Changeset 9407 for branches/2017/dev_merge_2017/DOC/tex_sub/chap_LBC.tex
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branches/2017/dev_merge_2017/DOC/tex_sub/chap_LBC.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Lateral Boundary Condition (LBC)} 7 \label{ LBC}7 \label{chap:LBC} 8 8 \minitoc 9 9 … … 18 18 % ================================================================ 19 19 \section{Boundary condition at the coast (\protect\np{rn\_shlat})} 20 \label{ LBC_coast}20 \label{sec:LBC_coast} 21 21 %--------------------------------------------nam_lbc------------------------------------------------------- 22 22 \forfile{../namelists/namlbc} 23 23 %-------------------------------------------------------------------------------------------------------------- 24 24 25 %The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt (no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip). They are handled automatically by the mask system (see \ S\ref{DOM_msk}).26 27 %OPA allows land and topography grid points in the computational domain due to the presence of continents or islands, and includes the use of a full or partial step representation of bottom topography. The computation is performed over the whole domain, i.e. we do not try to restrict the computation to ocean-only points. This choice has two motivations. Firstly, working on ocean only grid points overloads the code and harms the code readability. Secondly, and more importantly, it drastically reduces the vector portion of the computation, leading to a dramatic increase of CPU time requirement on vector computers. The current section describes how the masking affects the computation of the various terms of the equations with respect to the boundary condition at solid walls. The process of defining which areas are to be masked is described in \ S\ref{DOM_msk}.25 %The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt (no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip). They are handled automatically by the mask system (see \autoref{subsec:DOM_msk}). 26 27 %OPA allows land and topography grid points in the computational domain due to the presence of continents or islands, and includes the use of a full or partial step representation of bottom topography. The computation is performed over the whole domain, i.e. we do not try to restrict the computation to ocean-only points. This choice has two motivations. Firstly, working on ocean only grid points overloads the code and harms the code readability. Secondly, and more importantly, it drastically reduces the vector portion of the computation, leading to a dramatic increase of CPU time requirement on vector computers. The current section describes how the masking affects the computation of the various terms of the equations with respect to the boundary condition at solid walls. The process of defining which areas are to be masked is described in \autoref{subsec:DOM_msk}. 28 28 29 29 Options are defined through the \ngn{namlbc} namelist variables. … … 44 44 at $u$-points. Evaluating this quantity as, 45 45 46 \begin{equation} \label{ Eq_lbc_aaaa}46 \begin{equation} \label{eq:lbc_aaaa} 47 47 \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 48 48 }{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u … … 51 51 zero inside land and at the boundaries, since mask$_{u}$ is zero at solid boundaries 52 52 which in this case are defined at $u$-points (normal velocity $u$ remains zero at 53 the coast) ( Fig.~\ref{Fig_LBC_uv}).53 the coast) (\autoref{fig:LBC_uv}). 54 54 55 55 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 56 56 \begin{figure}[!t] \begin{center} 57 57 \includegraphics[width=0.90\textwidth]{Fig_LBC_uv} 58 \caption{ \protect\label{ Fig_LBC_uv}58 \caption{ \protect\label{fig:LBC_uv} 59 59 Lateral boundary (thick line) at T-level. The velocity normal to the boundary is set to zero.} 60 60 \end{center} \end{figure} … … 66 66 For example, at a given $T$-level, the lateral boundary (a coastline or an intersection 67 67 with the bottom topography) is made of segments joining $f$-points, and normal 68 velocity points are located between two $f-$points ( Fig.~\ref{Fig_LBC_uv}).68 velocity points are located between two $f-$points (\autoref{fig:LBC_uv}). 