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r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Space Domain (DOM)} 7 \label{ DOM}7 \label{chap:DOM} 8 8 \minitoc 9 9 … … 20 20 $\ $\newline % force a new line 21 21 22 Having defined the continuous equations in Chap.~\ref{PE} and chosen a time23 discretization Chap.~\ref{STP}, we need to choose a discretization on a grid,22 Having defined the continuous equations in \autoref{chap:PE} and chosen a time 23 discretization \autoref{chap:STP}, we need to choose a discretization on a grid, 24 24 and numerical algorithms. In the present chapter, we provide a general description 25 25 of the staggered grid used in \NEMO, and other information relevant to the main … … 32 32 % ================================================================ 33 33 \section{Fundamentals of the discretisation} 34 \label{ DOM_basics}34 \label{sec:DOM_basics} 35 35 36 36 % ------------------------------------------------------------------------------------------------------------- … … 38 38 % ------------------------------------------------------------------------------------------------------------- 39 39 \subsection{Arrangement of variables} 40 \label{ DOM_cell}40 \label{subsec:DOM_cell} 41 41 42 42 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 43 43 \begin{figure}[!tb] \begin{center} 44 44 \includegraphics[width=0.90\textwidth]{Fig_cell} 45 \caption{ \protect\label{ Fig_cell}45 \caption{ \protect\label{fig:cell} 46 46 Arrangement of variables. $t$ indicates scalar points where temperature, 47 47 salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$) … … 56 56 space directions. The arrangement of variables is the same in all directions. 57 57 It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector 58 points $(u, v, w)$ defined in the centre of each face of the cells ( Fig. \ref{Fig_cell}).58 points $(u, v, w)$ defined in the centre of each face of the cells (\autoref{fig:cell}). 59 59 This is the generalisation to three dimensions of the well-known ``C'' grid in 60 60 Arakawa's classification \citep{Mesinger_Arakawa_Bk76}. The relative and … … 66 66 by the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$. 67 67 The grid-points are located at integer or integer and a half value of $(i,j,k)$ as 68 indicated on Table \ref{Tab_cell}. In all the following, subscripts $u$, $v$, $w$,68 indicated on \autoref{tab:cell}. In all the following, subscripts $u$, $v$, $w$, 69 69 $f$, $uw$, $vw$ or $fw$ indicate the position of the grid-point where the scale 70 70 factors are defined. Each scale factor is defined as the local analytical value 71 provided by \ eqref{Eq_scale_factors}. As a result, the mesh on which partial71 provided by \autoref{eq:scale_factors}. As a result, the mesh on which partial 72 72 derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and 73 73 $\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity. … … 78 78 from their analytical expression. This preserves the symmetry of the discrete set 79 79 of equations and therefore satisfies many of the continuous properties (see 80 Appendix~\ref{Apdx_C}). A similar, related remark can be made about the domain80 \autoref{apdx:C}). A similar, related remark can be made about the domain 81 81 size: when needed, an area, volume, or the total ocean depth must be evaluated 82 as the sum of the relevant scale factors (see \ eqref{DOM_bar}) in the next section).82 as the sum of the relevant scale factors (see \autoref{eq:DOM_bar}) in the next section). 83 83 84 84 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 95 95 fw & $i+1/2$ & $j+1/2$ & $k+1/2$ \\ \hline 96 96 \end{tabular} 97 \caption{ \protect\label{ Tab_cell}97 \caption{ \protect\label{tab:cell} 98 98 Location of grid-points as a function of integer or integer and a half value of the column, 99 99 line or level. This indexing is only used for the writing of the semi-discrete equation. 