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branches/2016/dev_INGV_UKMO_2016/DOC/TexFiles/Chapters/Chap_DOM.tex
r5120 r7351 1 \documentclass[NEMO_book]{subfiles} 2 \begin{document} 1 3 % ================================================================ 2 % Chapter 2 �Space and Time Domain (DOM)4 % Chapter 2 ——— Space and Time Domain (DOM) 3 5 % ================================================================ 4 6 \chapter{Space Domain (DOM) } … … 40 42 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 41 43 \begin{figure}[!tb] \begin{center} 42 \includegraphics[width=0.90\textwidth]{ ./TexFiles/Figures/Fig_cell.pdf}44 \includegraphics[width=0.90\textwidth]{Fig_cell} 43 45 \caption{ \label{Fig_cell} 44 46 Arrangement of variables. $t$ indicates scalar points where temperature, … … 138 140 and $f$-points, and its divergence defined at $t$-points: 139 141 \begin{eqnarray} \label{Eq_DOM_curl} 140 \nabla \times {\rm 142 \nabla \times {\rm{\bf A}}\equiv & 141 143 \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right) &\ \mathbf{i} \\ 142 144 +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1 \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right) &\ \mathbf{j} \\ 143 145 +& \frac{1}{e_{1f} \,e_{2f} } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2 \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right) &\ \mathbf{k} 144 146 \end{eqnarray} 145 \begin{equation} \label{Eq_DOM_div} 146 \nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}\,e_{3t}}\left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] 147 +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right] 148 \end{equation} 149 150 In the special case of a pure $z$-coordinate system, \eqref{Eq_DOM_lap} and 151 \eqref{Eq_DOM_div} can be simplified. In this case, the vertical scale factor 152 becomes a function of the single variable $k$ and thus does not depend on the 153 horizontal location of a grid point. For example \eqref{Eq_DOM_div} reduces to: 154 \begin{equation*} 155 \nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}} \left( \delta_i \left[e_{2u}\,a_1 \right] 156 +\delta_j \left[e_{1v}\, a_2 \right] \right) 157 +\frac{1}{e_{3t}} \delta_k \left[ a_3 \right] 158 \end{equation*} 147 \begin{eqnarray} \label{Eq_DOM_div} 148 \nabla \cdot \rm{\bf A} \equiv 149 \frac{1}{e_{1t}\,e_{2t}\,e_{3t}} \left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] 150 +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right] 151 \end{eqnarray} 159 152 160 153 The vertical average over the whole water column denoted by an overbar becomes … … 183 176 Let $a$ and $b$ be two fields defined on the mesh, with value zero inside 184 177 continental area. Using integration by parts it can be shown that the differencing 185 operators ($\delta_i$, $\delta_j$ and $\delta_k$) are anti-symmetric linear186 operators,and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$,178 operators ($\delta_i$, $\delta_j$ and $\delta_k$) are skew-symmetric linear operators, 179 and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$, 187 180 $\overline{\,\cdot\,}^{\,k}$ and $\overline{\,\cdot\,}^{\,k}$) are symmetric linear 188 181 operators, $i.e.$ … … 210 203 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 211 204 \begin{figure}[!tb] \begin{center} 212 \includegraphics[width=0.90\textwidth]{ ./TexFiles/Figures/Fig_index_hor.pdf}205 \includegraphics[width=0.90\textwidth]{Fig_index_hor} 213 206 \caption{ \label{Fig_index_hor} 214 207 Horizontal integer indexing used in the \textsc{Fortran} code. The dashed area indicates … … 260 253 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 261 254 \begin{figure}[!pt] \begin{center} 262 \includegraphics[width=.90\textwidth]{ ./TexFiles/Figures/Fig_index_vert.pdf}255 \includegraphics[width=.90\textwidth]{Fig_index_vert} 263 256 \caption{ \label{Fig_index_vert} 264 257 Vertical integer indexing used in the \textsc{Fortran } code. Note that … … 358 351 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 359 352 \begin{figure}[!t] \begin{center} 360 \includegraphics[width=0.90\textwidth]{ ./TexFiles/Figures/Fig_zgr_e3.pdf}353 \includegraphics[width=0.90\textwidth]{Fig_zgr_e3} 361 354 \caption{ \label{Fig_zgr_e3} 362 355 Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical, … … 364 357 For both grids here, the same $w$-point depth has been chosen but in (a) the 365 358 $t$-points are set half way between $w$-points while in (b) they are defined from 366 an analytical function: $z(k)=5\,( i-1/2)^3 - 45\,(i-1/2)^2 + 140\,(i-1/2) - 150$.359 an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$. 