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r11179 r11315 18 18 %  domclo: closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled 19 19 20 \vfill 21 \begin{figure}[b] 22 \subsubsection*{Changes record} 23 \begin{tabular}{m{0.08\linewidth}m{0.32\linewidth}m{0.6\linewidth}} 24 Release & Author(s) & Modifications \\ 25 \hline 26 {\em 4.0} & {\em Simon M{\"u}ller \& Andrew Coward} & {\em Compatibility changes for v4.0. Major simplication has moved many of the options to external domain configuration tools. For now this information has been retained in an appendix } \\ 27 {\em 3.x} & {\em Sebastien Masson, Gurvan Madec \& Rashid Benshila } & {\em } \\ 28 \end{tabular} 29 \end{figure} 30 20 31 \newpage 21 32 22 33 Having defined the continuous equations in \autoref{chap:PE} and chosen a time discretization \autoref{chap:STP}, 23 we need to choose a discretization on a grid, and numerical algorithms.34 we need to choose a grid for spatial discretization and related numerical algorithms. 24 35 In the present chapter, we provide a general description of the staggered grid used in \NEMO, 25 and other information relevant to the main directory routines as well as the DOM (DOMain) directory.36 and other relevant information about the DOM (DOMain) sourcecode modules . 26 37 27 38 % ================================================================ … … 55 66 The numerical techniques used to solve the Primitive Equations in this model are based on the traditional, 56 67 centred secondorder finite difference approximation. 57 Special attention has been given to the homogeneity of the solution in the three spa cedirections.68 Special attention has been given to the homogeneity of the solution in the three spatial directions. 58 69 The arrangement of variables is the same in all directions. 59 70 It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector points $(u, v, w)$ defined in … … 71 82 Each scale factor is defined as the local analytical value provided by \autoref{eq:scale_factors}. 72 83 As a result, the mesh on which partial derivatives $\pd[]{\lambda}$, $\pd[]{\varphi}$ and 73 $\pd[]{z}$ are evaluated i na uniform mesh with a grid size of unity.84 $\pd[]{z}$ are evaluated is a uniform mesh with a grid size of unity. 74 85 Discrete partial derivatives are formulated by the traditional, centred second order finite difference approximation 75 86 while the scale factors are chosen equal to their local analytical value. … … 79 90 the continuous properties (see \autoref{apdx:C}). 80 91 A similar, related remark can be made about the domain size: 81 when needed, an area, volume, or the total ocean depth must be evaluated as the sum of the relevant scale factors92 when needed, an area, volume, or the total ocean depth must be evaluated as the product or sum of the relevant scale factors 82 93 (see \autoref{eq:DOM_bar} in the next section). 83 94 … … 87 98 \begin{tabular}{p{46pt}p{56pt}p{56pt}p{56pt}} 88 99 \hline 89 T& $i $ & $j $ & $k $ \\100 t & $i $ & $j $ & $k $ \\ 90 101 \hline 91 102 u & $i + 1/2$ & $j $ & $k $ \\ … … 107 118 \protect\label{tab:cell} 108 119 Location of gridpoints as a function of integer or integer and a half value of the column, line or level. 109 This indexing is only used for the writing of the semi discrete equation .110 In the code, the indexing uses integer values only and has a reverse direction in the vertical120 This indexing is only used for the writing of the semi discrete equations. 121 In the code, the indexing uses integer values only and is positive downwards in the vertical with $k=1$ at the surface. 111 122 (see \autoref{subsec:DOM_Num_Index}) 112 123 } 113 124 \end{center} 114 125 \end{table} 126 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 127 128 Note that the definition of the scale factors 129 (\ie as the analytical first derivative of the transformation that 130 results in $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) 131 is specific to the \NEMO model \citep{marti.madec.ea_JGR92}. 132 As an example, a scale factor in the $i$ direction is defined locally at a $t$point, 133 whereas many other models on a C grid choose to define such a scale factor as 134 the distance between the $u$points on each side of the $t$point. 135 Relying on an analytical transformation has two advantages: 136 firstly, there is no ambiguity in the scale factors appearing in the discrete equations, 137 since they are first introduced in the continuous equations; 138 secondly, analytical transformations encourage good practice by the definition of smoothly varying grids 139 (rather than allowing the user to set arbitrary jumps in thickness between adjacent layers) \citep{treguier.dukowicz.ea_JGR96}. 140 An example of the effect of such a choice is shown in \autoref{fig:zgr_e3}. 141 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 142 \begin{figure}[!t] 143 \begin{center} 144 \includegraphics[width=\textwidth]{Fig_zgr_e3} 145 \caption{ 146 \protect\label{fig:zgr_e3} 147 Comparison of (a) traditional definitions of gridpoint position and gridsize in the vertical, 148 and (b) analytically derived gridpoint position and scale factors. 149 For both grids here, the same $w$point depth has been chosen but 150 in (a) the $t$points are set half way between $w$points while 151 in (b) they are defined from an analytical function: 152 $z(k) = 5 \, (k  1/2)^3  45 \, (k  1/2)^2 + 140 \, (k  1/2)  150$. 153 Note the resulting difference between the value of the gridsize $\Delta_k$ and 154 those of the scale factor $e_k$. 155 } 156 \end{center} 157 \end{figure} 115 158 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 116 159 … … 132 175 Following \autoref{eq:PE_grad} and \autoref{eq:PE_lap}, the gradient of a variable $q$ defined at 133 176 a $t$point has its three components defined at $u$, $v$ and $w$points while 134 its Laplacian is defined at $t$point.177 its Laplacian is defined at the $t$point. 135 178 These operators have the following discrete forms in the curvilinear $s$coordinates system: 136 179 \[ … … 171 214 \end{equation} 172 215 173 The vertical average over the whole water column denoted by an overbar becomes for a quantity $q$ which174 is a masked field (i.e. equal to zero inside solid area):216 The vertical average over the whole water column is denoted by an overbar and is for 217 a masked field $q$ (\ie a quantity that is equal to zero inside solid areas): 175 218 \begin{equation} 176 219 \label{eq:DOM_bar} … … 178 221 \end{equation} 179 222 where $H_q$ is the ocean depth, which is the masked sum of the vertical scale factors at $q$ points, 180 $k^b$ and $k^o$ are the bottom and surface $k$indices, and the symbol $ k^o$ refers to a summation over223 $k^b$ and $k^o$ are the bottom and surface $k$indices, and the symbol $\sum \limits_k$ refers to a summation over 181 224 all grid points of the same type in the direction indicated by the subscript (here $k$). 182 225 … … 193 236 vector points $(u,v,w)$. 194 237 195 Let $a$ and $b$ be two fields defined on the mesh, with value zero inside continental area.196 Using integration by parts it can be shown that the differencing operators ($\delta_i$, $\delta_j$ and $\delta_k$)238 Let $a$ and $b$ be two fields defined on the mesh, with a value of zero inside continental areas. 239 It can be shown that the differencing operators ($\delta_i$, $\delta_j$ and $\delta_k$) 197 240 are skewsymmetric linear operators, and further that the averaging operators $\overline{\cdots}^{\, i}$, 198 241 $\overline{\cdots}^{\, j}$ and $\overline{\cdots}^{\, k}$) are symmetric linear operators, \ie … … 228 271 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 229 272 230 The array representation used in the \fortran code requires an integer indexing while231 the analytical definition of the mesh (see \autoref{subsec:DOM_cell}) is associated with the use of232 integer values for $t$points and both integer and integer and a half values for all the other points.233 Therefore a specific integer indexing must bedefined for points other than $t$points273 The array representation used in the \fortran code requires an integer indexing. 274 However, the analytical definition of the mesh (see \autoref{subsec:DOM_cell}) is associated with the use of 275 integer values for $t$points only while all the other points involve integer and a half values. 