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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5\chapter{Lateral Boundary Condition (LBC)}
6\label{chap:LBC}
7
8\thispagestyle{plain}
9
10\chaptertoc
11
12\paragraph{Changes record} ~\\
13
14{\footnotesize
15  \begin{tabularx}{\textwidth}{l||X|X}
16    Release & Author(s) & Modifications \\
17    \hline
18    {\em  next} & {\em Simon M{\" u}ller} & {\em Minor update of \autoref{subsec:LBC_bdy_tides}} \\[2mm]
19    {\em   4.0} & {\em ...} & {\em ...} \\
20    {\em   3.6} & {\em ...} & {\em ...} \\
21    {\em   3.4} & {\em ...} & {\em ...} \\
22    {\em <=3.4} & {\em ...} & {\em ...}
23  \end{tabularx}
24}
25
26\clearpage
27
28\cmtgm{Add here introduction to this chapter}
29
30%% =================================================================================================
31\section[Boundary condition at the coast (\forcode{rn_shlat})]{Boundary condition at the coast (\protect\np{rn_shlat}{rn\_shlat})}
32\label{sec:LBC_coast}
33
34\begin{listing}
35  \nlst{namlbc}
36  \caption{\forcode{&namlbc}}
37  \label{lst:namlbc}
38\end{listing}
39
40%The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt
41%(no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip).
42%They are handled automatically by the mask system (see \autoref{subsec:DOM_msk}).
43
44%OPA allows land and topography grid points in the computational domain due to the presence of continents or islands,
45%and includes the use of a full or partial step representation of bottom topography.
46%The computation is performed over the whole domain, \ie\ we do not try to restrict the computation to ocean-only points.
47%This choice has two motivations.
48%Firstly, working on ocean only grid points overloads the code and harms the code readability.
49%Secondly, and more importantly, it drastically reduces the vector portion of the computation,
50%leading to a dramatic increase of CPU time requirement on vector computers.
51%The current section describes how the masking affects the computation of the various terms of the equations
52%with respect to the boundary condition at solid walls.
53%The process of defining which areas are to be masked is described in \autoref{subsec:DOM_msk}.
54
55Options are defined through the \nam{lbc}{lbc} namelist variables.
56The discrete representation of a domain with complex boundaries (coastlines and bottom topography) leads to
57arrays that include large portions where a computation is not required as the model variables remain at zero.
58Nevertheless, vectorial supercomputers are far more efficient when computing over a whole array,
59and the readability of a code is greatly improved when boundary conditions are applied in
60an automatic way rather than by a specific computation before or after each computational loop.
61An efficient way to work over the whole domain while specifying the boundary conditions,
62is to use multiplication by mask arrays in the computation.
63A mask array is a matrix whose elements are $1$ in the ocean domain and $0$ elsewhere.
64A simple multiplication of a variable by its own mask ensures that it will remain zero over land areas.
65Since most of the boundary conditions consist of a zero flux across the solid boundaries,
66they can be simply applied by multiplying variables by the correct mask arrays,
67\ie\ the mask array of the grid point where the flux is evaluated.
68For example, the heat flux in the \textbf{i}-direction is evaluated at $u$-points.
69Evaluating this quantity as,
70
71\[
72  % \label{eq:LBC_aaaa}
73  \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT}
74  }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u
75\]
76(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is zero inside land and
77at the boundaries, since mask$_{u}$ is zero at solid boundaries which in this case are defined at $u$-points
78(normal velocity $u$ remains zero at the coast) (\autoref{fig:LBC_uv}).
79
80\begin{figure}[!t]
81  \centering
82  \includegraphics[width=0.66\textwidth]{LBC_uv}
83  \caption[Lateral boundary at $T$-level]{
84    Lateral boundary (thick line) at T-level.
85    The velocity normal to the boundary is set to zero.}
86  \label{fig:LBC_uv}
87\end{figure}
88
89For momentum the situation is a bit more complex as two boundary conditions must be provided along the coast
90(one each for the normal and tangential velocities).
91The boundary of the ocean in the C-grid is defined by the velocity-faces.
92For example, at a given $T$-level,
93the lateral boundary (a coastline or an intersection with the bottom topography) is made of
94segments joining $f$-points, and normal velocity points are located between two $f-$points (\autoref{fig:LBC_uv}).
95The boundary condition on the normal velocity (no flux through solid boundaries)
96can thus be easily implemented using the mask system.
97The boundary condition on the tangential velocity requires a more specific treatment.
