Changeset 10419 for NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics_zstar.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_model_basics_zstar.tex (modified) (12 diffs)

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

@@ -1 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
-% Chapter 1 ——— Model Basics
+% Chapter 1 Model Basics
 % ================================================================
 % ================================================================
@@ -20 +21 @@
 
 \colorbox{yellow}{  end of to be updated}
-\newline
 
 % from MOM4p1 documentation
 
 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 
-\begin{equation} \label{eq:PE_}
-   z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
-\end{equation}
+\[
+  % \label{eq:PE_}
+  z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
+\]
 
 This coordinate is closely related to the "eta" coordinate used in many atmospheric models
@@ -65 +66 @@
 the surface height, again so long as $\eta > -H$. 
 
-
-
 %%%
 %  essai update time splitting...
 %%%
 
-
 % ================================================================
 % Surface Pressure Gradient and Sea Surface Height
@@ -79 +77 @@
 %-----------------------------------------nam_dynspg----------------------------------------------------
 
-\nlst{nam_dynspg} 
+%\nlst{nam_dynspg} 
 %------------------------------------------------------------------------------------------------------------
 Options are defined through the \ngn{nam\_dynspg} namelist variables.
@@ -105 +103 @@
 describe the external gravity waves (typically a few ten seconds).
 The sea surface height is given by:
-\begin{equation} \label{eq:dynspg_ssh}
-\frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T} 
-e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u} 
-\right]+\delta _j \left[ {e_{1v} e_{3v} v} \right]} \right)} 
+\begin{equation}
+  \label{eq:dynspg_ssh}
+  \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T}
+    e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u}
+        \right]+\delta_j \left[ {e_{1v} e_{3v} v} \right]} \right)}
 \end{equation}
 
 where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs
 (if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$,
-and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water.
+and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water.
 The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter,
 i.e. the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity). 
 
 The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by:
-\begin{equation} \label{eq:dynspg_exp}
-\left\{ \begin{aligned}
- - \frac{1}                      {e_{1u}} \; \delta _{i+1/2} \left[  \,\eta\,  \right]    \\
- \\
- - \frac{1}                      {e_{2v}} \; \delta _{j+1/2} \left[  \,\eta\,  \right]  
-\end{aligned} \right.
+\begin{equation}
+  \label{eq:dynspg_exp}
+  \left\{
+    \begin{aligned}
+      - \frac{1}                    {e_{1u}} \; \delta_{i+1/2} \left[  \,\eta\,  \right] \\ \\
+      - \frac{1}                    {e_{2v}} \; \delta_{j+1/2} \left[  \,\eta\,  \right]
+    \end{aligned}
+  \right.
 \end{equation} 
 
-Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in
+Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in
 (\autoref{eq:dynspg_exp}). 
 
@@ -144 +145 @@
 
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
-\begin{figure}[!t]   \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts}
-\caption{    \protect\label{fig:DYN_dynspg_ts}
-  Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
-  after \citet{Griffies2004}.
-  Time increases to the right.
-  Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$.
-  The curved line represents a leap-frog time step,
-  and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
-  The vertically integrated forcing \textbf{M}(t) computed at
-  baroclinic time step t represents the interaction between the barotropic and baroclinic motions.
-  While keeping the total depth, tracer, and freshwater forcing fields fixed,
-  a leap-frog integration carries the surface height and vertically integrated velocity from
-  t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$.
-  Time averaging the barotropic fields over the N+1 time steps (endpoints included)
-  centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
-  A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of
-  the time averaged vertically integrated velocity taken from baroclinic time step t. }
-\end{center}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts}
+    \caption{
+      \protect\label{fig:DYN_dynspg_ts}
+      Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
+      after \citet{Griffies2004}.
+      Time increases to the right.
+      Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$.
+      The curved line represents a leap-frog time step,
+      and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
+      The vertically integrated forcing \textbf{M}(t) computed at
+      baroclinic time step t represents the interaction between the barotropic and baroclinic motions.
+      While keeping the total depth, tracer, and freshwater forcing fields fixed,
+      a leap-frog integration carries the surface height and vertically integrated velocity from
+      t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$.
+      Time averaging the barotropic fields over the N+1 time steps (endpoints included)
+      centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
+      A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of
+      the time averaged vertically integrated velocity taken from baroclinic time step t.
+    }
+  \end{center}
 \end{figure}
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
@@ -178 +182 @@
 the small barotropic time step $\Delta t$.
 We have
-\begin{equation} \label{eq:DYN_spg_ts_eta}
-\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
-   = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
-\end{equation}
-\begin{multline} \label{eq:DYN_spg_ts_u}
-\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\
-   = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) 
-   - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
-\end{multline}
+\[
+  % \label{eq:DYN_spg_ts_eta}
+  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
+  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
+\]
+\begin{multline*}
+  % \label{eq:DYN_spg_ts_u}
+  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\
+  = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
+    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
+\end{multline*}
 \
 
