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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\nlst{namlbc}
25%--------------------------------------------------------------------------------------------------------------
26
27%The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt
28%(no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip).
29%They are handled automatically by the mask system (see \autoref{subsec:DOM_msk}).
30
31%OPA allows land and topography grid points in the computational domain due to the presence of continents or islands,
32%and includes the use of a full or partial step representation of bottom topography.
33%The computation is performed over the whole domain, \ie\ we do not try to restrict the computation to ocean-only points.
34%This choice has two motivations.
35%Firstly, working on ocean only grid points overloads the code and harms the code readability.
36%Secondly, and more importantly, it drastically reduces the vector portion of the computation,
37%leading to a dramatic increase of CPU time requirement on vector computers.
38%The current section describes how the masking affects the computation of the various terms of the equations
39%with respect to the boundary condition at solid walls.
40%The process of defining which areas are to be masked is described in \autoref{subsec:DOM_msk}.
41
42Options are defined through the \nam{lbc} namelist variables.
43The discrete representation of a domain with complex boundaries (coastlines and bottom topography) leads to
44arrays that include large portions where a computation is not required as the model variables remain at zero.
45Nevertheless, vectorial supercomputers are far more efficient when computing over a whole array,
46and the readability of a code is greatly improved when boundary conditions are applied in
47an automatic way rather than by a specific computation before or after each computational loop.
48An efficient way to work over the whole domain while specifying the boundary conditions,
49is to use multiplication by mask arrays in the computation.
50A mask array is a matrix whose elements are $1$ in the ocean domain and $0$ elsewhere.
51A simple multiplication of a variable by its own mask ensures that it will remain zero over land areas.
52Since most of the boundary conditions consist of a zero flux across the solid boundaries,
53they can be simply applied by multiplying variables by the correct mask arrays,
54\ie\ the mask array of the grid point where the flux is evaluated.
55For example, the heat flux in the \textbf{i}-direction is evaluated at $u$-points.
56Evaluating this quantity as,
57
58\[
59  % \label{eq:LBC_aaaa}
60  \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT}
61  }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u
62\]
63(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is zero inside land and
64at the boundaries, since mask$_{u}$ is zero at solid boundaries which in this case are defined at $u$-points
65(normal velocity $u$ remains zero at the coast) (\autoref{fig:LBC_uv}).
66
67%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
68\begin{figure}[!t]
69  \begin{center}
70    \includegraphics[width=\textwidth]{Fig_LBC_uv}
71    \caption{
72      \protect\label{fig:LBC_uv}
73      Lateral boundary (thick line) at T-level.
74      The velocity normal to the boundary is set to zero.
75    }
76  \end{center}
77\end{figure}
78%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
79
80For momentum the situation is a bit more complex as two boundary conditions must be provided along the coast
81(one each for the normal and tangential velocities).
82The boundary of the ocean in the C-grid is defined by the velocity-faces.
83For example, at a given $T$-level,
84the lateral boundary (a coastline or an intersection with the bottom topography) is made of
85segments joining $f$-points, and normal velocity points are located between two $f-$points (\autoref{fig:LBC_uv}).
86The boundary condition on the normal velocity (no flux through solid boundaries)
87can thus be easily implemented using the mask system.
88The boundary condition on the tangential velocity requires a more specific treatment.
89This boundary condition influences the relative vorticity and momentum diffusive trends,
90and is required in order to compute the vorticity at the coast.
91Four different types of lateral boundary condition are available,
92controlled by the value of the \np{rn\_shlat} namelist parameter
93(The value of the mask$_{f}$ array along the coastline is set equal to this parameter).
94These are:
95
96%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
97\begin{figure}[!p]
98  \begin{center}
99    \includegraphics[width=\textwidth]{Fig_LBC_shlat}
100    \caption{
101      \protect\label{fig:LBC_shlat}
102      lateral boundary condition
103      (a) free-slip ($rn\_shlat=0$);
104      (b) no-slip ($rn\_shlat=2$);
105      (c) "partial" free-slip ($0<rn\_shlat<2$) and
106      (d) "strong" no-slip ($2<rn\_shlat$).