276 Therefore, a specific integer indexing has been defined for points other than $t$points 234 277 (\ie velocity and vorticity gridpoints). 235 Furthermore, the direction of the vertical indexing has been changed so that the surface level isat $k = 1$.278 Furthermore, the direction of the vertical indexing has been reversed and the surface level set at $k = 1$. 236 279 237 280 %  … … 253 296 \label{subsec:DOM_Num_Index_vertical} 254 297 255 In the vertical, the chosen indexing requires special attention since the $k$axis is reorientated downwardin256 the \fortran code compared to the indexingused in the semi discrete equations and298 In the vertical, the chosen indexing requires special attention since the direction of the $k$axis in 299 the \fortran code is the reverse of that used in the semi discrete equations and 257 300 given in \autoref{subsec:DOM_cell}. 258 The sea surface corresponds to the $w$level $k = 1$ which is the same index as$t$level just below301 The sea surface corresponds to the $w$level $k = 1$, which is the same index as the $t$level just below 259 302 (\autoref{fig:index_vert}). 260 The last $w$level ($k = jpk$) either corresponds to the ocean floor or is inside the bathymetry while 261 the last $t$level is always inside the bathymetry (\autoref{fig:index_vert}). 262 Note that for an increasing $k$ index, a $w$point and the $t$point just below have the same $k$ index, 263 in opposition to what is done in the horizontal plane where 264 it is the $t$point and the nearest velocity points in the direction of the horizontal axis that 265 have the same $i$ or $j$ index 303 The last $w$level ($k = jpk$) either corresponds to or is below the ocean floor while 304 the last $t$level is always outside the ocean domain (\autoref{fig:index_vert}). 305 Note that a $w$point and the directly underlaying $t$point have a common $k$ index (\ie $t$points and their 306 nearest $w$point neighbour in negative index direction), in contrast to the indexing on the horizontal plane where 307 the $t$point has the same index as the nearest velocity points in the positive direction of the respective horizontal axis index 266 308 (compare the dashed area in \autoref{fig:index_hor} and \autoref{fig:index_vert}). 267 309 Since the scale factors are chosen to be strictly positive, 268 a \textit{minus sign} appears in the \fortran code \textit{beforeall the vertical derivatives} of269 the discrete equations given in this documentation.310 a \textit{minus sign} is included in the \fortran implementations of \textit{all the vertical derivatives} of 311 the discrete equations given in this manual in order to accommodate the opposing vertical index directions in implementation and documentation. 270 312 271 313 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 276 318 \protect\label{fig:index_vert} 277 319 Vertical integer indexing used in the \fortran code. 278 Note that the $k$axis is orient ated downward.279 The dashed area indicates the cell in which variables contained in arrays have the same$k$index.320 Note that the $k$axis is oriented downward. 321 The dashed area indicates the cell in which variables contained in arrays have a common $k$index. 280 322 } 281 323 \end{center} … … 283 325 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 284 326 327 %  328 % Domain configuration 329 %  330 \section{Spatial domain configuration} 331 \label{subsec:DOM_config} 332 333 \nlst{namcfg} 334 335 Two typical methods are available to specify the spatial domain 336 configuration; they can be selected using parameter \np{ln\_read\_cfg} 337 parameter in namelist \ngn{namcfg}. 338 339 If \np{ln\_read\_cfg} is set to \forcode{.true.}, the domainspecific parameters 340 and fields are read from a netCDF input file, whose name (without its .nc 341 suffix) can be specified as the value of the \np{cn\_domcfg} parameter in 342 namelist \ngn{namcfg}. 343 344 If \np{ln\_read\_cfg} is set to \forcode{.false.}, the domainspecific 345 parameters and fields can be provided (\eg analytically computed) by subroutines 346 \mdl{usrdef\_hgr} and \mdl{usrdef\_zgr}. These subroutines can be supplied in 347 the \path{MY_SRC} directory of the configuration, and default versions that 348 configure the spatial domain for the GYRE reference configuration are present in 349 the \path{src/OCE/USR} directory. 350 351 In version 4.0 there are no longer any options for reading complex bathmetries and 352 performing a vertical discretization at runtime. Whilst it is occasionally convenient 353 to have a common bathymetry file and, for example, to run similar models with and 354 without partial bottom boxes and/or sigmacoordinates, supporting such choices leads to 355 overly complex code. Worse still is the difficulty of ensuring the model configurations 356 intended to be identical are indeed so when the model domain itself can be altered by runtime 357 selections. The code previously used to perform vertical discretization has be incorporated 358 into an external tool (\path{tools/DOMAINcfg}) which is briefly described in appendix F. 359 360 The next subsections summarise the parameter and fields related to the 361 configuration of the whole model domain. These represent the minimum information 362 that must be provided either via the \np{cn\_domcfg} file or set by code 363 inserted into usersupplied versions of the \mdl{usrdef\_*} subroutines. The 364 requirements are presented in three sections: the domain size 365 (\autoref{subsec:DOM_size}), the horizontal mesh 366 (\autoref{subsec:DOM_hgr}), and the vertical grid 367 (\autoref{subsec:DOM_zgr}). 368 285 369 %  286 370 % Domain Size 287 371 %  288 \subs ubsection{Domain size}372 \subsection{Domain size} 289 373 \label{subsec:DOM_size} 290 374 291 The total size of the computational domain is set by the parameters \np{jpiglo}, 292 \np{jpjglo} and \np{jpkglo} in the $i$, $j$ and $k$ directions respectively. 293 Parameters $jpi$ and $jpj$ refer to the size of each processor subdomain when 294 the code is run in parallel using domain decomposition (\key{mpp\_mpi} defined, 295 see \autoref{sec:LBC_mpp}). 296 297 % ================================================================ 298 % Domain: List of fields needed 299 % ================================================================ 300 \section{Needed fields} 301 \label{sec:DOM_fields} 302 The ocean mesh (\ie the position of all the scalar and vector points) is defined by the transformation that 303 gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 304 The gridpoints are located at integer or integer and a half values of as indicated in \autoref{tab:cell}. 305 The associated scale factors are defined using the analytical first derivative of the transformation 306 \autoref{eq:scale_factors}. 307 Necessary fields for configuration definition are: 308 309 \begin{itemize} 310 \item 311 Geographic position: 312 longitude with \texttt{glamt}, \texttt{glamu}, \texttt{glamv}, \texttt{glamf} and 313 latitude with \texttt{gphit}, \texttt{gphiu}, \texttt{gphiv}, \texttt{gphif} 314 (all respectively at T, U, V and F point) 315 \item 316 Coriolis parameter (if domain not on the sphere): \texttt{ff\_f} and \texttt{ff\_t} 317 (at T and F point) 318 \item 319 Scale factors: 320 \texttt{e1t}, \texttt{e1u}, \texttt{e1v} and \texttt{e1f} (on i direction), 321 \texttt{e2t}, \texttt{e2u}, \texttt{e2v} and \texttt{e2f} (on j direction) and 322 \texttt{ie1e2u\_v}, \texttt{e1e2u}, \texttt{e1e2v}. \\ 323 \texttt{e1e2u}, \texttt{e1e2v} are u and v surfaces (if gridsize reduction in some straits), 324 \texttt{ie1e2u\_v} is to flag set u and v surfaces are neither read nor computed. 325 \end{itemize} 326 327 These fields can be read in an domain input file which name is setted in \np{cn\_domcfg} parameter specified in 328 \ngn{namcfg}. 329 330 \nlst{namcfg} 331 332 Or they can be defined in an analytical way in \path{MY_SRC} directory of the configuration. 333 For Reference Configurations of NEMO input domain files are supplied by NEMO System Team. 334 For analytical definition of input fields two routines are supplied: \mdl{usrdef\_hgr} and \mdl{usrdef\_zgr}. 