@@ -197 +203 @@
 the duration of the barotropic time stepping over a single cycle.
 This is also the time that sets the barotropic time steps via 
-\begin{equation} \label{eq:DYN_spg_ts_t}
-t_n=\tau+n\Delta t   
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_t}
+  t_n=\tau+n\Delta t   
+\]
 with $n$ an integer.
 The density scaled surface pressure is evaluated via 
-\begin{equation} \label{eq:DYN_spg_ts_ps}
-p_s^{(b)}(\tau,t_{n}) = \begin{cases}
-   g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
-   g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case} 
-   \end{cases}
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_ps}
+  p_s^{(b)}(\tau,t_{n}) =
+  \begin{cases}
+    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
+    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
+  \end{cases}
+\]
 To get started, we assume the following initial conditions 
-\begin{equation} \label{eq:DYN_spg_ts_eta}
-\begin{split}
-\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
-\\
-\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 
-\end{split}
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_eta}
+  \begin{split}
+    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\
+    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
+  \end{split}
+\]
 with 
-\begin{equation} \label{eq:DYN_spg_ts_etaF}
- \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_etaF}
+  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
+\]
 the time averaged surface height taken from the previous barotropic cycle.
 Likewise,
-\begin{equation} \label{eq:DYN_spg_ts_u}
-\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\
-\\
-\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}   
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_u}
+  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\
+  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
+\]
 with 
-\begin{equation} \label{eq:DYN_spg_ts_u}
- \overline{\textbf{U}^{(b)}(\tau)} 
-   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_u}
+  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
+\]
 the time averaged vertically integrated transport.
 Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 
@@ -237 +247 @@
 Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to
 produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 
-\begin{equation} \label{eq:DYN_spg_ts_u}
-\textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 
-   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_u}
+  \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)}
+  = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
+\]
 The surface height on the new baroclinic time step is then determined via
 a baroclinic leap-frog using the following form 
-\begin{equation} \label{eq:DYN_spg_ts_ssh}
-\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]  
+\begin{equation}
+  \label{eq:DYN_spg_ts_ssh}
+  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
 \end{equation}
 
@@ -257 +269 @@
 reasonably good maintenance of tracer conservation properties (see ??) 
 
-\begin{equation} \label{eq:DYN_spg_ts_sshf}
-\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
+\begin{equation}
+  \label{eq:DYN_spg_ts_sshf}
+  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
 \end{equation}
 Another approach tried was 
 
-\begin{equation} \label{eq:DYN_spg_ts_sshf2}
-\eta^{F}(\tau-\Delta) = \eta(\tau) 
-   + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t)
-                + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right]
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_sshf2}
+  \eta^{F}(\tau-\Delta) = \eta(\tau)
+  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t)
+    + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right]
+\]
 
 which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
@@ -274 +288 @@
 be more conservative, and so is recommended. 
 
-
-
-
-
 %-------------------------------------------------------------
 % Filtered formulation 
@@ -304 +314 @@
 In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
 
+\biblio
 
 \end{document}