# Changeset 9364

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

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

File:
1 edited

### Legend:

Unmodified
 r6997 \minitoc \pagebreak \section{Choice of \ngn{namtra\_ldf} namelist parameters} \section{Choice of \protect\ngn{namtra\_ldf} namelist parameters} %-----------------------------------------nam_traldf------------------------------------------------------ \namdisplay{namtra_ldf} Additionally, we will sometimes write the contributions towards the fluxes $\vect{f}$ and $\vect{F}_\mathrm{iso}$ from the component fluxes $\vect{f}$ and $\vect{F}_{\mathrm{iso}}$ from the component $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, with $f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc. \begin{figure}[tb] \begin{center} \includegraphics[width=1.05\textwidth]{Fig_GRIFF_triad_fluxes} \caption{ \label{fig:triad:ISO_triad} \caption{ \protect\label{fig:triad:ISO_triad} (a) Arrangement of triads $S_i$ and tracer gradients to give lateral tracer flux from box $i,k$ to $i+1,k$ \begin{figure}[tb] \begin{center} \includegraphics[width=0.80\textwidth]{Fig_GRIFF_qcells} \caption{   \label{fig:triad:qcells} \caption{   \protect\label{fig:triad:qcells} Triad notation for quarter cells. $T$-cells are inside boxes, while the  $i+\half,k$ $u$-cell is shaded in green and the $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: %(Fig.~\ref{Fig_ISO_triad}): \begin{flalign} \label{Eq_iso_flux} \vect{F}_\mathrm{iso}(T) &\equiv \begin{flalign} \label{Eq_iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv \sum_{\substack{i_p,\,k_p}} \begin{pmatrix} \begin{subequations}\label{eq:triad:alltriadflux} \begin{flalign}\label{eq:triad:vect_isoflux} \vect{F}_\mathrm{iso}(T) &\equiv \vect{F}_{\mathrm{iso}}(T) &\equiv \sum_{\substack{i_p,\,k_p}} \begin{pmatrix} \begin{figure}[h] \begin{center} \includegraphics[width=0.60\textwidth]{Fig_GRIFF_bdry_triads} \caption{  \label{fig:triad:bdry_triads} \caption{  \protect\label{fig:triad:bdry_triads} (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer points (black dots), and $i+1/2,1$ $u$-point (blue square). Triad $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$ (yellow line) are still applied, giving diapycnal diffusive fluxes.\\ fluxes.\newline (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is masked. The associated lateral fluxes (grey-black dashed line) are masked if \np{botmix\_triad}=.false., but left unmasked, giving bottom mixing, if \np{botmix\_triad}=.true.} line) are masked if \protect\np{botmix\_triad}=.false., but left unmasked, giving bottom mixing, if \protect\np{botmix\_triad}=.true.} \end{center} \end{figure} % >>>>>>>>>>>>>>>>>>>>>>>>>>>> Fig.~\ref{fig:triad:MLB_triad}), we define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer, $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point) layer, $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that the potential density ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, where $i,k_{10}$ is $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is used to output the diagnosed mixed-layer depth $h_\mathrm{ML}=|z_{W}|_{k_\mathrm{ML}+1/2}$, the depth of the $w$-point above the $i,k_\mathrm{ML}$ tracer point. $h_{\mathrm{ML}}=|z_{W}|_{k_{\mathrm{ML}}+1/2}$, the depth of the $w$-point above the $i,k_{\mathrm{ML}}$ tracer point. \item We define basal' triad slopes ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ as the slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as the slopes of those triads whose vertical arms' go down from the $i,k_\mathrm{ML}$ tracer point to the $i,k_\mathrm{ML}-1$ tracer point $i,k_{\mathrm{ML}}$ tracer point to the $i,k_{\mathrm{ML}}-1$ tracer point below. This is to ensure that the vertical density gradients associated with these basal triad slopes ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ are ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are representative of the thermocline. The four basal triads defined in the bottom part of Fig.~\ref{fig:triad:MLB_triad} are then \begin{align} {\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p} &= {\:}^{k_\mathrm{ML}-k_p-1/2}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}, \label{eq:triad:Rbase} {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &= {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, \label{eq:triad:Rbase} \\ \intertext{with e.g.\ the green triad} {\:}_i{\mathbb{R}_\mathrm{base}}_{1/2}^{-1/2}&= {\:}^{k_\mathrm{ML}}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2}. \notag {\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{-1/2}&= {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}. \notag \end{align} The vertical flux associated with each of these triads passes through the $w$-point $i,k_\mathrm{ML}-1/2$ lying \emph{below} the $i,k_\mathrm{ML}$ tracer point, $i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, so it is this depth (one gridbox deeper than the diagnosed ML depth $z_\mathrm{ML})$ that sets the $h$ used to taper diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in \eqref{eq:triad:rmtilde}. \item Finally, we calculate the adjusted triads ${\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p}$ within the mixed ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within the mixed layer, by multiplying the appropriate ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ by the ratio of the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_\mathrm{base}}_{\,i}$. For ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ by the ratio of the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$. For instance the green triad centred on $i,k$ \begin{align} {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,1/2}^{-1/2} &= \frac{{z_w}_{k-1/2}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2} {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{-1/2} &= \frac{{z_w}_{k-1/2}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2} \notag \\ \intertext{and more generally} {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p} &= \frac{{z_w}_{k+k_p}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}.\label{eq:triad:RML} {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &= \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}.\label{eq:triad:RML} \end{align} \end{enumerate} % >>>>>>>>>>>>>>>>>>>>>>>>>>>> \begin{figure}[h] \fcapside {\caption{\label{fig:triad:MLB_triad} Definition of %  \fcapside { \caption{\protect\label{fig:triad:MLB_triad} Definition of mixed-layer depth and calculation of linearly tapered triads. The figure shows a water column at a given $i,j$ (simplified to $i$), with the ocean surface at the top. Tracer points are denoted by bullets, and black lines the edges of the tracer cells; $k$ increases upwards. \\ increases upwards. \newline \hspace{5 em}We define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer, $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point) $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, where $i,k_{10}$ is the tracer gridbox within which the depth layer by linearly tapering them from zero (at the surface) to the `basal' slopes, the slopes of the four triads passing through the $w$-point $i,k_\mathrm{ML}-1/2$ (blue square), ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$. Triads with $w$-point $i,k_{\mathrm{ML}}-1/2$ (blue square), ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$. Triads with different $i_p,k_p$, denoted by different colours, (e.g. the green triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}} triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.} %} {\includegraphics[width=0.60\textwidth]{Fig_GRIFF_MLB_triads}} \end{figure} not change the potential energy. This approach is similar to multiplying the iso-neutral  diffusion coefficient by $\tilde{r}_\mathrm{max}^{-2}\tilde{r}_i^{-2}$ for steep coefficient by $\tilde{r}_{\mathrm{max}}^{-2}\tilde{r}_i^{-2}$ for steep slopes, as suggested by \citet{Gerdes1991} (see also \citet{Griffies_Bk04}). Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$ \begin{flalign*} \begin{split} \textbf{F}_\mathrm{eiv}^T = \textbf{F}_{\mathrm{eiv}}^T = \begin{pmatrix} {e_{2}\,e_{3}\;  u^*}       \\ \begin{subequations}\label{eq:triad:allskewflux} \begin{flalign}\label{eq:triad:vect_skew_flux} \vect{F}_\mathrm{eiv}(T) &\equiv \vect{F}_{\mathrm{eiv}}(T) &\equiv \sum_{\substack{i_p,\,k_p}} \begin{pmatrix}