69 69 The boundary condition on the normal velocity (no flux through solid boundaries) 70 70 can thus be easily implemented using the mask system. The boundary condition … … 79 79 \begin{figure}[!p] \begin{center} 80 80 \includegraphics[width=0.90\textwidth]{Fig_LBC_shlat} 81 \caption{ \protect\label{ Fig_LBC_shlat}81 \caption{ \protect\label{fig:LBC_shlat} 82 82 lateral boundary condition (a) free-slip ($rn\_shlat=0$) ; (b) no-slip ($rn\_shlat=2$) 83 83 ; (c) "partial" free-slip ($0<rn\_shlat<2$) and (d) "strong" no-slip ($2<rn\_shlat$). … … 91 91 coastline is equal to the offshore velocity, $i.e.$ the normal derivative of the 92 92 tangential velocity is zero at the coast, so the vorticity: mask$_{f}$ array is set 93 to zero inside the land and just at the coast ( Fig.~\ref{Fig_LBC_shlat}-a).93 to zero inside the land and just at the coast (\autoref{fig:LBC_shlat}-a). 94 94 95 95 \item[no-slip boundary condition (\np{rn\_shlat}\forcode{ = 2}): ] the tangential velocity vanishes … … 98 98 evaluated as if the velocities at the closest land velocity gridpoint and the closest 99 99 ocean velocity gridpoint were of the same magnitude but in the opposite direction 100 ( Fig.~\ref{Fig_LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by:100 (\autoref{fig:LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by: 101 101 102 102 \begin{equation*} … … 106 106 the coastline provides a vorticity field computed with the no-slip boundary condition, 107 107 simply by multiplying it by the mask$_{f}$ : 108 \begin{equation} \label{ Eq_lbc_bbbb}108 \begin{equation} \label{eq:lbc_bbbb} 109 109 \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2} 110 110 \left[ {e_{2v} \,v} \right]-\delta _{j+1/2} \left[ {e_{1u} \,u} \right]} … … 115 115 velocity at the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral 116 116 friction but not strong enough to make the tangential velocity at the coast vanish 117 ( Fig.~\ref{Fig_LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$117 (\autoref{fig:LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$ 118 118 strictly inbetween $0$ and $2$. 119 119 120 120 \item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}): ] the viscous boundary 121 layer is assumed to be smaller than half the grid size ( Fig.~\ref{Fig_LBC_shlat}-d).121 layer is assumed to be smaller than half the grid size (\autoref{fig:LBC_shlat}-d). 122 122 The friction is thus larger than in the no-slip case. 123 123 … … 134 134 % ================================================================ 135 135 \section{Model domain boundary condition (\protect\np{jperio})} 136 \label{ LBC_jperio}136 \label{sec:LBC_jperio} 137 137 138 138 At the model domain boundaries several choices are offered: closed, cyclic east-west, … … 144 144 % ------------------------------------------------------------------------------------------------------------- 145 145 \subsection{Closed, cyclic, south symmetric (\protect\np{jperio}\forcode{= 0..2})} 146 \label{ LBC_jperio012}146 \label{subsec:LBC_jperio012} 147 147 148 148 The choice of closed, cyclic or symmetric model domain boundary condition is made … … 160 160 \item[For cyclic east-west boundary (\np{jperio}\forcode{ = 1})], first and last rows are set 161 161 to zero (closed) whilst the first column is set to the value of the last-but-one column 162 and the last column to the value of the second one ( Fig.~\ref{Fig_LBC_jperio}-a).162 and the last column to the value of the second one (\autoref{fig:LBC_jperio}-a). 163 163 Whatever flows out of the eastern (western) end of the basin enters the western 164 164 (eastern) end. Note that there is no option for north-south cyclic or for doubly … … 171 171 to the value of the third row while for most of $v$- and $f$-point arrays ($v$, $\zeta$, 172 172 $j\psi$, but \gmcomment{not sure why this is "but"} scalar arrays such as eddy coefficients) 173 the first row is set to minus the value of the second row ( Fig.~\ref{Fig_LBC_jperio}-b).173 the first row is set to minus the value of the second row (\autoref{fig:LBC_jperio}-b). 174 174 Note that this boundary condition is not yet available for the case of a massively 175 175 parallel computer (\textbf{key{\_}mpp} defined). … … 180 180 \begin{figure}[!t] \begin{center} 181 181 \includegraphics[width=1.0\textwidth]{Fig_LBC_jperio} 182 \caption{ \protect\label{ Fig_LBC_jperio}182 \caption{ \protect\label{fig:LBC_jperio} 183 183 setting of (a) east-west cyclic (b) symmetric across the equator boundary conditions.