367 360 Note the resulting difference between the value of the grid-size $\Delta_k$ and 368 361 those of the scale factor $e_k$. } … … 425 418 426 419 The choice of the grid must be consistent with the boundary conditions specified 427 by the parameter \np{jperio}(see {\S\ref{LBC}).420 by \np{jperio}, a parameter found in \ngn{namcfg} namelist (see {\S\ref{LBC}). 428 421 429 422 % ------------------------------------------------------------------------------------------------------------- … … 467 460 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 468 461 \begin{figure}[!tb] \begin{center} 469 \includegraphics[width=1.0\textwidth]{ ./TexFiles/Figures/Fig_z_zps_s_sps.pdf}462 \includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps} 470 463 \caption{ \label{Fig_z_zps_s_sps} 471 464 The ocean bottom as seen by the model: … … 475 468 (d) hybrid $s-z$ coordinate, 476 469 (e) hybrid $s-z$ coordinate with partial step, and 477 (f) same as (e) but with variable volume associated with the non-linear free surface.478 Note that the variable volume option (\key{vvl})can be used with any of the470 (f) same as (e) but in the non-linear free surface (\np{ln\_linssh}=false). 471 Note that the non-linear free surface can be used with any of the 479 472 5 coordinates (a) to (e).} 480 473 \end{center} \end{figure} 481 474 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 482 475 483 The choice of a vertical coordinate, even if it is made through a namelist parameter,476 The choice of a vertical coordinate, even if it is made through \ngn{namzgr} namelist parameters, 484 477 must be done once of all at the beginning of an experiment. It is not intended as an 485 478 option which can be enabled or disabled in the middle of an experiment. Three main … … 488 481 (\np{ln\_zps}~=~true), or generalized, $s$-coordinate (\np{ln\_sco}~=~true). 489 482 Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate 490 (Fig.~\ref{Fig_z_zps_s_sps}d and \ref{Fig_z_zps_s_sps}e). When using the variable 491 volume option \key{vvl} ($i.e.$ non-linear free surface), the coordinate follow the 492 time-variation of the free surface so that the transformation is time dependent: 493 $z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). This option can be used with full step 494 bathymetry or $s$-coordinate (hybrid and partial step coordinates have not 495 yet been tested in NEMO v2.3). If using $z$-coordinate with partial step bathymetry 496 (\np{ln\_zps}~=~true), ocean cavity beneath ice shelves can be open (\np{ln\_isfcav}~=~true). 483 (Fig.~\ref{Fig_z_zps_s_sps}d and \ref{Fig_z_zps_s_sps}e). By default a non-linear free surface is used: 484 the coordinate follow the time-variation of the free surface so that the transformation is time dependent: 485 $z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). When a linear free surface is assumed (\np{ln\_linssh}=true), 486 the vertical coordinate are fixed in time, but the seawater can move up and down across the z=0 surface 487 (in other words, the top of the ocean in not a rigid-lid). 488 The last choice in terms of vertical coordinate concerns the presence (or not) in the model domain 489 of ocean cavities beneath ice shelves. Setting \np{ln\_isfcav} to true allows to manage ocean cavities, 490 otherwise they are filled in. This option is currently only available in $z$- or $zps$-coordinate, 491 and partial step are also applied at the ocean/ice shelf interface. 497 492 498 493 Contrary to the horizontal grid, the vertical grid is computed in the code and no 499 494 provision is made for reading it from a file. The only input file is the bathymetry 500 (in meters) (\ifile{bathy\_meter}) 495 (in meters) (\ifile{bathy\_meter}). 501 496 \footnote{N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the 502 497 \ifile{bathy\_meter} file, so that the computation of the number of wet ocean point 503 498 in each water column is by-passed}. 499 If \np{ln\_isfcav}~=~true, an extra file input file describing the ice shelf draft 500 (in meters) (\ifile{isf\_draft\_meter}) is needed. 501 504 502 After reading the bathymetry, the algorithm for vertical grid definition differs 505 503 between the different options: … … 519 517 %%% 520 518 521 The arrays describing the grid point depths and vertical scale factors 522 are three dimensional arrays $(i,j,k)$ even in the case of $z$-coordinate with 523 full step bottom topography. In non-linear free surface (\key{vvl}), their knowledge 524 is required at \textit{before}, \textit{now} and \textit{after} time step, while they 525 do not vary in time in linear free surface case. 526 To improve the code readability while providing this flexibility, the vertical coordinate 527 and scale factors are defined as functions of 528 $(i,j,k)$ with "fs" as prefix (examples: \textit{fse3t\_b, fse3t\_n, fse3t\_a,} 529 for the \textit{before}, \textit{now} and \textit{after} scale factors at $t$-point) 530 that can be either three different arrays when \key{vvl} is defined, or a single fixed arrays. 