335 They are an example of GYRE configuration parameters, and they are available in \path{src/OCE/USR} directory, 336 they provide the horizontal and vertical mesh. 337 %  338 % Needed fields 339 %  340 %\subsection{List of needed fields to build DOMAIN} 341 %\label{subsec:DOM_fields_list} 342 375 The total size of the computational domain is set by the parameters 376 \np{jpiglo}, \np{jpjglo} and \np{jpkglo} for the $i$, $j$ and $k$ 377 directions, respectively. Note, that the variables \forcode{jpi} and \forcode{jpj} 378 refer to the size of each processor subdomain when the code is run in 379 parallel using domain decomposition (\key{mpp\_mpi} defined, see 380 \autoref{sec:LBC_mpp}). 381 382 The name of the configuration is set through parameter \np{cn\_cfg}, 383 and the nominal resolution through parameter \np{nn\_cfg} (unless in 384 the input file both of variables \forcode{ORCA} and \forcode{ORCA_index} 385 are present, in which case \np{cn\_cfg} and \np{nn\_cfg} are set from these 386 values accordingly). 387 388 The global lateral boundary condition type is selected from 8 options 389 using parameter \np{jperio}. See \autoref{sec:LBC_jperio} for 390 details on the available options and the corresponding values for 391 \np{jperio}. 343 392 344 393 % ================================================================ 345 394 % Domain: Horizontal Grid (mesh) 346 395 % ================================================================ 347 \section[Horizontal grid mesh (\textit{domhgr.F90})] 348 {Horizontal grid mesh (\protect\mdl{domhgr})} 349 \label{sec:DOM_hgr} 350 351 %  352 % Coordinates and scale factors 353 %  354 \subsection{Coordinates and scale factors} 355 \label{subsec:DOM_hgr_coord_e} 356 357 The ocean mesh (\ie the position of all the scalar and vector points) is defined by 358 the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 359 The gridpoints are located at integer or integer and a half values of as indicated in \autoref{tab:cell}. 360 The associated scale factors are defined using the analytical first derivative of the transformation 361 \autoref{eq:scale_factors}. 362 These definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, 363 which provide the horizontal and vertical meshes, respectively. 364 This section deals with the horizontal mesh parameters. 365 366 In a horizontal plane, the location of all the model grid points is defined from 367 the analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a function of $(i,j)$. 368 The horizontal scale factors are calculated using \autoref{eq:scale_factors}. 369 For example, when the longitude and latitude are function of a single value 370 ($i$ and $j$, respectively) (geographical configuration of the mesh), 371 the horizontal mesh definition reduces to define the wanted $\lambda(i)$, $\varphi(j)$, 372 and their derivatives $\lambda'(i) \ \varphi'(j)$ in the \mdl{domhgr} module. 373 The model computes the gridpoint positions and scale factors in the horizontal plane as follows: 374 \begin{align*} 375 \lambda_t &\equiv \text{glamt} = \lambda (i ) 376 &\varphi_t &\equiv \text{gphit} = \varphi (j ) \\ 377 \lambda_u &\equiv \text{glamu} = \lambda (i + 1/2) 378 &\varphi_u &\equiv \text{gphiu} = \varphi (j ) \\ 379 \lambda_v &\equiv \text{glamv} = \lambda (i ) 380 &\varphi_v &\equiv \text{gphiv} = \varphi (j + 1/2) \\ 381 \lambda_f &\equiv \text{glamf} = \lambda (i + 1/2) 382 &\varphi_f &\equiv \text{gphif} = \varphi (j + 1/2) \\ 383 e_{1t} &\equiv \text{e1t} = r_a \lambda'(i ) \; \cos\varphi(j )  384 &e_{2t} &\equiv \text{e2t} = r_a \varphi'(j )  \\ 385 e_{1u} &\equiv \text{e1t} = r_a \lambda'(i + 1/2) \; \cos\varphi(j )  386 &e_{2u} &\equiv \text{e2t} = r_a \varphi'(j )  \\ 387 e_{1v} &\equiv \text{e1t} = r_a \lambda'(i ) \; \cos\varphi(j + 1/2)  388 &e_{2v} &\equiv \text{e2t} = r_a \varphi'(j + 1/2)  \\ 389 e_{1f} &\equiv \text{e1t} = r_a \lambda'(i + 1/2) \; \cos\varphi(j + 1/2)  390 &e_{2f} &\equiv \text{e2t} = r_a \varphi'(j + 1/2)  391 \end{align*} 392 where the last letter of each computational name indicates the grid point considered and 393 $r_a$ is the earth radius (defined in \mdl{phycst} along with all universal constants). 394 Note that the horizontal position of and scale factors at $w$points are exactly equal to those of $t$points, 395 thus no specific arrays are defined at $w$points. 396 397 Note that the definition of the scale factors 398 (\ie as the analytical first derivative of the transformation that 399 gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) 400 is specific to the \NEMO model \citep{marti.madec.ea_JGR92}. 401 As an example, $e_{1t}$ is defined locally at a $t$point, 402 whereas many other models on a C grid choose to define such a scale factor as 403 the distance between the $U$points on each side of the $t$point. 404 Relying on an analytical transformation has two advantages: 405 firstly, there is no ambiguity in the scale factors appearing in the discrete equations, 406 since they are first introduced in the continuous equations; 407 secondly, analytical transformations encourage good practice by the definition of smoothly varying grids 408 (rather than allowing the user to set arbitrary jumps in thickness between adjacent layers) \citep{treguier.dukowicz.ea_JGR96}. 409 An example of the effect of such a choice is shown in \autoref{fig:zgr_e3}. 410 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 411 \begin{figure}[!t] 412 \begin{center} 413 \includegraphics[width=\textwidth]{Fig_zgr_e3} 414 \caption{ 415 \protect\label{fig:zgr_e3} 416 Comparison of (a) traditional definitions of gridpoint position and gridsize in the vertical, 417 and (b) analytically derived gridpoint position and scale factors. 418 For both grids here, the same $w$point depth has been chosen but 419 in (a) the $t$points are set half way between $w$points while 420 in (b) they are defined from an analytical function: 421 $z(k) = 5 \, (k  1/2)^3  45 \, (k  1/2)^2 + 140 \, (k  1/2)  150$. 422 Note the resulting difference between the value of the gridsize $\Delta_k$ and 423 those of the scale factor $e_k$. 424 } 425 \end{center} 426 \end{figure} 427 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 428 429 %  430 % Choice of horizontal grid 431 %  432 \subsection{Choice of horizontal grid} 433 \label{subsec:DOM_hgr_msh_choice} 434 435 %  436 % Grid files 437 %  438 \subsection{Output grid files} 439 \label{subsec:DOM_hgr_files} 440 441 All the arrays relating to a particular ocean model configuration (gridpoint position, scale factors, masks) 442 can be saved in files if \np{nn\_msh} $\not = 0$ (namelist variable in \ngn{namdom}). 443 This can be particularly useful for plots and offline diagnostics. 444 In some cases, the user may choose to make a local modification of a scale factor in the code. 445 This is the case in global configurations when restricting the width of a specific strait 446 (usually a onegridpoint strait that happens to be too wide due to insufficient model resolution). 447 An example is Gibraltar Strait in the ORCA2 configuration. 448 When such modifications are done, 449 the output grid written when \np{nn\_msh} $\not = 0$ is no more equal to the input grid. 396 \subsection{Horizontal grid mesh (\protect\mdl{domhgr})} 397 \label{subsec:DOM_hgr} 398 399 % ================================================================ 400 % Domain: List of hgrrelated fields needed 401 % ================================================================ 402 \subsubsection{Required fields} 403 \label{sec:DOM_hgr_fields} 404 The explicit specification of a range of meshrelated fields are required for the definition of a configuration. These include: 405 406 \begin{Verbatim}[fontsize=\tiny] 407 int jpiglo, jpjglo, jpkglo /* global domain sizes */ 408 int jperio /* lateral global domain b.c. */ 409 double glamt, glamu, glamv, glamf /* geographic longitude (t,u,v and f points respectively) */ 410 double gphit, gphiu, gphiv, gphif /* geographic latitude */ 411 double e1t, e1u, e1v, e1f /* horizontal scale factors */ 412 double e2t, e2u, e2v, e2f /* horizontal scale factors */ 413 \end{Verbatim} 414 415 The values of the geographic longitude and latitude arrays at indices $i,j$ correspond to the analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a function of $(i,j)$, evaluated at the values as specified in Table \autoref{tab:cell} for the respective gridpoint position. The calculation of the values of the horizontal scale factor arrays in general additionally involves partial derivatives of $\lambda$ and $\varphi$ with respect to $i$ and $j$, evaluated for the same arguments as $\lambda$ and $\varphi$. 