98This boundary condition influences the relative vorticity and momentum diffusive trends,
99and is required in order to compute the vorticity at the coast.
100Four different types of lateral boundary condition are available,
101controlled by the value of the \np{rn_shlat}{rn\_shlat} namelist parameter
102(The value of the mask$_{f}$ array along the coastline is set equal to this parameter).
103These are:
104
105\begin{figure}[!p]
106  \centering
107  \includegraphics[width=0.66\textwidth]{LBC_shlat}
108  \caption[Lateral boundary conditions]{
109    Lateral boundary conditions
110    (a) free-slip                       (\protect\np[=0]{rn_shlat}{rn\_shlat});
111    (b) no-slip                         (\protect\np[=2]{rn_shlat}{rn\_shlat});
112    (c) "partial" free-slip (\forcode{0<}\protect\np[<2]{rn_shlat}{rn\_shlat}) and
113    (d) "strong" no-slip    (\forcode{2<}\protect\np{rn_shlat}{rn\_shlat}).
114    Implied "ghost" velocity inside land area is display in grey.}
115  \label{fig:LBC_shlat}
116\end{figure}
117
118\begin{description}
119
120\item [free-slip boundary condition ({\np[=0]{rn_shlat}{rn\_shlat}})] the tangential velocity at
121  the coastline is equal to the offshore velocity,
122  \ie\ the normal derivative of the tangential velocity is zero at the coast,
123  so the vorticity: mask$_{f}$ array is set to zero inside the land and just at the coast
124  (\autoref{fig:LBC_shlat}-a).
125
126\item [no-slip boundary condition ({\np[=2]{rn_shlat}{rn\_shlat}})] the tangential velocity vanishes at the coastline.
127  Assuming that the tangential velocity decreases linearly from
128  the closest ocean velocity grid point to the coastline,
129  the normal derivative is evaluated as if the velocities at the closest land velocity gridpoint and
130  the closest ocean velocity gridpoint were of the same magnitude but in the opposite direction
131  (\autoref{fig:LBC_shlat}-b).
132  Therefore, the vorticity along the coastlines is given by:
133
134  \[
135    \zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ ,
136  \]
137  where $u$ and $v$ are masked fields.
138  Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with
139  the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ :
140  \[
141    % \label{eq:LBC_bbbb}
142    \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2}
143        \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]}
144    \right)\;\mbox{mask}_f
145  \]
146
147\item ["partial" free-slip boundary condition (0$<$\np{rn_shlat}{rn\_shlat}$<$2)] the tangential velocity at
148  the coastline is smaller than the offshore velocity, \ie\ there is a lateral friction but
149  not strong enough to make the tangential velocity at the coast vanish (\autoref{fig:LBC_shlat}-c).
150  This can be selected by providing a value of mask$_{f}$ strictly inbetween $0$ and $2$.
151
152\item ["strong" no-slip boundary condition (2$<$\np{rn_shlat}{rn\_shlat})] the viscous boundary layer is assumed to
153  be smaller than half the grid size (\autoref{fig:LBC_shlat}-d).
154  The friction is thus larger than in the no-slip case.
155
156\end{description}
157
158Note that when the bottom topography is entirely represented by the $s$-coordinates (pure $s$-coordinate),
159the lateral boundary condition on tangential velocity is of much less importance as
160it is only applied next to the coast where the minimum water depth can be quite shallow.
161
162%% =================================================================================================
163\section[Model domain boundary condition (\forcode{jperio})]{Model domain boundary condition (\protect\jp{jperio})}
164\label{sec:LBC_jperio}
165
166At the model domain boundaries several choices are offered:
167closed, cyclic east-west, cyclic north-south, a north-fold, and combination closed-north fold or
168bi-cyclic east-west and north-fold.
169The north-fold boundary condition is associated with the 3-pole ORCA mesh.
170
171%% =================================================================================================
172\subsection[Closed, cyclic (\forcode{=0,1,2,7})]{Closed, cyclic (\protect\jp{jperio}\forcode{=0,1,2,7})}
173\label{subsec:LBC_jperio012}
174
175The choice of closed or cyclic model domain boundary condition is made by
176setting \jp{jperio} to 0, 1, 2 or 7 in namelist \nam{cfg}{cfg}.
177Each time such a boundary condition is needed, it is set by a call to routine \mdl{lbclnk}.
178The computation of momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to $j=jpj-1$,
179\ie\ in the model interior.
180To choose a lateral model boundary condition is to specify the first and last rows and columns of
181the model variables.
