABLE---------------------------------------------------
-\begin{table}[tb]   \begin{center}   \begin{tabular}{|l|l|l|l|}
-\hline
-Variable description             & Model variable  & Units  & point \\  \hline
-i-component of the surface current  & ssu\_m & $m.s^{-1}$   & U \\   \hline
-j-component of the surface current  & ssv\_m & $m.s^{-1}$   & V \\   \hline
-Sea surface temperature          & sst\_m & \r{}$K$      & T \\   \hline
-Sea surface salinty              & sss\_m & $psu$        & T \\   \hline
-\end{tabular}
-\caption{  \protect\label{tab:ssm}
-  Ocean variables provided by the ocean to the surface module (SBC).
-  The variable are averaged over nn{\_}fsbc time step,
-  $i.e.$ the frequency of computation of surface fluxes.}
-\end{center}   \end{table}
+\begin{table}[tb]
+  \begin{center}
+    \begin{tabular}{|l|l|l|l|}
+      \hline
+      Variable description             & Model variable  & Units  & point \\  \hline
+      i-component of the surface current  & ssu\_m & $m.s^{-1}$   & U \\   \hline
+      j-component of the surface current  & ssv\_m & $m.s^{-1}$   & V \\   \hline
+      Sea surface temperature          & sst\_m & \r{}$K$      & T \\   \hline
+      Sea surface salinty              & sss\_m & $psu$        & T \\   \hline
+    \end{tabular}
+    \caption{
+      \protect\label{tab:ssm}
+      Ocean variables provided by the ocean to the surface module (SBC).
+      The variable are averaged over nn{\_}fsbc time step,
+      $i.e.$ the frequency of computation of surface fluxes.
+    }
+  \end{center}
+\end{table}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -239 +244 @@
 
 %--------------------------------------------------TABLE--------------------------------------------------
-\begin{table}[htbp] 
-\begin{center}
-\begin{tabular}{|l|c|c|c|}
-\hline
-                         & daily or weekLLL          & monthly                   &   yearly          \\   \hline
-\np{clim}\forcode{ = .false.} & fn\_yYYYYmMMdDD.nc  &   fn\_yYYYYmMM.nc   &   fn\_yYYYY.nc  \\   \hline
-\np{clim}\forcode{ = .true.}        & not possible                  &  fn\_m??.nc             &   fn                \\   \hline
-\end{tabular}
-\end{center}
-\caption{ \protect\label{tab:fldread}
-  naming nomenclature for climatological or interannual input file, as a function of the Open/close frequency.
-  The stem name is assumed to be 'fn'.
-  For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week
-  ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon').
-  The 'YYYY', 'MM' and 'DD' should be replaced by the actual year/month/day, always coded with 4 or 2 digits.
-  Note that (1) in mpp, if the file is split over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc',
-  where 'PPPP' is the process number coded with 4 digits;
-  (2) when using AGRIF, the prefix '\_N' is added to files, where 'N'  is the child grid number.}
-\end{table}
+  \begin{table}[htbp]
+    \begin{center}
+      \begin{tabular}{|l|c|c|c|}
+        \hline
+        & daily or weekLLL         & monthly                   &   yearly          \\   \hline
+        \np{clim}\forcode{ = .false.}  & fn\_yYYYYmMMdDD.nc  &   fn\_yYYYYmMM.nc   &   fn\_yYYYY.nc  \\   \hline
+        \np{clim}\forcode{ = .true.}         & not possible                  &  fn\_m??.nc             &   fn                \\   \hline
+      \end{tabular}
+    \end{center}
+    \caption{
+      \protect\label{tab:fldread}
+      naming nomenclature for climatological or interannual input file, as a function of the Open/close frequency.
+      The stem name is assumed to be 'fn'.
+      For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week
+      ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon').
+      The 'YYYY', 'MM' and 'DD' should be replaced by the actual year/month/day, always coded with 4 or 2 digits.
+      Note that (1) in mpp, if the file is split over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc',
+      where 'PPPP' is the process number coded with 4 digits;
+      (2) when using AGRIF, the prefix '\_N' is added to files, where 'N'  is the child grid number.
+    }
+  \end{table}
 %--------------------------------------------------------------------------------------------------------------
   
