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branches/UKMO/dev_r5518_GC3p0_package/DOC/TexFiles/Chapters/Chap_DOM.tex
r5120 r6440 1 1 % ================================================================ 2 % Chapter 2 �Space and Time Domain (DOM)2 % Chapter 2 ——— Space and Time Domain (DOM) 3 3 % ================================================================ 4 4 \chapter{Space Domain (DOM) } … … 138 138 and $f$-points, and its divergence defined at $t$-points: 139 139 \begin{eqnarray} \label{Eq_DOM_curl} 140 \nabla \times {\rm 140 \nabla \times {\rm{\bf A}}\equiv & 141 141 \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} \\ 142 142 +& \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} \\ … … 183 183 Let $a$ and $b$ be two fields defined on the mesh, with value zero inside 184 184 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 $\overline{\,\cdot\,}^{\,i}$,185 operators ($\delta_i$, $\delta_j$ and $\delta_k$) are skew-symmetric linear operators, 186 and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$, 187 187 $\overline{\,\cdot\,}^{\,k}$ and $\overline{\,\cdot\,}^{\,k}$) are symmetric linear 188 188 operators, $i.e.$ … … 364 364 For both grids here, the same $w$-point depth has been chosen but in (a) the 365 365 $t$-points are set half way between $w$-points while in (b) they are defined from 366 an analytical function: $z(k)=5\,( i-1/2)^3 - 45\,(i-1/2)^2 + 140\,(i-1/2) - 150$.366 an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$. 367 367 Note the resulting difference between the value of the grid-size $\Delta_k$ and 368 368 those of the scale factor $e_k$. } … … 425 425 426 426 The choice of the grid must be consistent with the boundary conditions specified 427 by the parameter \np{jperio}(see {\S\ref{LBC}).427 by \np{jperio}, a parameter found in \ngn{namcfg} namelist (see {\S\ref{LBC}). 428 428 429 429 % ------------------------------------------------------------------------------------------------------------- … … 481 481 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 482 482 483 The choice of a vertical coordinate, even if it is made through a namelist parameter,483 The choice of a vertical coordinate, even if it is made through \ngn{namzgr} namelist parameters, 484 484 must be done once of all at the beginning of an experiment. It is not intended as an 485 485 option which can be enabled or disabled in the middle of an experiment. Three main … … 494 494 bathymetry or $s$-coordinate (hybrid and partial step coordinates have not 495 495 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). 496 (\np{ln\_zps}~=~true), ocean cavity beneath ice shelves can be open (\np{ln\_isfcav}~=~true) 497 and partial step are also applied at the ocean/ice shelf interface. 497 498 498 499 Contrary to the horizontal grid, the vertical grid is computed in the code and no 499 500 provision is made for reading it from a file. The only input file is the bathymetry 500 (in meters) (\ifile{bathy\_meter}) 501 (in meters) (\ifile{bathy\_meter}). 501 502 \footnote{N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the 502 503 \ifile{bathy\_meter} file, so that the computation of the number of wet ocean point … … 540 541 541 542 Three options are possible for defining the bathymetry, according to the 542 namelist variable \np{nn\_bathy} :543 namelist variable \np{nn\_bathy} (found in \ngn{namdom} namelist): 543 544 \begin{description} 544 545 \item[\np{nn\_bathy} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$ … … 548 549 domain width at the central latitude. This is meant for the "EEL-R5" configuration, 549 550 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 product551 \item[\np{nn\_bathy} = 1] read a bathymetry and ice shelf draft (if needed). 552 The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters) 553 at each grid point of the model grid. The bathymetry is usually built by interpolating a standard bathymetry product 553 554 ($e.g.$ ETOPO2) onto the horizontal ocean mesh. Defining the bathymetry also 554 555 defines the coastline: where the bathymetry is zero, no model levels are defined 555 556 (all levels are masked). 