Changeset 9364

2018-02-28T12:35:14+01:00 (2 years ago)

Fix mathrm syntax, equations in multline math env is still not accepted

1 edited


  • branches/2017/dev_merge_2017/DOC/TexFiles/Chapters/Annex_ISO.tex

    r6997 r9364  
    11 \section{Choice of \ngn{namtra\_ldf} namelist parameters} 
     11\section{Choice of \protect\ngn{namtra\_ldf} namelist parameters} 
    113113Additionally, we will sometimes write the contributions towards the 
    114 fluxes $\vect{f}$ and $\vect{F}_\mathrm{iso}$ from the component 
     114fluxes $\vect{f}$ and $\vect{F}_{\mathrm{iso}}$ from the component 
    115115$R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, with 
    116116$f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc. 
    193193\begin{figure}[tb] \begin{center} 
    194194    \includegraphics[width=1.05\textwidth]{Fig_GRIFF_triad_fluxes} 
    195     \caption{ \label{fig:triad:ISO_triad} 
     195    \caption{ \protect\label{fig:triad:ISO_triad} 
    196196      (a) Arrangement of triads $S_i$ and tracer gradients to 
    197197           give lateral tracer flux from box $i,k$ to $i+1,k$ 
    257257\begin{figure}[tb] \begin{center} 
    258258    \includegraphics[width=0.80\textwidth]{Fig_GRIFF_qcells} 
    259     \caption{   \label{fig:triad:qcells} 
     259    \caption{   \protect\label{fig:triad:qcells} 
    260260    Triad notation for quarter cells. $T$-cells are inside 
    261261      boxes, while the  $i+\half,k$ $u$-cell is shaded in green and the 
    360360$w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: 
    362 \begin{flalign} \label{Eq_iso_flux} \vect{F}_\mathrm{iso}(T) &\equiv 
     362\begin{flalign} \label{Eq_iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv 
    363363  \sum_{\substack{i_p,\,k_p}} 
    364364  \begin{pmatrix} 
    507507  \begin{flalign}\label{eq:triad:vect_isoflux} 
    508     \vect{F}_\mathrm{iso}(T) &\equiv 
     508    \vect{F}_{\mathrm{iso}}(T) &\equiv 
    509509    \sum_{\substack{i_p,\,k_p}} 
    510510    \begin{pmatrix} 
    661661\begin{figure}[h] \begin{center} 
    662662    \includegraphics[width=0.60\textwidth]{Fig_GRIFF_bdry_triads} 
    663     \caption{  \label{fig:triad:bdry_triads} 
     663    \caption{  \protect\label{fig:triad:bdry_triads} 
    664664      (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer 
    665665      points (black dots), and $i+1/2,1$ $u$-point (blue square). Triad 
    669669      $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$ 
    670670      (yellow line) are still applied, giving diapycnal diffusive 
    671       fluxes.\\ 
     671      fluxes.\newline 
    672672      (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and 
    673673      $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ 
    674674      or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point 
    675675      is masked. The associated lateral fluxes (grey-black dashed 
    676       line) are masked if \np{botmix\_triad}=.false., but left 
    677       unmasked, giving bottom mixing, if \np{botmix\_triad}=.true.} 
     676      line) are masked if \protect\np{botmix\_triad}=.false., but left 
     677      unmasked, giving bottom mixing, if \protect\np{botmix\_triad}=.true.} 
    678678 \end{center} \end{figure} 
    679679% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    758758Fig.~\ref{fig:triad:MLB_triad}), we define the mixed-layer by setting 
    759759the vertical index of the tracer point immediately below the mixed 
    760 layer, $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point) 
     760layer, $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) 
    761761such that the potential density 
    762762${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, where $i,k_{10}$ is 
    767767$\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is 
    768768used to output the diagnosed mixed-layer depth 
    769 $h_\mathrm{ML}=|z_{W}|_{k_\mathrm{ML}+1/2}$, the depth of the $w$-point 
    770 above the $i,k_\mathrm{ML}$ tracer point. 
     769$h_{\mathrm{ML}}=|z_{W}|_{k_{\mathrm{ML}}+1/2}$, the depth of the $w$-point 
     770above the $i,k_{\mathrm{ML}}$ tracer point. 
    772772\item We define `basal' triad slopes 
    773 ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ as the slopes 
     773${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as the slopes 
    774774of those triads whose vertical `arms' go down from the 
    775 $i,k_\mathrm{ML}$ tracer point to the $i,k_\mathrm{ML}-1$ tracer point 
     775$i,k_{\mathrm{ML}}$ tracer point to the $i,k_{\mathrm{ML}}-1$ tracer point 
    776776below. This is to ensure that the vertical density gradients 
    777777associated with these basal triad slopes 
    778 ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ are 
     778${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are 
    779779representative of the thermocline. The four basal triads defined in the bottom part 
    780780of Fig.~\ref{fig:triad:MLB_triad} are then 