} 184 184 \end{center} \end{figure} … … 189 189 % ------------------------------------------------------------------------------------------------------------- 190 190 \subsection{North-fold (\protect\np{jperio}\forcode{ = 3..6})} 191 \label{ LBC_north_fold}191 \label{subsec:LBC_north_fold} 192 192 193 193 The north fold boundary condition has been introduced in order to handle the north 194 194 boundary of a three-polar ORCA grid. Such a grid has two poles in the northern hemisphere 195 ( Fig.\ref{Fig_MISC_ORCA_msh}, and thus requires a specific treatment illustrated in Fig.\ref{Fig_North_Fold_T}.195 (\autoref{fig:MISC_ORCA_msh}, and thus requires a specific treatment illustrated in \autoref{fig:North_Fold_T}. 196 196 Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition. 197 197 … … 199 199 \begin{figure}[!t] \begin{center} 200 200 \includegraphics[width=0.90\textwidth]{Fig_North_Fold_T} 201 \caption{ \protect\label{ Fig_North_Fold_T}201 \caption{ \protect\label{fig:North_Fold_T} 202 202 North fold boundary with a $T$-point pivot and cyclic east-west boundary condition 203 203 ($jperio=4$), as used in ORCA 2, 1/4, and 1/12. Pink shaded area corresponds … … 210 210 % ==================================================================== 211 211 \section{Exchange with neighbouring processors (\protect\mdl{lbclnk}, \protect\mdl{lib\_mpp})} 212 \label{ LBC_mpp}212 \label{sec:LBC_mpp} 213 213 214 214 For massively parallel processing (mpp), a domain decomposition method is used. … … 261 261 \begin{figure}[!t] \begin{center} 262 262 \includegraphics[width=0.90\textwidth]{Fig_mpp} 263 \caption{ \protect\label{ Fig_mpp}263 \caption{ \protect\label{fig:mpp} 264 264 Positioning of a sub-domain when massively parallel processing is used. } 265 265 \end{center} \end{figure} … … 279 279 \begin{eqnarray} 280 280 jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci \nonumber \\ 281 jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj \label{ Eq_lbc_jpi}281 jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj \label{eq:lbc_jpi} 282 282 \end{eqnarray} 283 283 where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis. … … 287 287 An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$, 288 288 a global array (whole domain) by the relationship: 289 \begin{equation} \label{ Eq_lbc_nimpp}289 \begin{equation} \label{eq:lbc_nimpp} 290 290 T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k), 291 291 \end{equation} … … 315 315 global ocean where more than 50 \% of points are land points. For this reason, a 316 316 pre-processing tool can be used to choose the mpp domain decomposition with a 317 maximum number of only land points processors, which can then be eliminated ( Fig. \ref{Fig_mppini2})317 maximum number of only land points processors, which can then be eliminated (\autoref{fig:mppini2}) 318 318 (For example, the mpp\_optimiz tools, available from the DRAKKAR web site). 319 319 This optimisation is dependent on the specific bathymetry employed. The user … … 335 335 \begin{figure}[!ht] \begin{center} 336 336 \includegraphics[width=0.90\textwidth]{Fig_mppini2} 337 \caption { \protect\label{ Fig_mppini2}337 \caption { \protect\label{fig:mppini2} 338 338 Example of Atlantic domain defined for the CLIPPER projet. Initial grid is 339 339 composed of 773 x 1236 horizontal points. … … 350 350 % ==================================================================== 351 351 \section{Unstructured open boundary conditions (BDY)} 352 \label{ LBC_bdy}352 \label{sec:LBC_bdy} 353 353 354 354 %-----------------------------------------nambdy-------------------------------------------- … … 384 384 %---------------------------------------------- 385 385 \subsection{Namelists} 386 \label{ BDY_namelist}386 \label{subsec:BDY_namelist} 387 387 388 388 The BDY module is activated by setting \np{ln\_bdy} to true. … … 400 400 a file and the second is defined in a namelist. For more details of 401 401 the definition of the boundary geometry see section 402 \ ref{BDY_geometry}.402 \autoref{subsec:BDY_geometry}. 