182
183\begin{description}
184
185\item [For closed boundary (\jp{jperio}\forcode{=0})], solid walls are imposed at all model boundaries:
186  first and last rows and columns are set to zero.
187
188\item [For cyclic east-west boundary (\jp{jperio}\forcode{=1})], first and last rows are set to zero (closed) whilst the first column is set to
189  the value of the last-but-one column and the last column to the value of the second one
190  (\autoref{fig:LBC_jperio}-a).
191  Whatever flows out of the eastern (western) end of the basin enters the western (eastern) end.
192
193\item [For cyclic north-south boundary (\jp{jperio}\forcode{=2})], first and last columns are set to zero (closed) whilst the first row is set to
194  the value of the last-but-one row and the last row to the value of the second one
195  (\autoref{fig:LBC_jperio}-a).
196  Whatever flows out of the northern (southern) end of the basin enters the southern (northern) end.
197
198\item [Bi-cyclic east-west and north-south boundary (\jp{jperio}\forcode{=7})] combines cases 1 and 2.
199
200\end{description}
201
202\begin{figure}[!t]
203  \centering
204  \includegraphics[width=0.66\textwidth]{LBC_jperio}
205  \caption[Setting of east-west cyclic and symmetric across the Equator boundary conditions]{
206    Setting of (a) east-west cyclic (b) symmetric across the Equator boundary conditions}
207  \label{fig:LBC_jperio}
208\end{figure}
209
210%% =================================================================================================
211\subsection[North-fold (\forcode{=3,6})]{North-fold (\protect\jp{jperio}\forcode{=3,6})}
212\label{subsec:LBC_north_fold}
213
214The north fold boundary condition has been introduced in order to handle the north boundary of
215a three-polar ORCA grid.
216Such a grid has two poles in the northern hemisphere (\autoref{fig:CFGS_ORCA_msh},
217and thus requires a specific treatment illustrated in \autoref{fig:LBC_North_Fold_T}.
218Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition.
219
220\begin{figure}[!t]
221  \centering
222  \includegraphics[width=0.66\textwidth]{LBC_North_Fold_T}
223  \caption[North fold boundary in ORCA 2\deg, 1/4\deg and 1/12\deg]{
224    North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$),
225    as used in ORCA 2\deg, 1/4\deg and 1/12\deg.
226    Pink shaded area corresponds to the inner domain mask (see text).}
227  \label{fig:LBC_North_Fold_T}
228\end{figure}
229
230%% =================================================================================================
231\section[Exchange with neighbouring processors (\textit{lbclnk.F90}, \textit{lib\_mpp.F90})]{Exchange with neighbouring processors (\protect\mdl{lbclnk}, \protect\mdl{lib\_mpp})}
232\label{sec:LBC_mpp}
233
234\begin{listing}
235  \nlst{nammpp}
236  \caption{\forcode{&nammpp}}
237  \label{lst:nammpp}
238\end{listing}
239
240For massively parallel processing (mpp), a domain decomposition method is used.
241The basic idea of the method is to split the large computation domain of a numerical experiment into several smaller domains and
242solve the set of equations by addressing independent local problems.
243Each processor has its own local memory and computes the model equation over a subdomain of the whole model domain.
244The subdomain boundary conditions are specified through communications between processors which are organized by
245explicit statements (message passing method).
246The present implementation is largely inspired by Guyon's work [Guyon 1995].
247
248The parallelization strategy is defined by the physical characteristics of the ocean model.
249Second order finite difference schemes lead to local discrete operators that
250depend at the very most on one neighbouring point.
251The only non-local computations concern the vertical physics
252(implicit diffusion, turbulent closure scheme, ...).
253Therefore, a pencil strategy is used for the data sub-structuration:
254the 3D initial domain is laid out on local processor memories following a 2D horizontal topological splitting.
255Each sub-domain computes its own surface and bottom boundary conditions and
256has a side wall overlapping interface which defines the lateral boundary conditions for
257computations in the inner sub-domain.
258The overlapping area consists of the two rows at each edge of the sub-domain.
259After a computation, a communication phase starts:
260each processor sends to its neighbouring processors the update values of the points corresponding to
261the interior overlapping area to its neighbouring sub-domain (\ie\ the innermost of the two overlapping rows).
262Communications are first done according to the east-west direction and next according to the north-south direction.
263There is no specific communications for the corners.
264The communication is done through the Message Passing Interface (MPI) and requires \key{mpp\_mpi}.
