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Changeset 10414 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_ASM.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/chap_ASM.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    … …  
    1112\minitoc 
    1213 
    13  
    1414\newpage 
    15 $\ $\newline    % force a new line 
    1615 
    1716The ASM code adds the functionality to apply increments to the model variables: temperature, salinity, 
    … …  
    4544is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH as 
    4645additional tendency terms to the prognostic equations: 
    47 \begin{align}     \label{eq:wa_traj_iau} 
    48 {\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})]  
    49 \; + \; F_{i} \delta \tilde{\bf x}^{a}  
    50 \end{align} 
     46\begin{align*} 
     47  % \label{eq:wa_traj_iau} 
     48  {\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})] \; + \; F_{i} \delta \tilde{\bf x}^{a} 
     49\end{align*} 
    5150where $F_{i}$ is a weighting function for applying the increments $\delta\tilde{\bf x}^{a}$ defined such that 
    5251$\sum_{i=1}^{N} F_{i}=1$. 
    … …  
    5857In addition, two different weighting functions have been implemented. 
    5958The first function employs constant weights,  
    60 \begin{align}    \label{eq:F1_i} 
    61 F^{(1)}_{i} 
    62 =\left\{ \begin{array}{ll} 
    63    0     &    {\rm if} \; \; \; t_{i} < t_{m}                \\ 
    64    1/M &    {\rm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\ 
    65    0     &    {\rm if} \; \; \; t_{i} > t_{n} 
    66   \end{array} \right.  
     59\begin{align} 
     60  \label{eq:F1_i} 
     61  F^{(1)}_{i} 
     62  =\left\{ 
     63  \begin{array}{ll} 
     64    0     &    {\rm if} \; \; \; t_{i} < t_{m}                \\ 
     65    1/M &    {\rm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\ 
     66    0     &    {\rm if} \; \; \; t_{i} > t_{n} 
     67  \end{array} 
     68            \right.  
    6769\end{align} 
    6870where $M = m-n$. 
    6971The second function employs peaked hat-like weights in order to give maximum weight in the centre of the sub-window, 
    7072with the weighting reduced linearly to a small value at the window end-points: 
    71 \begin{align}     \label{eq:F2_i} 
    72 F^{(2)}_{i} 
    73 =\left\{ \begin{array}{ll} 
    74    0                           &    {\rm if} \; \; \; t_{i}       <     t_{m}                        \\ 
    75    \alpha \, i               &    {\rm if} \; \; \; t_{m}    \leq t_{i}    \leq   t_{M/2}   \\ 
    76    \alpha \, (M - i +1) &    {\rm if} \; \; \; t_{M/2}  <    t_{i}    \leq   t_{n}       \\ 
    77    0                            &   {\rm if} \; \; \; t_{i}        >    t_{n} 
    78   \end{array} \right. 
     73\begin{align} 
     74  \label{eq:F2_i} 
     75  F^{(2)}_{i} 
     76  =\left\{ 
     77  \begin{array}{ll} 
     78    0                           &    {\rm if} \; \; \; t_{i}       <     t_{m}                        \\ 
     79    \alpha \, i               &    {\rm if} \; \; \; t_{m}    \leq t_{i}    \leq   t_{M/2}   \\ 
     80    \alpha \, (M - i +1) &    {\rm if} \; \; \; t_{M/2}  <    t_{i}    \leq   t_{n}       \\ 
     81    0                            &   {\rm if} \; \; \; t_{i}        >    t_{n} 
     82  \end{array} 
     83                                   \right. 
    7984\end{align} 
    8085where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even.  
    … …  
    8994The velocity increments may be initialized by the iterative application of a divergence damping operator. 
    9095In iteration step $n$ new estimates of velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by: 
    91 \begin{equation} \label{eq:asm_dmp} 
    92 \left\{ \begin{aligned} 
    93  u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta_{i+1/2} \left( {A_D 
    94 \;\chi^{n-1}_I } \right) \\ 
    95 \\ 
    96  v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta_{j+1/2} \left( {A_D 
    97 \;\chi^{n-1}_I } \right) \\ 
    98 \end{aligned} \right., 
     96\begin{equation} 
     97  \label{eq:asm_dmp} 
     98  \left\{ 
     99    \begin{aligned} 
     100      u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta_{i+1/2} \left( {A_D 
     101          \;\chi^{n-1}_I } \right) \\ \\ 
     102      v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta_{j+1/2} \left( {A_D 
     103          \;\chi^{n-1}_I } \right) \\ 
     104    \end{aligned} 
     105  \right., 
    99106\end{equation} 
    100107where 
    101 \begin{equation} \label{eq:asm_div} 
    102 \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
    103                 \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] 
    104                        +\delta_j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). 
    105 \end{equation} 
     108\[ 
     109  % \label{eq:asm_div} 
     110  \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
     111  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] 
     112      +\delta_j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). 
     113\] 
    106114By the application of \autoref{eq:asm_dmp} and \autoref{eq:asm_dmp} the divergence is filtered in each iteration, 
    107115and the vorticity is left unchanged. 
    … …  
    169177\end{clines} 
    170178 
     179\biblio 
     180 
    171181\end{document} 
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