Changeset 12065 for NEMO/branches/2019/dev_r10742_ENHANCE-12_SimonM-Tides/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

Timestamp:

2019-12-05T12:06:36+01:00 (4 years ago)

Author:

smueller

Message:

Synchronizing with ^{/NEMO/trunk@12055 (ticket #2194)}

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NEMO/branches/2019/dev_r10742_ENHANCE-12_SimonM-Tides/doc

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• . (modified) (2 props)
• latex (modified) (1 prop)
• latex/NEMO (modified) (2 props)
• latex/NEMO/subfiles (modified) (1 prop)
• latex/NEMO/subfiles/chap_model_basics_zstar.tex (modified) (15 diffs)

NEMO/branches/2019/dev_r10742_ENHANCE-12_SimonM-Tides/doc/latex/NEMO/subfiles

@@ -1 @@
-*.aux
-*.bbl
-*.blg
-*.dvi
-*.fdb*
-*.fls
-*.idx
+*.ind
 *.ilg
-*.ind
-*.log
-*.maf
-*.mtc*
-*.out
-*.pdf
-*.toc
-_minted-*

@@ -1 @@
-*.aux
-*.bbl
-*.blg
-*.dvi
-*.fdb*
-*.fls
-*.idx
+*.ind
 *.ilg
-*.ind
-*.log
-*.maf
-*.mtc*
-*.out
-*.pdf
-*.toc
-_minted-*

NEMO/branches/2019/dev_r10742_ENHANCE-12_SimonM-Tides/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

@@ -2 +2 @@
 
 \begin{document}
-% ================================================================
-% Chapter 1 Model Basics
-% ================================================================
-% ================================================================
-% Curvilinear \zstar- \sstar-coordinate System
-% ================================================================
+
 \chapter{ essai \zstar \sstar}
+
+\thispagestyle{plain}
+
+\chaptertoc
+
+\paragraph{Changes record} ~\\
+
+{\footnotesize
+  \begin{tabularx}{\textwidth}{l||X|X}
+    Release & Author(s) & Modifications \\
+    \hline
+    {\em   4.0} & {\em ...} & {\em ...} \\
+    {\em   3.6} & {\em ...} & {\em ...} \\
+    {\em   3.4} & {\em ...} & {\em ...} \\
+    {\em <=3.4} & {\em ...} & {\em ...}
+  \end{tabularx}
+}
+
+\clearpage
+
+%% =================================================================================================
 \section{Curvilinear \zstar- or \sstar coordinate system}
 
-% -------------------------------------------------------------------------------------------------------------
-% ????
-% -------------------------------------------------------------------------------------------------------------
-
 \colorbox{yellow}{ to be updated }
 
 In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account.
-These coordinates systems is presented in a report \citep{Levier2007} available on the \NEMO web site. 
+These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site.
 
 \colorbox{yellow}{  end of to be updated}
@@ -24 +36 @@
 % from MOM4p1 documentation
 
-To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 
-\[
-  % \label{eq:PE_}
+To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate
+\[
+  % \label{eq:MBZ_PE_}
   z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
 \]
@@ -40 +52 @@
 the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces.
 These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to
-terrain following sigma models discussed in \autoref{subsec:PE_sco}. 
+terrain following sigma models discussed in \autoref{subsec:MB_sco}.
 Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in
 an unforced ocean starting from rest, regardless the bottom topography.
@@ -49 +61 @@
 neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models
 (see Chapters 13-16 of Griffies (2004) for a discussion of neutral physics in $z$-models,
-as well as  \autoref{sec:LDF_slp} in this document for treatment in \NEMO). 
+as well as  \autoref{sec:LDF_slp} in this document for treatment in \NEMO).
 
 The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$.
 Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$.
-This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. 
+This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$.
 
 Because $z^\star$ has a time independent range, all grid cells have static increments ds,
-and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H. 
+and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H.
 The $z^\star$ coordinate is therefore invisible to undulations of the free surface,
 since it moves along with the free surface.
@@ -64 +76 @@
 Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that
 allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of
-the surface height, again so long as $\eta > -H$. 
-
-%%%
+the surface height, again so long as $\eta > -H$.
+
 %  essai update time splitting...
-%%%
-
-% ================================================================
-% Surface Pressure Gradient and Sea Surface Height
-% ================================================================
-\section{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})}
-\label{sec:DYN_hpg_spg}
-%-----------------------------------------nam_dynspg----------------------------------------------------
-
-%\nlst{nam_dynspg} 
-%------------------------------------------------------------------------------------------------------------
-Options are defined through the \ngn{nam\_dynspg} namelist variables.
-The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}).
+
+%% =================================================================================================
+\section[Surface pressure gradient and sea surface heigth (\textit{dynspg.F90})]{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})}
+\label{sec:MBZ_dyn_hpg_spg}
+
+%\nlst{nam_dynspg}
+Options are defined through the \nam{_dynspg}{\_dynspg} namelist variables.
+The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}).
 The main distinction is between the fixed volume case (linear free surface or rigid lid) and
 the variable volume case (nonlinear free surface, \key{vvl} is active).
-In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}),
+In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}),
 the vertical scale factors $e_{3}$ are fixed in time,
-while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent.
+while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent.
 With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
 which imposes a very small time step when an explicit time stepping is used.
 Two methods are proposed to allow a longer time step for the three-dimensional equations:
-the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:PE_flt}),
+the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}),
 and the split-explicit free surface described below.
 The extra term introduced in the filtered method is calculated implicitly,
 so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
 