107      Implied "ghost" velocity inside land area is display in grey.
108    }
109  \end{center}
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  \begin{center}
210    \includegraphics[width=\textwidth]{Fig_LBC_jperio}
211    \caption{
212      \protect\label{fig:LBC_jperio}
213      setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.
214    }
215  \end{center}
216\end{figure}
217%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
218
219% -------------------------------------------------------------------------------------------------------------
220%        North fold (\textit{jperio = 3 }to $6)$
221% -------------------------------------------------------------------------------------------------------------
222\subsection[North-fold (\forcode{jperio=[3-6]})]
223{North-fold (\protect\jp{jperio}\forcode{=[3-6]})}
224\label{subsec:LBC_north_fold}
225
226The north fold boundary condition has been introduced in order to handle the north boundary of
227a three-polar ORCA grid.
228Such a grid has two poles in the northern hemisphere (\autoref{fig:CFGS_ORCA_msh},
229and thus requires a specific treatment illustrated in \autoref{fig:LBC_North_Fold_T}.
230Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition.
231
232%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
233\begin{figure}[!t]
234  \begin{center}
235    \includegraphics[width=\textwidth]{Fig_North_Fold_T}
236    \caption{
237      \protect\label{fig:LBC_North_Fold_T}
238      North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$),
239      as used in ORCA 2, 1/4, and 1/12.
240      Pink shaded area corresponds to the inner domain mask (see text).
241    }
242  \end{center}
243\end{figure}
244%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
245
246% ====================================================================
247% Exchange with neighbouring processors
248% ====================================================================
249\section[Exchange with neighbouring processors (\textit{lbclnk.F90}, \textit{lib\_mpp.F90})]
250{Exchange with neighbouring processors (\protect\mdl{lbclnk}, \protect\mdl{lib\_mpp})}
251\label{sec:LBC_mpp}
252
253%-----------------------------------------nammpp--------------------------------------------
254
255\nlst{nammpp}
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  \begin{center}
294    \includegraphics[width=\textwidth]{Fig_mpp}
295    \caption{
296      \protect\label{fig:LBC_mpp}
297      Positioning of a sub-domain when massively parallel processing is used.
298    }
299  \end{center}
300\end{figure}
301%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
302
303In \NEMO, the splitting is regular and arithmetic.
304The total number of subdomains corresponds to the number of MPI processes allocated to \NEMO\ when the model is launched
305(\ie\ mpirun -np x ./nemo will automatically give x subdomains).
306The i-axis is divided by \np{jpni} and the j-axis by \np{jpnj}.
307These parameters are defined in \nam{mpp} namelist.
308If \np{jpni} and \np{jpnj} are < 1, they will be automatically redefined in the code to give the best domain decomposition
309(see bellow).
310
311Each processor is independent and without message passing or synchronous process, programs run alone and access just its own local memory.
312For this reason,
313the main model dimensions are now the local dimensions of the subdomain (pencil) that are named \jp{jpi}, \jp{jpj}, \jp{jpk}.
314These dimensions include the internal domain and the overlapping rows.
315The number of rows to exchange (known as the halo) is usually set to one (nn\_hls=1, in \mdl{par\_oce},
316and must be kept to one until further notice).
317The whole domain dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}.
318The relationship between the whole domain and a sub-domain is:
319\begin{gather*}
320  jpi = ( jpiglo-2\times nn\_hls + (jpni-1) ) / jpni + 2\times nn\_hls \\
321  jpj = ( jpjglo-2\times nn\_hls + (jpnj-1) ) / jpnj + 2\times nn\_hls
322\end{gather*}
323
324One 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.
325
326An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
327a global array (whole domain) by the relationship:
328\[
329  % \label{eq:LBC_nimpp}
330  T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
331\]
332with $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.
333
334The 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).\\
335
336The \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:
337\[
338  N_{mpi} = jpni \times jpnj - N_{land} + N_{useless}
339\]
340$N_{land}$ 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.