416 417 \subsubsection{Optional fields} 418 \begin{Verbatim}[fontsize=\tiny] 419 /* Optional: */ 420 int ORCA, ORCA_index /* configuration name, configuration resolution */ 421 double e1e2u, e1e2v /* U and V surfaces (if grid size reduction in some straits) */ 422 double ff_f, ff_t /* Coriolis parameter (if not on the sphere) */ 423 \end{Verbatim} 424 425 NEMO can support the local reduction of key strait widths by altering individual values of 426 e1u or e1v at the appropriate locations. This is particularly useful for locations such as 427 Gibraltar or Indonesian Throughflow pinchpoints (see \autoref{sec:MISC_strait} for 428 illustrated examples). The key is to reduce the faces of $T$cell (\ie change the value of 429 the horizontal scale factors at $u$ or $v$point) but not the volume of the cells. Doing 430 otherwise can lead to numerical instability issues. In normal operation the surface areas 431 are computed from $\texttt{e1u} * \texttt{e2u}$ and $\texttt{e1v} * \texttt{e2v}$ but in 432 cases where a gridsize reduction is required, the unaltered surface areas at $u$ and $v$ 433 grid points (\texttt{e1e2u} and \texttt{e1e2v}, respectively) must be read or precomputed 434 in \mdl{usrdef\_hgr}. If these arrays are present in the \np{cn\_domcfg} file they are 435 read and the internal computation is suppressed. Versions of \mdl{usrdef\_hgr} which set 436 their own values of \texttt{e1e2u} and \texttt{e1e2v} should set the surfacearea 437 computation flag: \texttt{ie1e2u\_v} to a nonzero value to suppress their recomputation. 438 439 \smallskip 440 Similar logic applies to the other optional fields: \texttt{ff\_f} and \texttt{ff\_t} 441 which can be used to provide the Coriolis parameter at F and Tpoints respectively if the 442 mesh is not on a sphere. If present these fields will be read and used and the normal 443 calculation ($2*\Omega*\sin(\varphi)$) suppressed. Versions of \mdl{usrdef\_hgr} which set 444 their own values of \texttt{ff\_f} and \texttt{ff\_t} should set the Coriolis computation 445 flag: \texttt{iff} to a nonzero value to suppress their recomputation. 446 447 Note that longitudes, latitudes, and scale factors at $w$ points are exactly 448 equal to those of $t$ points, thus no specific arrays are defined at $w$ points. 449 450 450 451 451 % ================================================================ 452 452 % Domain: Vertical Grid (domzgr) 453 453 % ================================================================ 454 \s ection[Vertical grid (\textit{domzgr.F90})]454 \subsection[Vertical grid (\textit{domzgr.F90})] 455 455 {Vertical grid (\protect\mdl{domzgr})} 456 \label{sec:DOM_zgr} 457 %nam_zgr & namdom 458 % 459 %\nlst{namzgr} 460 461 \nlst{namdom} 456 \label{subsec:DOM_zgr} 457 %namdom 458 \nlst{namdom} 462 459 % 463 460 464 Variables are defined through the \ngn{namzgr} and \ngn{namdom} namelists.465 461 In the vertical, the model mesh is determined by four things: 466 (1) the bathymetry given in meters; 467 (2) the number of levels of the model (\jp{jpk}); 468 (3) the analytical transformation $z(i,j,k)$ and the vertical scale factors (derivatives of the transformation); and 469 (4) the masking system, \ie the number of wet model levels at each 470 $(i,j)$ column of points. 462 \begin{enumerate} 463 \item the bathymetry given in meters; 464 \item the number of levels of the model (\jp{jpk}); 465 \item the analytical transformation $z(i,j,k)$ and the vertical scale factors (derivatives of the transformation); and 466 \item the masking system, \ie the number of wet model levels at each 467 $(i,j)$ location of the horizontal grid. 468 \end{enumerate} 471 469 472 470 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 489 487 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 490 488 491 The choice of a vertical coordinate, even if it is made through \ngn{namzgr} namelist parameters, 492 must be done once of all at the beginning of an experiment. 493 It is not intended as an option which can be enabled or disabled in the middle of an experiment. 494 Three main choices are offered (\autoref{fig:z_zps_s_sps}): 495 $z$coordinate with full step bathymetry (\np{ln\_zco}\forcode{ = .true.}), 496 $z$coordinate with partial step bathymetry (\np{ln\_zps}\forcode{ = .true.}), 497 or generalized, $s$coordinate (\np{ln\_sco}\forcode{ = .true.}). 498 Hybridation of the three main coordinates are available: 499 $sz$ or $szps$ coordinate (\autoref{fig:z_zps_s_sps} and \autoref{fig:z_zps_s_sps}). 500 By default a nonlinear free surface is used: the coordinate follow the timevariation of the free surface so that 501 the transformation is time dependent: $z(i,j,k,t)$ (\autoref{fig:z_zps_s_sps}). 502 When a linear free surface is assumed (\np{ln\_linssh}\forcode{ = .true.}), 503 the vertical coordinate are fixed in time, but the seawater can move up and down across the $z_0$ surface 504 (in other words, the top of the ocean in not a rigidlid). 505 The last choice in terms of vertical coordinate concerns the presence (or not) in 506 the model domain of ocean cavities beneath ice shelves. 507 Setting \np{ln\_isfcav} to true allows to manage ocean cavities, otherwise they are filled in. 508 This option is currently only available in $z$ or $zps$coordinate, 509 and partial step are also applied at the ocean/ice shelf interface. 510 511 Contrary to the horizontal grid, the vertical grid is computed in the code and no provision is made for 512 reading it from a file. 513 The only input file is the bathymetry (in meters) (\ifile{bathy\_meter}) 514 \footnote{ 515 N.B. in full step $z$coordinate, a \ifile{bathy\_level} file can replace the \ifile{bathy\_meter} file, 516 so that the computation of the number of wet ocean point in each water column is bypassed}. 517 If \np{ln\_isfcav}\forcode{ = .true.}, an extra file input file (\ifile{isf\_draft\_meter}) describing 518 the ice shelf draft (in meters) is needed. 519 520 After reading the bathymetry, the algorithm for vertical grid definition differs between the different options: 521 \begin{description} 522 \item[\textit{zco}] 523 set a reference coordinate transformation $z_0(k)$, and set $z(i,j,k,t) = z_0(k)$. 524 \item[\textit{zps}] 525 set a reference coordinate transformation $z_0(k)$, and calculate the thickness of the deepest level at 526 each $(i,j)$ point using the bathymetry, to obtain the final threedimensional depth and scale factor arrays. 527 \item[\textit{sco}] 528 smooth the bathymetry to fulfill the hydrostatic consistency criteria and 529 set the threedimensional transformation. 530 \item[\textit{sz} and \textit{szps}] 531 smooth the bathymetry to fulfill the hydrostatic consistency criteria and 532 set the threedimensional transformation $z(i,j,k)$, 533 and possibly introduce masking of extra land points to better fit the original bathymetry file. 534 \end{description} 535 %%% 536 \gmcomment{ add the description of the smoothing: envelop topography...} 537 %%% 538 539 Unless a linear free surface is used (\np{ln\_linssh}\forcode{ = .false.}), 540 the arrays describing the grid point depths and vertical scale factors are three set of 541 three dimensional arrays $(i,j,k)$ defined at \textit{before}, \textit{now} and \textit{after} time step. 542 The time at which they are defined is indicated by a suffix: $\_b$, $\_n$, or $\_a$, respectively. 543 They are updated at each model time step using a fixed reference coordinate system which 544 computer names have a $\_0$ suffix. 545 When the linear free surface option is used (\np{ln\_linssh}\forcode{ = .true.}), \textit{before}, 546 \textit{now} and \textit{after} arrays are simply set one for all to their reference counterpart. 547 548 %  549 % Meter Bathymetry 550 %  551 \subsection{Meter bathymetry} 552 \label{subsec:DOM_bathy} 553 554 Three options are possible for defining the bathymetry, according to the namelist variable \np{nn\_bathy} 555 (found in \ngn{namdom} namelist): 556 \begin{description} 557 \item[\np{nn\_bathy}\forcode{ = 0}]: 558 a flatbottom domain is defined. 559 The total depth $z_w (jpk)$ is given by the coordinate transformation. 