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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4% ================================================================
5% Chapter — Lateral Boundary Condition (LBC)
6% ================================================================
7\chapter{Lateral Boundary Condition (LBC)}
8\label{chap:LBC}
9
10\chaptertoc
11
12\newpage
13
14%gm% add here introduction to this chapter
15
16% ================================================================
17% Boundary Condition at the Coast
18% ================================================================
19\section[Boundary condition at the coast (\texttt{rn\_shlat})]
20{Boundary condition at the coast (\protect\np{rn\_shlat})}
21\label{sec:LBC_coast}
22%--------------------------------------------namlbc-------------------------------------------------------
23
24\begin{listing}
25  \nlst{namlbc}
26  \caption{\texttt{namlbc}}
27  \label{lst:namlbc}
28\end{listing}
29%--------------------------------------------------------------------------------------------------------------
30
31%The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt
32%(no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip).
33%They are handled automatically by the mask system (see \autoref{subsec:DOM_msk}).
34
35%OPA allows land and topography grid points in the computational domain due to the presence of continents or islands,
36%and includes the use of a full or partial step representation of bottom topography.
37%The computation is performed over the whole domain, \ie\ we do not try to restrict the computation to ocean-only points.
38%This choice has two motivations.
39%Firstly, working on ocean only grid points overloads the code and harms the code readability.
40%Secondly, and more importantly, it drastically reduces the vector portion of the computation,
41%leading to a dramatic increase of CPU time requirement on vector computers.
42%The current section describes how the masking affects the computation of the various terms of the equations
43%with respect to the boundary condition at solid walls.
44%The process of defining which areas are to be masked is described in \autoref{subsec:DOM_msk}.
45
46Options are defined through the \nam{lbc} namelist variables.
47The discrete representation of a domain with complex boundaries (coastlines and bottom topography) leads to
48arrays that include large portions where a computation is not required as the model variables remain at zero.
49Nevertheless, vectorial supercomputers are far more efficient when computing over a whole array,
50and the readability of a code is greatly improved when boundary conditions are applied in
51an automatic way rather than by a specific computation before or after each computational loop.
52An efficient way to work over the whole domain while specifying the boundary conditions,
53is to use multiplication by mask arrays in the computation.
54A mask array is a matrix whose elements are $1$ in the ocean domain and $0$ elsewhere.
55A simple multiplication of a variable by its own mask ensures that it will remain zero over land areas.
56Since most of the boundary conditions consist of a zero flux across the solid boundaries,
57they can be simply applied by multiplying variables by the correct mask arrays,
58\ie\ the mask array of the grid point where the flux is evaluated.
59For example, the heat flux in the \textbf{i}-direction is evaluated at $u$-points.
60Evaluating this quantity as,
61
62\[
63  % \label{eq:LBC_aaaa}
64  \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT}
65  }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u
66\]
67(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is zero inside land and
68at the boundaries, since mask$_{u}$ is zero at solid boundaries which in this case are defined at $u$-points
69(normal velocity $u$ remains zero at the coast) (\autoref{fig:LBC_uv}).
70
71%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
72\begin{figure}[!t]
73  \centering
74  \includegraphics[width=0.66\textwidth]{Fig_LBC_uv}
75  \caption[Lateral boundary at $T$-level]{
76    Lateral boundary (thick line) at T-level.
77    The velocity normal to the boundary is set to zero.}
78  \label{fig:LBC_uv}
79\end{figure}
80%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
81
82For momentum the situation is a bit more complex as two boundary conditions must be provided along the coast
83(one each for the normal and tangential velocities).
84The boundary of the ocean in the C-grid is defined by the velocity-faces.
85For example, at a given $T$-level,
86the lateral boundary (a coastline or an intersection with the bottom topography) is made of
87segments joining $f$-points, and normal velocity points are located between two $f-$points (\autoref{fig:LBC_uv}).
88The boundary condition on the normal velocity (no flux through solid boundaries)
89can thus be easily implemented using the mask system.
90The boundary condition on the tangential velocity requires a more specific treatment.
91This boundary condition influences the relative vorticity and momentum diffusive trends,
92and is required in order to compute the vorticity at the coast.
93Four different types of lateral boundary condition are available,
94controlled by the value of the \np{rn\_shlat} namelist parameter
95(The value of the mask$_{f}$ array along the coastline is set equal to this parameter).
96These are:
97
98%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
99\begin{figure}[!p]
100  \centering
101  \includegraphics[width=0.66\textwidth]{Fig_LBC_shlat}
102  \caption[Lateral boundary conditions]{
103    Lateral boundary conditions
104    (a) free-slip                       (\protect\np{rn\_shlat}\forcode{=0});
105    (b) no-slip                         (\protect\np{rn\_shlat}\forcode{=2});
106    (c) "partial" free-slip (\forcode{0<}\protect\np{rn\_shlat}\forcode{<2}) and
107    (d) "strong" no-slip    (\forcode{2<}\protect\np{rn\_shlat}).
108    Implied "ghost" velocity inside land area is display in grey.}
109  \label{fig:LBC_shlat}
110\end{figure}
111%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
112
113\begin{description}
114
115\item[free-slip boundary condition (\np{rn\_shlat}\forcode{=0}):] the tangential velocity at
116  the coastline is equal to the offshore velocity,
117  \ie\ the normal derivative of the tangential velocity is zero at the coast,
118  so the vorticity: mask$_{f}$ array is set to zero inside the land and just at the coast
119  (\autoref{fig:LBC_shlat}-a).