341
342If 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.
343In 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.
344
345Processors 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 $N_{useless}$.
346
347When 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}).
348
349%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
350\begin{figure}[!ht]
351  \begin{center}
352    \includegraphics[width=\textwidth]{Fig_mppini2}
353    \caption[Atlantic domain]{
354      \protect\label{fig:LBC_mppini2}
355      Example of Atlantic domain defined for the CLIPPER projet.
356      Initial grid is composed of 773 x 1236 horizontal points.
357      (a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
358      52 subdomains are land areas.
359      (b) 52 subdomains are eliminated (white rectangles) and
360      the resulting number of processors really used during the computation is jpnij=128.
361    }
362  \end{center}
363\end{figure}
364%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
365
366
367% ====================================================================
368% Unstructured open boundaries BDY
369% ====================================================================
370\section{Unstructured open boundary conditions (BDY)}
371\label{sec:LBC_bdy}
372
373%-----------------------------------------nambdy--------------------------------------------
374
375\nlst{nambdy}
376%-----------------------------------------------------------------------------------------------
377%-----------------------------------------nambdy_dta--------------------------------------------
378
379\nlst{nambdy_dta}
380%-----------------------------------------------------------------------------------------------
381
382Options are defined through the \nam{bdy} and \nam{bdy\_dta} namelist variables.
383The BDY module is the core implementation of open boundary conditions for regional configurations on
384ocean temperature, salinity, barotropic-baroclinic velocities, ice-snow concentration, thicknesses, temperatures, salinity and melt ponds concentration and thickness.
385
386The BDY module was modelled on the OBC module (see \NEMO\ 3.4) and shares many features and
387a similar coding structure \citep{chanut_rpt05}.
388The specification of the location of the open boundary is completely flexible and
389allows any type of setup, from regular boundaries to irregular contour (it includes the possibility to set an open boundary able to follow an isobath).
390Boundary data files used with versions of \NEMO\ prior to Version 3.4 may need to be re-ordered to work with this version.
391See the section on the Input Boundary Data Files for details.
392
393%----------------------------------------------
394\subsection{Namelists}
395\label{subsec:LBC_bdy_namelist}
396
397The BDY module is activated by setting \np{ln\_bdy}\forcode{=.true.} .
398It is possible to define more than one boundary ``set'' and apply different boundary conditions to each set.
399The number of boundary sets is defined by \np{nb\_bdy}.
400Each boundary set can be either defined as a series of straight line segments directly in the namelist
401(\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).
402The coordinates.bdy file is analagous to the usual \NEMO\ ``\ifile{coordinates}'' file.
403In the example above, there are two boundary sets, the first of which is defined via a file and
404the second is defined in the namelist.
405For more details of the definition of the boundary geometry see section \autoref{subsec:LBC_bdy_geometry}.
406
407For each boundary set a boundary condition has to be chosen for the barotropic solution
408(``u2d'':sea-surface height and barotropic velocities), for the baroclinic velocities (``u3d''),
409for the active tracers \footnote{The BDY module does not deal with passive tracers at this version} (``tra''), and for sea-ice (``ice'').
410For 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).\\
411
412The choice of algorithm is currently as follows:
413
414\begin{description}
415\item[\forcode{'none'}:] No boundary condition applied.
416  So the solution will ``see'' the land points around the edge of the edge of the domain.
417\item[\forcode{'specified'}:] Specified boundary condition applied (only available for baroclinic velocity and tracer variables).
418\item[\forcode{'neumann'}:] Value at the boundary are duplicated (No gradient). Only available for baroclinic velocity and tracer variables.
419\item[\forcode{'frs'}:] Flow Relaxation Scheme (FRS) available for all variables.
420\item[\forcode{'Orlanski'}:] Orlanski radiation scheme (fully oblique) for barotropic, baroclinic and tracer variables.
421\item[\forcode{'Orlanski_npo'}:] Orlanski radiation scheme for barotropic, baroclinic and tracer variables.