560 The domain can either be a closed basin or a periodic channel depending on the parameter \np{jperio}. 561 \item[\np{nn\_bathy}\forcode{ = 1}]: 562 a domain with a bump of topography one third of the domain width at the central latitude. 563 This is meant for the "EELR5" configuration, a periodic or open boundary channel with a seamount. 564 \item[\np{nn\_bathy}\forcode{ = 1}]: 565 read a bathymetry and ice shelf draft (if needed). 566 The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters) at 567 each grid point of the model grid. 568 The bathymetry is usually built by interpolating a standard bathymetry product (\eg ETOPO2) onto 569 the horizontal ocean mesh. 570 Defining the bathymetry also defines the coastline: where the bathymetry is zero, 571 no model levels are defined (all levels are masked). 572 573 The \ifile{isfdraft\_meter} file (Netcdf format) provides the ice shelf draft (positive, in meters) at 574 each grid point of the model grid. 575 This file is only needed if \np{ln\_isfcav}\forcode{ = .true.}. 576 Defining the ice shelf draft will also define the ice shelf edge and the grounding line position. 577 \end{description} 578 579 When a global ocean is coupled to an atmospheric model it is better to represent all large water bodies 580 (\eg great lakes, Caspian sea...) even if the model resolution does not allow their communication with 581 the rest of the ocean. 582 This is unnecessary when the ocean is forced by fixed atmospheric conditions, 583 so these seas can be removed from the ocean domain. 584 The user has the option to set the bathymetry in closed seas to zero (see \autoref{sec:MISC_closea}), 585 but the code has to be adapted to the user's configuration. 586 587 %  588 % zcoordinate and reference coordinate transformation 589 %  590 \subsection[$Z$coordinate (\forcode{ln_zco = .true.}) and ref. coordinate] 591 {$Z$coordinate (\protect\np{ln\_zco}\forcode{ = .true.}) and reference coordinate} 592 \label{subsec:DOM_zco} 593 594 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 595 \begin{figure}[!tb] 596 \begin{center} 597 \includegraphics[width=\textwidth]{Fig_zgr} 598 \caption{ 599 \protect\label{fig:zgr} 600 Default vertical mesh for ORCA2: 30 ocean levels (L30). 601 Vertical level functions for (a) Tpoint depth and (b) the associated scale factor as computed from 602 \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$coordinate. 603 } 604 \end{center} 605 \end{figure} 606 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 607 608 The reference coordinate transformation $z_0(k)$ defines the arrays $gdept_0$ and $gdepw_0$ for $t$ and $w$points, 609 respectively. 610 As indicated on \autoref{fig:index_vert} \jp{jpk} is the number of $w$levels. 611 $gdepw_0(1)$ is the ocean surface. 612 There are at most \jp{jpk}1 $t$points inside the ocean, 613 the additional $t$point at $jk = jpk$ is below the sea floor and is not used. 614 The vertical location of $w$ and $t$levels is defined from the analytic expression of the depth $z_0(k)$ whose 615 analytical derivative with respect to $k$ provides the vertical scale factors. 616 The user must provide the analytical expression of both $z_0$ and its first derivative with respect to $k$. 617 This is done in routine \mdl{domzgr} through statement functions, 618 using parameters provided in the \ngn{namcfg} namelist. 619 620 It is possible to define a simple regular vertical grid by giving zero stretching (\np{ppacr}\forcode{ = 0}). 621 In that case, the parameters \jp{jpk} (number of $w$levels) and 622 \np{pphmax} (total ocean depth in meters) fully define the grid. 623 624 For climaterelated studies it is often desirable to concentrate the vertical resolution near the ocean surface. 625 The following function is proposed as a standard for a $z$coordinate (with either full or partial steps): 626 \begin{gather} 627 \label{eq:DOM_zgr_ana_1} 628 z_0 (k) = h_{sur}  h_0 \; k  \; h_1 \; \log \big[ \cosh ((k  h_{th}) / h_{cr}) \big] \\ 629 e_3^0(k) = \lt  h_0  h_1 \; \tanh \big[ (k  h_{th}) / h_{cr} \big] \rt 630 \end{gather} 631 where $k = 1$ to \jp{jpk} for $w$levels and $k = 1$ to $k = 1$ for $T$levels. 632 Such an expression allows us to define a nearly uniform vertical location of levels at the ocean top and bottom with 633 a smooth hyperbolic tangent transition in between (\autoref{fig:zgr}). 634 635 If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same. 636 However, definition of $e_3^0$ at $t$ and $w$points is respectively changed to: 637 \begin{equation} 638 \label{eq:DOM_zgr_ana_2} 639 \begin{split} 640 e_3^T(k) &= z_W (k + 1)  z_W (k ) \\ 641 e_3^W(k) &= z_T (k )  z_T (k  1) 642 \end{split} 643 \end{equation} 644 This formulation decrease the selfgenerated circulation into the ice shelf cavity 645 (which can, in extreme case, leads to blow up).\\ 646 647 The most used vertical grid for ORCA2 has $10~m$ ($500~m$) resolution in the surface (bottom) layers and 648 a depth which varies from 0 at the sea surface to a minimum of $5000~m$. 649 This leads to the following conditions: 650 \begin{equation} 651 \label{eq:DOM_zgr_coef} 652 \begin{array}{ll} 653 e_3 (1 + 1/2) = 10. & z(1 ) = 0. \\ 654 e_3 (jpk  1/2) = 500. & z(jpk) = 5000. 655 \end{array} 656 \end{equation} 657 658 With the choice of the stretching $h_{cr} = 3$ and the number of levels \jp{jpk}~$= 31$, 659 the four coefficients $h_{sur}$, $h_0$, $h_1$, and $h_{th}$ in 660 \autoref{eq:DOM_zgr_ana_2} have been determined such that 661 \autoref{eq:DOM_zgr_coef} is satisfied, through an optimisation procedure using a bisection method. 662 For the first standard ORCA2 vertical grid this led to the following values: 663 $h_{sur} = 4762.96$, $h_0 = 255.58, h_1 = 245.5813$, and $h_{th} = 21.43336$. 664 The resulting depths and scale factors as a function of the model levels are shown in 665 \autoref{fig:zgr} and given in \autoref{tab:orca_zgr}. 666 Those values correspond to the parameters \np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist. 667 668 Rather than entering parameters $h_{sur}$, $h_0$, and $h_1$ directly, it is possible to recalculate them. 669 In that case the user sets \np{ppsur}~$=$~\np{ppa0}~$=$~\np{ppa1}~$= 999999$., 670 in \ngn{namcfg} namelist, and specifies instead the four following parameters: 489 The choice of a vertical coordinate is made when setting up the configuration; 490 it is not intended to be an option which can be changed in the middle of an 491 experiment. The one exception to this statement being the choice of linear or 492 nonlinear free surface. In v4.0 the linear free surface option is implemented 493 as a special case of the nonlinear free surface. This is computationally 494 wasteful since it uses the structures for timevarying 3D metrics for fields 495 that (in the linear free surface case) are fixed. However, the linear 496 freesurface is rarely used and implementing it this way means a single configuration 497 file can support both options. 498 499 By default a nonlinear free surface is used (\np{ln\_linssh} set to \forcode{ = 500 .false.} in \ngn{namdom}): the coordinate follow the timevariation of the free 501 surface so that the transformation is time dependent: $z(i,j,k,t)$ 502 (\eg \autoref{fig:z_zps_s_sps}f). When a linear free surface is assumed 503 (\np{ln\_linssh} set to \forcode{ = .true.} in \ngn{namdom}), the vertical 504 coordinates are fixed in time, but the seawater can move up and down across the 505 $z_0$ surface (in other words, the top of the ocean in not a rigid lid). 506 507 Note that settings: \np{ln\_zco}, \np{ln\_zps}, \np{ln\_sco} and \np{ln\_isfcav} mentioned 508 in the following sections appear to be namelist options but they are no longer truly 509 namelist options for NEMO. Their value is written to and read from the domain configuration file 510 and they should be treated as fixed parameters for a particular configuration. They are 511 namelist options for the \forcode{DOMAINcfg} tool that can be used to build the 512 configuration file and serve both to provide a record of the choices made whilst building the 513 configuration and to trigger appropriate code blocks within NEMO. 514 These values should not be altered in the \np{cn\_domcfg} file. 515 516 \medskip 517 The decision on these choices must be made when the \np{cn\_domcfg} file is constructed. 