100 100 In the code, the indexing uses integer values only and has a reverse direction 101 in the vertical (see \ S\ref{DOM_Num_Index})}101 in the vertical (see \autoref{subsec:DOM_Num_Index})} 102 102 \end{center} 103 103 \end{table} … … 108 108 % ------------------------------------------------------------------------------------------------------------- 109 109 \subsection{Discrete operators} 110 \label{ DOM_operators}110 \label{subsec:DOM_operators} 111 111 112 112 Given the values of a variable $q$ at adjacent points, the differencing and 113 113 averaging operators at the midpoint between them are: 114 \begin{subequations} \label{ Eq_di_mi}114 \begin{subequations} \label{eq:di_mi} 115 115 \begin{align} 116 116 \delta _i [q] &= \ \ q(i+1/2) - q(i-1/2) \\ … … 120 120 121 121 Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and 122 $k+1/2$. Following \ eqref{Eq_PE_grad} and \eqref{Eq_PE_lap}, the gradient of a122 $k+1/2$. Following \autoref{eq:PE_grad} and \autoref{eq:PE_lap}, the gradient of a 123 123 variable $q$ defined at a $t$-point has its three components defined at $u$-, $v$- 124 124 and $w$-points while its Laplacien is defined at $t$-point. These operators have 125 125 the following discrete forms in the curvilinear $s$-coordinate system: 126 \begin{equation} \label{ Eq_DOM_grad}126 \begin{equation} \label{eq:DOM_grad} 127 127 \nabla q\equiv \frac{1}{e_{1u} } \delta _{i+1/2 } [q] \;\,\mathbf{i} 128 128 + \frac{1}{e_{2v} } \delta _{j+1/2 } [q] \;\,\mathbf{j} 129 129 + \frac{1}{e_{3w}} \delta _{k+1/2} [q] \;\,\mathbf{k} 130 130 \end{equation} 131 \begin{multline} \label{ Eq_DOM_lap}131 \begin{multline} \label{eq:DOM_lap} 132 132 \Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 133 133 \;\left( \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right] … … 136 136 \end{multline} 137 137 138 Following \ eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$138 Following \autoref{eq:PE_curl} and \autoref{eq:PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ 139 139 defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, 140 140 and $f$-points, and its divergence defined at $t$-points: 141 \begin{eqnarray} \label{ Eq_DOM_curl}141 \begin{eqnarray} \label{eq:DOM_curl} 142 142 \nabla \times {\rm{\bf A}}\equiv & 143 143 \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} \\ … … 145 145 +& \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} 146 146 \end{eqnarray} 147 \begin{eqnarray} \label{ Eq_DOM_div}147 \begin{eqnarray} \label{eq:DOM_div} 148 148 \nabla \cdot \rm{\bf A} \equiv 149 149 \frac{1}{e_{1t}\,e_{2t}\,e_{3t}} \left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] … … 153 153 The vertical average over the whole water column denoted by an overbar becomes 154 154 for a quantity $q$ which is a masked field (i.e. equal to zero inside solid area): 155 \begin{equation} \label{ DOM_bar}155 \begin{equation} \label{eq:DOM_bar} 156 156 \bar q = \frac{1}{H} \int_{k^b}^{k^o} {q\;e_{3q} \,dk} 157 157 \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} } … … 163 163 164 164 In continuous form, the following properties are satisfied: 165 \begin{equation} \label{ Eq_DOM_curl_grad}165 \begin{equation} \label{eq:DOM_curl_grad} 166 166 \nabla \times \nabla q ={\rm {\bf {0}}} 167 167 \end{equation} 168 \begin{equation} \label{ Eq_DOM_div_curl}168 \begin{equation} \label{eq:DOM_div_curl} 169 169 \nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0 170 170 \end{equation} … … 181 181 operators, $i.e.$ 182 182 \begin{align} 183 \label{ DOM_di_adj}183 \label{eq:DOM_di_adj} 184 184 \sum\limits_i { a_i \;\delta _i \left[ b \right]} 185 185 &\equiv -\sum\limits_i {\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} } \\ 186 \label{ DOM_mi_adj}186 \label{eq:DOM_mi_adj} 187 187 \sum\limits_i { a_i \;\overline b^{\,i}} 188 188 & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} } … … 192 192 $\delta_i^*=\delta_{i+1/2}$ and 193 193 ${(\overline{\,\cdot \,}^{\,i})}^*= \overline{\,\cdot\,}^{\,i+1/2}$, respectively. 194 These two properties will be used extensively in the Appendix~\ref{Apdx_C} to194 These two properties will be used extensively in the \autoref{apdx:C} to 195 195 demonstrate integral conservative properties of the discrete formulation chosen. 