120
121\item[no-slip boundary condition (\np{rn\_shlat}\forcode{=2}):] the tangential velocity vanishes at the coastline.
122  Assuming that the tangential velocity decreases linearly from
123  the closest ocean velocity grid point to the coastline,
124  the normal derivative is evaluated as if the velocities at the closest land velocity gridpoint and
125  the closest ocean velocity gridpoint were of the same magnitude but in the opposite direction
126  (\autoref{fig:LBC_shlat}-b).
127  Therefore, the vorticity along the coastlines is given by:
128
129  \[
130    \zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ ,
131  \]
132  where $u$ and $v$ are masked fields.
133  Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with
134  the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ :
135  \[
136    % \label{eq:LBC_bbbb}
137    \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2}
138        \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]}
139    \right)\;\mbox{mask}_f
140  \]
141
142\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2):] the tangential velocity at
143  the coastline is smaller than the offshore velocity, \ie\ there is a lateral friction but
144  not strong enough to make the tangential velocity at the coast vanish (\autoref{fig:LBC_shlat}-c).
145  This can be selected by providing a value of mask$_{f}$ strictly inbetween $0$ and $2$.
146
147\item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}):] the viscous boundary layer is assumed to
148  be smaller than half the grid size (\autoref{fig:LBC_shlat}-d).
149  The friction is thus larger than in the no-slip case.
150
151\end{description}
152
153Note that when the bottom topography is entirely represented by the $s$-coordinates (pure $s$-coordinate),
154the lateral boundary condition on tangential velocity is of much less importance as
155it is only applied next to the coast where the minimum water depth can be quite shallow.
156
157
158% ================================================================
159% Boundary Condition around the Model Domain
160% ================================================================
161\section[Model domain boundary condition (\texttt{jperio})]
162{Model domain boundary condition (\protect\jp{jperio})}
163\label{sec:LBC_jperio}
164
165At the model domain boundaries several choices are offered:
166closed, cyclic east-west, cyclic north-south, a north-fold, and combination closed-north fold or
167bi-cyclic east-west and north-fold.
168The north-fold boundary condition is associated with the 3-pole ORCA mesh.
169
170% -------------------------------------------------------------------------------------------------------------
171%        Closed, cyclic (\jp{jperio}\forcode{ = 0..2})
172% -------------------------------------------------------------------------------------------------------------
173\subsection[Closed, cyclic (\forcode{jperio=[0127]})]
174{Closed, cyclic (\protect\jp{jperio}\forcode{=[0127]})}
175\label{subsec:LBC_jperio012}
176
177The choice of closed or cyclic model domain boundary condition is made by
178setting \jp{jperio} to 0, 1, 2 or 7 in namelist \nam{cfg}.
179Each time such a boundary condition is needed, it is set by a call to routine \mdl{lbclnk}.
180The computation of momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to $j=jpj-1$,
181\ie\ in the model interior.
182To choose a lateral model boundary condition is to specify the first and last rows and columns of
183the model variables.
184
185\begin{description}
186
187\item[For closed boundary (\jp{jperio}\forcode{=0})],
188  solid walls are imposed at all model boundaries:
189  first and last rows and columns are set to zero.
190
191\item[For cyclic east-west boundary (\jp{jperio}\forcode{=1})],
192  first and last rows are set to zero (closed) whilst the first column is set to
193  the value of the last-but-one column and the last column to the value of the second one
194  (\autoref{fig:LBC_jperio}-a).
195  Whatever flows out of the eastern (western) end of the basin enters the western (eastern) end.
196
197\item[For cyclic north-south boundary (\jp{jperio}\forcode{=2})],
198  first and last columns are set to zero (closed) whilst the first row is set to
199  the value of the last-but-one row and the last row to the value of the second one
200  (\autoref{fig:LBC_jperio}-a).
201  Whatever flows out of the northern (southern) end of the basin enters the southern (northern) end.
202
203\item[Bi-cyclic east-west and north-south boundary (\jp{jperio}\forcode{=7})] combines cases 1 and 2.
204
205\end{description}
206
207%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
208\begin{figure}[!t]
209  \centering
210  \includegraphics[width=0.66\textwidth]{Fig_LBC_jperio}
211  \caption[Setting of east-west cyclic and symmetric across the Equator boundary conditions]{
212    Setting of (a) east-west cyclic (b) symmetric across the Equator boundary conditions}
213  \label{fig:LBC_jperio}
214\end{figure}
215%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
216
217% -------------------------------------------------------------------------------------------------------------
218%        North fold (\textit{jperio = 3 }to $6)$
219% -------------------------------------------------------------------------------------------------------------
220\subsection[North-fold (\forcode{jperio=[3-6]})]
221{North-fold (\protect\jp{jperio}\forcode{=[3-6]})}
222\label{subsec:LBC_north_fold}
223
224The north fold boundary condition has been introduced in order to handle the north boundary of
225a three-polar ORCA grid.
