Changeset 9364
 Timestamp:
 20180228T12:35:14+01:00 (2 years ago)
 File:

 1 edited
Legend:
 Unmodified
 Added
 Removed

branches/2017/dev_merge_2017/DOC/TexFiles/Chapters/Annex_ISO.tex
r6997 r9364 9 9 \minitoc 10 10 \pagebreak 11 \section{Choice of \ ngn{namtra\_ldf} namelist parameters}11 \section{Choice of \protect\ngn{namtra\_ldf} namelist parameters} 12 12 %nam_traldf 13 13 \namdisplay{namtra_ldf} … … 112 112 \end{equation} 113 113 Additionally, we will sometimes write the contributions towards the 114 fluxes $\vect{f}$ and $\vect{F}_ \mathrm{iso}$ from the component114 fluxes $\vect{f}$ and $\vect{F}_{\mathrm{iso}}$ from the component 115 115 $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, with 116 116 $f_{ij}=R_{ij}e_i^{1}\partial T/\partial x_i$ (no summation) etc. … … 193 193 \begin{figure}[tb] \begin{center} 194 194 \includegraphics[width=1.05\textwidth]{Fig_GRIFF_triad_fluxes} 195 \caption{ \ label{fig:triad:ISO_triad}195 \caption{ \protect\label{fig:triad:ISO_triad} 196 196 (a) Arrangement of triads $S_i$ and tracer gradients to 197 197 give lateral tracer flux from box $i,k$ to $i+1,k$ … … 257 257 \begin{figure}[tb] \begin{center} 258 258 \includegraphics[width=0.80\textwidth]{Fig_GRIFF_qcells} 259 \caption{ \ label{fig:triad:qcells}259 \caption{ \protect\label{fig:triad:qcells} 260 260 Triad notation for quarter cells. $T$cells are inside 261 261 boxes, while the $i+\half,k$ $u$cell is shaded in green and the … … 360 360 $w$points as sums of the triad fluxes that cross the $u$ and $w$faces: 361 361 %(Fig.~\ref{Fig_ISO_triad}): 362 \begin{flalign} \label{Eq_iso_flux} \vect{F}_ \mathrm{iso}(T) &\equiv362 \begin{flalign} \label{Eq_iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv 363 363 \sum_{\substack{i_p,\,k_p}} 364 364 \begin{pmatrix} … … 506 506 \begin{subequations}\label{eq:triad:alltriadflux} 507 507 \begin{flalign}\label{eq:triad:vect_isoflux} 508 \vect{F}_ \mathrm{iso}(T) &\equiv508 \vect{F}_{\mathrm{iso}}(T) &\equiv 509 509 \sum_{\substack{i_p,\,k_p}} 510 510 \begin{pmatrix} … … 661 661 \begin{figure}[h] \begin{center} 662 662 \includegraphics[width=0.60\textwidth]{Fig_GRIFF_bdry_triads} 663 \caption{ \ label{fig:triad:bdry_triads}663 \caption{ \protect\label{fig:triad:bdry_triads} 664 664 (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer 665 665 points (black dots), and $i+1/2,1$ $u$point (blue square). Triad … … 669 669 $\triad[u]{i}{1}{F}{1/2}{1/2}$ and $\triad[u]{i+1}{1}{F}{1/2}{1/2}$ 670 670 (yellow line) are still applied, giving diapycnal diffusive 671 fluxes.\ \671 fluxes.\newline 672 672 (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and 673 673 $\triad{i+1}{k}{R}{1/2}{1/2}$ are masked when either of the $i,k+1$ 674 674 or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$point 675 675 is masked. The associated lateral fluxes (greyblack dashed 676 line) are masked if \ np{botmix\_triad}=.false., but left677 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.} 678 678 \end{center} \end{figure} 679 679 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 758 758 Fig.~\ref{fig:triad:MLB_triad}), we define the mixedlayer by setting 759 759 the vertical index of the tracer point immediately below the mixed 760 layer, $k_ \mathrm{ML}$, as the maximum $k$ (shallowest tracer point)760 layer, $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) 761 761 such that the potential density 762 762 ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, where $i,k_{10}$ is … … 767 767 $\Delta\rho_c=0.01\;\mathrm{kg\:m^{3}}$ for ML triad tapering as is 768 768 used to output the diagnosed mixedlayer depth 769 $h_ \mathrm{ML}=z_{W}_{k_\mathrm{ML}+1/2}$, the depth of the $w$point770 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 770 above the $i,k_{\mathrm{ML}}$ tracer point. 771 771 772 772 \item We define `basal' triad slopes 773 ${\:}_i{\mathbb{R}_ \mathrm{base}}_{\,i_p}^{k_p}$ as the slopes773 ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as the slopes 774 774 of those triads whose vertical `arms' go down from the 775 $i,k_ \mathrm{ML}$ tracer point to the $i,k_\mathrm{ML}1$ tracer point775 $i,k_{\mathrm{ML}}$ tracer point to the $i,k_{\mathrm{ML}}1$ tracer point 776 776 below. This is to ensure that the vertical density gradients 777 777 associated with these basal triad slopes 778 ${\:}_i{\mathbb{R}_ \mathrm{base}}_{\,i_p}^{k_p}$ are778 ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are 779 779 representative of the thermocline. The four basal triads defined in the bottom part 780 780 of Fig.