-%-------------------------------------------------------------
 % Explicit
-%-------------------------------------------------------------
-\subsubsection{Explicit (\protect\key{dynspg\_exp})}
-\label{subsec:DYN_spg_exp}
+%% =================================================================================================
+\subsubsection[Explicit (\texttt{\textbf{key\_dynspg\_exp}})]{Explicit (\protect\key{dynspg\_exp})}
+\label{subsec:MBZ_dyn_spg_exp}
 
 In the explicit free surface formulation, the model time step is chosen small enough to
@@ -104 +109 @@
 The sea surface height is given by:
 \begin{equation}
-  \label{eq:dynspg_ssh}
+  \label{eq:MBZ_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}
@@ -114 +119 @@
 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,
-(\ie the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity). 
+(\ie\ the velocity appearing in (\autoref{eq:DYN_spg_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}
+  \label{eq:MBZ_dynspg_exp}
   \left\{
     \begin{aligned}
@@ -125 +130 @@
     \end{aligned}
   \right.
-\end{equation} 
+\end{equation}
 
 Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in
-(\autoref{eq:dynspg_exp}). 
-
-%-------------------------------------------------------------
+(\autoref{eq:DYN_spg_exp}).
+
 % Split-explicit time-stepping
-%-------------------------------------------------------------
-\subsubsection{Split-explicit time-stepping (\protect\key{dynspg\_ts})}
-\label{subsec:DYN_spg_ts}
-%--------------------------------------------namdom----------------------------------------------------
-
-\nlst{namdom} 
-%--------------------------------------------------------------------------------------------------------------
-The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}.
+%% =================================================================================================
+\subsubsection[Split-explicit time-stepping (\texttt{\textbf{key\_dynspg\_ts}})]{Split-explicit time-stepping (\protect\key{dynspg\_ts})}
+\label{subsec:MBZ_dyn_spg_ts}
+
+The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}.
 The general idea is to solve the free surface equation with a small time step,
 while the three dimensional prognostic variables are solved with a longer time step that
-is a multiple of \np{rdtbt} in the \ngn{namdom} namelist (Figure III.3). 
-
-%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
+is a multiple of \np{rdtbt}{rdtbt} in the \nam{dom}{dom} namelist (Figure III.3).
+
 \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}
+  \centering
+  %\includegraphics[width=0.66\textwidth]{MBZ_DYN_dynspg_ts}
+  \caption[Schematic of the split-explicit time stepping scheme for
+  the barotropic and baroclinic modes, after \citet{Griffies2004?}]{
+    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.}
+  \label{fig:MBZ_dyn_dynspg_ts}
 \end{figure}
-%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
 
 The split-explicit formulation has a damping effect on external gravity waves,
-which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in
-the case of an analytical barotropic Kelvin wave. 
+which is weaker than the filtered free surface but still significant as shown by \citet{levier.treguier.ea_rpt07} in
+the case of an analytical barotropic Kelvin wave.
 
 %from griffies book: .....   copy past !
@@ -183 +182 @@
 We have
 \[
-  % \label{eq:DYN_spg_ts_eta}
+  % \label{eq:MBZ_dyn_spg_ts_eta}
   \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
-  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
+  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
 \]
 \begin{multline*}
-  % \label{eq:DYN_spg_ts_u}
+  % \label{eq:MBZ_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})
@@ -202 +201 @@
 the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for
 the duration of the barotropic time stepping over a single cycle.
-This is also the time that sets the barotropic time steps via 
-\[
-  % \label{eq:DYN_spg_ts_t}
-  t_n=\tau+n\Delta t   
+This is also the time that sets the barotropic time steps via
+\[
+  % \label{eq:MBZ_dyn_spg_ts_t}
+  t_n=\tau+n\Delta t
 \]
 with $n$ an integer.
-The density scaled surface pressure is evaluated via 
-\[
-  % \label{eq:DYN_spg_ts_ps}
+The density scaled surface pressure is evaluated via
+\[
+  % \label{eq:MBZ_dyn_spg_ts_ps}
   p_s^{(b)}(\tau,t_{n}) =
   \begin{cases}
@@ -217 +216 @@
   \end{cases}
 \]
-To get started, we assume the following initial conditions 
-\[
-  % \label{eq:DYN_spg_ts_eta}
+To get started, we assume the following initial conditions
+\[
+  % \label{eq:MBZ_dyn_spg_ts_eta}
   \begin{split}
     \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\
@@ -225 +224 @@
   \end{split}
 \]
-with 
-\[
-  % \label{eq:DYN_spg_ts_etaF}
+with
+\[
+  % \label{eq:MBZ_dyn_spg_ts_etaF}
   \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
 \]
@@ -233 +232 @@
 Likewise,
 \[
-  % \label{eq:DYN_spg_ts_u}
+  % \label{eq:MBZ_dyn_spg_ts_u}
   \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\
   \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
 \]
-with 
-\[
-  % \label{eq:DYN_spg_ts_u}
+with
+\[
+  % \label{eq:MBZ_dyn_spg_ts_u}
   \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
 \]
 the time averaged vertically integrated transport.
-Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 
+Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
 
