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Changeset 10414 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_B.tex – NEMO

Ignore:
Timestamp:
2018-12-19T00:02:00+01:00 (5 years ago)
Author:
nicolasmartin
Message:
  • Comment \label commands on maths environments for unreferenced equations and adapt the unnumbered math container accordingly (mainly switch to shortanded LateX syntax with \[ ... \])
  • Add a code trick to build subfile with its own bibliography
  • Fix right path for main LaTeX document in first line of subfiles (\documentclass[...]{subfiles})
  • Rename abstract_foreword.tex to foreword.tex
  • Fix some non-ASCII codes inserted here or there in LaTeX (\[0-9]*)
  • Made a first iteration on the indentation and alignement within math, figure and table environments to improve source code readability
File:
1 edited

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

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    4 % Chapter Ñ Appendix B : Diffusive Operators 
     5% Chapter Appendix B : Diffusive Operators 
    56% ================================================================ 
    67\chapter{Appendix B : Diffusive Operators} 
    78\label{apdx:B} 
     9 
    810\minitoc 
    911 
    10  
    1112\newpage 
    12 $\ $\newline    % force a new ligne 
    1313 
    1414% ================================================================ 
    … …  
    1919 
    2020\subsubsection*{In z-coordinates} 
     21 
    2122In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: 
    22 \begin{align} \label{apdx:B1} 
    23  &D^T = \frac{1}{e_1 \, e_2}      \left[ 
    24   \left. \frac{\partial}{\partial i} \left(  \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right. 
    25                        \left. 
    26 + \left. \frac{\partial}{\partial j} \left(  \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z   \right)   \right|_z      \right] 
    27 + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) 
     23\begin{align} 
     24  \label{apdx:B1} 
     25  &D^T = \frac{1}{e_1 \, e_2}      \left[ 
     26    \left. \frac{\partial}{\partial i} \left(   \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right. 
     27    \left. 
     28    + \left. \frac{\partial}{\partial j} \left(  \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z   \right)   \right|_z      \right] 
     29    + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) 
    2830\end{align} 
    2931 
    3032\subsubsection*{In generalized vertical coordinates} 
     33 
    3134In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{apdx:A_s_slope} and 
    3235the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$. 
    3336The diffusion operator is given by: 
    3437 
    35 \begin{equation} \label{apdx:B2} 
    36 D^T = \left. \nabla \right|_s \cdot 
    37            \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
    38 \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} 
    39  1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\ 
    40  0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\ 
    41  {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1 
    42 ^2+\sigma _2 ^2} \hfill \\ 
    43 \end{array} }} \right) 
     38\begin{equation} 
     39  \label{apdx:B2} 
     40  D^T = \left. \nabla \right|_s \cdot 
     41  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
     42  \;\;\text{where} \;\Re =\left( {{ 
     43        \begin{array}{*{20}c} 
     44          1 \hfill & 0 \hfill & {-\sigma_1 } \hfill \\ 
     45          0 \hfill & 1 \hfill & {-\sigma_2 } \hfill \\ 
     46          {-\sigma_1 } \hfill & {-\sigma_2 } \hfill & {\varepsilon +\sigma_1 
     47                                                      ^2+\sigma_2 ^2} \hfill \\ 
     48        \end{array} 
     49      }} \right) 
    4450\end{equation} 
    4551or in expanded form: 
    46 \begin{subequations} 
    47 \begin{align*} {\begin{array}{*{20}l} 
    48 D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left. 
    49 {\frac{\partial }{\partial i}\left( {\frac{1}{e_1}\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _1 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.  \\ 
    50 &\qquad \quad \ \ \ +e_1\,e_3\,A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma _2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\ 
    51 &\qquad \quad \ \ \ + e_1\,e_2\,A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma _1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right. 
    52  \left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right] 
    53 \end{array} } 
     52\begin{align*} 
     53  { 
     54  \begin{array}{*{20}l} 
     55    D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left. 
     56          {\frac{\partial }{\partial i}\left( {\frac{1}{e_1}\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma_1 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.  \\ 
     57        &\qquad \quad \ \ \ +e_1\,e_3\,A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma_2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\ 
     58        &\qquad \quad \ \ \ + e_1\,e_2\,A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma_1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma_2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right. 
     59          \left. {\left. {+\left( {\varepsilon +\sigma_1^2+\sigma_2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right] 
     60  \end{array} 
     61          } 
    5462\end{align*} 
    55 \end{subequations} 
    5663 
    5764Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption. 