~\ref{fig:triad:MLB_triad} are then 781 781 \begin{align} 782 {\:}_i{\mathbb{R}_ \mathrm{base}}_{\,i_p}^{k_p} &=783 {\:}^{k_ \mathrm{ML}k_p1/2}_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_p1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, \label{eq:triad:Rbase} 784 784 \\ 785 785 \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}. \notag786 {\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{1/2}&= 787 {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{1/2}. \notag 788 788 \end{align} 789 789 The 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, 791 791 so it is this depth 792 792 \begin{equation} … … 795 795 \end{equation} 796 796 (one gridbox deeper than the 797 diagnosed ML depth $z_ \mathrm{ML})$ that sets the $h$ used to taper797 diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper 798 798 the slopes in \eqref{eq:triad:rmtilde}. 799 799 \item Finally, we calculate the adjusted triads 800 ${\:}_i^k{\mathbb{R}_ \mathrm{ML}}_{\,i_p}^{k_p}$ within the mixed800 ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within the mixed 801 801 layer, by multiplying the appropriate 802 ${\:}_i{\mathbb{R}_ \mathrm{base}}_{\,i_p}^{k_p}$ by the ratio of803 the depth of the $w$point ${z_w}_{k+k_p}$ to ${z_ \mathrm{base}}_{\,i}$. For802 ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ by the ratio of 803 the depth of the $w$point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$. For 804 804 instance the green triad centred on $i,k$ 805 805 \begin{align} 806 {\:}_i^k{\mathbb{R}_ \mathrm{ML}}_{\,1/2}^{1/2} &=807 \frac{{z_w}_{k1/2}}{{z_ \mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{1/2}806 {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{1/2} &= 807 \frac{{z_w}_{k1/2}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{1/2} 808 808 \notag \\ 809 809 \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} &= 811 \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}.\label{eq:triad:RML} 812 812 \end{align} 813 813 \end{enumerate} … … 815 815 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 816 816 \begin{figure}[h] 817 \fcapside {\caption{\label{fig:triad:MLB_triad} Definition of 817 % \fcapside { 818 \caption{\protect\label{fig:triad:MLB_triad} Definition of 818 819 mixedlayer depth and calculation of linearly tapered 819 820 triads. The figure shows a water column at a given $i,j$ 820 821 (simplified to $i$), with the ocean surface at the top. Tracer points are denoted by 821 822 bullets, and black lines the edges of the tracer cells; $k$ 822 increases upwards. \ \823 increases upwards. \newline 823 824 \hspace{5 em}We define the mixedlayer by setting the vertical index 824 825 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) 826 827 such that ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, 827 828 where $i,k_{10}$ is the tracer gridbox within which the depth … … 829 830 layer by linearly tapering them from zero (at the surface) to 830 831 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 with832 $w$point $i,k_{\mathrm{ML}}1/2$ (blue square), 833 ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$. Triads with 833 834 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.} 836 %} 835 837 {\includegraphics[width=0.60\textwidth]{Fig_GRIFF_MLB_triads}} 836 838 \end{figure} … … 865 867 not change the potential energy. 866 868 This approach is similar to multiplying the isoneutral diffusion 867 coefficient by $\tilde{r}_ \mathrm{max}^{2}\tilde{r}_i^{2}$ for steep869 coefficient by $\tilde{r}_{\mathrm{max}}^{2}\tilde{r}_i^{2}$ for steep 868 870 slopes, as suggested by \citet{Gerdes1991} (see also \citet{Griffies_Bk04}). 869 871 Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$ … … 931 933 \begin{flalign*} 932 934 \begin{split} 933 \textbf{F}_ \mathrm{eiv}^T =935 \textbf{F}_{\mathrm{eiv}}^T = 934 936 \begin{pmatrix} 935 937 {e_{2}\,e_{3}\; u^*} \\ … … 1000 1002 \begin{subequations}\label{eq:triad:allskewflux} 1001 1003 \begin{flalign}\label{eq:triad:vect_skew_flux} 1002 \vect{F}_ \mathrm{eiv}(T) &\equiv1004 \vect{F}_{\mathrm{eiv}}(T) &\equiv 1003 1005 \sum_{\substack{i_p,\,k_p}} 1004 1006 \begin{pmatrix}
Note: See TracChangeset
for help on using the changeset viewer.