531 These functions are defined in the file \hf{domzgr\_substitute} of the DOM directory. 532 They are used throughout the code, and replaced by the corresponding arrays at 533 the time of pre-processing (CPP capability). 519 Unless a linear free surface is used (\np{ln\_linssh}=false), the arrays describing 520 the grid point depths and vertical scale factors are three set of three dimensional arrays $(i,j,k)$ 521 defined at \textit{before}, \textit{now} and \textit{after} time step. The time at which they are 522 defined is indicated by a suffix:$\_b$, $\_n$, or $\_a$, respectively. They are updated at each model time step 523 using a fixed reference coordinate system which computer names have a $\_0$ suffix. 524 When the linear free surface option is used (\np{ln\_linssh}=true), \textit{before}, \textit{now} 525 and \textit{after} arrays are simply set one for all to their reference counterpart. 526 534 527 535 528 % ------------------------------------------------------------------------------------------------------------- … … 540 533 541 534 Three options are possible for defining the bathymetry, according to the 542 namelist variable \np{nn\_bathy} :535 namelist variable \np{nn\_bathy} (found in \ngn{namdom} namelist): 543 536 \begin{description} 544 537 \item[\np{nn\_bathy} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$ … … 548 541 domain width at the central latitude. This is meant for the "EEL-R5" configuration, 549 542 a periodic or open boundary channel with a seamount. 550 \item[\np{nn\_bathy} = 1] read a bathymetry . The \ifile{bathy\_meter} file (Netcdf format)551 provides the ocean depth (positive, in meters) at each grid point of the model grid. 552 The bathymetry is usually built by interpolating a standard bathymetry product543 \item[\np{nn\_bathy} = 1] read a bathymetry and ice shelf draft (if needed). 544 The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters) 545 at each grid point of the model grid. The bathymetry is usually built by interpolating a standard bathymetry product 553 546 ($e.g.$ ETOPO2) onto the horizontal ocean mesh. Defining the bathymetry also 554 547 defines the coastline: where the bathymetry is zero, no model levels are defined 555 548 (all levels are masked). 549 550 The \ifile{isfdraft\_meter} file (Netcdf format) provides the ice shelf draft (positive, in meters) 551 at each grid point of the model grid. This file is only needed if \np{ln\_isfcav}~=~true. 552 Defining the ice shelf draft will also define the ice shelf edge and the grounding line position. 556 553 \end{description} 557 554 … … 573 570 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 574 571 \begin{figure}[!tb] \begin{center} 575 \includegraphics[width=0.90\textwidth]{ ./TexFiles/Figures/Fig_zgr.pdf}572 \includegraphics[width=0.90\textwidth]{Fig_zgr} 576 573 \caption{ \label{Fig_zgr} 577 574 Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for … … 610 607 (Fig.~\ref{Fig_zgr}). 611 608 609 If the ice shelf cavities are opened (\np{ln\_isfcav}=~true~}), the definition of $z_0$ is the same. 610 However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to: 611 \begin{equation} \label{DOM_zgr_ana} 612 \begin{split} 613 e_3^T(k) &= z_W (k+1) - z_W (k) \\ 614 e_3^W(k) &= z_T (k) - z_T (k-1) \\ 615 \end{split} 616 \end{equation} 617 This formulation decrease the self-generated circulation into the ice shelf cavity 618 (which can, in extreme case, leads to blow up).\\ 619 620 612 621 The most used vertical grid for ORCA2 has $10~m$ ($500~m)$ resolution in the 613 622 surface (bottom) layers and a depth which varies from 0 at the sea surface to a … … 721 730 usually 10\%, of the default thickness $e_{3t}(jk)$). 722 731 723 \colorbox{yellow}{Add a figure here of pstep especially at last ocean level}732 \gmcomment{ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level } } 724 733 725 734 % ------------------------------------------------------------------------------------------------------------- … … 749 758 depth, since a mixed step-like and bottom-following representation of the 750 759 topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e) or an envelop bathymetry can be defined (Fig.~\ref{Fig_z_zps_s_sps}f). 751 The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects the sea bed and becomes a pseudo z-coordinate. The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} as the minimum and maximum depths at which the terrain-following vertical coordinate is calculated. 