    … …  
    6471any loss of generality: 
    6572 
    66 \begin{subequations} 
    67 \begin{align*} {\begin{array}{*{20}l} 
    68 D^T&=\frac{1}{e_1\,e_2} \left. {\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z 
    69                      +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right)     \\ 
    70  \\ 
    71 % 
    72 &=\frac{1}{e_1\,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s 
    73                                                     -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ 
    74 & \qquad \qquad \left. { -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right] 
    75 \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\ 
    76  \\ 
    77 % 
    78 &=\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{e_2 }{e_1}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ 
    79 &  \qquad \qquad \quad \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 -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) \\ 
    80 &  \qquad \qquad \quad \shoveright{ -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\quad} \right] }\\ 
    81 \end{array} }     \\ 
    82 % 
    83  {\begin{array}{*{20}l} 
    84 \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:} 
    85 % 
    86 & =\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. \\ 
    87 & \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) \\ 
    88 & \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] }\\ 
    89 \\ 
    90 &=\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. \\ 
    91 & \qquad \qquad \quad \left. {+\frac{e_2 \,\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}}  \right|_s -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\ 
    92 & \qquad \qquad \quad-e_2 \,\sigma _1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\ 
    93 & \qquad \qquad \quad\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s} \left( {\frac{\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]} 
    94 \end{array} } \\ 
    95 {\begin{array}{*{20}l} 
    96 % 
    97 \intertext{using the same remark as just above, it becomes:} 
    98 % 
    99 &= \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.\;\;\; \\ 
    100 & \qquad \qquad \quad+\frac{e_1 \,e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}\;\frac{\partial \sigma _1 }{\partial s} - \frac {\sigma _1 }{e_3} A^{lT} \;\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s}\;\frac{\partial T}{\partial s} \\ 
    101 & \qquad \qquad \quad-e_2 \left( {A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s +\frac{\partial }{\partial s}\left( {\sigma _1 A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)-\frac{\partial \sigma _1 }{\partial s}\;A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
    102 & \qquad \qquad \quad\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] } 
    103  \end{array} } \\ 
    104 {\begin{array}{*{20}l} 
    105 % 
    106 \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, 
    107 the last term of the first line and the first term of the last line cancel, while 
    108 the second line reduces to a single vertical derivative, so it becomes:} 
    109 % 
    110 & =\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. \\ 
    111 & \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]} 
    112  \\ 
    113 % 
    114 \intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} 
    115 \end{array} } \\ 
    116 % 
    117 {\frac{1}{e_1\,e_2\,e_3}} 
    118 \left( {{\begin{array}{*{30}c} 
    119 {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\ 
    120 {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\ 
    121 \end{array}}}\right) 
    122 \cdot \left[ {A^{lT} 
    123 \left( {{\begin{array}{*{30}c} 
    124  {1} \hfill & {-\sigma_1 } \hfill \\ 
    125  {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\ 
    126 \end{array} }} \right) 
    127 \cdot 
    128 \left( {{\begin{array}{*{30}c} 
    129 {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\ 
    130 {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\ 
    131 \end{array}}}       \right) \left( T \right)} \right] 
     73\begin{align*} 
     74  { 
     75  \begin{array}{*{20}l} 
     76    D^T&=\frac{1}{e_1\,e_2} \left. {\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z 
     77         +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ \\ 
     78         % 
     79       &=\frac{1}{e_1\,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s 
     80         -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ 
     81       & \qquad \qquad \left. { -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right] 
     82         \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\ \\ 
     83         % 
     84       &=\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{e_2 }{e_1}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ 
     85       &  \qquad \qquad \quad \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 -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) \\ 
     86       &  \qquad \qquad \quad \shoveright{ -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\quad} \right] }\\ 
     87  \end{array} 
     88  }      \\ 
     89  % 
     90  { 
     91  \begin{array}{*{20}l} 
     92    \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma_1 }{\partial s}$, it becomes:} 
     93    % 
     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. \\ 
     95    & \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) \\ 
     96    & \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] }\\ 
     97    \\ 
     98    &=\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. \\ 
     99    & \qquad \qquad \quad \left. {+\frac{e_2 \,\sigma_1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}}  \right|_s -e_2 A^{lT}\;\frac{\partial \sigma_1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\ 