226Such a grid has two poles in the northern hemisphere (\autoref{fig:CFGS_ORCA_msh},
227and thus requires a specific treatment illustrated in \autoref{fig:LBC_North_Fold_T}.
228Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition.
229
230%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
231\begin{figure}[!t]
232  \centering
233  \includegraphics[width=0.66\textwidth]{Fig_North_Fold_T}
234  \caption[North fold boundary in ORCA 2\deg, 1/4\deg and 1/12\deg]{
235    North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$),
236    as used in ORCA 2\deg, 1/4\deg and 1/12\deg.
237    Pink shaded area corresponds to the inner domain mask (see text).}
238  \label{fig:LBC_North_Fold_T}
239\end{figure}
240%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
241
242% ====================================================================
243% Exchange with neighbouring processors
244% ====================================================================
245\section[Exchange with neighbouring processors (\textit{lbclnk.F90}, \textit{lib\_mpp.F90})]
246{Exchange with neighbouring processors (\protect\mdl{lbclnk}, \protect\mdl{lib\_mpp})}
247\label{sec:LBC_mpp}
248
249%-----------------------------------------nammpp--------------------------------------------
250
251\begin{listing}
252  \nlst{nammpp}
253  \caption{\texttt{nammpp}}
254  \label{lst:nammpp}
255\end{listing}
256%-----------------------------------------------------------------------------------------------
257
258For massively parallel processing (mpp), a domain decomposition method is used.
259The basic idea of the method is to split the large computation domain of a numerical experiment into several smaller domains and
260solve the set of equations by addressing independent local problems.
261Each processor has its own local memory and computes the model equation over a subdomain of the whole model domain.
262The subdomain boundary conditions are specified through communications between processors which are organized by
263explicit statements (message passing method).
264The present implementation is largely inspired by Guyon's work [Guyon 1995].
265
266The parallelization strategy is defined by the physical characteristics of the ocean model.
267Second order finite difference schemes lead to local discrete operators that
268depend at the very most on one neighbouring point.
269The only non-local computations concern the vertical physics
270(implicit diffusion, turbulent closure scheme, ...).
271Therefore, a pencil strategy is used for the data sub-structuration:
272the 3D initial domain is laid out on local processor memories following a 2D horizontal topological splitting.
273Each sub-domain computes its own surface and bottom boundary conditions and
274has a side wall overlapping interface which defines the lateral boundary conditions for
275computations in the inner sub-domain.
276The overlapping area consists of the two rows at each edge of the sub-domain.
277After a computation, a communication phase starts:
278each processor sends to its neighbouring processors the update values of the points corresponding to
279the interior overlapping area to its neighbouring sub-domain (\ie\ the innermost of the two overlapping rows).
280Communications are first done according to the east-west direction and next according to the north-south direction.
281There is no specific communications for the corners.
282The communication is done through the Message Passing Interface (MPI) and requires \key{mpp\_mpi}.
283Use also \key{mpi2} if MPI3 is not available on your computer.
284The data exchanges between processors are required at the very place where
285lateral domain boundary conditions are set in the mono-domain computation:
286the \rou{lbc\_lnk} routine (found in \mdl{lbclnk} module) which manages such conditions is interfaced with
287routines found in \mdl{lib\_mpp} module.
288The output file \textit{communication\_report.txt} provides the list of which routines do how
289many communications during 1 time step of the model.\\
290
291%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
292\begin{figure}[!t]
293  \centering
294  \includegraphics[width=0.66\textwidth]{Fig_mpp}
295  \caption{Positioning of a sub-domain when massively parallel processing is used}
296  \label{fig:LBC_mpp}
297\end{figure}
298%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
299
300In \NEMO, the splitting is regular and arithmetic.
301The total number of subdomains corresponds to the number of MPI processes allocated to \NEMO\ when the model is launched
302(\ie\ mpirun -np x ./nemo will automatically give x subdomains).
303The i-axis is divided by \np{jpni} and the j-axis by \np{jpnj}.
304These parameters are defined in \nam{mpp} namelist.
305If \np{jpni} and \np{jpnj} are < 1, they will be automatically redefined in the code to give the best domain decomposition
306(see bellow).
307
308Each processor is independent and without message passing or synchronous process, programs run alone and access just its own local memory.
309For this reason,
310the main model dimensions are now the local dimensions of the subdomain (pencil) that are named \jp{jpi}, \jp{jpj}, \jp{jpk}.
311These dimensions include the internal domain and the overlapping rows.
312The number of rows to exchange (known as the halo) is usually set to one (nn\_hls=1, in \mdl{par\_oce},
313and must be kept to one until further notice).
314The whole domain dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}.
315The relationship between the whole domain and a sub-domain is:
316\begin{gather*}
317  jpi = ( jpiglo-2\times nn\_hls + (jpni-1) ) / jpni + 2\times nn\_hls \\
318  jpj = ( jpjglo-2\times nn\_hls + (jpnj-1) ) / jpnj + 2\times nn\_hls
319\end{gather*}
320
321One also defines variables nldi and nlei which correspond to the internal domain bounds, and the variables nimpp and njmpp which are the position of the (1,1) grid-point in the global domain (\autoref{fig:LBC_mpp}). Note that since the version 4, there is no more extra-halo area as defined in \autoref{fig:LBC_mpp} so \jp{jpi} is now always equal to nlci and \jp{jpj} equal to nlcj.