196 196 … … 199 199 % ------------------------------------------------------------------------------------------------------------- 200 200 \subsection{Numerical indexing} 201 \label{ DOM_Num_Index}201 \label{subsec:DOM_Num_Index} 202 202 203 203 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 204 204 \begin{figure}[!tb] \begin{center} 205 205 \includegraphics[width=0.90\textwidth]{Fig_index_hor} 206 \caption{ \protect\label{ Fig_index_hor}206 \caption{ \protect\label{fig:index_hor} 207 207 Horizontal integer indexing used in the \textsc{Fortran} code. The dashed area indicates 208 208 the cell in which variables contained in arrays have the same $i$- and $j$-indices} … … 211 211 212 212 The array representation used in the \textsc{Fortran} code requires an integer 213 indexing while the analytical definition of the mesh (see \ S\ref{DOM_cell}) is213 indexing while the analytical definition of the mesh (see \autoref{subsec:DOM_cell}) is 214 214 associated with the use of integer values for $t$-points and both integer and 215 215 integer and a half values for all the other points. Therefore a specific integer … … 222 222 % ----------------------------------- 223 223 \subsubsection{Horizontal indexing} 224 \label{ DOM_Num_Index_hor}225 226 The indexing in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}.224 \label{subsec:DOM_Num_Index_hor} 225 226 The indexing in the horizontal plane has been chosen as shown in \autoref{fig:index_hor}. 227 227 For an increasing $i$ index ($j$ index), the $t$-point and the eastward $u$-point 228 (northward $v$-point) have the same index (see the dashed area in Fig.\ref{Fig_index_hor}).228 (northward $v$-point) have the same index (see the dashed area in \autoref{fig:index_hor}). 229 229 A $t$-point and its nearest northeast $f$-point have the same $i$-and $j$-indices. 230 230 … … 233 233 % ----------------------------------- 234 234 \subsubsection{Vertical indexing} 235 \label{ DOM_Num_Index_vertical}235 \label{subsec:DOM_Num_Index_vertical} 236 236 237 237 In the vertical, the chosen indexing requires special attention since the 238 238 $k$-axis is re-orientated downward in the \textsc{Fortran} code compared 239 to the indexing used in the semi-discrete equations and given in \ S\ref{DOM_cell}.239 to the indexing used in the semi-discrete equations and given in \autoref{subsec:DOM_cell}. 240 240 The sea surface corresponds to the $w$-level $k=1$ which is the same index 241 as $t$-level just below ( Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$)241 as $t$-level just below (\autoref{fig:index_vert}). The last $w$-level ($k=jpk$) 242 242 either corresponds to the ocean floor or is inside the bathymetry while the last 243 $t$-level is always inside the bathymetry ( Fig.\ref{Fig_index_vert}). Note that243 $t$-level is always inside the bathymetry (\autoref{fig:index_vert}). Note that 244 244 for an increasing $k$ index, a $w$-point and the $t$-point just below have the 245 245 same $k$ index, in opposition to what is done in the horizontal plane where 246 246 it is the $t$-point and the nearest velocity points in the direction of the horizontal 247 247 axis that have the same $i$ or $j$ index (compare the dashed area in 248 Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are248 \autoref{fig:index_hor} and \autoref{fig:index_vert}). Since the scale factors are 249 249 chosen to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran} 250 250 code \emph{before all the vertical derivatives} of the discrete equations given in … … 254 254 \begin{figure}[!pt] \begin{center} 255 255 \includegraphics[width=.90\textwidth]{Fig_index_vert} 256 \caption{ \protect\label{ Fig_index_vert}256 \caption{ \protect\label{fig:index_vert} 257 257 Vertical integer indexing used in the \textsc{Fortran } code. Note that 258 258 the $k$-axis is orientated downward. The dashed area indicates the cell in … … 265 265 % ----------------------------------- 266 266 \subsubsection{Domain size} 267 \label{ DOM_size}267 \label{subsec:DOM_size} 268 268 269 269 The total size of the computational domain is set by the parameters \np{jpiglo}, … … 273 273 %%% 274 274 Parameters $jpi$ and $jpj$ refer to the size of each processor subdomain when the code is 275 run in parallel using domain decomposition (\key{mpp\_mpi} defined, see \ S\ref{LBC_mpp}).275 run in parallel using domain decomposition (\key{mpp\_mpi} defined, see \autoref{sec:LBC_mpp}). 