518 Three main choices are offered (\autoref{fig:z_zps_s_sps}ac): 519 671 520 \begin{itemize} 672 \item 673 \np{ppacr}~$= h_{cr}$: stretching factor (nondimensional). 674 The larger \np{ppacr}, the smaller the stretching. 675 Values from $3$ to $10$ are usual. 676 \item 677 \np{ppkth}~$= h_{th}$: is approximately the model level at which maximum stretching occurs 678 (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk}) 679 \item 680 \np{ppdzmin}: minimum thickness for the top layer (in meters). 681 \item 682 \np{pphmax}: total depth of the ocean (meters). 521 \item $z$coordinate with full step bathymetry (\np{ln\_zco}\forcode{ = .true.}), 522 \item $z$coordinate with partial step ($zps$) bathymetry (\np{ln\_zps}\forcode{ = .true.}), 523 \item Generalized, $s$coordinate (\np{ln\_sco}\forcode{ = .true.}). 683 524 \end{itemize} 684 As an example, for the $45$ layers used in the DRAKKAR configuration those parameters are: 685 \jp{jpk}~$= 46$, \np{ppacr}~$= 9$, \np{ppkth}~$= 23.563$, \np{ppdzmin}~$= 6~m$, \np{pphmax}~$= 5750~m$. 686 687 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 688 \begin{table} 689 \begin{center} 690 \begin{tabular}{crrrr} 691 \hline 692 \textbf{LEVEL} & \textbf{gdept\_1d} & \textbf{gdepw\_1d} & \textbf{e3t\_1d } & \textbf{e3w\_1d} \\ 693 \hline 694 1 & \textbf{ 5.00} & 0.00 & \textbf{ 10.00} & 10.00 \\ 695 \hline 696 2 & \textbf{ 15.00} & 10.00 & \textbf{ 10.00} & 10.00 \\ 697 \hline 698 3 & \textbf{ 25.00} & 20.00 & \textbf{ 10.00} & 10.00 \\ 699 \hline 700 4 & \textbf{ 35.01} & 30.00 & \textbf{ 10.01} & 10.00 \\ 701 \hline 702 5 & \textbf{ 45.01} & 40.01 & \textbf{ 10.01} & 10.01 \\ 703 \hline 704 6 & \textbf{ 55.03} & 50.02 & \textbf{ 10.02} & 10.02 \\ 705 \hline 706 7 & \textbf{ 65.06} & 60.04 & \textbf{ 10.04} & 10.03 \\ 707 \hline 708 8 & \textbf{ 75.13} & 70.09 & \textbf{ 10.09} & 10.06 \\ 709 \hline 710 9 & \textbf{ 85.25} & 80.18 & \textbf{ 10.17} & 10.12 \\ 711 \hline 712 10 & \textbf{ 95.49} & 90.35 & \textbf{ 10.33} & 10.24 \\ 713 \hline 714 11 & \textbf{ 105.97} & 100.69 & \textbf{ 10.65} & 10.47 \\ 715 \hline 716 12 & \textbf{ 116.90} & 111.36 & \textbf{ 11.27} & 10.91 \\ 717 \hline 718 13 & \textbf{ 128.70} & 122.65 & \textbf{ 12.47} & 11.77 \\ 719 \hline 720 14 & \textbf{ 142.20} & 135.16 & \textbf{ 14.78} & 13.43 \\ 721 \hline 722 15 & \textbf{ 158.96} & 150.03 & \textbf{ 19.23} & 16.65 \\ 723 \hline 724 16 & \textbf{ 181.96} & 169.42 & \textbf{ 27.66} & 22.78 \\ 725 \hline 726 17 & \textbf{ 216.65} & 197.37 & \textbf{ 43.26} & 34.30 \\ 727 \hline 728 18 & \textbf{ 272.48} & 241.13 & \textbf{ 70.88} & 55.21 \\ 729 \hline 730 19 & \textbf{ 364.30} & 312.74 & \textbf{ 116.11} & 90.99 \\ 731 \hline 732 20 & \textbf{ 511.53} & 429.72 & \textbf{ 181.55} & 146.43 \\ 733 \hline 734 21 & \textbf{ 732.20} & 611.89 & \textbf{ 261.03} & 220.35 \\ 735 \hline 736 22 & \textbf{ 1033.22} & 872.87 & \textbf{ 339.39} & 301.42 \\ 737 \hline 738 23 & \textbf{ 1405.70} & 1211.59 & \textbf{ 402.26} & 373.31 \\ 739 \hline 740 24 & \textbf{ 1830.89} & 1612.98 & \textbf{ 444.87} & 426.00 \\ 741 \hline 742 25 & \textbf{ 2289.77} & 2057.13 & \textbf{ 470.55} & 459.47 \\ 743 \hline 744 26 & \textbf{ 2768.24} & 2527.22 & \textbf{ 484.95} & 478.83 \\ 745 \hline 746 27 & \textbf{ 3257.48} & 3011.90 & \textbf{ 492.70} & 489.44 \\ 747 \hline 748 28 & \textbf{ 3752.44} & 3504.46 & \textbf{ 496.78} & 495.07 \\ 749 \hline 750 29 & \textbf{ 4250.40} & 4001.16 & \textbf{ 498.90} & 498.02 \\ 751 \hline 752 30 & \textbf{ 4749.91} & 4500.02 & \textbf{ 500.00} & 499.54 \\ 753 \hline 754 31 & \textbf{ 5250.23} & 5000.00 & \textbf{ 500.56} & 500.33 \\ 755 \hline 756 \end{tabular} 757 \end{center} 758 \caption{ 759 \protect\label{tab:orca_zgr} 760 Default vertical mesh in $z$coordinate for 30 layers ORCA2 configuration as computed from 761 \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef} 762 } 763 \end{table} 764 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 765 766 %  767 % zcoordinate with partial step 768 %  769 \subsection[$Z$coordinate with partial step (\forcode{ln_zps = .true.})] 770 {$Z$coordinate with partial step (\protect\np{ln\_zps}\forcode{ = .true.})} 771 \label{subsec:DOM_zps} 772 %namdom 773 774 \nlst{namdom} 775 % 776 777 In $z$coordinate partial step, 778 the depths of the model levels are defined by the reference analytical function $z_0(k)$ as described in 779 the previous section, \textit{except} in the bottom layer. 780 The thickness of the bottom layer is allowed to vary as a function of geographical location $(\lambda,\varphi)$ to 781 allow a better representation of the bathymetry, especially in the case of small slopes 782 (where the bathymetry varies by less than one level thickness from one grid point to the next). 783 The reference layer thicknesses $e_{3t}^0$ have been defined in the absence of bathymetry. 784 With partial steps, layers from 1 to \jp{jpk}2 can have a thickness smaller than $e_{3t}(jk)$. 785 The model deepest layer (\jp{jpk}1) is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$: 786 the maximum thickness allowed is $2*e_{3t}(jpk  1)$. 787 This has to be kept in mind when specifying values in \ngn{namdom} namelist, 788 as the maximum depth \np{pphmax} in partial steps: 789 for example, with \np{pphmax}~$= 5750~m$ for the DRAKKAR 45 layer grid, 790 the maximum ocean depth allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk  1)$ being $250~m$). 791 Two variables in the namdom namelist are used to define the partial step vertical grid. 792 The mimimum water thickness (in meters) allowed for a cell partially filled with bathymetry at level jk is 793 the minimum of \np{rn\_e3zps\_min} (thickness in meters, usually $20~m$) or $e_{3t}(jk)*$\np{rn\_e3zps\_rat} 794 (a fraction, usually 10\%, of the default thickness $e_{3t}(jk)$). 795 796 \gmcomment{ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level } } 797 798 %  799 % scoordinate 800 %  801 \subsection[$S$coordinate (\forcode{ln_sco = .true.})] 802 {$S$coordinate (\protect\np{ln\_sco}\forcode{ = .true.})} 803 \label{subsec:DOM_sco} 804 %nam_zgr_sco 805 % 806 %\nlst{namzgr_sco} 807 % 808 Options are defined in \ngn{namzgr\_sco}. 809 In $s$coordinate (\np{ln\_sco}\forcode{ = .true.}), the depth and thickness of the model levels are defined from 810 the product of a depth field and either a stretching function or its derivative, respectively: 811 812 \begin{align*} 813 % \label{eq:DOM_sco_ana} 814 z(k) &= h(i,j) \; z_0 (k) \\ 815 e_3(k) &= h(i,j) \; z_0'(k) 816 \end{align*} 817 818 where $h$ is the depth of the last $w$level ($z_0(k)$) defined at the $t$point location in the horizontal and 819 $z_0(k)$ is a function which varies from $0$ at the sea surface to $1$ at the ocean bottom. 820 The depth field $h$ is not necessary the ocean depth, 821 since a mixed steplike and bottomfollowing representation of the topography can be used 822 (\autoref{fig:z_zps_s_sps}) or an envelop bathymetry can be defined (\autoref{fig:z_zps_s_sps}). 823 The namelist parameter \np{rn\_rmax} determines the slope at which 824 the terrainfollowing coordinate intersects the sea bed and becomes a pseudo zcoordinate. 825 The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} as 826 the minimum and maximum depths at which the terrainfollowing vertical coordinate is calculated. 827 828 Options for stretching the coordinate are provided as examples, 829 but care must be taken to ensure that the vertical stretch used is appropriate for the application. 830 831 The original default NEMO scoordinate stretching is available if neither of the other options are specified as true 832 (\np{ln\_s\_SH94}\forcode{ = .false.} and \np{ln\_s\_SF12}\forcode{ = .false.}). 833 This uses a depth independent $\tanh$ function for the stretching \citep{madec.delecluse.ea_JPO96}: 834 835 \[ 836 z = s_{min} + C (s) (H  s_{min}) 837 % \label{eq:SH94_1} 838 \] 839 840 where $s_{min}$ is the depth at which the $s$coordinate stretching starts and 841 allows a $z$coordinate to placed on top of the stretched coordinate, 842 and $z$ is the depth (negative down from the asea surface). 843 \begin{gather*} 844 s =  \frac{k}{n  1} \quad \text{and} \quad 0 \leq k \leq n  1 845 % \label{eq:DOM_s} 846 \\ 847 % \label{eq:DOM_sco_function} 848 C(s) = \frac{[\tanh(\theta \, (s + b))  \tanh(\theta \, b)]}{2 \; \sinh(\theta)} 849 \end{gather*} 850 851 A stretching function, 852 modified from the commonly used \citet{song.haidvogel_JCP94} stretching (\np{ln\_s\_SH94}\forcode{ = .true.