752 753 Options for stretching the coordinate are provided as examples, but care must be taken to ensure that the vertical stretch used is appropriate for the application. 754 755 The original default NEMO s-coordinate stretching is available if neither of the other options are specified as true (\np{ln\_sco\_SH94}~=~false and \np{ln\_sco\_SF12}~=~false.) This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}: 760 The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects 761 the sea bed and becomes a pseudo z-coordinate. 762 The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} 763 as the minimum and maximum depths at which the terrain-following vertical coordinate is calculated. 764 765 Options for stretching the coordinate are provided as examples, but care must be taken to ensure 766 that the vertical stretch used is appropriate for the application. 767 768 The original default NEMO s-coordinate stretching is available if neither of the other options 769 are specified as true (\np{ln\_s\_SH94}~=~false and \np{ln\_s\_SF12}~=~false). 770 This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}: 756 771 757 772 \begin{equation} … … 760 775 \end{equation} 761 776 762 where $s_{min}$ is the depth at which the s-coordinate stretching starts and allows a z-coordinate to placed on top of the stretched coordinate, and z is the depth (negative down from the asea surface). 777 where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and 778 allows a $z$-coordinate to placed on top of the stretched coordinate, 779 and $z$ is the depth (negative down from the asea surface). 763 780 764 781 \begin{equation} … … 775 792 \end{equation} 776 793 777 A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94} stretching (\np{ln\_sco\_SH94}~=~true), is also available and is more commonly used for shelf seas modelling: 794 A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94} 795 stretching (\np{ln\_s\_SH94}~=~true), is also available and is more commonly used for shelf seas modelling: 778 796 779 797 \begin{equation} … … 785 803 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 786 804 \begin{figure}[!ht] \begin{center} 787 \includegraphics[width=1.0\textwidth]{ ./TexFiles/Figures/Fig_sco_function.pdf}805 \includegraphics[width=1.0\textwidth]{Fig_sco_function} 788 806 \caption{ \label{Fig_sco_function} 789 807 Examples of the stretching function applied to a seamount; from left to right: … … 792 810 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 793 811 794 where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from pure $\sigma$ to the stretched coordinate, and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) are the surface and 795 bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 812 where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from 813 pure $\sigma$ to the stretched coordinate, and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) 814 are the surface and bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 796 815 $0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom 797 816 increase of the vertical resolution (Fig.~\ref{Fig_sco_function}). 798 817 799 Another example has been provided at version 3.5 (\np{ln\_sco\_SF12}) that allows a fixed surface resolution in an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}. In this case the a stretching function $\gamma$ is defined such that: 818 Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows 819 a fixed surface resolution in an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}. 820 In this case the a stretching function $\gamma$ is defined such that: 800 821 801 822 \begin{equation} … … 815 836 \end{equation} 816 837 817 This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and user prescribed surface (\np{rn\_zs}) and bottom depths. The bottom cell depth in this example is given as a function of water depth: 838 This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of 839 the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards 840 the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and user prescribed surface (\np{rn\_zs}) 841 and bottom depths. The bottom cell depth in this example is given as a function of water depth: 818 842 819 843 \begin{equation} \label{DOM_zb} … … 825 849 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 826 850 \begin{figure}[!ht] 827 \includegraphics[width=1.0\textwidth]{ ./TexFiles/Figures/FIG_DOM_compare_coordinates_surface.pdf}851 \includegraphics[width=1.0\textwidth]{FIG_DOM_compare_coordinates_surface} 828 852 \caption{A comparison of the \citet{Song_Haidvogel_JCP94} $S$-coordinate (solid lines), a 50 level $Z$-coordinate (contoured surfaces) and the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in the surface 100m for a idealised bathymetry that goes from 50m to 5500m depth. For clarity every third coordinate surface is shown.