322
323An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
324a global array (whole domain) by the relationship:
325\[
326  % \label{eq:LBC_nimpp}
327  T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
328\]
329with $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.
330
331The 1-d arrays $mig(1:\jp{jpi})$ and $mjg(1:\jp{jpj})$, defined in \rou{dom\_glo} routine (\mdl{domain} module), should be used to get global domain indices from local domain indices. The 1-d arrays, $mi0(1:\jp{jpiglo})$, $mi1(1:\jp{jpiglo})$ and $mj0(1:\jp{jpjglo})$, $mj1(1:\jp{jpjglo})$ have the reverse purpose and should be used to define loop indices expressed in global domain indices (see examples in \mdl{dtastd} module).\\
332
333The \NEMO\ model computes equation terms with the help of mask arrays (0 on land points and 1 on sea points). It is therefore possible that an MPI subdomain contains only land points. To save ressources, we try to supress from the computational domain as much land subdomains as possible. For example if $N_{mpi}$ processes are allocated to NEMO, the domain decomposition will be given by the following equation:
334\[
335  N_{mpi} = jpni \times jpnj - N_{land} + N_{useless}
336\]
337$N_{land}$ is the total number of land subdomains in the domain decomposition defined by \np{jpni} and \np{jpnj}. $N_{useless}$ is the number of land subdomains that are kept in the compuational domain in order to make sure that $N_{mpi}$ MPI processes are indeed allocated to a given subdomain. The values of $N_{mpi}$, \np{jpni}, \np{jpnj}$N_{land}$ and $N_{useless}$ are printed in the output file \texttt{ocean.output}. $N_{useless}$ must, of course, be as small as possible to limit the waste of ressources. A warning is issued in  \texttt{ocean.output} if $N_{useless}$ is not zero. Note that non-zero value of $N_{useless}$ is uselly required when using AGRIF as, up to now, the parent grid and each of the child grids must use all the $N_{mpi}$ processes.
338
339If the domain decomposition is automatically defined (when \np{jpni} and \np{jpnj} are < 1), the decomposition chosen by the model will minimise the sub-domain size (defined as $max_{all domains}(jpi \times jpj)$) and maximize the number of eliminated land subdomains. This means that no other domain decomposition (a set of \np{jpni} and \np{jpnj} values) will use less processes than $(jpni  \times  jpnj - N_{land})$ and get a smaller subdomain size.
340In order to specify $N_{mpi}$ properly (minimize $N_{useless}$), you must run the model once with \np{ln\_list} activated. In this case, the model will start the initialisation phase, print the list of optimum decompositions ($N_{mpi}$, \np{jpni} and \np{jpnj}) in \texttt{ocean.output} and directly abort. The maximum value of $N_{mpi}$ tested in this list is given by $max(N_{MPI\_tasks}, jpni \times jpnj)$. For example, run the model on 40 nodes with ln\_list activated and $jpni = 10000$ and $jpnj = 1$, will print the list of optimum domains decomposition from 1 to about 10000.
341
342Processors are numbered from 0 to $N_{mpi} - 1$. Subdomains containning some ocean points are numbered first from 0 to $jpni * jpnj - N_{land} -1$. The remaining $N_{useless}$ land subdomains are numbered next, which means that, for a given (\np{jpni}, \np{jpnj}), the numbers attributed to he ocean subdomains do not vary with $N_{useless}$.
343
344When land processors are eliminated, the value corresponding to these locations in the model output files is undefined. \np{ln\_mskland} must be activated in order avoid Not a Number values in output files. Note that it is better to not eliminate land processors when creating a meshmask file (\ie\ when setting a non-zero value to \np{nn\_msh}).
345
346%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
347\begin{figure}[!ht]
348  \centering
349  \includegraphics[width=0.66\textwidth]{Fig_mppini2}
350  \caption[Atlantic domain defined for the CLIPPER projet]{
351    Example of Atlantic domain defined for the CLIPPER projet.
352    Initial grid is composed of 773 x 1236 horizontal points.
353    (a) the domain is split onto 9 $times$ 20 subdomains (jpni=9, jpnj=20).
354    52 subdomains are land areas.
355    (b) 52 subdomains are eliminated (white rectangles) and
356    the resulting number of processors really used during the computation is jpnij=128.}
357  \label{fig:LBC_mppini2}
358\end{figure}
359%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
360
361
362% ====================================================================
363% Unstructured open boundaries BDY
364% ====================================================================
365\section{Unstructured open boundary conditions (BDY)}
366\label{sec:LBC_bdy}
367
368%-----------------------------------------nambdy--------------------------------------------
369
370\begin{listing}
371  \nlst{nambdy}
372  \caption{\texttt{nambdy}}
373  \label{lst:nambdy}
374\end{listing}
375%-----------------------------------------------------------------------------------------------
376%-----------------------------------------nambdy_dta--------------------------------------------
377
378\begin{listing}
379  \nlst{nambdy_dta}
380  \caption{\texttt{nambdy\_dta}}
381  \label{lst:nambdy_dta}
382\end{listing}
383%-----------------------------------------------------------------------------------------------
384
385Options are defined through the \nam{bdy} and \nam{bdy\_dta} namelist variables.