}), 853 is also available and is more commonly used for shelf seas modelling: 854 855 \[ 856 C(s) = (1  b) \frac{\sinh(\theta s)}{\sinh(\theta)} 857 + b \frac{\tanh \lt[ \theta \lt(s + \frac{1}{2} \rt) \rt]  \tanh \lt( \frac{\theta}{2} \rt)} 858 { 2 \tanh \lt( \frac{\theta}{2} \rt)} 859 % \label{eq:SH94_2} 860 \] 861 862 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 863 \begin{figure}[!ht] 864 \begin{center} 865 \includegraphics[width=\textwidth]{Fig_sco_function} 866 \caption{ 867 \protect\label{fig:sco_function} 868 Examples of the stretching function applied to a seamount; 869 from left to right: surface, surface and bottom, and bottom intensified resolutions 870 } 871 \end{center} 872 \end{figure} 873 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 874 875 where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from pure $\sigma$ to 876 the stretched coordinate, and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) are the surface and 877 bottom control parameters such that $0 \leqslant \theta \leqslant 20$, and $0 \leqslant b \leqslant 1$. 878 $b$ has been designed to allow surface and/or bottom increase of the vertical resolution 879 (\autoref{fig:sco_function}). 880 881 Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows a fixed surface resolution in 882 an analytical terrainfollowing stretching \citet{siddorn.furner_OM13}. 883 In this case the a stretching function $\gamma$ is defined such that: 884 885 \begin{equation} 886 z =  \gamma h \quad \text{with} \quad 0 \leq \gamma \leq 1 887 % \label{eq:z} 888 \end{equation} 889 890 The function is defined with respect to $\sigma$, the unstretched terrainfollowing coordinate: 891 892 \begin{gather*} 893 % \label{eq:DOM_gamma_deriv} 894 \gamma = A \lt( \sigma  \frac{1}{2} (\sigma^2 + f (\sigma)) \rt) 895 + B \lt( \sigma^3  f (\sigma) \rt) + f (\sigma) \\ 896 \intertext{Where:} 897 % \label{eq:DOM_gamma} 898 f(\sigma) = (\alpha + 2) \sigma^{\alpha + 1}  (\alpha + 1) \sigma^{\alpha + 2} 899 \quad \text{and} \quad \sigma = \frac{k}{n  1} 900 \end{gather*} 901 902 This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of 903 the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards 904 the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and 905 user prescribed surface (\np{rn\_zs}) and bottom depths. 906 The bottom cell depth in this example is given as a function of water depth: 907 908 \[ 909 % \label{eq:DOM_zb} 910 Z_b = h a + b 911 \] 912 913 where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively. 914 915 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 916 \begin{figure}[!ht] 917 \includegraphics[width=\textwidth]{Fig_DOM_compare_coordinates_surface} 918 \caption{ 919 A comparison of the \citet{song.haidvogel_JCP94} $S$coordinate (solid lines), 920 a 50 level $Z$coordinate (contoured surfaces) and 921 the \citet{siddorn.furner_OM13} $S$coordinate (dashed lines) in the surface $100~m$ for 922 a idealised bathymetry that goes from $50~m$ to $5500~m$ depth. 923 For clarity every third coordinate surface is shown. 924 } 925 \label{fig:fig_compare_coordinates_surface} 926 \end{figure} 927 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 928 929 This gives a smooth analytical stretching in computational space that is constrained to 930 given specified surface and bottom grid cell thicknesses in real space. 931 This is not to be confused with the hybrid schemes that 932 superimpose geopotential coordinates on terrain following coordinates thus 933 creating a nonanalytical vertical coordinate that 934 therefore may suffer from large gradients in the vertical resolutions. 935 This stretching is less straightforward to implement than the \citet{song.haidvogel_JCP94} stretching, 936 but has the advantage of resolving diurnal processes in deep water and has generally flatter slopes. 937 938 As with the \citet{song.haidvogel_JCP94} stretching the stretch is only applied at depths greater than 939 the critical depth $h_c$. 940 In this example two options are available in depths shallower than $h_c$, 941 with pure sigma being applied if the \np{ln\_sigcrit} is true and pure zcoordinates if it is false 942 (the zcoordinate being equal to the depths of the stretched coordinate at $h_c$). 943 944 Minimising the horizontal slope of the vertical coordinate is important in terrainfollowing systems as 945 large slopes lead to hydrostatic consistency. 946 A hydrostatic consistency parameter diagnostic following \citet{haney_JPO91} has been implemented, 947 and is output as part of the model mesh file at the start of the run. 948 949 %  950 % z* or s*coordinate 951 %  952 \subsection[\zstar or \sstarcoordinate (\forcode{ln_linssh = .false.})] 953 {\zstar or \sstarcoordinate (\protect\np{ln\_linssh}\forcode{ = .false.})} 954 \label{subsec:DOM_zgr_star} 955 956 This option is described in the Report by Levier \textit{et al.} (2007), available on the \NEMO web site. 957 958 %gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances 525 526 Additionally, hybrid combinations of the three main coordinates are available: 527 $sz$ or $szps$ coordinate (\autoref{fig:z_zps_s_sps}d and \autoref{fig:z_zps_s_sps}e). 528 529 A further choice related to vertical coordinate concerns the presence (or not) of ocean 530 cavities beneath ice shelves within the model domain. A setting of \np{ln\_isfcav} as 531 \forcode{.true.} indicates that the domain contains ocean cavities, otherwise the top, 532 wet layer of the ocean will always be at the ocean surface. This option is currently only 533 available for $z$ or $zps$coordinates. In the latter case, partial steps are also applied 534 at the ocean/ice shelf interface. 535 536 Within the model, the arrays describing the grid point depths and vertical scale factors 537 are three set of three dimensional arrays $(i,j,k)$ defined at \textit{before}, 538 \textit{now} and \textit{after} time step. The time at which they are defined is 539 indicated by a suffix: $\_b$, $\_n$, or $\_a$, respectively. They are updated at each 540 model time step. The initial fixed reference coordinate system is held in variable names 541 with a $\_0$ suffix. When the linear free surface option is used 542 (\np{ln\_linssh}\forcode{ = .true.}), \textit{before}, \textit{now} and \textit{after} 543 arrays are initially set to their reference counterpart and remain fixed. 544 545 \subsubsection{Required fields} 546 \label{sec:DOM_zgr_fields} 547 The explicit specification of a range of fields related to the vertical grid are required for the definition of a configuration. These include: 548 549 \begin{Verbatim}[fontsize=\tiny] 550 int ln_zco, ln_zps, ln_sco /* flags for zcoord, zcoord with partial steps and scoord */ 551 int ln_isfcav /* flag for ice shelf cavities */ 552 double e3t_1d, e3w_1d /* reference vertical scale factors at T and W points */ 553 double e3t_0, e3u_0, e3v_0, e3f_0, e3w_0 /* vertical scale factors 3D coordinate at T,U,V,F and W points */ 554 double e3uw_0, e3vw_0 /* vertical scale factors 3D coordinate at UW and VW points */ 555 int bottom_level, top_level /* last wet Tpoints, 1st wet Tpoints (for ice shelf cavities) */ 556 /* For reference: */ 557 float bathy_metry /* bathymetry used in setting top and bottom levels */ 558 \end{Verbatim} 559 560 This set of vertical metrics is sufficient to describe the initial depth and thickness of 561 every gridcell in the model regardless of the choice of vertical coordinate. With constant 562 zlevels, e3 metrics will be uniform across each horizontal level. In the partial step 563 case each e3 at the \np{bottom\_level} (and, possibly, \np{top\_level} if ice cavities are 564 present) may vary from its horizontal neighbours. And, in scoordinates, variations can 565 occur throughout the water column. With the nonlinear freesurface, all the coordinates 566 behave more like the scoordinate in that variations occurr throughout the water column 567 with displacements related to the sea surface height. These variations are typically much 568 smaller than those arising from bottom fitted coordinates. The values for vertical metrics 569 supplied in the domain configuration file can be considered as those arising from a flat 570 sea surface with zero elevation. 