403 403 404 404 For each boundary set a boundary … … 457 457 %---------------------------------------------- 458 458 \subsection{Flow relaxation scheme} 459 \label{ BDY_FRS_scheme}459 \label{subsec:BDY_FRS_scheme} 460 460 461 461 The Flow Relaxation Scheme (FRS) \citep{Davies_QJRMS76,Engerdahl_Tel95}, … … 463 463 externally-specified values over a zone next to the edge of the model 464 464 domain. Given a model prognostic variable $\Phi$ 465 \begin{equation} \label{ Eq_bdy_frs1}465 \begin{equation} \label{eq:bdy_frs1} 466 466 \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N 467 467 \end{equation} … … 472 472 to adding a relaxation term to the prognostic equation for $\Phi$ of 473 473 the form: 474 \begin{equation} \label{ Eq_bdy_frs2}474 \begin{equation} \label{eq:bdy_frs2} 475 475 -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right) 476 476 \end{equation} 477 477 where the relaxation time scale $\tau$ is given by a function of 478 478 $\alpha$ and the model time step $\Delta t$: 479 \begin{equation} \label{ Eq_bdy_frs3}479 \begin{equation} \label{eq:bdy_frs3} 480 480 \tau = \frac{1-\alpha}{\alpha} \,\rdt 481 481 \end{equation} … … 487 487 488 488 The function $\alpha$ is specified as a $tanh$ function: 489 \begin{equation} \label{ Eq_bdy_frs4}489 \begin{equation} \label{eq:bdy_frs4} 490 490 \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right), \quad d=1,N 491 491 \end{equation} … … 495 495 %---------------------------------------------- 496 496 \subsection{Flather radiation scheme} 497 \label{ BDY_flather_scheme}497 \label{subsec:BDY_flather_scheme} 498 498 499 499 The \citet{Flather_JPO94} scheme is a radiation condition on the normal, depth-mean 500 500 transport across the open boundary. It takes the form 501 \begin{equation} \label{ Eq_bdy_fla1}501 \begin{equation} \label{eq:bdy_fla1} 502 502 U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right), 503 503 \end{equation} … … 510 510 external depth-mean normal velocity, plus a correction term that 511 511 allows gravity waves generated internally to exit the model boundary. 512 Note that the sea-surface height gradient in \ eqref{Eq_bdy_fla1}512 Note that the sea-surface height gradient in \autoref{eq:bdy_fla1} 513 513 is a spatial gradient across the model boundary, so that $\eta_{e}$ is 514 514 defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the … … 518 518 %---------------------------------------------- 519 519 \subsection{Boundary geometry} 520 \label{ BDY_geometry}520 \label{subsec:BDY_geometry} 521 521 522 522 Each open boundary set is defined as a list of points. The information … … 529 529 further away from the edge of the model domain. A set of $nbi$, $nbj$, 530 530 and $nbr$ arrays is defined for each of the $T$, $U$ and $V$ 531 grids. Figure \ ref{Fig_LBC_bdy_geom} shows an example of an irregular531 grids. Figure \autoref{fig:LBC_bdy_geom} shows an example of an irregular 532 532 boundary. 533 533 … … 545 545 546 546 The boundary geometry may also be defined from a 547 ``\ifile{coordinates.bdy}'' file. Figure \ ref{Fig_LBC_nc_header}547 ``\ifile{coordinates.bdy}'' file. Figure \autoref{fig:LBC_nc_header} 548 548 gives an example of the header information from such a file. The file 549 549 should contain the index arrays for each of the $T$, $U$ and $V$ … … 566 566 \begin{figure}[!t] \begin{center} 567 567 \includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom} 568 \caption { \protect\label{ Fig_LBC_bdy_geom}568 \caption { \protect\label{fig:LBC_bdy_geom} 569 569 Example of geometry of unstructured open boundary} 570 570 \end{center} \end{figure} … … 573 573 %---------------------------------------------- 574 574 \subsection{Input boundary data files} 575 \label{ BDY_data}575 \label{subsec:BDY_data} 576 576 577 577 The data files contain the data arrays … … 607 607 \begin{figure}[!t] \begin{center} 608 608 \includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header} 609 \caption { \protect\label{ Fig_LBC_nc_header}610 Example of the header for a \ ifile{coordinates.bdy} file}609 \caption { \protect\label{fig:LBC_nc_header} 610 Example of the header for a \protect\ifile{coordinates.bdy} file} 611 611 \end{center} \end{figure} 612 612 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 614 614 %---------------------------------------------- 615 615 \subsection{Volume correction} 616 \label{ BDY_vol_corr}616 \label{subsec:BDY_vol_corr} 617 617 618 618 There is an option to force the total volume in the regional model to be constant, … … 631 631 %---------------------------------------------- 632 632 \subsection{Tidal harmonic forcing} 633 \label{ BDY_tides}633 \label{subsec:BDY_tides} 634 634 635 635 %-----------------------------------------nambdy_tide--------------------------------------------
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