@@ -378 +385 @@
 
 Symbolically, the algorithm used is:
-\begin{equation}
-f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
-\end{equation}
+\[
+  f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
+\]
 where function idx() transforms a one dimensional index src(k) into a two dimensional index,
 and wgt(1) corresponds to variable "wgt01" for example.
@@ -391 +398 @@
 The symbolic algorithm used to calculate values on the model grid is now:
 
-\begin{equation*} \begin{split}
-f_{m}(i,j) =  f_{m}(i,j) +& \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}     
-              +   \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }    \\
-              +& \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} }   
-              +   \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} }
-\end{split}
-\end{equation*}
+\[
+  \begin{split}
+    f_{m}(i,j) =  f_{m}(i,j) +& \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
+    +   \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }    \\
+    +& \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} }
+    +   \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} }
+  \end{split}
+\]
 The gradients here are taken with respect to the horizontal indices and not distances since
 the spatial dependency has been absorbed into the weights.
@@ -638 +646 @@
 
 %--------------------------------------------------TABLE--------------------------------------------------
-\begin{table}[htbp]   \label{tab:CORE}
-\begin{center}
-\begin{tabular}{|l|c|c|c|}
-\hline
-Variable desciption              & Model variable  & Units   & point \\    \hline
-i-component of the 10m air velocity & utau      & $m.s^{-1}$         & T  \\  \hline
-j-component of the 10m air velocity & vtau      & $m.s^{-1}$         & T  \\  \hline
-10m air temperature              & tair      & \r{}$K$            & T   \\ \hline
-Specific humidity             & humi      & \%              & T \\      \hline
-Incoming long wave radiation     & qlw    & $W.m^{-2}$         & T \\      \hline
-Incoming short wave radiation    & qsr    & $W.m^{-2}$         & T \\      \hline
-Total precipitation (liquid + solid)   & precip & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
-Solid precipitation              & snow      & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
-\end{tabular}
-\end{center}
+\begin{table}[htbp]
+  \label{tab:CORE}
+  \begin{center}
+    \begin{tabular}{|l|c|c|c|}
+      \hline
+      Variable desciption              & Model variable  & Units   & point \\    \hline
+      i-component of the 10m air velocity & utau      & $m.s^{-1}$         & T  \\  \hline
+      j-component of the 10m air velocity & vtau      & $m.s^{-1}$         & T  \\  \hline
+      10m air temperature              & tair      & \r{}$K$            & T   \\ \hline
+      Specific humidity             & humi      & \%              & T \\      \hline
+      Incoming long wave radiation     & qlw    & $W.m^{-2}$         & T \\      \hline
+      Incoming short wave radiation    & qsr    & $W.m^{-2}$         & T \\      \hline
+      Total precipitation (liquid + solid)   & precip & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
+      Solid precipitation              & snow      & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
+    \end{tabular}
+  \end{center}
 \end{table}
 %--------------------------------------------------------------------------------------------------------------
@@ -695 +704 @@
 