386The BDY module is the core implementation of open boundary conditions for regional configurations on
387ocean temperature, salinity, barotropic-baroclinic velocities, ice-snow concentration, thicknesses, temperatures, salinity and melt ponds concentration and thickness.
388
389The BDY module was modelled on the OBC module (see \NEMO\ 3.4) and shares many features and
390a similar coding structure \citep{chanut_rpt05}.
391The specification of the location of the open boundary is completely flexible and
392allows any type of setup, from regular boundaries to irregular contour (it includes the possibility to set an open boundary able to follow an isobath).
393Boundary data files used with versions of \NEMO\ prior to Version 3.4 may need to be re-ordered to work with this version.
394See the section on the Input Boundary Data Files for details.
395
396%----------------------------------------------
397\subsection{Namelists}
398\label{subsec:LBC_bdy_namelist}
399
400The BDY module is activated by setting \np{ln\_bdy}\forcode{=.true.} .
401It is possible to define more than one boundary ``set'' and apply different boundary conditions to each set.
402The number of boundary sets is defined by \np{nb\_bdy}.
403Each boundary set can be either defined as a series of straight line segments directly in the namelist
404(\np{ln\_coords\_file}\forcode{=.false.}, and a namelist block \nam{bdy\_index} must be included for each set) or read in from a file (\np{ln\_coords\_file}\forcode{=.true.}, and a ``\ifile{coordinates.bdy}'' file must be provided).
405The coordinates.bdy file is analagous to the usual \NEMO\ ``\ifile{coordinates}'' file.
406In the example above, there are two boundary sets, the first of which is defined via a file and
407the second is defined in the namelist.
408For more details of the definition of the boundary geometry see section \autoref{subsec:LBC_bdy_geometry}.
409
410For each boundary set a boundary condition has to be chosen for the barotropic solution
411(``u2d'':sea-surface height and barotropic velocities), for the baroclinic velocities (``u3d''),
412for the active tracers \footnote{The BDY module does not deal with passive tracers at this version} (``tra''), and for sea-ice (``ice'').
413For each set of variables one has to choose an algorithm and the boundary data (set resp. by \np{cn\_tra} and \np{nn\_tra\_dta} for tracers).\\
414
415The choice of algorithm is currently as follows:
416
417\begin{description}
418\item[\forcode{'none'}:] No boundary condition applied.
419  So the solution will ``see'' the land points around the edge of the edge of the domain.
420\item[\forcode{'specified'}:] Specified boundary condition applied (only available for baroclinic velocity and tracer variables).
421\item[\forcode{'neumann'}:] Value at the boundary are duplicated (No gradient). Only available for baroclinic velocity and tracer variables.
422\item[\forcode{'frs'}:] Flow Relaxation Scheme (FRS) available for all variables.
423\item[\forcode{'Orlanski'}:] Orlanski radiation scheme (fully oblique) for barotropic, baroclinic and tracer variables.
424\item[\forcode{'Orlanski_npo'}:] Orlanski radiation scheme for barotropic, baroclinic and tracer variables.
425\item[\forcode{'flather'}:] Flather radiation scheme for the barotropic variables only.
426\end{description}
427
428The boundary data is either set to initial conditions
429(\np{nn\_tra\_dta}\forcode{=0}) or forced with external data from a file (\np{nn\_tra\_dta}\forcode{=1}).
430In case the 3d velocity data contain the total velocity (ie, baroclinic and barotropic velocity),
431the bdy code can derived baroclinic and barotropic velocities by setting \np{ln\_full\_vel}\forcode{=.true.}
432For the barotropic solution there is also the option to use tidal harmonic forcing either by
433itself (\np{nn\_dyn2d\_dta}\forcode{=2}) or in addition to other external data (\np{nn\_dyn2d\_dta}\forcode{=3}).\\
434If not set to initial conditions, sea-ice salinity, temperatures and melt ponds data at the boundary can either be read in a file or defined as constant (by \np{rn\_ice\_sal}, \np{rn\_ice\_tem}, \np{rn\_ice\_apnd}, \np{rn\_ice\_hpnd}). Ice age is constant and defined by \np{rn\_ice\_age}.
435
436If external boundary data is required then the \nam{bdy\_dta} namelist must be defined.
437One \nam{bdy\_dta} namelist is required for each boundary set, adopting the same order of indexes in which the boundary sets are defined in nambdy.
438In the example given, two boundary sets have been defined. The first one is reading data file in the \nam{bdy\_dta} namelist shown above
439and the second one is using data from intial condition (no namelist block needed).
440The boundary data is read in using the fldread module,
441so the \nam{bdy\_dta} namelist is in the format required for fldread.
442For each required variable, the filename, the frequency of the files and
443the frequency of the data in the files are given.
444Also whether or not time-interpolation is required and whether the data is climatological (time-cyclic) data.
445For sea-ice salinity, temperatures and melt ponds, reading the files are skipped and constant values are used if filenames are defined as {'NOT USED'}.\\
446
447There is currently an option to vertically interpolate the open boundary data onto the native grid at run-time.
448If \np{nn\_bdy\_jpk}$<-1$, it is assumed that the lateral boundary data are already on the native grid.