265Use also \key{mpi2} if MPI3 is not available on your computer.
266The data exchanges between processors are required at the very place where
267lateral domain boundary conditions are set in the mono-domain computation:
268the \rou{lbc\_lnk} routine (found in \mdl{lbclnk} module) which manages such conditions is interfaced with
269routines found in \mdl{lib\_mpp} module.
270The output file \textit{communication\_report.txt} provides the list of which routines do how
271many communications during 1 time step of the model.\\
272
273\begin{figure}[!t]
274  \centering
275  \includegraphics[width=0.66\textwidth]{LBC_mpp}
276  \caption{Positioning of a sub-domain when massively parallel processing is used}
277  \label{fig:LBC_mpp}
278\end{figure}
279
280In \NEMO, the splitting is regular and arithmetic.
281The total number of subdomains corresponds to the number of MPI processes allocated to \NEMO\ when the model is launched
282(\ie\ mpirun -np x ./nemo will automatically give x subdomains).
283The i-axis is divided by \np{jpni}{jpni} and the j-axis by \np{jpnj}{jpnj}.
284These parameters are defined in \nam{mpp}{mpp} namelist.
285If \np{jpni}{jpni} and \np{jpnj}{jpnj} are < 1, they will be automatically redefined in the code to give the best domain decomposition
286(see bellow).
287
288Each processor is independent and without message passing or synchronous process, programs run alone and access just its own local memory.
289For this reason,
290the main model dimensions are now the local dimensions of the subdomain (pencil) that are named \jp{jpi}, \jp{jpj}, \jp{jpk}.
291These dimensions include the internal domain and the overlapping rows.
292The number of rows to exchange (known as the halo) is usually set to one (nn\_hls=1, in \mdl{par\_oce},
293and must be kept to one until further notice).
294The whole domain dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}.
295The relationship between the whole domain and a sub-domain is:
296\begin{gather*}
297  jpi = ( jpiglo-2\times nn\_hls + (jpni-1) ) / jpni + 2\times nn\_hls \\
298  jpj = ( jpjglo-2\times nn\_hls + (jpnj-1) ) / jpnj + 2\times nn\_hls
299\end{gather*}
300
301One 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.
302
303An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
304a global array (whole domain) by the relationship:
305\[
306  % \label{eq:LBC_nimpp}
307  T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
308\]
309with $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.
310
311The 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).\\
312
313The \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:
314\[
315  N_{mpi} = jpni \times jpnj - N_{land} + N_{useless}
316\]
317$N_{land}$ is the total number of land subdomains in the domain decomposition defined by \np{jpni}{jpni} and \np{jpnj}{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}{jpni}, \np{jpnj}{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.
318
319If the domain decomposition is automatically defined (when \np{jpni}{jpni} and \np{jpnj}{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}{jpni} and \np{jpnj}{jpnj} values) will use less processes than $(jpni  \times  jpnj - N_{land})$ and get a smaller subdomain size.
320In order to specify $N_{mpi}$ properly (minimize $N_{useless}$), you must run the model once with \np{ln_list}{ln\_list} activated. In this case, the model will start the initialisation phase, print the list of optimum decompositions ($N_{mpi}$, \np{jpni}{jpni} and \np{jpnj}{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.
321
322Processors 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}{jpni}, \np{jpnj}{jpnj}), the numbers attributed to he ocean subdomains do not vary with $N_{useless}$.
323
324When land processors are eliminated, the value corresponding to these locations in the model output files is undefined. \np{ln_mskland}{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}{nn\_msh}).
325
326\begin{figure}[!ht]
327  \centering
328  \includegraphics[width=0.66\textwidth]{LBC_mppini2}
329  \caption[Atlantic domain defined for the CLIPPER projet]{
330    Example of Atlantic domain defined for the CLIPPER projet.
331    Initial grid is composed of 773 x 1236 horizontal points.
332    (a) the domain is split onto 9 $times$ 20 subdomains (jpni=9, jpnj=20).
333    52 subdomains are land areas.
334    (b) 52 subdomains are eliminated (white rectangles) and
335    the resulting number of processors really used during the computation is jpnij=128.}
336  \label{fig:LBC_mppini2}
337\end{figure}
338
339%% =================================================================================================
340\section{Unstructured open boundary conditions (BDY)}
341\label{sec:LBC_bdy}
342
343\begin{listing}
344  \nlst{nambdy}
345  \caption{\forcode{&nambdy}}
346  \label{lst:nambdy}
347\end{listing}
348
349\begin{listing}
350  \nlst{nambdy_dta}
351  \caption{\forcode{&nambdy_dta}}
352  \label{lst:nambdy_dta}
353\end{listing}
354
355Options are defined through the \nam{bdy}{bdy} and \nam{bdy_dta}{bdy\_dta} namelist variables.