 %--------------------------------------------------TABLE--------------------------------------------------
-\begin{table}[htbp]   \label{tab:CLIO}
-\begin{center}
-\begin{tabular}{|l|l|l|l|}
-\hline
-Variable desciption           & Model variable  & Units           & point \\  \hline
-i-component of the ocean stress     & utau         & $N.m^{-2}$         & U \\   \hline
-j-component of the ocean stress     & vtau         & $N.m^{-2}$         & V \\   \hline
-Wind speed module             & vatm         & $m.s^{-1}$         & T \\   \hline
-10m air temperature              & tair         & \r{}$K$            & T \\   \hline
-Specific humidity                & humi         & \%              & T \\   \hline
-Cloud cover                   &           & \%              & T \\   \hline
-Total precipitation (liquid + solid)   & precip    & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
-Solid precipitation              & snow         & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
-\end{tabular}
-\end{center}
+\begin{table}[htbp]
+  \label{tab:CLIO}
+  \begin{center}
+    \begin{tabular}{|l|l|l|l|}
+      \hline
+      Variable desciption           & Model variable  & Units           & point \\  \hline
+      i-component of the ocean stress     & utau         & $N.m^{-2}$         & U \\   \hline
+      j-component of the ocean stress     & vtau         & $N.m^{-2}$         & V \\   \hline
+      Wind speed module             & vatm         & $m.s^{-1}$         & T \\   \hline
+      10m air temperature              & tair         & \r{}$K$            & T \\   \hline
+      Specific humidity                & humi         & \%              & T \\   \hline
+      Cloud cover                   &           & \%              & T \\   \hline
+      Total precipitation (liquid + solid)   & precip    & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
+      Solid precipitation              & snow         & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
+    \end{tabular}
+  \end{center}
 \end{table}
 %--------------------------------------------------------------------------------------------------------------
@@ -773 +783 @@
 When used to force the dynamics, the atmospheric pressure is further transformed into
 an equivalent inverse barometer sea surface height, $\eta_{ib}$, using:
-\begin{equation} \label{eq:SBC_ssh_ib}
-   \eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right) 
-\end{equation}
+\[
+  % \label{eq:SBC_ssh_ib}
+  \eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right)
+\]
 where $P_{atm}$ is the atmospheric pressure and $P_o$ a reference atmospheric pressure.
 A value of $101,000~N/m^2$ is used unless \np{ln\_ref\_apr} is set to true.
@@ -806 +817 @@
 is activated if \np{ln\_tide} and \np{ln\_tide\_pot} are both set to \np{.true.} in \ngn{nam\_tide}.
 This translates as an additional barotropic force in the momentum equations \ref{eq:PE_dyn} such that:
-\begin{equation}     \label{eq:PE_dyn_tides}
-\frac{\partial {\rm {\bf U}}_h }{\partial t}= ...
-+g\nabla (\Pi_{eq} + \Pi_{sal}) 
-\end{equation} 
+\[
+  % \label{eq:PE_dyn_tides}
+  \frac{\partial {\rm {\bf U}}_h }{\partial t}= ...
+  +g\nabla (\Pi_{eq} + \Pi_{sal})
+\]
 where $\Pi_{eq}$ stands for the equilibrium tidal forcing and $\Pi_{sal}$ a self-attraction and loading term (SAL). 
  
@@ -816 +828 @@
 For the three types of tidal frequencies it reads: \\
 Long period tides :
-\begin{equation}
-\Pi_{eq}(l)=A_{l}(1+k-h)(\frac{1}{2}-\frac{3}{2}sin^{2}\phi)cos(\omega_{l}t+V_{l})
-\end{equation}
+\[
+  \Pi_{eq}(l)=A_{l}(1+k-h)(\frac{1}{2}-\frac{3}{2}sin^{2}\phi)cos(\omega_{l}t+V_{l})
+\]
 diurnal tides :
-\begin{equation}
-\Pi_{eq}(l)=A_{l}(1+k-h)(sin 2\phi)cos(\omega_{l}t+\lambda+V_{l})
-\end{equation}
+\[
+  \Pi_{eq}(l)=A_{l}(1+k-h)(sin 2\phi)cos(\omega_{l}t+\lambda+V_{l})
+\]
 Semi-diurnal tides:
-\begin{equation}
-\Pi_{eq}(l)=A_{l}(1+k-h)(cos^{2}\phi)cos(\omega_{l}t+2\lambda+V_{l})
-\end{equation}
+\[
+  \Pi_{eq}(l)=A_{l}(1+k-h)(cos^{2}\phi)cos(\omega_{l}t+2\lambda+V_{l})
+\]
 Here $A_{l}$ is the amplitude, $\omega_{l}$ is the frequency, $\phi$ the latitude, $\lambda$ the longitude,
 $V_{0l}$ a phase shift with respect to Greenwich meridian and $t$ the time.
@@ -837 +849 @@
 (\np{ln\_read\_load=.true.}) or use a ``scalar approximation'' (\np{ln\_scal\_load=.true.}).
 In the latter case, it reads:\\
-\begin{equation}
-\Pi_{sal} = \beta \eta
-\end{equation}
+\[
+  \Pi_{sal} = \beta \eta
+\]
 where $\beta$ (\np{rn\_scal\_load}, $\approx0.09$) is a spatially constant scalar,
 often chosen to minimize tidal prediction errors.
@@ -1207 +1219 @@
 \label{subsec:SBC_wave_cdgw}
 