 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 $\tau + \Delta \tau$ 
-\[
-  % \label{eq:DYN_spg_ts_u}
+produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$
+\[
+  % \label{eq:MBZ_dyn_spg_ts_u}
   \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)}
   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
 \]
 The surface height on the new baroclinic time step is then determined via
-a baroclinic leap-frog using the following form 
+a baroclinic leap-frog using the following form
 \begin{equation}
-  \label{eq:DYN_spg_ts_ssh}
+  \label{eq:MBZ_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}
@@ -261 +260 @@
 The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
 the mass/volume budgets and the tracer budgets.
-More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 
- 
+More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
+
 In general, some form of time filter is needed to maintain integrity of the surface height field due to
-the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}.
+the leap-frog splitting mode in equation \autoref{eq:MBZ_dyn_spg_ts_ssh}.
 We have tried various forms of such filtering,
 with the following method discussed in Griffies et al. (2001) chosen due to its stability and
-reasonably good maintenance of tracer conservation properties (see ??) 
+reasonably good maintenance of tracer conservation properties (see ??)
 
 \begin{equation}
-  \label{eq:DYN_spg_ts_sshf}
+  \label{eq:MBZ_dyn_spg_ts_sshf}
   \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
 \end{equation}
-Another approach tried was 
-
-\[
-  % \label{eq:DYN_spg_ts_sshf2}
+Another approach tried was
+
+\[
+  % \label{eq:MBZ_dyn_spg_ts_sshf2}
   \eta^{F}(\tau-\Delta) = \eta(\tau)
   + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t)
@@ -285 +284 @@
 This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and
 surface height time filtering (see ?? for more complete discussion).
-However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to
-be more conservative, and so is recommended. 
-
-%-------------------------------------------------------------
-% Filtered formulation 
-%-------------------------------------------------------------
-\subsubsection{Filtered formulation (\protect\key{dynspg\_flt})}
-\label{subsec:DYN_spg_flt}
-
-The filtered formulation follows the \citet{Roullet2000} implementation.
+However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:MBZ_dyn_spg_ts_sshf} was found to
+be more conservative, and so is recommended.
+
+% Filtered formulation
+%% =================================================================================================
+\subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})]{Filtered formulation (\protect\key{dynspg\_flt})}
+\label{subsec:MBZ_dyn_spg_flt}
+
+The filtered formulation follows the \citet{Roullet2000?} implementation.
 The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly.
 The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
-The amplitude of the extra term is given by the namelist variable \np{rnu}.
-The default value is 1, as recommended by \citet{Roullet2000}
-
-\colorbox{red}{\np{rnu}\forcode{ = 1} to be suppressed from namelist !}
-
-%-------------------------------------------------------------
-% Non-linear free surface formulation 
-%-------------------------------------------------------------
-\subsection{Non-linear free surface formulation (\protect\key{vvl})}
-\label{subsec:DYN_spg_vvl}
+The amplitude of the extra term is given by the namelist variable \np{rnu}{rnu}.
+The default value is 1, as recommended by \citet{Roullet2000?}
+
+\colorbox{red}{\np[=1]{rnu}{rnu} to be suppressed from namelist !}
+
+% Non-linear free surface formulation
+%% =================================================================================================
+\subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})]{Non-linear free surface formulation (\protect\key{vvl})}
+\label{subsec:MBZ_dyn_spg_vvl}
 
 In the non-linear free surface formulation, the variations of volume are fully taken into account.
-This option is presented in a report \citep{Levier2007} available on the NEMO web site.
+This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site.
 The three time-stepping methods (explicit, split-explicit and filtered) are the same as in
-\autoref{DYN_spg_linear} except that the ocean depth is now time-dependent.
+\autoref{?:DYN_spg_linear?} except that the ocean depth is now time-dependent.
 In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
 
-\biblio
-
-\pindex
+\subinc{\input{../../global/epilogue}}
 
 \end{document}

@@ -1 @@
-*.aux
-*.bbl
-*.blg
-*.dvi
-*.fdb*
-*.fls
-*.idx
+*.ind
 *.ilg
-*.ind
-*.log
-*.maf
-*.mtc*
-*.out
-*.pdf
-*.toc
-_minted-*