Changeset 11630 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex

Timestamp:

2019-10-01T19:37:49+02:00 (5 years ago)

Author:

nicolasmartin

Message:

Review of TRA chapter

File:

• NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex (modified) (6 diffs)

NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex

@@ -208 +208 @@
 \]
 Two strategies can be considered for the surface pressure term:
-\begin{enumerate*}[label={(\alph*)}]
+\begin{enumerate*}[label=(\textit{\alph*})]
 \item introduce of a new variable $\eta$, the free-surface elevation,
 for which a prognostic equation can be established and solved;
@@ -486 +486 @@
 \item [Flux form of the momentum equations]
   % \label{eq:MB_dyn_flux}
-  \begin{multline*}
+  \begin{alignat*}{2}
     % \label{eq:MB_dyn_flux_u}
-    \pd[u]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, v \, u)]{j} \rt) \\
-    - \frac{1}{e_3} \pd[(w \, u)]{k} - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_u^{\vect U} + F_u^{\vect U}
-  \end{multline*}
-  \begin{multline*}
+    \pd[u]{t} = &+ \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, v \, u)]{j} \rt) - \frac{1}{e_3} \pd[(w \, u)]{k} \\
+    &- \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_u^{\vect U} + F_u^{\vect U}
+  \end{alignat*}
+  \begin{alignat*}{2}
     % \label{eq:MB_dyn_flux_v}
-    \pd[v]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\
-    - \frac{1}{e_3} \pd[(w \, v)]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_v^{\vect U} + F_v^{\vect U}
-  \end{multline*}
+    \pd[v]{t} = &- \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) - \frac{1}{e_3} \pd[(w \, v)]{k} \\
+    &- \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_v^{\vect U} + F_v^{\vect U}
+  \end{alignat*}
   where $\zeta$, the relative vorticity, is given by \autoref{eq:MB_curl_Uh} and
   $p_s$, the surface pressure, is given by:
@@ -650 +650 @@
   \end{gather*}
 \item [Flux form of the momentum equation]
-  \begin{multline*}
+  \begin{alignat*}{2}
     % \label{eq:MB_sco_u_flux}
-    \frac{1}{e_3} \pd[(e_3 \, u)]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[(e_2 \, e_3 \, u \, u)]{i} + \pd[(e_1 \, e_3 \, v \, u)]{j} \rt) \\
-    - \frac{1}{e_3} \pd[(\omega \, u)]{k} - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U}
-  \end{multline*}
-  \begin{multline*}
+    \frac{1}{e_3} \pd[(e_3 \, u)]{t} = &+ \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[(e_2 \, e_3 \, u \, u)]{i} + \pd[(e_1 \, e_3 \, v \, u)]{j} \rt) - \frac{1}{e_3} \pd[(\omega \, u)]{k} \\
+    &- \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U}
+  \end{alignat*}
+  \begin{alignat*}{2}
   % \label{eq:MB_sco_v_flux}
-    \frac{1}{e_3} \pd[(e_3 \, v)]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[( e_2 \; e_3 \, u \, v)]{i} + \pd[(e_1 \; e_3 \, v \, v)]{j} \rt) \\
-    - \frac{1}{e_3} \pd[(\omega \, v)]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U}
-  \end{multline*}
+    \frac{1}{e_3} \pd[(e_3 \, v)]{t} = &- \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[( e_2 \; e_3 \, u \, v)]{i} + \pd[(e_1 \; e_3 \, v \, v)]{j} \rt) - \frac{1}{e_3} \pd[(\omega \, v)]{k} \\
+    &- \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U}
+  \end{alignat*}
   where the relative vorticity, $\zeta$, the surface pressure gradient,
   and the hydrostatic pressure have the same expressions as in $z$-coordinates although
@@ -694 +694 @@
   \includegraphics[width=0.66\textwidth]{Fig_z_zstar}
   \caption[Curvilinear z-coordinate systems (\{non-\}linear free-surface cases and re-scaled \zstar)]{
-    \begin{enumerate*}[label={(\alph*)}]
+    \begin{enumerate*}[label=(\textit{\alph*})]
     \item $z$-coordinate in linear free-surface case;
     \item $z$-coordinate in non-linear free surface case;
@@ -1067 +1067 @@
 where $ \vect U^\ast = \lt( u^\ast,v^\ast,w^\ast \rt)$ is a non-divergent,
 eddy-induced transport velocity. This velocity field is defined by:
-\begin{gather*}
+\[
   % \label{eq:MB_eiv}
   u^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_1 \rt) \quad
-  v^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_2 \rt) \\
+  v^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_2 \rt) \quad
   w^\ast = - \frac{1}{e_1 e_2} \lt[   \pd[]{i} \lt( A^{eiv} \; e_2 \, \tilde{r}_1 \rt)
                                      + \pd[]{j} \lt( A^{eiv} \; e_1 \, \tilde{r}_2 \rt) \rt]
-\end{gather*}
+\]
 where $A^{eiv}$ is the eddy induced velocity coefficient
 (or equivalently the isoneutral thickness diffusivity coefficient),
@@ -1130 +1130 @@
 the $u$ and $v$-fields are considered as independent scalar fields,
 so that the diffusive operator is given by:
-\begin{gather*}
+\[
   % \label{eq:MB_lapU_iso}
-    D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \\
+    D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \quad
     D_v^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla v \rt)
-\end{gather*}
+\]
 where $\Re$ is given by \autoref{eq:MB_iso_tensor}.
 It is the same expression as those used for diffusive operator on tracers.