} 829 853 \label{fig_compare_coordinates_surface} … … 840 864 % z*- or s*-coordinate 841 865 % ------------------------------------------------------------------------------------------------------------- 842 \subsection{$z^*$- or $s^*$-coordinate (add \key{vvl}) } 843 \label{DOM_zgr_vvl} 844 845 This option is described in the Report by Levier \textit{et al.} (2007), available on 846 the \NEMO web site. 866 \subsection{$z^*$- or $s^*$-coordinate (\np{ln\_linssh}=false) } 867 \label{DOM_zgr_star} 868 869 This option is described in the Report by Levier \textit{et al.} (2007), available on the \NEMO web site. 847 870 848 871 %gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances … … 860 883 gives the number of ocean levels ($i.e.$ those that are not masked) at each 861 884 $t$-point. mbathy is computed from the meter bathymetry using the definiton of 862 gdept as the number of $t$-points which gdept $\leq$ bathy. 885 gdept as the number of $t$-points which gdept $\leq$ bathy. 863 886 864 887 Modifications of the model bathymetry are performed in the \textit{bat\_ctl} … … 866 889 that do not communicate with another ocean point at the same level are eliminated. 867 890 868 From the \textit{mbathy} array, the mask fields are defined as follows: 891 As for the representation of bathymetry, a 2D integer array, misfdep, is created. 892 misfdep defines the level of the first wet $t$-point. All the cells between $k=1$ and $misfdep(i,j)-1$ are masked. 893 By default, misfdep(:,:)=1 and no cells are masked. 894 895 In case of ice shelf cavities, modifications of the model bathymetry and ice shelf draft into 896 the cavities are performed in the \textit{zgr\_isf} routine. The compatibility between ice shelf draft and bathymetry is checked. 897 All the locations where the isf cavity is thinnest than \np{rn\_isfhmin} meters are grounded ($i.e.$ masked). 898 If only one cell on the water column is opened at $t$-, $u$- or $v$-points, the bathymetry or the ice shelf draft is dug to fit this constrain. 899 If the incompatibility is too strong (need to dig more than 1 cell), the cell is masked.\\ 900 901 From the \textit{mbathy} and \textit{misfdep} array, the mask fields are defined as follows: 869 902 \begin{align*} 870 tmask(i,j,k) &= \begin{cases} \; 1& \text{ if $k\leq mbathy(i,j)$ } \\ 871 \; 0& \text{ if $k\leq mbathy(i,j)$ } \end{cases} \\ 903 tmask(i,j,k) &= \begin{cases} \; 0& \text{ if $k < misfdep(i,j) $ } \\ 904 \; 1& \text{ if $misfdep(i,j) \leq k\leq mbathy(i,j)$ } \\ 905 \; 0& \text{ if $k > mbathy(i,j)$ } \end{cases} \\ 872 906 umask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 873 907 vmask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i,j+1,k) \\ 874 908 fmask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 875 & \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) 909 & \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 910 wmask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i,j,k-1) \text{ with } wmask(i,j,1) = tmask(i,j,1) 876 911 \end{align*} 877 912 878 Note that \textit{wmask} is not defined as it is exactly equal to \textit{tmask} with 879 the numerical indexing used (\S~\ref{DOM_Num_Index}). Moreover, the 880 specification of closed lateral boundaries requires that at least the first and last 913 Note that, without ice shelves cavities, masks at $t-$ and $w-$points are identical with 914 the numerical indexing used (\S~\ref{DOM_Num_Index}). Nevertheless, $wmask$ are required 915 with oceean cavities to deal with the top boundary (ice shelf/ocean interface) 916 exactly in the same way as for the bottom boundary. 917 918 The specification of closed lateral boundaries requires that at least the first and last 881 919 rows and columns of the \textit{mbathy} array are set to zero. In the particular 882 920 case of an east-west cyclical boundary condition, \textit{mbathy} has its last … … 884 922 (and so too the mask arrays) (see \S~\ref{LBC_jperio}). 885 923 886 %%%887 \gmcomment{ \colorbox{yellow}{Add one word on tricky trick !} mbathy in further modified in zdfbfr{\ldots}. }888 %%%889 924 890 925 % ================================================================ … … 910 945 (typical of the tropical ocean), see \rou{istate\_t\_s} subroutine called from \mdl{istate} module. 911 946 \end{description} 947 \end{document}
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