571 572 The \np{bottom\_level} and \np{top\_level} 2D arrays define the \np{bottom\_level} and top 573 wet levels in each grid column. Without ice cavities, \np{top\_level} is essentially a land 574 mask (0 on land; 1 everywhere else). With ice cavities, \np{top\_level} determines the 575 first wet point below the overlying ice shelf. 576 577 959 578 960 579 %  961 580 % level bathymetry and mask 962 581 %  963 \subs ection{Level bathymetry and mask}582 \subsubsection{Level bathymetry and mask} 964 583 \label{subsec:DOM_msk} 965 584 966 Whatever the vertical coordinate used, the model offers the possibility of representing the bottom topography with 967 steps that follow the face of the model cells (step like topography) \citep{madec.delecluse.ea_JPO96}. 968 The distribution of the steps in the horizontal is defined in a 2D integer array, mbathy, which 969 gives the number of ocean levels (\ie those that are not masked) at each $t$point. 970 mbathy is computed from the meter bathymetry using the definiton of gdept as the number of $t$points which 971 gdept $\leq$ bathy. 972 973 Modifications of the model bathymetry are performed in the \textit{bat\_ctl} routine (see \mdl{domzgr} module) after 974 mbathy is computed. 975 Isolated grid points that do not communicate with another ocean point at the same level are eliminated. 976 977 As for the representation of bathymetry, a 2D integer array, misfdep, is created. 978 misfdep defines the level of the first wet $t$point. 979 All the cells between $k = 1$ and $misfdep(i,j)  1$ are masked. 980 By default, $misfdep(:,:) = 1$ and no cells are masked. 981 982 In case of ice shelf cavities, modifications of the model bathymetry and ice shelf draft into 983 the cavities are performed in the \textit{zgr\_isf} routine. 984 The compatibility between ice shelf draft and bathymetry is checked. 985 All the locations where the isf cavity is thinnest than \np{rn\_isfhmin} meters are grounded (\ie masked). 986 If only one cell on the water column is opened at $t$, $u$ or $v$points, 987 the bathymetry or the ice shelf draft is dug to fit this constrain. 988 If the incompatibility is too strong (need to dig more than 1 cell), the cell is masked. 989 990 From the \textit{mbathy} and \textit{misfdep} array, the mask fields are defined as follows: 585 586 From \np{top\_level} and \np{bottom\_level} fields, the mask fields are defined as follows: 991 587 \begin{alignat*}{2} 992 588 tmask(i,j,k) &= & & 993 589 \begin{cases} 994 0 &\text{if $ k < misfdep(i,j)$} \\995 1 &\text{if $ misfdep(i,j) \leq k \leq mbathy(i,j)$} \\996 0 &\text{if $ k > mbathy(i,j)$}590 0 &\text{if $ k < top\_level(i,j)$} \\ 591 1 &\text{if $bottom\_level(i,j) \leq k \leq top\_level(i,j)$} \\ 592 0 &\text{if $ k > bottom\_level(i,j)$} 997 593 \end{cases} 998 594 \\ … … 1010 606 exactly in the same way as for the bottom boundary. 1011 607 1012 The specification of closed lateral boundaries requires that at least 1013 the first and last rows and columns of the \textit{mbathy} array are set to zero. 1014 In the particular case of an eastwest cyclical boundary condition, \textit{mbathy} has its last column equal to 1015 the second one and its first column equal to the last but one (and so too the mask arrays) 1016 (see \autoref{fig:LBC_jperio}). 608 %% The specification of closed lateral boundaries requires that at least 609 %% the first and last rows and columns of the \textit{mbathy} array are set to zero. 610 %% In the particular case of an eastwest cyclical boundary condition, \textit{mbathy} has its last column equal to 611 %% the second one and its first column equal to the last but one (and so too the mask arrays) 612 %% (see \autoref{fig:LBC_jperio}). 613 614 615 % 616 % Closed seas 617 % 618 \subsection{Closed seas} \label{subsec:DOM_closea} 619 620 When a global ocean is coupled to an atmospheric model it is better to represent all large 621 water bodies (\eg great lakes, Caspian sea...) even if the model resolution does not allow 622 their communication with the rest of the ocean. This is unnecessary when the ocean is 623 forced by fixed atmospheric conditions, so these seas can be removed from the ocean 624 domain. The user has the option to set the bathymetry in closed seas to zero (see 625 \autoref{sec:MISC_closea}) and to optionally decide on the fate of any freshwater 626 imbalance over the area. The options are explained in \autoref{sec:MISC_closea} but it 627 should be noted here that a successful use of these options requires appropriate mask 628 fields to be present in the domain configuration file. Among the possibilities are: 629 630 \begin{Verbatim}[fontsize=\tiny] 631 int closea_mask /* nonzero values in closed sea areas for optional masking */ 632 int closea_mask_rnf /* nonzero values in closed sea areas with runoff locations (precip only) */ 633 int closea_mask_emp /* nonzero values in closed sea areas with runoff locations (total emp) */ 634 \end{Verbatim} 635 636 %  637 % Grid files 638 %  639 \subsection{Output grid files} 640 \label{subsec:DOM_meshmask} 641 642 \nlst{namcfg} 643 644 Most of the arrays relating to a particular ocean model configuration dicussed in this 645 chapter (gridpoint position, scale factors) can be saved in a file if namelist parameter 646 \np{ln\_write\_cfg} (namelist \ngn{namcfg}) is set to \forcode{.true.}; the output 647 filename is set thorugh parameter \np{cn\_domcfg\_out}. This is only really useful 648 if the fields are computed in subroutines \mdl{usrdef\_hgr} or \mdl{usrdef\_zgr} and 649 checking or confirmation is required. 650 651 \nlst{namdom} 652 653 Alternatively, all the arrays relating to a particular ocean model configuration 654 (gridpoint position, scale factors, depths and masks) can be saved in a file called 655 \texttt{mesh\_mask} if namelist parameter \np{ln\_meshmask} (namelist \ngn{namdom}) is set 656 to \forcode{.true.}. This file contains additional fields that can be useful for 657 postprocessing applications 1017 658 1018 659 % ================================================================ … … 1023 664 \label{sec:DTA_tsd} 1024 665 %namtsd 1025 1026 666 \nlst{namtsd} 1027 667 % 1028 668 1029 Options are defined in \ngn{namtsd}. 1030 By default, the ocean start from rest (the velocity field is set to zero) and the initialization of temperature and 1031 salinity fields is controlled through the \np{ln\_tsd\_ini} namelist parameter. 669 Basic initial state options are defined in \ngn{namtsd}. By default, the ocean starts 670 from rest (the velocity field is set to zero) and the initialization of temperature and 671 salinity fields is controlled through the \np{ln\_tsd\_init} namelist parameter. 672 1032 673 \begin{description} 1033 \item[\np{ln\_tsd\_init}\forcode{ 1034 use a T and S input files that can be given on the model grid itself or on their native input data grid.1035 In the latter case,1036 the data will be interpolated onthefly both in the horizontal and the vertical to the model grid1037 (see \autoref{subsec:SBC_iof}).1038 The information relative to the input files are given in the \np{sn\_tem} and \np{sn\_sal} structures.1039 The computation is done in the \mdl{dtatsd} module. 1040 \item[\np{ln\_tsd\_init}\forcode{ = .false.}] 1041 use constant salinity value of $35.5~psu$ and an analytical profile of temperature1042 (typical of the tropical ocean), see \rou{istate\_t\_s} subroutine called from \mdl{istate} module.674 \item[\np{ln\_tsd\_init}\forcode{= .true.}] 675 Use T and S input files that can be given on the model grid itself or on their native 676 input data grids. In the latter case, the data will be interpolated onthefly both in 677 the horizontal and the vertical to the model grid (see \autoref{subsec:SBC_iof}). The 678 information relating to the input files are specified in the \np{sn\_tem} and 679 \np{sn\_sal} structures. The computation is done in the \mdl{dtatsd} module. 680 \item[\np{ln\_tsd\_init}\forcode{= .false.}] 681 Initial values for T and S are set via a user supplied \rou{usr\_def\_istate} routine 682 contained in \mdl{userdef\_istate}. The default version sets horizontally uniform T and 683 profiles as used in the GYRE configuration (see \autoref{sec:CFG_gyre}). 1043 684 \end{description} 1044 685
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