276 276 277 277 … … 282 282 % ================================================================ 283 283 \section{Needed fields} 284 \label{ DOM_fields}284 \label{sec:DOM_fields} 285 285 The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined 286 286 by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 287 287 The grid-points are located at integer or integer and a half values of as indicated 288 in Table~\ref{Tab_cell}. The associated scale factors are defined using the289 analytical first derivative of the transformation \ eqref{Eq_scale_factors}.288 in \autoref{tab:cell}. The associated scale factors are defined using the 289 analytical first derivative of the transformation \autoref{eq:scale_factors}. 290 290 Necessary fields for configuration definition are: \\ 291 291 Geographic position : … … 316 316 % ------------------------------------------------------------------------------------------------------------- 317 317 %\subsection{List of needed fields to build DOMAIN} 318 %\label{ DOM_fields_list}318 %\label{subsec:DOM_fields_list} 319 319 320 320 … … 323 323 % ================================================================ 324 324 \section{Horizontal grid mesh (\protect\mdl{domhgr})} 325 \label{ DOM_hgr}325 \label{sec:DOM_hgr} 326 326 327 327 % ------------------------------------------------------------------------------------------------------------- … … 329 329 % ------------------------------------------------------------------------------------------------------------- 330 330 \subsection{Coordinates and scale factors} 331 \label{ DOM_hgr_coord_e}331 \label{subsec:DOM_hgr_coord_e} 332 332 333 333 The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined 334 334 by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 335 335 The grid-points are located at integer or integer and a half values of as indicated 336 in Table~\ref{Tab_cell}. The associated scale factors are defined using the337 analytical first derivative of the transformation \ eqref{Eq_scale_factors}. These336 in \autoref{tab:cell}. The associated scale factors are defined using the 337 analytical first derivative of the transformation \autoref{eq:scale_factors}. These 338 338 definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, which 339 339 provide the horizontal and vertical meshes, respectively. This section deals with … … 343 343 analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a 344 344 function of $(i,j)$. The horizontal scale factors are calculated using 345 \ eqref{Eq_scale_factors}. For example, when the longitude and latitude are345 \autoref{eq:scale_factors}. For example, when the longitude and latitude are 346 346 function of a single value ($i$ and $j$, respectively) (geographical configuration 347 347 of the mesh), the horizontal mesh definition reduces to define the wanted … … 382 382 allowing the user to set arbitrary jumps in thickness between adjacent layers) 383 383 \citep{Treguier1996}. An example of the effect of such a choice is shown in 384 Fig.~\ref{Fig_zgr_e3}.384 \autoref{fig:zgr_e3}. 385 385 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 386 386 \begin{figure}[!t] \begin{center} 387 387 \includegraphics[width=0.90\textwidth]{Fig_zgr_e3} 388 \caption{ \protect\label{ Fig_zgr_e3}388 \caption{ \protect\label{fig:zgr_e3} 389 389 Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical, 390 390 and (b) analytically derived grid-point position and scale factors. … … 401 401 % ------------------------------------------------------------------------------------------------------------- 402 402 \subsection{Choice of horizontal grid} 403 \label{ DOM_hgr_msh_choice}403 \label{subsec:DOM_hgr_msh_choice} 404 404 405 405 … … 408 408 % ------------------------------------------------------------------------------------------------------------- 409 409 \subsection{Output grid files} 410 \label{ DOM_hgr_files}410 \label{subsec:DOM_hgr_files} 411 411 412 412 All the arrays relating to a particular ocean model configuration (grid-point … … 426 426 % ================================================================ 427 427 \section{Vertical grid (\protect\mdl{domzgr})} 428 \label{ DOM_zgr}428 \label{sec:DOM_zgr} 429 429 %-----------------------------------------nam_zgr & namdom------------------------------------------- 430 430 %\forfile{../namelists/namzgr} … … 444 444 \begin{figure}[!tb] \begin{center} 445 445 \includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps} 446 \caption{ \protect\label{ Fig_z_zps_s_sps}446 \caption{ \protect\label{fig:z_zps_s_sps} 447 447 The ocean bottom as seen by the model: 448 448 (a) $z$-coordinate with full step, … … 451 451 (d) hybrid $s-z$ coordinate, 452 452 (e) hybrid $s-z$ coordinate with partial step, and 453 (f) same as (e) but in the non-linear free surface (\ np{ln\_linssh}\forcode{ = .false.