449However, if \np{nn\_bdy\_jpk} is set to the number of vertical levels present in the boundary data,
450a bilinear interpolation onto the native grid will be triggered at runtime.
451For this to be successful the additional variables: $gdept$, $gdepu$, $gdepv$, $e3t$, $e3u$ and $e3v$, are required to be present in the lateral boundary files.
452These correspond to the depths and scale factors of the input data,
453the latter used to make any adjustment to the velocity fields due to differences in the total water depths between the two vertical grids.\\
454
455In the example of given namelists, two boundary sets are defined.
456The first set is defined via a file and applies FRS conditions to temperature and salinity and
457Flather conditions to the barotropic variables. No condition specified for the baroclinic velocity and sea-ice.
458External data is provided in daily files (from a large-scale model).
459Tidal harmonic forcing is also used.
460The second set is defined in a namelist.
461FRS conditions are applied on temperature and salinity and climatological data is read from initial condition files.
462
463%----------------------------------------------
464\subsection{Flow relaxation scheme}
465\label{subsec:LBC_bdy_FRS_scheme}
466
467The Flow Relaxation Scheme (FRS) \citep{davies_QJRMS76,engedahl_T95},
468applies a simple relaxation of the model fields to externally-specified values over
469a zone next to the edge of the model domain.
470Given a model prognostic variable $\Phi$
471\[
472  % \label{eq:LBC_bdy_frs1}
473  \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
474\]
475where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified external field,
476$d$ gives the discrete distance from the model boundary and
477$\alpha$ is a parameter that varies from $1$ at $d=1$ to a small value at $d=N$.
478It can be shown that this scheme is equivalent to adding a relaxation term to
479the prognostic equation for $\Phi$ of the form:
480\[
481  % \label{eq:LBC_bdy_frs2}
482  -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
483\]
484where the relaxation time scale $\tau$ is given by a function of $\alpha$ and the model time step $\Delta t$:
485\[
486  % \label{eq:LBC_bdy_frs3}
487  \tau = \frac{1-\alpha}{\alpha\,\rdt
488\]
489Thus the model solution is completely prescribed by the external conditions at the edge of the model domain and
490is relaxed towards the external conditions over the rest of the FRS zone.
491The application of a relaxation zone helps to prevent spurious reflection of
492outgoing signals from the model boundary.
493
494The function $\alpha$ is specified as a $tanh$ function:
495\[
496  % \label{eq:LBC_bdy_frs4}
497  \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
498\]
499The width of the FRS zone is specified in the namelist as \np{nn\_rimwidth}.
500This is typically set to a value between 8 and 10.
501
502%----------------------------------------------
503\subsection{Flather radiation scheme}
504\label{subsec:LBC_bdy_flather_scheme}
505
506The \citet{flather_JPO94} scheme is a radiation condition on the normal,
507depth-mean transport across the open boundary.
508It takes the form
509\begin{equation}
510  \label{eq:LBC_bdy_fla1}
511  U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
512\end{equation}
513where $U$ is the depth-mean velocity normal to the boundary and $\eta$ is the sea surface height,
514both from the model.
515The subscript $e$ indicates the same fields from external sources.
516The speed of external gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the water column.
517The depth-mean normal velocity along the edge of the model domain is set equal to
518the external depth-mean normal velocity,
519plus a correction term that allows gravity waves generated internally to exit the model boundary.
520Note that the sea-surface height gradient in \autoref{eq:LBC_bdy_fla1} is a spatial gradient across the model boundary,
521so that $\eta_{e}$ is defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the $T$ points with $nbr=2$.
522$U$ and $U_{e}$ are defined on the $U$ or $V$ points with $nbr=1$, \ie\ between the two $T$ grid points.
523
524%----------------------------------------------
525\subsection{Orlanski radiation scheme}
526\label{subsec:LBC_bdy_orlanski_scheme}
527
528The Orlanski scheme is based on the algorithm described by \citep{marchesiello.mcwilliams.ea_OM01}, hereafter MMS.
529
530The adaptive Orlanski condition solves a wave plus relaxation equation at the boundary:
531\begin{equation}
532  \label{eq:LBC_wave_continuous}
533  \frac{\partial\phi}{\partial t} + c_x \frac{\partial\phi}{\partial x} + c_y \frac{\partial\phi}{\partial y} =
534  -\frac{1}{\tau}(\phi - \phi^{ext})
535\end{equation}
536
537where $\phi$ is the model field, $x$ and $y$ refer to the normal and tangential directions to the boundary respectively, and the phase
538velocities are diagnosed from the model fields as:
539
540\begin{equation}
541  \label{eq:LBC_cx}
542  c_x = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial x}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}
543\end{equation}
544\begin{equation}
545  \label{eq:LBC_cy}
546  c_y = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial y}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}
547\end{equation}
548
549(As noted by MMS, this is a circular diagnosis of the phase speeds which only makes sense on a discrete grid).
550Equation (\autoref{eq:LBC_wave_continuous}) is defined adaptively depending on the sign of the phase velocity normal to the boundary $c_x$.