356The BDY module is the core implementation of open boundary conditions for regional configurations on
357ocean temperature, salinity, barotropic-baroclinic velocities, ice-snow concentration, thicknesses, temperatures, salinity and melt ponds concentration and thickness.
358
359The BDY module was modelled on the OBC module (see \NEMO\ 3.4) and shares many features and
360a similar coding structure \citep{chanut_trpt05}.
361The specification of the location of the open boundary is completely flexible and
362allows any type of setup, from regular boundaries to irregular contour (it includes the possibility to set an open boundary able to follow an isobath).
363Boundary data files used with versions of \NEMO\ prior to Version 3.4 may need to be re-ordered to work with this version.
364See the section on the Input Boundary Data Files for details.
365
366%% =================================================================================================
367\subsection{Namelists}
368\label{subsec:LBC_bdy_namelist}
369
370The BDY module is activated by setting \np[=.true.]{ln_bdy}{ln\_bdy} .
371It is possible to define more than one boundary ``set'' and apply different boundary conditions to each set.
372The number of boundary sets is defined by \np{nb_bdy}{nb\_bdy}.
373Each boundary set can be either defined as a series of straight line segments directly in the namelist
374(\np[=.false.]{ln_coords_file}{ln\_coords\_file}, and a namelist block \forcode{&nambdy_index} must be included for each set) or read in from a file (\np[=.true.]{ln_coords_file}{ln\_coords\_file}, and a ``\ifile{coordinates.bdy}'' file must be provided).
375The coordinates.bdy file is analagous to the usual \NEMO\ ``\ifile{coordinates}'' file.
376In the example above, there are two boundary sets, the first of which is defined via a file and
377the second is defined in the namelist.
378For more details of the definition of the boundary geometry see section \autoref{subsec:LBC_bdy_geometry}.
379
380For each boundary set a boundary condition has to be chosen for the barotropic solution
381(``u2d'':sea-surface height and barotropic velocities), for the baroclinic velocities (``u3d''),
382for the active tracers \footnote{The BDY module does not deal with passive tracers at this version} (``tra''), and for sea-ice (``ice'').
383For each set of variables one has to choose an algorithm and the boundary data (set resp. by \np{cn_tra}{cn\_tra} and \np{nn_tra_dta}{nn\_tra\_dta} for tracers).\\
384
385The choice of algorithm is currently as follows:
386
387\begin{description}
388\item [\forcode{'none'}:] No boundary condition applied.
389  So the solution will ``see'' the land points around the edge of the edge of the domain.
390\item [\forcode{'specified'}:] Specified boundary condition applied (only available for baroclinic velocity and tracer variables).
391\item [\forcode{'neumann'}:] Value at the boundary are duplicated (No gradient). Only available for baroclinic velocity and tracer variables.
392\item [\forcode{'frs'}:] Flow Relaxation Scheme (FRS) available for all variables.
393\item [\forcode{'Orlanski'}:] Orlanski radiation scheme (fully oblique) for barotropic, baroclinic and tracer variables.
394\item [\forcode{'Orlanski_npo'}:] Orlanski radiation scheme for barotropic, baroclinic and tracer variables.
395\item [\forcode{'flather'}:] Flather radiation scheme for the barotropic variables only.
396\end{description}
397
398The boundary data is either set to initial conditions
399(\np[=0]{nn_tra_dta}{nn\_tra\_dta}) or forced with external data from a file (\np[=1]{nn_tra_dta}{nn\_tra\_dta}).
400In case the 3d velocity data contain the total velocity (ie, baroclinic and barotropic velocity),
401the bdy code can derived baroclinic and barotropic velocities by setting \np[=.true.]{ln_full_vel}{ln\_full\_vel}
402For the barotropic solution there is also the option to use tidal harmonic forcing either by
403itself (\np[=2]{nn_dyn2d_dta}{nn\_dyn2d\_dta}) or in addition to other external data (\np[=3]{nn_dyn2d_dta}{nn\_dyn2d\_dta}).\\
404If 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}{rn\_ice\_sal}, \np{rn_ice_tem}{rn\_ice\_tem}, \np{rn_ice_apnd}{rn\_ice\_apnd}, \np{rn_ice_hpnd}{rn\_ice\_hpnd}). Ice age is constant and defined by \np{rn_ice_age}{rn\_ice\_age}.