422\item[\forcode{'flather'}:] Flather radiation scheme for the barotropic variables only.
423\end{description}
424
425The boundary data is either set to initial conditions
426(\np{nn\_tra\_dta}\forcode{=0}) or forced with external data from a file (\np{nn\_tra\_dta}\forcode{=1}).
427In case the 3d velocity data contain the total velocity (ie, baroclinic and barotropic velocity),
428the bdy code can derived baroclinic and barotropic velocities by setting \np{ln\_full\_vel}\forcode{=.true.}
429For the barotropic solution there is also the option to use tidal harmonic forcing either by
430itself (\np{nn\_dyn2d\_dta}\forcode{=2}) or in addition to other external data (\np{nn\_dyn2d\_dta}\forcode{=3}).\\
431If 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}.
432
433If external boundary data is required then the \nam{bdy\_dta} namelist must be defined.
434One \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.
435In the example given, two boundary sets have been defined. The first one is reading data file in the \nam{bdy\_dta} namelist shown above
436and the second one is using data from intial condition (no namelist block needed).
437The boundary data is read in using the fldread module,
438so the \nam{bdy\_dta} namelist is in the format required for fldread.
439For each required variable, the filename, the frequency of the files and
440the frequency of the data in the files are given.
441Also whether or not time-interpolation is required and whether the data is climatological (time-cyclic) data.
442For sea-ice salinity, temperatures and melt ponds, reading the files are skipped and constant values are used if filenames are defined as {'NOT USED'}.\\
443
444There is currently an option to vertically interpolate the open boundary data onto the native grid at run-time.
445If \np{nn\_bdy\_jpk}$<-1$, it is assumed that the lateral boundary data are already on the native grid.
446However, if \np{nn\_bdy\_jpk} is set to the number of vertical levels present in the boundary data,
447a bilinear interpolation onto the native grid will be triggered at runtime.
448For this to be successful the additional variables: $gdept$, $gdepu$, $gdepv$, $e3t$, $e3u$ and $e3v$, are required to be present in the lateral boundary files.
449These correspond to the depths and scale factors of the input data,
450the latter used to make any adjustment to the velocity fields due to differences in the total water depths between the two vertical grids.\\
451
452In the example of given namelists, two boundary sets are defined.
453The first set is defined via a file and applies FRS conditions to temperature and salinity and
454Flather conditions to the barotropic variables. No condition specified for the baroclinic velocity and sea-ice.
455External data is provided in daily files (from a large-scale model).
456Tidal harmonic forcing is also used.
457The second set is defined in a namelist.
458FRS conditions are applied on temperature and salinity and climatological data is read from initial condition files.
459
460%----------------------------------------------
461\subsection{Flow relaxation scheme}
462\label{subsec:LBC_bdy_FRS_scheme}
463
464The Flow Relaxation Scheme (FRS) \citep{davies_QJRMS76,engedahl_T95},
465applies a simple relaxation of the model fields to externally-specified values over
466a zone next to the edge of the model domain.
467Given a model prognostic variable $\Phi$
468\[
469  % \label{eq:LBC_bdy_frs1}
470  \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
471\]
472where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified external field,
473$d$ gives the discrete distance from the model boundary and
474$\alpha$ is a parameter that varies from $1$ at $d=1$ to a small value at $d=N$.
475It can be shown that this scheme is equivalent to adding a relaxation term to
476the prognostic equation for $\Phi$ of the form:
477\[
478  % \label{eq:LBC_bdy_frs2}
479  -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
480\]
481where the relaxation time scale $\tau$ is given by a function of $\alpha$ and the model time step $\Delta t$:
482\[
483  % \label{eq:LBC_bdy_frs3}
484  \tau = \frac{1-\alpha}{\alpha\,\rdt
485\]
486Thus the model solution is completely prescribed by the external conditions at the edge of the model domain and
487is relaxed towards the external conditions over the rest of the FRS zone.