551For $c_x$ outward, we have
552
553\begin{equation}
554\tau = \tau_{out}
555\end{equation}
556
557For $c_x$ inward, the radiation equation is not applied:
558
559\begin{equation}
560  \label{eq:LBC_tau_in}
561  \tau = \tau_{in}\,\,\,;\,\,\, c_x = c_y = 0
562\end{equation}
563
564Generally the relaxation time scale at inward propagation points (\np{rn\_time\_dmp}) is set much shorter than the time scale at outward propagation
565points (\np{rn\_time\_dmp\_out}) so that the solution is constrained more strongly by the external data at inward propagation points.
566See \autoref{subsec:LBC_bdy_relaxation} for detailed on the spatial shape of the scaling.\\
567The ``normal propagation of oblique radiation'' or NPO approximation (called \forcode{'orlanski_npo'}) involves assuming
568that $c_y$ is zero in equation (\autoref{eq:LBC_wave_continuous}), but including
569this term in the denominator of equation (\autoref{eq:LBC_cx}). Both versions of the scheme are options in BDY. Equations
570(\autoref{eq:LBC_wave_continuous}) - (\autoref{eq:LBC_tau_in}) correspond to equations (13) - (15) and (2) - (3) in MMS.\\
571
572%----------------------------------------------
573\subsection{Relaxation at the boundary}
574\label{subsec:LBC_bdy_relaxation}
575
576In addition to a specific boundary condition specified as \np{cn\_tra} and \np{cn\_dyn3d}, relaxation on baroclinic velocities and tracers variables are available.
577It is control by the namelist parameter \np{ln\_tra\_dmp} and \np{ln\_dyn3d\_dmp} for each boundary set.
578
579The relaxation time scale value (\np{rn\_time\_dmp} and \np{rn\_time\_dmp\_out}, $\tau$) are defined at the boundaries itself.
580This time scale ($\alpha$) is weighted by the distance ($d$) from the boundary over \np{nn\_rimwidth} cells ($N$):
581
582\[
583  \alpha = \frac{1}{\tau}(\frac{N+1-d}{N})^2,       \quad d=1,N
584\]
585
586The same scaling is applied in the Orlanski damping.
587
588%----------------------------------------------
589\subsection{Boundary geometry}
590\label{subsec:LBC_bdy_geometry}
591
592Each open boundary set is defined as a list of points.
593The information is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$ structure.
594The $nbi$ and $nbj$ arrays define the local $(i,j)$ indexes of each point in the boundary zone and
595the $nbr$ array defines the discrete distance from the boundary: $nbr=1$ means that
596the boundary point is next to the edge of the model domain, while $nbr>1$ means that
597the boundary point is increasingly further away from the edge of the model domain.
598A set of $nbi$, $nbj$, and $nbr$ arrays is defined for each of the $T$, $U$ and $V$ grids.
599\autoref{fig:LBC_bdy_geom} shows an example of an irregular boundary.
600
601The boundary geometry for each set may be defined in a namelist nambdy\_index or
602by reading in a ``\ifile{coordinates.bdy}'' file.
603The nambdy\_index namelist defines a series of straight-line segments for north, east, south and west boundaries.
604One nambdy\_index namelist block is needed for each boundary condition defined by indexes.
605For the northern boundary, \texttt{nbdysegn} gives the number of segments,
606\jp{jpjnob} gives the $j$ index for each segment and \jp{jpindt} and
607\jp{jpinft} give the start and end $i$ indices for each segment with similar for the other boundaries.
608These segments define a list of $T$ grid points along the outermost row of the boundary ($nbr\,=\, 1$).
609The code deduces the $U$ and $V$ points and also the points for $nbr\,>\, 1$ if \np{nn\_rimwidth}\forcode{>1}.
610
611The boundary geometry may also be defined from a ``\ifile{coordinates.bdy}'' file.
612\autoref{fig:LBC_nc_header} gives an example of the header information from such a file, based on the description of geometrical setup given above.
613The file should contain the index arrays for each of the $T$, $U$ and $V$ grids.
614The arrays must be in order of increasing $nbr$.
615Note that the $nbi$, $nbj$ values in the file are global values and are converted to local values in the code.
616Typically this file will be used to generate external boundary data via interpolation and so
617will also contain the latitudes and longitudes of each point as shown.
618However, this is not necessary to run the model.
619
620For some choices of irregular boundary the model domain may contain areas of ocean which
621are not part of the computational domain.
622For example, if an open boundary is defined along an isobath, say at the shelf break,
623then the areas of ocean outside of this boundary will need to be masked out.
624This can be done by reading a mask file defined as \np{cn\_mask\_file} in the nam\_bdy namelist.
625Only one mask file is used even if multiple boundary sets are defined.
626
627%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
628\begin{figure}[!t]
629  \centering
630  \includegraphics[width=0.66\textwidth]{Fig_LBC_bdy_geom}
631  \caption[Geometry of unstructured open boundary]{Example of geometry of unstructured open boundary}
632  \label{fig:LBC_bdy_geom}
633\end{figure}
634%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
635
636%----------------------------------------------
637\subsection{Input boundary data files}
638\label{subsec:LBC_bdy_data}
639
640The data files contain the data arrays in the order in which the points are defined in the $nbi$ and $nbj$ arrays.
641The data arrays are dimensioned on:
642a time dimension;
643$xb$ which is the index of the boundary data point in the horizontal;
644and $yb$ which is a degenerate dimension of 1 to enable the file to be read by the standard \NEMO\ I/O routines.