-The neutral surface drag coefficient provided from an external data source ($i.e.$ a wave
-model), 
+The neutral surface drag coefficient provided from an external data source ($i.e.$ a wave model), 
 can be used by setting the logical variable \np{ln\_cdgw} \forcode{= .true.} in \ngn{namsbc} namelist. 
 Then using the routine \rou{turb\_ncar} and starting from the neutral drag coefficent provided, 
@@ -1231 +1242 @@
 The Stokes drift velocity $\mathbf{U}_{st}$ in deep water can be computed from the wave spectrum and may be written as: 
 
-\begin{equation} \label{eq:sbc_wave_sdw}
-\mathbf{U}_{st} = \frac{16{\pi^3}} {g} 
-                \int_0^\infty \int_{-\pi}^{\pi} (cos{\theta},sin{\theta}) {f^3}
-                \mathrm{S}(f,\theta) \mathrm{e}^{2kz}\,\mathrm{d}\theta {d}f
-\end{equation}
+\[
+  % \label{eq:sbc_wave_sdw}
+  \mathbf{U}_{st} = \frac{16{\pi^3}} {g}
+  \int_0^\infty \int_{-\pi}^{\pi} (cos{\theta},sin{\theta}) {f^3}
+  \mathrm{S}(f,\theta) \mathrm{e}^{2kz}\,\mathrm{d}\theta {d}f
+\]
 
 where: ${\theta}$ is the wave direction, $f$ is the wave intrinsic frequency, 
@@ -1255 +1267 @@
 \citet{Breivik_al_JPO2014}:
 
-\begin{equation} \label{eq:sbc_wave_sdw_0a}
-\mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \frac{\mathrm{e}^{-2k_ez}} {1-8k_ez} 
-\end{equation}
+\[
+  % \label{eq:sbc_wave_sdw_0a}
+  \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \frac{\mathrm{e}^{-2k_ez}} {1-8k_ez} 
+\]
 
 where $k_e$ is the effective wave number which depends on the Stokes transport $T_{st}$ defined as follows:
 
-\begin{equation} \label{eq:sbc_wave_sdw_0b}
-k_e = \frac{|\mathbf{U}_{\left.st\right|_{z=0}}|} {|T_{st}|} 
-\quad \text{and }\
-T_{st} = \frac{1}{16} \bar{\omega} H_s^2 
-\end{equation}
+\[
+  % \label{eq:sbc_wave_sdw_0b}
+  k_e = \frac{|\mathbf{U}_{\left.st\right|_{z=0}}|} {|T_{st}|}
+  \quad \text{and }\
+  T_{st} = \frac{1}{16} \bar{\omega} H_s^2 
+\]
 
 where $H_s$ is the significant wave height and $\omega$ is the wave frequency.
@@ -1273 +1287 @@
 \citep{Breivik_al_OM2016}:
 
-\begin{equation} \label{eq:sbc_wave_sdw_1}
-\mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \Big[exp(2k_pz)-\beta \sqrt{-2 \pi k_pz} 
-\textit{ erf } \Big(\sqrt{-2 k_pz}\Big)\Big]
-\end{equation}
+\[
+  % \label{eq:sbc_wave_sdw_1}
+  \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \Big[exp(2k_pz)-\beta \sqrt{-2 \pi k_pz}
+  \textit{ erf } \Big(\sqrt{-2 k_pz}\Big)\Big]
+\]
 
 where $erf$ is the complementary error function and $k_p$ is the peak wavenumber.
@@ -1288 +1303 @@
 and its effect on the evolution of the sea-surface height ${\eta}$ is considered as follows: 
 
-\begin{equation} \label{eq:sbc_wave_eta_sdw}
-\frac{\partial{\eta}}{\partial{t}} = 
--\nabla_h \int_{-H}^{\eta} (\mathbf{U} + \mathbf{U}_{st}) dz 
-\end{equation}
+\[
+  % \label{eq:sbc_wave_eta_sdw}
+  \frac{\partial{\eta}}{\partial{t}} =
+  -\nabla_h \int_{-H}^{\eta} (\mathbf{U} + \mathbf{U}_{st}) dz
+\]
 