488The application of a relaxation zone helps to prevent spurious reflection of
489outgoing signals from the model boundary.
490
491The function $\alpha$ is specified as a $tanh$ function:
492\[
493  % \label{eq:LBC_bdy_frs4}
494  \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
495\]
496The width of the FRS zone is specified in the namelist as \np{nn\_rimwidth}.
497This is typically set to a value between 8 and 10.
498
499%----------------------------------------------
500\subsection{Flather radiation scheme}
501\label{subsec:LBC_bdy_flather_scheme}
502
503The \citet{flather_JPO94} scheme is a radiation condition on the normal,
504depth-mean transport across the open boundary.
505It takes the form
506\begin{equation}
507  \label{eq:LBC_bdy_fla1}
508  U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
509\end{equation}
510where $U$ is the depth-mean velocity normal to the boundary and $\eta$ is the sea surface height,
511both from the model.
512The subscript $e$ indicates the same fields from external sources.
513The speed of external gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the water column.
514The depth-mean normal velocity along the edge of the model domain is set equal to
515the external depth-mean normal velocity,
516plus a correction term that allows gravity waves generated internally to exit the model boundary.
517Note that the sea-surface height gradient in \autoref{eq:LBC_bdy_fla1} is a spatial gradient across the model boundary,
518so that $\eta_{e}$ is defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the $T$ points with $nbr=2$.
519$U$ and $U_{e}$ are defined on the $U$ or $V$ points with $nbr=1$, \ie\ between the two $T$ grid points.
520
521%----------------------------------------------
522\subsection{Orlanski radiation scheme}
523\label{subsec:LBC_bdy_orlanski_scheme}
524
525The Orlanski scheme is based on the algorithm described by \citep{marchesiello.mcwilliams.ea_OM01}, hereafter MMS.
526
527The adaptive Orlanski condition solves a wave plus relaxation equation at the boundary:
528\begin{equation}
529  \label{eq:LBC_wave_continuous}
530  \frac{\partial\phi}{\partial t} + c_x \frac{\partial\phi}{\partial x} + c_y \frac{\partial\phi}{\partial y} =
531  -\frac{1}{\tau}(\phi - \phi^{ext})
532\end{equation}
533
534where $\phi$ is the model field, $x$ and $y$ refer to the normal and tangential directions to the boundary respectively, and the phase
535velocities are diagnosed from the model fields as:
536
537\begin{equation}
538  \label{eq:LBC_cx}
539  c_x = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial x}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}
540\end{equation}
541\begin{equation}
542  \label{eq:LBC_cy}
543  c_y = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial y}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}
544\end{equation}
545
546(As noted by MMS, this is a circular diagnosis of the phase speeds which only makes sense on a discrete grid).
547Equation (\autoref{eq:LBC_wave_continuous}) is defined adaptively depending on the sign of the phase velocity normal to the boundary $c_x$.
548For $c_x$ outward, we have
549
550\begin{equation}
551\tau = \tau_{out}
552\end{equation}
553
554For $c_x$ inward, the radiation equation is not applied:
555
556\begin{equation}
557  \label{eq:LBC_tau_in}
558  \tau = \tau_{in}\,\,\,;\,\,\, c_x = c_y = 0
559\end{equation}
560
561Generally the relaxation time scale at inward propagation points (\np{rn\_time\_dmp}) is set much shorter than the time scale at outward propagation
562points (\np{rn\_time\_dmp\_out}) so that the solution is constrained more strongly by the external data at inward propagation points.
563See \autoref{subsec:LBC_bdy_relaxation} for detailed on the spatial shape of the scaling.\\
564The ``normal propagation of oblique radiation'' or NPO approximation (called \forcode{'orlanski_npo'}) involves assuming
565that $c_y$ is zero in equation (\autoref{eq:LBC_wave_continuous}), but including
566this term in the denominator of equation (\autoref{eq:LBC_cx}). Both versions of the scheme are options in BDY. Equations
567(\autoref{eq:LBC_wave_continuous}) - (\autoref{eq:LBC_tau_in}) correspond to equations (13) - (15) and (2) - (3) in MMS.\\
568
569%----------------------------------------------
570\subsection{Relaxation at the boundary}
571\label{subsec:LBC_bdy_relaxation}
572
573In addition to a specific boundary condition specified as \np{cn\_tra} and \np{cn\_dyn3d}, relaxation on baroclinic velocities and tracers variables are available.