}).453 (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}\forcode{ = .false.}). 454 454 Note that the non-linear free surface can be used with any of the 455 455 5 coordinates (a) to (e).} … … 460 460 must be done once of all at the beginning of an experiment. It is not intended as an 461 461 option which can be enabled or disabled in the middle of an experiment. Three main 462 choices are offered ( Fig.~\ref{Fig_z_zps_s_sps}a to c): $z$-coordinate with full step462 choices are offered (\autoref{fig:z_zps_s_sps}a to c): $z$-coordinate with full step 463 463 bathymetry (\np{ln\_zco}\forcode{ = .true.}), $z$-coordinate with partial step bathymetry 464 464 (\np{ln\_zps}\forcode{ = .true.}), or generalized, $s$-coordinate (\np{ln\_sco}\forcode{ = .true.}). 465 465 Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate 466 ( 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:466 (\autoref{fig:z_zps_s_sps}d and \autoref{fig:z_zps_s_sps}e). By default a non-linear free surface is used: 467 467 the coordinate follow the time-variation of the free surface so that the transformation is time dependent: 468 $z(i,j,k,t)$ ( Fig.~\ref{Fig_z_zps_s_sps}f). When a linear free surface is assumed (\np{ln\_linssh}\forcode{ = .true.}),468 $z(i,j,k,t)$ (\autoref{fig:z_zps_s_sps}f). When a linear free surface is assumed (\np{ln\_linssh}\forcode{ = .true.}), 469 469 the vertical coordinate are fixed in time, but the seawater can move up and down across the z=0 surface 470 470 (in other words, the top of the ocean in not a rigid-lid). … … 513 513 % ------------------------------------------------------------------------------------------------------------- 514 514 \subsection{Meter bathymetry} 515 \label{ DOM_bathy}515 \label{subsec:DOM_bathy} 516 516 517 517 Three options are possible for defining the bathymetry, according to the … … 541 541 This is unnecessary when the ocean is forced by fixed atmospheric conditions, 542 542 so these seas can be removed from the ocean domain. The user has the option 543 to set the bathymetry in closed seas to zero (see \ S\ref{MISC_closea}), but the543 to set the bathymetry in closed seas to zero (see \autoref{sec:MISC_closea}), but the 544 544 code has to be adapted to the user's configuration. 545 545 … … 549 549 \subsection[$Z$-coordinate (\protect\np{ln\_zco}\forcode{ = .true.}) and ref. coordinate] 550 550 {$Z$-coordinate (\protect\np{ln\_zco}\forcode{ = .true.}) and reference coordinate} 551 \label{ DOM_zco}551 \label{subsec:DOM_zco} 552 552 553 553 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 554 554 \begin{figure}[!tb] \begin{center} 555 555 \includegraphics[width=0.90\textwidth]{Fig_zgr} 556 \caption{ \protect\label{ Fig_zgr}556 \caption{ \protect\label{fig:zgr} 557 557 Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for 558 558 (a) T-point depth and (b) the associated scale factor as computed 559 from \ eqref{DOM_zgr_ana} using \eqref{DOM_zgr_coef} in $z$-coordinate.}559 from \autoref{eq:DOM_zgr_ana} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.} 560 560 \end{center} \end{figure} 561 561 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 563 563 The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$ 564 564 and $gdepw_0$ for $t$- and $w$-points, respectively. As indicated on 565 Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the565 \autoref{fig:index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the 566 566 ocean surface. There are at most \jp{jpk}-1 $t$-points inside the ocean, the 567 567 additional $t$-point at $jk=jpk$ is below the sea floor and is not used. … … 579 579 near the ocean surface. The following function is proposed as a standard for a 580 580 $z$-coordinate (with either full or partial steps): 581 \begin{equation} \label{ DOM_zgr_ana}581 \begin{equation} \label{eq:DOM_zgr_ana} 582 582 \begin{split} 583 583 z_0 (k) &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ … … 588 588 expression allows us to define a nearly uniform vertical location of levels at the 589 589 ocean top and bottom with a smooth hyperbolic tangent transition in between 590 ( Fig.