405
406If external boundary data is required then the \nam{bdy_dta}{bdy\_dta} namelist must be defined.
407One \nam{bdy_dta}{bdy\_dta} namelist is required for each boundary set, adopting the same order of indexes in which the boundary sets are defined in nambdy.
408In the example given, two boundary sets have been defined. The first one is reading data file in the \nam{bdy_dta}{bdy\_dta} namelist shown above
409and the second one is using data from intial condition (no namelist block needed).
410The boundary data is read in using the fldread module,
411so the \nam{bdy_dta}{bdy\_dta} namelist is in the format required for fldread.
412For each required variable, the filename, the frequency of the files and
413the frequency of the data in the files are given.
414Also whether or not time-interpolation is required and whether the data is climatological (time-cyclic) data.
415For sea-ice salinity, temperatures and melt ponds, reading the files are skipped and constant values are used if filenames are defined as {'NOT USED'}.\\
416
417There is currently an option to vertically interpolate the open boundary data onto the native grid at run-time.
418If \np{nn_bdy_jpk}{nn\_bdy\_jpk}$<-1$, it is assumed that the lateral boundary data are already on the native grid.
419However, if \np{nn_bdy_jpk}{nn\_bdy\_jpk} is set to the number of vertical levels present in the boundary data,
420a bilinear interpolation onto the native grid will be triggered at runtime.
421For this to be successful the additional variables: $gdept$, $gdepu$, $gdepv$, $e3t$, $e3u$ and $e3v$, are required to be present in the lateral boundary files.
422These correspond to the depths and scale factors of the input data,
423the latter used to make any adjustment to the velocity fields due to differences in the total water depths between the two vertical grids.\\
424
425In the example of given namelists, two boundary sets are defined.
426The first set is defined via a file and applies FRS conditions to temperature and salinity and
427Flather conditions to the barotropic variables. No condition specified for the baroclinic velocity and sea-ice.
428External data is provided in daily files (from a large-scale model).
429Tidal harmonic forcing is also used.
430The second set is defined in a namelist.
431FRS conditions are applied on temperature and salinity and climatological data is read from initial condition files.
432
433%% =================================================================================================
434\subsection{Flow relaxation scheme}
435\label{subsec:LBC_bdy_FRS_scheme}
436
437The Flow Relaxation Scheme (FRS) \citep{davies_QJRMS76,engedahl_T95},
438applies a simple relaxation of the model fields to externally-specified values over
439a zone next to the edge of the model domain.
440Given a model prognostic variable $\Phi$
441\[
442  % \label{eq:LBC_bdy_frs1}
443  \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
444\]
445where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified external field,
446$d$ gives the discrete distance from the model boundary and
447$\alpha$ is a parameter that varies from $1$ at $d=1$ to a small value at $d=N$.
448It can be shown that this scheme is equivalent to adding a relaxation term to
449the prognostic equation for $\Phi$ of the form:
450\[
451  % \label{eq:LBC_bdy_frs2}
452  -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
453\]
454where the relaxation time scale $\tau$ is given by a function of $\alpha$ and the model time step $\Delta t$:
455\[
456  % \label{eq:LBC_bdy_frs3}
457  \tau = \frac{1-\alpha}{\alpha\,\rdt
458\]
459Thus the model solution is completely prescribed by the external conditions at the edge of the model domain and
460is relaxed towards the external conditions over the rest of the FRS zone.
461The application of a relaxation zone helps to prevent spurious reflection of
462outgoing signals from the model boundary.
463
464The function $\alpha$ is specified as a $tanh$ function:
465\[
466  % \label{eq:LBC_bdy_frs4}
467  \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
468\]
469The width of the FRS zone is specified in the namelist as \np{nn_rimwidth}{nn\_rimwidth}.
470This is typically set to a value between 8 and 10.
471
472%% =================================================================================================
473\subsection{Flather radiation scheme}
474\label{subsec:LBC_bdy_flather_scheme}
475
476The \citet{flather_JPO94} scheme is a radiation condition on the normal,
477depth-mean transport across the open boundary.
478It takes the form
479\begin{equation}
480  \label{eq:LBC_bdy_fla1}
481  U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
482\end{equation}
483where $U$ is the depth-mean velocity normal to the boundary and $\eta$ is the sea surface height,
484both from the model.
485The subscript $e$ indicates the same fields from external sources.