574It is control by the namelist parameter \np{ln\_tra\_dmp} and \np{ln\_dyn3d\_dmp} for each boundary set.
575
576The relaxation time scale value (\np{rn\_time\_dmp} and \np{rn\_time\_dmp\_out}, $\tau$) are defined at the boundaries itself.
577This time scale ($\alpha$) is weighted by the distance ($d$) from the boundary over \np{nn\_rimwidth} cells ($N$):
578
579\[
580  \alpha = \frac{1}{\tau}(\frac{N+1-d}{N})^2,       \quad d=1,N
581\]
582
583The same scaling is applied in the Orlanski damping.
584
585%----------------------------------------------
586\subsection{Boundary geometry}
587\label{subsec:LBC_bdy_geometry}
588
589Each open boundary set is defined as a list of points.
590The information is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$ structure.
591The $nbi$ and $nbj$ arrays define the local $(i,j)$ indexes of each point in the boundary zone and
592the $nbr$ array defines the discrete distance from the boundary: $nbr=1$ means that
593the boundary point is next to the edge of the model domain, while $nbr>1$ means that
594the boundary point is increasingly further away from the edge of the model domain.
595A set of $nbi$, $nbj$, and $nbr$ arrays is defined for each of the $T$, $U$ and $V$ grids.
596Figure \autoref{fig:LBC_bdy_geom} shows an example of an irregular boundary.
597
598The boundary geometry for each set may be defined in a namelist nambdy\_index or
599by reading in a ``\ifile{coordinates.bdy}'' file.
600The nambdy\_index namelist defines a series of straight-line segments for north, east, south and west boundaries.
601One nambdy\_index namelist block is needed for each boundary condition defined by indexes.
602For the northern boundary, \texttt{nbdysegn} gives the number of segments,
603\jp{jpjnob} gives the $j$ index for each segment and \jp{jpindt} and
604\jp{jpinft} give the start and end $i$ indices for each segment with similar for the other boundaries.
605These segments define a list of $T$ grid points along the outermost row of the boundary ($nbr\,=\, 1$).
606The code deduces the $U$ and $V$ points and also the points for $nbr\,>\, 1$ if \np{nn\_rimwidth}\forcode{>1}.
607
608The boundary geometry may also be defined from a ``\ifile{coordinates.bdy}'' file.
609Figure \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.
610The file should contain the index arrays for each of the $T$, $U$ and $V$ grids.
611The arrays must be in order of increasing $nbr$.
612Note that the $nbi$, $nbj$ values in the file are global values and are converted to local values in the code.
613Typically this file will be used to generate external boundary data via interpolation and so
614will also contain the latitudes and longitudes of each point as shown.
615However, this is not necessary to run the model.
616
617For some choices of irregular boundary the model domain may contain areas of ocean which
618are not part of the computational domain.
619For example, if an open boundary is defined along an isobath, say at the shelf break,
620then the areas of ocean outside of this boundary will need to be masked out.
621This can be done by reading a mask file defined as \np{cn\_mask\_file} in the nam\_bdy namelist.
622Only one mask file is used even if multiple boundary sets are defined.
623
624%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
625\begin{figure}[!t]
626  \begin{center}
627    \includegraphics[width=\textwidth]{Fig_LBC_bdy_geom}
628    \caption {
629      \protect\label{fig:LBC_bdy_geom}
630      Example of geometry of unstructured open boundary
631    }
632  \end{center}
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  \begin{center}
667    \includegraphics[width=\textwidth]{Fig_LBC_nc_header}
668    \caption {
669      \protect\label{fig:LBC_nc_header}
670      Example of the header for a \protect\ifile{coordinates.bdy} file
671    }
672  \end{center}
673\end{figure}
674%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
675
676%----------------------------------------------
677\subsection{Volume correction}
678\label{subsec:LBC_bdy_vol_corr}
679
680There is an option to force the total volume in the regional model to be constant.