 The tracer advection equation is also modified in order for Eulerian ocean models to properly account 
@@ -1299 +1315 @@
 can be formulated as follows: 
 
-\begin{equation} \label{eq:sbc_wave_tra_sdw}
-\frac{\partial{c}}{\partial{t}} = 
-- (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c}
-\end{equation}
+\[
+  % \label{eq:sbc_wave_tra_sdw}
+  \frac{\partial{c}}{\partial{t}} =
+  - (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c}
+\]
 
 
@@ -1333 +1350 @@
 So the atmospheric stress felt by the ocean circulation $\tau_{oc,a}$ can be expressed as: 
 
-\begin{equation} \label{eq:sbc_wave_tauoc}
-\tau_{oc,a} = \tau_a - \tau_w
-\end{equation}
+\[
+  % \label{eq:sbc_wave_tauoc}
+  \tau_{oc,a} = \tau_a - \tau_w
+\]
 
 where $\tau_a$ is the atmospheric surface stress;
 $\tau_w$ is the atmospheric stress going into the waves defined as:
 
-\begin{equation} \label{eq:sbc_wave_tauw}
-\tau_w = \rho g \int {\frac{dk}{c_p} (S_{in}+S_{nl}+S_{diss})}
-\end{equation}
+\[
+  % \label{eq:sbc_wave_tauw}
+  \tau_w = \rho g \int {\frac{dk}{c_p} (S_{in}+S_{nl}+S_{diss})}
+\]
 
 where: $c_p$ is the phase speed of the gravity waves,
@@ -1375 +1394 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]    \begin{center}
-\includegraphics[width=0.8\textwidth]{Fig_SBC_diurnal}
-\caption{ \protect\label{fig:SBC_diurnal}
-  Example of recontruction of the diurnal cycle variation of short wave flux from daily mean values.
-  The reconstructed diurnal cycle (black line) is chosen as
-  the mean value of the analytical cycle (blue line) over a time step,
-  not as the mid time step value of the analytically cycle (red square).
-  From \citet{Bernie_al_CD07}.}
-\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.8\textwidth]{Fig_SBC_diurnal}
+    \caption{
+      \protect\label{fig:SBC_diurnal}
+      Example of recontruction of the diurnal cycle variation of short wave flux from daily mean values.
+      The reconstructed diurnal cycle (black line) is chosen as
+      the mean value of the analytical cycle (blue line) over a time step,
+      not as the mid time step value of the analytically cycle (red square).
+      From \citet{Bernie_al_CD07}.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -1409 +1432 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]  \begin{center}
-\includegraphics[width=0.7\textwidth]{Fig_SBC_dcy}
-\caption{ \protect\label{fig:SBC_dcy}
-  Example of recontruction of the diurnal cycle variation of short wave flux from
-  daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm).
-  The display is on (i,j) plane. }
-\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.7\textwidth]{Fig_SBC_dcy}
+    \caption{
+      \protect\label{fig:SBC_dcy}
+      Example of recontruction of the diurnal cycle variation of short wave flux from
+      daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm).
+      The display is on (i,j) plane.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -1455 +1482 @@
 On forced mode using a flux formulation (\np{ln\_flx}\forcode{ = .true.}),
 a feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$:
-\begin{equation} \label{eq:sbc_dmp_q}
-Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right)
-\end{equation}
+\[
+  % \label{eq:sbc_dmp_q}
+  Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right)
+\]
 where SST is a sea surface temperature field (observed or climatological),
 $T$ is the model surface layer temperature and
@@ -1467 +1495 @@
 Converted into an equivalent freshwater flux, it takes the following expression :
 
-\begin{equation} \label{eq:sbc_dmp_emp}
-\textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)}
-                                             {\left.S\right|_{k=1}}
+\begin{equation}
+  \label{eq:sbc_dmp_emp}
+  \textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)}
+  {\left.S\right|_{k=1}}
 \end{equation}
 
@@ -1599 +1628 @@
 % in ocean-ice models. 
 
+\biblio
 
 \end{document}