645The 3D fields also have a depth dimension.
646
647From Version 3.4 there are new restrictions on the order in which the boundary points are defined
648(and therefore restrictions on the order of the data in the file).
649In particular:
650
651\begin{enumerate}
652\item The data points must be in order of increasing $nbr$,
653  ie. all the $nbr=1$ points, then all the $nbr=2$ points etc.
654\item All the data for a particular boundary set must be in the same order.
655  (Prior to 3.4 it was possible to define barotropic data in a different order to
656  the data for tracers and baroclinic velocities).
657\end{enumerate}
658
659These restrictions mean that data files used with versions of the
660model prior to Version 3.4 may not work with Version 3.4 onwards.
661A \fortran\ utility {\itshape bdy\_reorder} exists in the TOOLS directory which
662will re-order the data in old BDY data files.
663
664%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
665\begin{figure}[!t]
666  \centering
667  \includegraphics[width=0.66\textwidth]{Fig_LBC_nc_header}
668  \caption[Header for a \protect\ifile{coordinates.bdy} file]{
669    Example of the header for a \protect\ifile{coordinates.bdy} file}
670  \label{fig:LBC_nc_header}
671\end{figure}
672%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
673
674%----------------------------------------------
675\subsection{Volume correction}
676\label{subsec:LBC_bdy_vol_corr}
677
678There is an option to force the total volume in the regional model to be constant.
679This is controlled  by the \np{ln\_vol} parameter in the namelist.
680A value of \np{ln\_vol}\forcode{=.false.} indicates that this option is not used.
681Two options to control the volume are available (\np{nn\_volctl}).
682If \np{nn\_volctl}\forcode{=0} then a correction is applied to the normal barotropic velocities around the boundary at
683each timestep to ensure that the integrated volume flow through the boundary is zero.
684If \np{nn\_volctl}\forcode{=1} then the calculation of the volume change on
685the timestep includes the change due to the freshwater flux across the surface and
686the correction velocity corrects for this as well.
687
688If more than one boundary set is used then volume correction is
689applied to all boundaries at once.
690
691%----------------------------------------------
692\subsection{Tidal harmonic forcing}
693\label{subsec:LBC_bdy_tides}
694
695%-----------------------------------------nambdy_tide--------------------------------------------
696
697\begin{listing}
698  \nlst{nambdy_tide}
699  \caption{\texttt{nambdy\_tide}}
700  \label{lst:nambdy_tide}
701\end{listing}
702%-----------------------------------------------------------------------------------------------
703
704Tidal forcing at open boundaries requires the activation of surface
705tides (i.e., in \nam{\_tide}, \np{ln\_tide} needs to be set to
706\forcode{.true.} and the required constituents need to be activated by
707including their names in the \np{clname} array; see
708\autoref{sec:SBC_tide}). Specific options related to the reading in of
709the complex harmonic amplitudes of elevation (SSH) and barotropic
710velocity (u,v) at open boundaries are defined through the
711\nam{bdy\_tide} namelist parameters.\\
712
713The tidal harmonic data at open boundaries can be specified in two
714different ways, either on a two-dimensional grid covering the entire
715model domain or along open boundary segments; these two variants can
716be selected by setting \np{ln\_bdytide\_2ddta } to \forcode{.true.} or
717\forcode{.false.}, respectively. In either case, the real and
718imaginary parts of SSH and the two barotropic velocity components for
719each activated tidal constituent \textit{tcname} have to be provided
720separately: when two-dimensional data is used, variables
721\textit{tcname\_z1} and \textit{tcname\_z2} for real and imaginary SSH,
722respectively, are expected in input file \np{filtide} with suffix
723\ifile{\_grid\_T}, variables \textit{tcname\_u1} and
724\textit{tcname\_u2} for real and imaginary u, respectively, are
725expected in input file \np{filtide} with suffix \ifile{\_grid\_U}, and
726\textit{tcname\_v1} and \textit{tcname\_v2} for real and imaginary v,
727respectively, are expected in input file \np{filtide} with suffix
728\ifile{\_grid\_V}; when data along open boundary segments is used,
729variables \textit{z1} and \textit{z2} (real and imaginary part of SSH)
730are expected to be available from file \np{filtide} with suffix
731\ifile{tcname\_grid\_T}, variables \textit{u1} and \textit{u2} (real
732and imaginary part of u) are expected to be available from file
733\np{filtide} with suffix \ifile{tcname\_grid\_U}, and variables
734\textit{v1} and \textit{v2} (real and imaginary part of v) are
735expected to be available from file \np{filtide} with suffix
736\ifile{tcname\_grid\_V}. If \np{ln\_bdytide\_conj} is set to
737\forcode{.true.}, the data is expected to be in complex conjugate
738form.
739
740Note that the barotropic velocity components are assumed to be defined
741on the native model grid and should be rotated accordingly when they
742are converted from their definition on a different source grid. To do
743so, the u, v amplitudes and phases can be converted into tidal
744ellipses, the grid rotation added to the ellipse inclination, and then
745converted back (care should be taken regarding conventions of the
746direction of rotation). %, e.g. anticlockwise or clockwise.
747
748\biblio
749
750\pindex
751
752\end{document}
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