    782   {\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p} &= 
    783  {\:}^{k_\mathrm{ML}-k_p-1/2}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}, \label{eq:triad:Rbase} 
     782  {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &= 
     783 {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, \label{eq:triad:Rbase} 
    785785\intertext{with e.g.\ the green triad} 
    786 {\:}_i{\mathbb{R}_\mathrm{base}}_{1/2}^{-1/2}&= 
    787 {\:}^{k_\mathrm{ML}}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2}. \notag 
     787{\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}. \notag 
    789789The vertical flux associated with each of these triads passes through the $w$-point 
    790 $i,k_\mathrm{ML}-1/2$ lying \emph{below} the $i,k_\mathrm{ML}$ tracer point, 
     790$i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, 
    791791so it is this depth 
    796796(one gridbox deeper than the 
    797 diagnosed ML depth $z_\mathrm{ML})$ that sets the $h$ used to taper 
     797diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper 
    798798the slopes in \eqref{eq:triad:rmtilde}. 
    799799\item Finally, we calculate the adjusted triads 
    800 ${\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p}$ within the mixed 
     800${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within the mixed 
    801801layer, by multiplying the appropriate 
    802 ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ by the ratio of 
    803 the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_\mathrm{base}}_{\,i}$. For 
     802${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ by the ratio of 
     803the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$. For 
    804804instance the green triad centred on $i,k$ 
    806   {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,1/2}^{-1/2} &= 
    807 \frac{{z_w}_{k-1/2}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2} 
     806  {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{-1/2} &= 
    808808\notag \\ 
    809809\intertext{and more generally} 
    810  {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p} &= 
    811 \frac{{z_w}_{k+k_p}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}.\label{eq:triad:RML} 
     810 {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &= 
    815815% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    817   \fcapside {\caption{\label{fig:triad:MLB_triad} Definition of 
     817%  \fcapside { 
     818    \caption{\protect\label{fig:triad:MLB_triad} Definition of 
    818819      mixed-layer depth and calculation of linearly tapered 
    819820      triads. The figure shows a water column at a given $i,j$ 
    820821      (simplified to $i$), with the ocean surface at the top. Tracer points are denoted by 
    821822      bullets, and black lines the edges of the tracer cells; $k$ 
    822       increases upwards. \\ 
     823      increases upwards. \newline 
    823824      \hspace{5 em}We define the mixed-layer by setting the vertical index 
    824825      of the tracer point immediately below the mixed layer, 
    825       $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point) 
     826      $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) 
    826827      such that ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, 
    827828      where $i,k_{10}$ is the tracer gridbox within which the depth 
    829830      layer by linearly tapering them from zero (at the surface) to 
    830831      the `basal' slopes, the slopes of the four triads passing through the 
    831       $w$-point $i,k_\mathrm{ML}-1/2$ (blue square), 
    832       ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$. Triads with 
     832      $w$-point $i,k_{\mathrm{ML}}-1/2$ (blue square), 
     833      ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$. Triads with 
    833834    different $i_p,k_p$, denoted by different colours, (e.g. the green 
    834     triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}} 
     835    triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.} 
    835837  {\includegraphics[width=0.60\textwidth]{Fig_GRIFF_MLB_triads}} 
    865867not change the potential energy. 
    866868This approach is similar to multiplying the iso-neutral  diffusion 
    867 coefficient by $\tilde{r}_\mathrm{max}^{-2}\tilde{r}_i^{-2}$ for steep 
     869coefficient by $\tilde{r}_{\mathrm{max}}^{-2}\tilde{r}_i^{-2}$ for steep 
    868870slopes, as suggested by \citet{Gerdes1991} (see also \citet{Griffies_Bk04}). 
    869871Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$ 
    933 \textbf{F}_\mathrm{eiv}^T = 
     935\textbf{F}_{\mathrm{eiv}}^T = 
    935937           {e_{2}\,e_{3}\;  u^*}       \\ 
    10011003  \begin{flalign}\label{eq:triad:vect_skew_flux} 
    1002     \vect{F}_\mathrm{eiv}(T) &\equiv 
     1004    \vect{F}_{\mathrm{eiv}}(T) &\equiv 
    10031005    \sum_{\substack{i_p,\,k_p}} 
    10041006    \begin{pmatrix} 
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