557 558 The \ifile{isfdraft\_meter} file (Netcdf format) provides the ice shelf draft (positive, in meters) 559 at each grid point of the model grid. This file is only needed if \np{ln\_isfcav}~=~true. 560 Defining the ice shelf draft will also define the ice shelf edge and the grounding line position. 556 561 \end{description} 557 562 … … 610 615 (Fig.~\ref{Fig_zgr}). 611 616 617 If the ice shelf cavities are opened (\np{ln\_isfcav}=~true~}), the definition of $z_0$ is the same. 618 However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to: 619 \begin{equation} \label{DOM_zgr_ana} 620 \begin{split} 621 e_3^T(k) &= z_W (k+1) - z_W (k) \\ 622 e_3^W(k) &= z_T (k) - z_T (k-1) \\ 623 \end{split} 624 \end{equation} 625 This formulation decrease the self-generated circulation into the ice shelf cavity 626 (which can, in extreme case, leads to blow up).\\ 627 628 612 629 The most used vertical grid for ORCA2 has $10~m$ ($500~m)$ resolution in the 613 630 surface (bottom) layers and a depth which varies from 0 at the sea surface to a … … 721 738 usually 10\%, of the default thickness $e_{3t}(jk)$). 722 739 723 \colorbox{yellow}{Add a figure here of pstep especially at last ocean level}740 \gmcomment{ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level } } 724 741 725 742 % ------------------------------------------------------------------------------------------------------------- … … 860 877 gives the number of ocean levels ($i.e.$ those that are not masked) at each 861 878 $t$-point. mbathy is computed from the meter bathymetry using the definiton of 862 gdept as the number of $t$-points which gdept $\leq$ bathy. 879 gdept as the number of $t$-points which gdept $\leq$ bathy. 863 880 864 881 Modifications of the model bathymetry are performed in the \textit{bat\_ctl} 865 882 routine (see \mdl{domzgr} module) after mbathy is computed. Isolated grid points 866 that do not communicate with another ocean point at the same level are eliminated. 883 that do not communicate with another ocean point at the same level are eliminated.\\ 884 885 As for the representation of bathymetry, a 2D integer array, misfdep, is created. 886 misfdep defines the level of the first wet $t$-point. All the cells between $k=1$ and $misfdep(i,j)-1$ are masked. 887 By default, misfdep(:,:)=1 and no cells are masked. 888 889 In case of ice shelf cavities (\np{ln\_isfcav}~=~true), modifications of the model bathymetry and ice shelf draft in 890 the cavities are performed through the \textit{zgr\_isf} routine. The compatibility between ice shelf draft and bathymetry is checked: 891 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 have a 2-level water column 892 (i.e. two unmasked levels). If the incompatibility is too strong (i.e. need to dig more than one cell), the entire water column is masked.\\ 867 893 868 894 From the \textit{mbathy} array, the mask fields are defined as follows: 869 895 \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} \\ 896 tmask(i,j,k) &= \begin{cases} \; 0& \text{ if $k < misfdep(i,j) $ } \\ 897 \; 1& \text{ if $misfdep(i,j) \leq k\leq mbathy(i,j)$ } \\ 898 \; 0& \text{ if $k > mbathy(i,j)$ } \end{cases} \\ 872 899 umask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 873 900 vmask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i,j+1,k) \\ 874 901 fmask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 875 & \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) 902 & \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 903 wmask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i,j,k-1) \text{ with } wmask(i,j,1) = tmask(i,j,1) 876 904 \end{align*} 877 905 878 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 906 Note, wmask is now defined. It allows, in case of ice shelves, 907 to deal with the top boundary (ice shelf/ocean interface) exactly in the same way as for the bottom boundary. 908 909 The specification of closed lateral boundaries requires that at least the first and last 881 910 rows and columns of the \textit{mbathy} array are set to zero. In the particular 882 911 case of an east-west cyclical boundary condition, \textit{mbathy} has its last
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