486The speed of external gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the water column.
487The depth-mean normal velocity along the edge of the model domain is set equal to
488the external depth-mean normal velocity,
489plus a correction term that allows gravity waves generated internally to exit the model boundary.
490Note that the sea-surface height gradient in \autoref{eq:LBC_bdy_fla1} is a spatial gradient across the model boundary,
491so that $\eta_{e}$ is defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the $T$ points with $nbr=2$.
492$U$ and $U_{e}$ are defined on the $U$ or $V$ points with $nbr=1$, \ie\ between the two $T$ grid points.
493
494%% =================================================================================================
495\subsection{Orlanski radiation scheme}
496\label{subsec:LBC_bdy_orlanski_scheme}
497
498The Orlanski scheme is based on the algorithm described by \citep{marchesiello.mcwilliams.ea_OM01}, hereafter MMS.
499
500The adaptive Orlanski condition solves a wave plus relaxation equation at the boundary:
501\begin{equation}
502  \label{eq:LBC_wave_continuous}
503  \frac{\partial\phi}{\partial t} + c_x \frac{\partial\phi}{\partial x} + c_y \frac{\partial\phi}{\partial y} =
504  -\frac{1}{\tau}(\phi - \phi^{ext})
505\end{equation}
506
507where $\phi$ is the model field, $x$ and $y$ refer to the normal and tangential directions to the boundary respectively, and the phase
508velocities are diagnosed from the model fields as:
509
510\begin{equation}
511  \label{eq:LBC_cx}
512  c_x = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial x}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}
513\end{equation}
514\begin{equation}
515  \label{eq:LBC_cy}
516  c_y = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial y}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}
517\end{equation}
518
519(As noted by MMS, this is a circular diagnosis of the phase speeds which only makes sense on a discrete grid).
520Equation (\autoref{eq:LBC_wave_continuous}) is defined adaptively depending on the sign of the phase velocity normal to the boundary $c_x$.
521For $c_x$ outward, we have
522
523\begin{equation}
524\tau = \tau_{out}
525\end{equation}
526
527For $c_x$ inward, the radiation equation is not applied:
528
529\begin{equation}
530  \label{eq:LBC_tau_in}
531  \tau = \tau_{in}\,\,\,;\,\,\, c_x = c_y = 0
532\end{equation}
533
534Generally the relaxation time scale at inward propagation points (\np{rn_time_dmp}{rn\_time\_dmp}) is set much shorter than the time scale at outward propagation
535points (\np{rn_time_dmp_out}{rn\_time\_dmp\_out}) so that the solution is constrained more strongly by the external data at inward propagation points.
536See \autoref{subsec:LBC_bdy_relaxation} for detailed on the spatial shape of the scaling.\\
537The ``normal propagation of oblique radiation'' or NPO approximation (called \forcode{'orlanski_npo'}) involves assuming
538that $c_y$ is zero in equation (\autoref{eq:LBC_wave_continuous}), but including
539this term in the denominator of equation (\autoref{eq:LBC_cx}). Both versions of the scheme are options in BDY. Equations
540(\autoref{eq:LBC_wave_continuous}) - (\autoref{eq:LBC_tau_in}) correspond to equations (13) - (15) and (2) - (3) in MMS.\\
541
542%% =================================================================================================
543\subsection{Relaxation at the boundary}
544\label{subsec:LBC_bdy_relaxation}
545
546In addition to a specific boundary condition specified as \np{cn_tra}{cn\_tra} and \np{cn_dyn3d}{cn\_dyn3d}, relaxation on baroclinic velocities and tracers variables are available.
547It is control by the namelist parameter \np{ln_tra_dmp}{ln\_tra\_dmp} and \np{ln_dyn3d_dmp}{ln\_dyn3d\_dmp} for each boundary set.
548
549The relaxation time scale value (\np{rn_time_dmp}{rn\_time\_dmp} and \np{rn_time_dmp_out}{rn\_time\_dmp\_out}, $\tau$) are defined at the boundaries itself.
550This time scale ($\alpha$) is weighted by the distance ($d$) from the boundary over \np{nn_rimwidth}{nn\_rimwidth} cells ($N$):
551
552\[
553  \alpha = \frac{1}{\tau}(\frac{N+1-d}{N})^2,       \quad d=1,N
554\]
555
556The same scaling is applied in the Orlanski damping.
557
558%% =================================================================================================
559\subsection{Boundary geometry}
560\label{subsec:LBC_bdy_geometry}
561
562Each open boundary set is defined as a list of points.