     100    & \qquad \qquad \quad-e_2 \,\sigma_1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma_1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\ 
     101    & \qquad \qquad \quad\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma_1 } \right)}{\partial s} \left( {\frac{\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]} 
     102  \end{array} 
     103      } \\ 
     104  { 
     105  \begin{array}{*{20}l} 
     106    % 
     107    \intertext{using the same remark as just above, it becomes:} 
     108    % 
     109    &= \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.\;\;\; \\ 
     110    & \qquad \qquad \quad+\frac{e_1 \,e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}\;\frac{\partial \sigma_1 }{\partial s} - \frac {\sigma_1 }{e_3} A^{lT} \;\frac{\partial \left( {e_1 \,e_2 \,\sigma_1 } \right)}{\partial s}\;\frac{\partial T}{\partial s} \\ 
     111    & \qquad \qquad \quad-e_2 \left( {A^{lT}\;\frac{\partial \sigma_1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s +\frac{\partial }{\partial s}\left( {\sigma_1 A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)-\frac{\partial \sigma_1 }{\partial s}\;A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
     112    & \qquad \qquad \quad\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma_1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] } 
     113  \end{array} 
     114      } \\ 
     115  { 
     116  \begin{array}{*{20}l} 
     117    % 
     118    \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, 
     119    the last term of the first line and the first term of the last line cancel, while 
     120    the second line reduces to a single vertical derivative, so it becomes:} 
     121  % 
     122    & =\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. \\ 
     123    & \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]} \\ 
     124    % 
     125    \intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} 
     126  \end{array} 
     127  } \\ 
     128  % 
     129  {\frac{1}{e_1\,e_2\,e_3}} 
     130  \left( {{ 
     131  \begin{array}{*{30}c} 
     132    {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\ 
     133    {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\ 
     134  \end{array}}} 
     135  \right) 
     136  \cdot \left[ {A^{lT} 
     137  \left( {{ 
     138  \begin{array}{*{30}c} 
     139    {1} \hfill & {-\sigma_1 } \hfill \\ 
     140    {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\ 
     141  \end{array} 
     142  }} \right) 
     143  \cdot 
     144  \left( {{ 
     145  \begin{array}{*{30}c} 
     146    {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\ 
     147    {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\ 
     148  \end{array} 
     149  }}       \right) \left( T \right)} \right] 
    132150\end{align*} 
    133 \end{subequations} 
    134 \addtocounter{equation}{-2} 
     151%\addtocounter{equation}{-2} 
    135152 
    136153% ================================================================ 
    … …  
    147164takes the following form \citep{Redi_JPO82}: 
    148165 
    149 \begin{equation} \label{apdx:B3} 
    150 \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} 
    151 \left[ {{\begin{array}{*{20}c} 
    152  {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\ 
    153  {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\ 
    154  {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ 
    155 \end{array} }} \right] 
     166\begin{equation} 
     167  \label{apdx:B3} 
     168  \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} 
     169  \left[ {{ 
     170        \begin{array}{*{20}c} 
     171          {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\ 
     172          {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\ 
     173          {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ 
     174        \end{array} 
     175      }} \right] 
    156176\end{equation} 
    157177where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials: 
    158178\[ 
    159 a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
    160 \qquad , \qquad 
    161 a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
    162 \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
     179  a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
     180  \qquad , \qquad 
     181  a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
     182  \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
    163183\] 
    164184 
    165185In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, 
    166186so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}: 
    167 \begin{subequations} \label{apdx:B4} 
    168 \begin{equation} \label{apdx:B4a} 
    169 {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = 
    170 \left[ {{\begin{array}{*{20}c} 
    171  1 \hfill & 0 \hfill & {-a_1 } \hfill \\ 
    172  0 \hfill & 1 \hfill & {-a_2 } \hfill \\ 
    173  {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ 
    174 \end{array} }} \right], 
    175 \end{equation} 
    176 and the iso/dianeutral diffusive operator in $z$-coordinates is then 
    177 \begin{equation}\label{apdx:B4b} 
    178  D^T = \left. \nabla \right|_z \cdot 
    179            \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\ 
    180 \end{equation} 
     187\begin{subequations} 
     188  \label{apdx:B4} 
     189  \begin{equation} 
     190    \label{apdx:B4a} 
     191    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = 
     192    \left[ {{ 
     193          \begin{array}{*{20}c} 
     194            1 \hfill & 0 \hfill & {-a_1 } \hfill \\ 
     195            0 \hfill & 1 \hfill & {-a_2 } \hfill \\ 
     196            {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ 
     197          \end{array} 
     198        }} \right], 
     199  \end{equation} 
     200  and the iso/dianeutral diffusive operator in $z$-coordinates is then 