~\ref{Fig_zgr}).590 (\autoref{fig:zgr}). 591 591 592 592 If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same. 593 593 However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to: 594 \begin{equation} \label{ DOM_zgr_ana}594 \begin{equation} \label{eq:DOM_zgr_ana} 595 595 \begin{split} 596 596 e_3^T(k) &= z_W (k+1) - z_W (k) \\ … … 605 605 surface (bottom) layers and a depth which varies from 0 at the sea surface to a 606 606 minimum of $-5000~m$. This leads to the following conditions: 607 \begin{equation} \label{ DOM_zgr_coef}607 \begin{equation} \label{eq:DOM_zgr_coef} 608 608 \begin{split} 609 609 e_3 (1+1/2) &=10. \\ … … 616 616 With the choice of the stretching $h_{cr} =3$ and the number of levels 617 617 \jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in 618 \ eqref{DOM_zgr_ana} have been determined such that \eqref{DOM_zgr_coef} is618 \autoref{eq:DOM_zgr_ana} have been determined such that \autoref{eq:DOM_zgr_coef} is 619 619 satisfied, through an optimisation procedure using a bisection method. For the first 620 620 standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$, 621 621 $h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$. The resulting depths and 622 scale factors as a function of the model levels are shown in Fig.~\ref{Fig_zgr} and623 given in Table \ref{Tab_orca_zgr}. Those values correspond to the parameters622 scale factors as a function of the model levels are shown in \autoref{fig:zgr} and 623 given in \autoref{tab:orca_zgr}. Those values correspond to the parameters 624 624 \np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist. 625 625 … … 675 675 31 & \textbf{5250.23}& 5000.00 & \textbf{500.56} & 500.33 \\ \hline 676 676 \end{tabular} \end{center} 677 \caption{ \protect\label{ Tab_orca_zgr}677 \caption{ \protect\label{tab:orca_zgr} 678 678 Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed 679 from \ eqref{DOM_zgr_ana} using the coefficients given in \eqref{DOM_zgr_coef}}679 from \autoref{eq:DOM_zgr_ana} using the coefficients given in \autoref{eq:DOM_zgr_coef}} 680 680 \end{table} 681 681 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 685 685 % ------------------------------------------------------------------------------------------------------------- 686 686 \subsection{$Z$-coordinate with partial step (\protect\np{ln\_zps}\forcode{ = .true.})} 687 \label{ DOM_zps}687 \label{subsec:DOM_zps} 688 688 %--------------------------------------------namdom------------------------------------------------------- 689 689 \forfile{../namelists/namdom} … … 717 717 % ------------------------------------------------------------------------------------------------------------- 718 718 \subsection{$S$-coordinate (\protect\np{ln\_sco}\forcode{ = .true.})} 719 \label{ DOM_sco}719 \label{subsec:DOM_sco} 720 720 %------------------------------------------nam_zgr_sco--------------------------------------------------- 721 721 %\forfile{../namelists/namzgr_sco} … … 726 726 function or its derivative, respectively: 727 727 728 \begin{equation} \label{ DOM_sco_ana}728 \begin{equation} \label{eq:DOM_sco_ana} 729 729 \begin{split} 730 730 z(k) &= h(i,j) \; z_0(k) \\ … … 737 737 surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean 738 738 depth, since a mixed step-like and bottom-following representation of the 739 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).739 topography can be used (\autoref{fig:z_zps_s_sps}d-e) or an envelop bathymetry can be defined (\autoref{fig:z_zps_s_sps}f). 