Changeset 11331 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_B.tex

Timestamp:

2019-07-23T15:02:57+02:00 (5 years ago)

Author:

davestorkey

Message:

Documentation: small corrections and tidy up of Appendix B.

File:

• NEMO/trunk/doc/latex/NEMO/subfiles/annex_B.tex (modified) (11 diffs)

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

@@ -62 +62 @@
 \end{align*}
 
-Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption.
+\autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption.
 Indeed, for the special case $k=z$ and thus $e_3 =1$,
 we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} and
@@ -94 +94 @@
     & =\frac{1}{e_1\,e_2\,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2\,e_3 }{e_1}\,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. -\, {e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\
     & \qquad \qquad \quad -e_2 A^{lT}\;\frac{\partial \sigma_1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\
-    & \qquad \qquad \quad\shoveright{ \left. { +e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial z}} \right)\;\;\;} \right] }\\
+    & \qquad \qquad \quad\shoveright{ \left. { +e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] }\\
     \\
     &=\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. {-\frac{\partial }{\partial i}\left( {e_2 \,\sigma_1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\
@@ -105 +105 @@
   \begin{array}{*{20}l}
     %
-    \intertext{using the same remark as just above, it becomes:}
+    \intertext{Using the same remark as just above, it becomes:}
     %
     &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma_1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.\;\;\; \\
@@ -117 +117 @@
     %
     \intertext{Since the horizontal scale factors do not depend on the vertical coordinate,
-    the last term of the first line and the first term of the last line cancel, while
-    the second line reduces to a single vertical derivative, so it becomes:}
+    the two terms on the second line cancel, while
+    the third line reduces to a single vertical derivative, so it becomes:}
   %
     & =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma_1 \,A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\
     & \qquad \qquad \quad \shoveright{ \left. {+\frac{\partial }{\partial s}\left( {-e_2 \,\sigma_1 \,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s +A^{lT}\frac{e_1 \,e_2 }{e_3 }\;\left( {\varepsilon +\sigma_1 ^2} \right)\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\
     %
-    \intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
-  \end{array}
-  } \\
+    \intertext{In other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
+  \end{array}
+  } \\ 
   %
   {\frac{1}{e_1\,e_2\,e_3}}
@@ -169 +169 @@
   \left[ {{
         \begin{array}{*{20}c}
-          {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
-          {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
-          {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
+          {1+a_2 ^2 +\varepsilon a_1 ^2} \hfill & {-a_1 a_2 (1-\varepsilon)} \hfill & {-a_1 (1-\varepsilon) } \hfill \\
+          {-a_1 a_2 (1-\varepsilon) } \hfill & {1+a_1 ^2 +\varepsilon a_2 ^2} \hfill & {-a_2 (1-\varepsilon)} \hfill \\
+          {-a_1 (1-\varepsilon)} \hfill & {-a_2 (1-\varepsilon)} \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
         \end{array}
       }} \right]
@@ -182 +182 @@
   \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
 \]
-
-In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean,
-so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}:
+and, as before, $\epsilon = A^{vT} / A^{lT}$.
+
+In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean,
+so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0) 
+and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}:
 \begin{subequations}
   \label{apdx:B4}
@@ -284 +286 @@
 \]
 
-To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious.
+To prove \autoref{apdx:B_ldfiso_s} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious.
 An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that
 the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as
@@ -306 +308 @@
 the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates.
 The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces,
-in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates,
+in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in
 \autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
 
@@ -316 +318 @@
 \label{sec:B_3}
 
-The second order momentum diffusion operator (Laplacian) in the $z$-coordinate is found by
+The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by
 applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector,
 to the horizontal velocity vector:
@@ -361 +363 @@
 \end{align*}
 Using \autoref{eq:PE_div}, the definition of the horizontal divergence,
-the third componant of the second vector is obviously zero and thus :
+the third component of the second vector is obviously zero and thus :
 \[
   \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
@@ -386 +388 @@
   & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
     -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
-    +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\
+    +\frac{1}{e_3} \frac{\partial}{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right)      \\
   D^{\textbf{U}}_v
   & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
     +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
-    +\frac{1}{e_3} \frac{\partial v}{\partial k}
+    +\frac{1}{e_3} \frac{\partial}{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right)  
 \end{align*}