     201  \begin{equation} 
     202    \label{apdx:B4b} 
     203    D^T = \left. \nabla \right|_z \cdot 
     204    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\ 
     205  \end{equation} 
    181206\end{subequations} 
    182  
    183207 
    184208Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to 
    … …  
    192216Written out explicitly, 
    193217 
    194 \begin{multline} \label{apdx:B_ldfiso} 
    195  D^T=\frac{1}{e_1 e_2 }\left\{ 
    196  {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2}{e_1}\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]} 
    197  {+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1}{e_2}\frac{\partial T}{\partial j}-a_2 \frac{e_1}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\ 
    198 \shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\ 
     218\begin{multline} 
     219  \label{apdx:B_ldfiso} 
     220  D^T=\frac{1}{e_1 e_2 }\left\{ 
     221    {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2}{e_1}\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]} 
     222    {+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1}{e_2}\frac{\partial T}{\partial j}-a_2 \frac{e_1}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\ 
     223  \shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\ 
    199224\end{multline} 
    200  
    201225 
    202226The isopycnal diffusion operator \autoref{apdx:B4}, 
    … …  
    205229Let us demonstrate the second one: 
    206230\[ 
    207 \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv 
    208           = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, 
     231  \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv 
     232  = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, 
    209233\] 
    210234and since 
    211 \begin{subequations} 
    212 \begin{align*} {\begin{array}{*{20}l} 
    213 \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} 
    214 \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 
    215 \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( 
    216 {\frac{\partial T}{\partial j}} \right)^2} \right. \\ 
    217 &\qquad \qquad \qquad 
    218 { \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)\left( {\frac{\partial T}{\partial k}} \right)^2} \right]} \\ 
    219 &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial 
    220           T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial 
    221           j}-a_2 \frac{\partial T}{\partial k}} \right)^2} 
    222   +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\ 
    223 & \geq 0 
    224 \end{array} } 
     235\begin{align*} 
     236  { 
     237  \begin{array}{*{20}l} 
     238    \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} 
     239    \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 
     240             \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( 
     241             {\frac{\partial T}{\partial j}} \right)^2} \right. \\ 
     242           &\qquad \qquad \qquad 
     243             { \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)\left( {\frac{\partial T}{\partial k}} \right)^2} \right]} \\ 
     244           &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial 
     245             T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial 
     246             j}-a_2 \frac{\partial T}{\partial k}} \right)^2} 
     247             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\ 
     248           & \geq 0 
     249  \end{array} 
     250             } 
    225251\end{align*} 
    226 \end{subequations} 
    227 \addtocounter{equation}{-1} 
     252%\addtocounter{equation}{-1} 
    228253the property becomes obvious. 
    229254 
    … …  
    236261The resulting operator then takes the simple form 
    237262 
    238 \begin{equation} \label{apdx:B_ldfiso_s} 
    239 D^T = \left. \nabla \right|_s \cdot 
    240            \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
    241 \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} 
    242  1 \hfill & 0 \hfill & {-r _1 } \hfill \\ 
    243  0 \hfill & 1 \hfill & {-r _2 } \hfill \\ 
    244  {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1 
    245 ^2+r _2 ^2} \hfill \\ 
    246 \end{array} }} \right), 
     263\begin{equation} 
     264  \label{apdx:B_ldfiso_s} 
     265  D^T = \left. \nabla \right|_s \cdot 
     266  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
     267  \;\;\text{where} \;\Re =\left( {{ 
     268        \begin{array}{*{20}c} 
     269          1 \hfill & 0 \hfill & {-r _1 } \hfill \\ 
     270          0 \hfill & 1 \hfill & {-r _2 } \hfill \\ 
     271          {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1 
     272                                              ^2+r _2 ^2} \hfill \\ 
     273        \end{array} 
     274      }} \right), 
    247275\end{equation} 
    248276 
    … …  
    250278relative to $s$-coordinate surfaces: 
    251279\[ 
    252 r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} 
    253 \qquad , \qquad 
    254 r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
    255 \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. 
     280  r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} 
     281  \qquad , \qquad 
     282  r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
     283  \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. 
    256284\] 
    257285 
    … …  
    260288the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as 
    261289 
    262 \begin{equation} \label{apdx:B5} 
    263 D^T = \left. \nabla \right|_\rho \cdot 
    264            \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\ 
    265 \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} 
    266  1 \hfill & 0 \hfill &0 \hfill \\ 
    267  0 \hfill & 1 \hfill & 0 \hfill \\ 
    268 0 \hfill & 0 \hfill & \varepsilon \hfill \\ 
    269 \end{array} }} \right). 