740 740 The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects 741 741 the sea bed and becomes a pseudo z-coordinate. … … 764 764 \end{equation} 765 765 766 \begin{equation} \label{ DOM_sco_function}766 \begin{equation} \label{eq:DOM_sco_function} 767 767 \begin{split} 768 768 C(s) &= \frac{ \left[ \tanh{ \left( \theta \, (s+b) \right)} … … 784 784 \begin{figure}[!ht] \begin{center} 785 785 \includegraphics[width=1.0\textwidth]{Fig_sco_function} 786 \caption{ \protect\label{ Fig_sco_function}786 \caption{ \protect\label{fig:sco_function} 787 787 Examples of the stretching function applied to a seamount; from left to right: 788 788 surface, surface and bottom, and bottom intensified resolutions} … … 794 794 are the surface and bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 795 795 $0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom 796 increase of the vertical resolution ( Fig.~\ref{Fig_sco_function}).796 increase of the vertical resolution (\autoref{fig:sco_function}). 797 797 798 798 Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows … … 807 807 The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate: 808 808 809 \begin{equation} \label{ DOM_gamma_deriv}809 \begin{equation} \label{eq:DOM_gamma_deriv} 810 810 \gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right) 811 811 \end{equation} 812 812 813 813 Where: 814 \begin{equation} \label{ DOM_gamma}814 \begin{equation} \label{eq:DOM_gamma} 815 815 f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1} 816 816 \end{equation} … … 821 821 and bottom depths. The bottom cell depth in this example is given as a function of water depth: 822 822 823 \begin{equation} \label{ DOM_zb}823 \begin{equation} \label{eq:DOM_zb} 824 824 Z_b= h a + b 825 825 \end{equation} … … 831 831 \includegraphics[width=1.0\textwidth]{FIG_DOM_compare_coordinates_surface} 832 832 \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.} 833 \label{fig _compare_coordinates_surface}833 \label{fig:fig_compare_coordinates_surface} 834 834 \end{figure} 835 835 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 845 845 % ------------------------------------------------------------------------------------------------------------- 846 846 \subsection{$Z^*$- or $S^*$-coordinate (\protect\np{ln\_linssh}\forcode{ = .false.}) } 847 \label{ DOM_zgr_star}847 \label{subsec:DOM_zgr_star} 848 848 849 849 This option is described in the Report by Levier \textit{et al.} (2007), available on the \NEMO web site. … … 855 855 % ------------------------------------------------------------------------------------------------------------- 856 856 \subsection{Level bathymetry and mask} 857 \label{ DOM_msk}857 \label{subsec:DOM_msk} 858 858 859 859 Whatever the vertical coordinate used, the model offers the possibility of … … 892 892 893 893 Note that, without ice shelves cavities, masks at $t-$ and $w-$points are identical with 894 the numerical indexing used (\ S~\ref{DOM_Num_Index}). Nevertheless, $wmask$ are required894 the numerical indexing used (\autoref{subsec:DOM_Num_Index}). Nevertheless, $wmask$ are required 895 895 with oceean cavities to deal with the top boundary (ice shelf/ocean interface) 896 896 exactly in the same way as for the bottom boundary. … … 900 900 case of an east-west cyclical boundary condition, \textit{mbathy} has its last 901 901 column equal to the second one and its first column equal to the last but one 902 (and so too the mask arrays) (see \ S~\ref{LBC_jperio}).902 (and so too the mask arrays) (see \autoref{fig:LBC_jperio}). 903 903 904 904 … … 907 907 % ================================================================ 908 908 \section{Initial state (\protect\mdl{istate} and \protect\mdl{dtatsd})} 909 \label{ DTA_tsd}909 \label{sec:DTA_tsd} 910 910 %-----------------------------------------namtsd------------------------------------------- 911 911 \forfile{../namelists/namtsd} … … 918 918 \item[\np{ln\_tsd\_init}\forcode{ = .true.}] use a T and S input files that can be given on the model grid itself or 919 919 on their native input data grid. In the latter case, the data will be interpolated on-the-fly both in the 920 horizontal and the vertical to the model grid (see \ S~\ref{SBC_iof}). The information relative to the920 horizontal and the vertical to the model grid (see \autoref{subsec:SBC_iof}). The information relative to the 921 921 input files are given in the \np{sn\_tem} and \np{sn\_sal} structures. 922 922 The computation is done in the \mdl{dtatsd} module.
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