Changeset 6289 for trunk/DOC/TexFiles/Chapters/Chap_Model_Basics.tex

Timestamp:

2016-02-05T00:47:05+01:00 (8 years ago)

Author:

gm

Message:

#1673 DOC of the trunk - Update, see associated wiki page for description

File:

• trunk/DOC/TexFiles/Chapters/Chap_Model_Basics.tex (modified) (8 diffs)

trunk/DOC/TexFiles/Chapters/Chap_Model_Basics.tex

@@ -257 +257 @@
 
 %\newpage
-%$\ $\newline    % force a new ligne
+%$\ $\newline    % force a new line
 
 % ================================================================
@@ -988 +988 @@
 \label{PE_zco_tilde}
 
-The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM10s}.
+The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM11}.
 It is available in \NEMO since the version 3.4. Nevertheless, it is currently not robust enough 
 to be used in all possible configurations. Its use is therefore not recommended.
-We 
+
 
 \newpage 
@@ -1113 +1113 @@
 
 All these parameterisations of subgrid scale physics have advantages and 
-drawbacks. There are not all available in \NEMO. In the $z$-coordinate 
-formulation, five options are offered for active tracers (temperature and 
-salinity): second order geopotential operator, second order isoneutral 
-operator, \citet{Gent1990} parameterisation, fourth order 
-geopotential operator, and various slightly diffusive advection schemes. 
-The same options are available for momentum, except 
-\citet{Gent1990} parameterisation which only involves tracers. In the
-$s$-coordinate formulation, additional options are offered for tracers: second 
-order operator acting along $s-$surfaces, and for momentum: fourth order 
-operator acting along $s-$surfaces (see \S\ref{LDF}).
-
-\subsubsection{Lateral second order tracer diffusive operator}
-
-The lateral second order tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}):
+drawbacks. There are not all available in \NEMO. For active tracers (temperature and 
+salinity) the main ones are: Laplacian and bilaplacian operators acting along 
+geopotential or iso-neutral surfaces, \citet{Gent1990} parameterisation, 
+and various slightly diffusive advection schemes. 
+For momentum, the main ones are: Laplacian and bilaplacian operators acting along 
+geopotential surfaces, and UBS advection schemes when flux form is chosen for the momentum advection.
+
+\subsubsection{Lateral Laplacian tracer diffusive operator}
+
+The lateral Laplacian tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}):
 \begin{equation} \label{Eq_PE_iso_tensor}
 D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
@@ -1158 +1154 @@
 but have similar expressions in $z$- and $s$-coordinates. In $z$-coordinates:
 \begin{equation} \label{Eq_PE_iso_slopes}
-r_1 =\frac{e_3 }{e_1 }  \left( {\frac{\partial \rho }{\partial i}} \right)
-                  \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \ , \quad
-r_1 =\frac{e_3 }{e_1 }  \left( {\frac{\partial \rho }{\partial i}} \right)
-                  \left( {\frac{\partial \rho }{\partial k}} \right)^{-1},
-\end{equation}
-while in $s$-coordinates $\partial/\partial k$ is replaced by
-$\partial/\partial s$.
+r_1 =\frac{e_3 }{e_1 }  \left( \pd[\rho]{i} \right) \left( \pd[\rho]{k} \right)^{-1} \, \quad
+r_2 =\frac{e_3 }{e_2 }  \left( \pd[\rho]{j} \right) \left( \pd[\rho]{k} \right)^{-1} \,
+\end{equation}
+while in $s$-coordinates $\pd[]{k}$ is replaced by $\pd[]{s}$.
 
 \subsubsection{Eddy induced velocity}
@@ -1181 +1174 @@
  w^\ast &=  -\frac{1}{e_1 e_2 }\left[ 
                       \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right)
-                    +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right)      \right]
+                     +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right)      \right]
    \end{split}
 \end{equation}
@@ -1190 +1183 @@
 \begin{align} \label{Eq_PE_slopes_eiv}
 \tilde{r}_n = \begin{cases}
-   r_n                  &      \text{in $z$-coordinate}    \\
+   r_n            &      \text{in $z$-coordinate}    \\
    r_n + \sigma_n &      \text{in \textit{z*} and $s$-coordinates}  
-                   \end{cases}
+              \end{cases}
 \quad \text{where } n=1,2
 \end{align}
@@ -1200 +1193 @@
 to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}).
 
-\subsubsection{Lateral fourth order tracer diffusive operator}
-
-The lateral fourth order tracer diffusive operator is defined by:
+\subsubsection{Lateral bilaplacian tracer diffusive operator}
+
+The lateral bilaplacian tracer diffusive operator is defined by:
 \begin{equation} \label{Eq_PE_bilapT}
-D^{lT}=\Delta \left( \;\Delta T \right) 
+D^{lT}= - \Delta \left( \;\Delta T \right) 
 \qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right)
  \end{equation}
-It is the second order operator given by \eqref{Eq_PE_iso_tensor} applied twice with 
+It is the Laplacian operator given by \eqref{Eq_PE_iso_tensor} applied twice with 
 the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. 
 
 
-\subsubsection{Lateral second order momentum diffusive operator}
-
-The second order momentum diffusive operator along $z$- or $s$-surfaces is found by 
+\subsubsection{Lateral Laplacian momentum diffusive operator}
+
+The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by 
 applying \eqref{Eq_PE_lap_vector} to the horizontal velocity vector (see Appendix~\ref{Apdx_B}):
 \begin{equation} \label{Eq_PE_lapU}
@@ -1247 +1240 @@
 of the Equator in a geographical coordinate system \citep{Lengaigne_al_JGR03}.
 
-\subsubsection{lateral fourth order momentum diffusive operator}
-
-As for tracers, the fourth order momentum diffusive operator along $z$ or $s$-surfaces 
-is a re-entering second order operator \eqref{Eq_PE_lapU} or \eqref{Eq_PE_lapU_iso} 
-with the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.
-
+\subsubsection{lateral bilaplacian momentum diffusive operator}
+
+As for tracers, the bilaplacian order momentum diffusive operator is a 
+re-entering Laplacian operator with the harmonic eddy diffusion coefficient 
+set to the square root of the biharmonic one. Nevertheless it is currently 
+not available in the iso-neutral case.
+