     290\begin{equation} 
     291  \label{apdx:B5} 
     292  D^T = \left. \nabla \right|_\rho \cdot 
     293  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\ 
     294  \;\;\text{where} \;\Re =\left( {{ 
     295        \begin{array}{*{20}c} 
     296          1 \hfill & 0 \hfill &0 \hfill \\ 
     297          0 \hfill & 1 \hfill & 0 \hfill \\ 
     298          0 \hfill & 0 \hfill & \varepsilon \hfill \\ 
     299        \end{array} 
     300      }} \right). 
    270301\end{equation} 
    271302Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives 
    … …  
    289320to the horizontal velocity vector: 
    290321\begin{align*} 
    291 \Delta {\textbf{U}}_h 
    292 &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- 
    293 \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)    \\ 
    294 \\ 
    295 &=\left( {{\begin{array}{*{20}c} 
    296  {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\ 
    297  {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\ 
    298  {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\ 
    299 \end{array} }} \right)-\left( {{\begin{array}{*{20}c} 
    300  {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 
    301 }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial 
    302 u}{\partial k}} \right)} \hfill \\ 
    303  {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 
    304 }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta 
    305 }{\partial i}} \hfill \\ 
    306  {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 
    307 }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial 
    308 j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} 
    309 \hfill \\ 
    310 \end{array} }} \right) 
    311 \\ 
    312 \\ 
    313 &=\left( {{\begin{array}{*{20}c} 
    314 {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\ 
    315 {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\ 
    316 0 \\ 
    317 \end{array} }} \right) 
    318 +\frac{1}{e_3 } 
    319 \left( {{\begin{array}{*{20}c} 
    320 {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\ 
    321 {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\ 
    322 {\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\ 
    323 \end{array} }} \right) 
     322  \Delta {\textbf{U}}_h 
     323  &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- 
     324    \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\ 
     325  &=\left( {{ 
     326    \begin{array}{*{20}c} 
     327      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\ 
     328      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\ 
     329      {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\ 
     330    \end{array} 
     331  }} \right) 
     332  -\left( {{ 
     333  \begin{array}{*{20}c} 
     334    {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 
     335    }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial 
     336    u}{\partial k}} \right)} \hfill \\ 
     337    {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 
     338    }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta 
     339    }{\partial i}} \hfill \\ 
     340    {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 
     341    }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial 
     342    j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} 
     343    \hfill \\ 
     344  \end{array} 
     345  }} \right) \\ \\ 
     346  &=\left( {{ 
     347    \begin{array}{*{20}c} 
     348      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\ 
     349      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\ 
     350      0 \\ 
     351    \end{array} 
     352  }} \right) 
     353  +\frac{1}{e_3 } 
     354  \left( {{ 
     355  \begin{array}{*{20}c} 
     356    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\ 
     357    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\ 
     358    {\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\ 
     359  \end{array} 
     360  }} \right) 
    324361\end{align*} 
    325362Using \autoref{eq:PE_div}, the definition of the horizontal divergence, 
    326363the third componant of the second vector is obviously zero and thus : 
    327364\[ 
    328 \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) 
     365  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) 
    329366\] 
    330367 
    … …  
    335372The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in 
    336373the $z$-coordinate therefore takes the following form: 
    337 \begin{equation} \label{apdx:B_Lap_U} 
    338  {\textbf{D}}^{\textbf{U}} = 
    339      \nabla _h \left( {A^{lm}\;\chi } \right) 
    340    - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) 
    341    + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } 
    342             \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ 
     374\begin{equation} 
     375  \label{apdx:B_Lap_U} 
     376  { 
     377    \textbf{D}}^{\textbf{U}} = 
     378  \nabla _h \left( {A^{lm}\;\chi } \right) 
     379  - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) 
     380  + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } 
     381      \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ 
    343382\end{equation} 
    344383that is, in expanded form: 
    345384\begin{align*} 
    346 D^{\textbf{U}}_u 
    347 & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i} 
    348      -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} 
    349      +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\ 
    350 D^{\textbf{U}}_v 
    351 & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j} 
    352      +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} 
    353      +\frac{1}{e_3} \frac{\partial v}{\partial k} 
     385  D^{\textbf{U}}_u 
     386  & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i} 
     387    -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} 
     388    +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\ 
     389  D^{\textbf{U}}_v 
     390  & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j} 
     391    +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} 
     392    +\frac{1}{e_3} \frac{\partial v}{\partial k} 
    354393\end{align*} 
    355394 
    … …  
    360399Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems, 
    361400that is a Laplacian diffusion is applied on momentum along the coordinate directions. 
     401 
     402\biblio 
     403 
    362404\end{document} 
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