681This is controlled  by the \np{ln\_vol} parameter in the namelist.
682A value of \np{ln\_vol}\forcode{=.false.} indicates that this option is not used.
683Two options to control the volume are available (\np{nn\_volctl}).
684If \np{nn\_volctl}\forcode{=0} then a correction is applied to the normal barotropic velocities around the boundary at
685each timestep to ensure that the integrated volume flow through the boundary is zero.
686If \np{nn\_volctl}\forcode{=1} then the calculation of the volume change on
687the timestep includes the change due to the freshwater flux across the surface and
688the correction velocity corrects for this as well.
689
690If more than one boundary set is used then volume correction is
691applied to all boundaries at once.
692
693%----------------------------------------------
694\subsection{Tidal harmonic forcing}
695\label{subsec:LBC_bdy_tides}
696
697%-----------------------------------------nambdy_tide--------------------------------------------
698
699\nlst{nambdy_tide}
700%-----------------------------------------------------------------------------------------------
701
702Tidal forcing at open boundaries requires the activation of surface
703tides (i.e., in \nam{\_tide}, \np{ln\_tide} needs to be set to
704\forcode{.true.} and the required constituents need to be activated by
705including their names in the \np{clname} array; see
706\autoref{sec:SBC_tide}). Specific options related to the reading in of
707the complex harmonic amplitudes of elevation (SSH) and barotropic
708velocity (u,v) at open boundaries are defined through the
709\nam{bdy\_tide} namelist parameters.\\
710
711The tidal harmonic data at open boundaries can be specified in two
712different ways, either on a two-dimensional grid covering the entire
713model domain or along open boundary segments; these two variants can
714be selected by setting \np{ln\_bdytide\_2ddta } to \forcode{.true.} or
715\forcode{.false.}, respectively. In either case, the real and
716imaginary parts of SSH and the two barotropic velocity components for
717each activated tidal constituent \textit{tcname} have to be provided
718separately: when two-dimensional data is used, variables
719\textit{tcname\_z1} and \textit{tcname\_z2} for real and imaginary SSH,
720respectively, are expected in input file \np{filtide} with suffix
721\ifile{\_grid\_T}, variables \textit{tcname\_u1} and
722\textit{tcname\_u2} for real and imaginary u, respectively, are
723expected in input file \np{filtide} with suffix \ifile{\_grid\_U}, and
724\textit{tcname\_v1} and \textit{tcname\_v2} for real and imaginary v,
725respectively, are expected in input file \np{filtide} with suffix
726\ifile{\_grid\_V}; when data along open boundary segments is used,
727variables \textit{z1} and \textit{z2} (real and imaginary part of SSH)
728are expected to be available from file \np{filtide} with suffix
729\ifile{tcname\_grid\_T}, variables \textit{u1} and \textit{u2} (real
730and imaginary part of u) are expected to be available from file
731\np{filtide} with suffix \ifile{tcname\_grid\_U}, and variables
732\textit{v1} and \textit{v2} (real and imaginary part of v) are
733expected to be available from file \np{filtide} with suffix
734\ifile{tcname\_grid\_V}. If \np{ln\_bdytide\_conj} is set to
735\forcode{.true.}, the data is expected to be in complex conjugate
736form.
737
738Note that the barotropic velocity components are assumed to be defined
739on the native model grid and should be rotated accordingly when they
740are converted from their definition on a different source grid. To do
741so, the u, v amplitudes and phases can be converted into tidal
742ellipses, the grid rotation added to the ellipse inclination, and then
743converted back (care should be taken regarding conventions of the
744direction of rotation). %, e.g. anticlockwise or clockwise.
745
746\biblio
747
748\pindex
749
750\end{document}
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