563The information is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$ structure.
564The $nbi$ and $nbj$ arrays define the local $(i,j)$ indexes of each point in the boundary zone and
565the $nbr$ array defines the discrete distance from the boundary: $nbr=1$ means that
566the boundary point is next to the edge of the model domain, while $nbr>1$ means that
567the boundary point is increasingly further away from the edge of the model domain.
568A set of $nbi$, $nbj$, and $nbr$ arrays is defined for each of the $T$, $U$ and $V$ grids.
569\autoref{fig:LBC_bdy_geom} shows an example of an irregular boundary.
570
571The boundary geometry for each set may be defined in a namelist \forcode{&nambdy_index} or
572by reading in a ``\ifile{coordinates.bdy}'' file.
573The \texttt{nambdy\_index} namelist defines a series of straight-line segments for north, east, south and west boundaries.
574One \texttt{nambdy\_index} namelist block is needed for each boundary condition defined by indexes.
575For the northern boundary, \texttt{nbdysegn} gives the number of segments,
576\jp{jpjnob} gives the $j$ index for each segment and \jp{jpindt} and
577\jp{jpinft} give the start and end $i$ indices for each segment with similar for the other boundaries.
578These segments define a list of $T$ grid points along the outermost row of the boundary ($nbr\,=\, 1$).
579The code deduces the $U$ and $V$ points and also the points for $nbr\,>\, 1$ if \np[>1]{nn_rimwidth}{nn\_rimwidth}.
580
581The boundary geometry may also be defined from a ``\ifile{coordinates.bdy}'' file.
582\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.
583The file should contain the index arrays for each of the $T$, $U$ and $V$ grids.
584The arrays must be in order of increasing $nbr$.
585Note that the $nbi$, $nbj$ values in the file are global values and are converted to local values in the code.
586Typically this file will be used to generate external boundary data via interpolation and so
587will also contain the latitudes and longitudes of each point as shown.
588However, this is not necessary to run the model.
589
590For some choices of irregular boundary the model domain may contain areas of ocean which
591are not part of the computational domain.
592For example, if an open boundary is defined along an isobath, say at the shelf break,
593then the areas of ocean outside of this boundary will need to be masked out.
594This can be done by reading a mask file defined as \np{cn_mask_file}{cn\_mask\_file} in the nam\_bdy namelist.
595Only one mask file is used even if multiple boundary sets are defined.
596
597\begin{figure}[!t]
598  \centering
599  \includegraphics[width=0.66\textwidth]{LBC_bdy_geom}
600  \caption[Geometry of unstructured open boundary]{Example of geometry of unstructured open boundary}
601  \label{fig:LBC_bdy_geom}
602\end{figure}
603
604%% =================================================================================================
605\subsection{Input boundary data files}
606\label{subsec:LBC_bdy_data}
607
608The data files contain the data arrays in the order in which the points are defined in the $nbi$ and $nbj$ arrays.
609The data arrays are dimensioned on:
610a time dimension;
611$xb$ which is the index of the boundary data point in the horizontal;
612and $yb$ which is a degenerate dimension of 1 to enable the file to be read by the standard \NEMO\ I/O routines.
613The 3D fields also have a depth dimension.
614
615From Version 3.4 there are new restrictions on the order in which the boundary points are defined
616(and therefore restrictions on the order of the data in the file).
617In particular:
618
619\begin{enumerate}
620\item The data points must be in order of increasing $nbr$,
621  ie. all the $nbr=1$ points, then all the $nbr=2$ points etc.
622\item All the data for a particular boundary set must be in the same order.
623  (Prior to 3.4 it was possible to define barotropic data in a different order to
624  the data for tracers and baroclinic velocities).
625\end{enumerate}
626
627These restrictions mean that data files used with versions of the
628model prior to Version 3.4 may not work with Version 3.4 onwards.
629A \fortran\ utility {\itshape bdy\_reorder} exists in the TOOLS directory which
630will re-order the data in old BDY data files.
631
632\begin{figure}[!t]
633  \centering
634  \includegraphics[width=0.66\textwidth]{LBC_nc_header}
635  \caption[Header for a \protect\ifile{coordinates.bdy} file]{
636    Example of the header for a \protect\ifile{coordinates.bdy} file}
637  \label{fig:LBC_nc_header}
638\end{figure}
639
640%% =================================================================================================
641\subsection{Volume correction}
642\label{subsec:LBC_bdy_vol_corr}
643
644There is an option to force the total volume in the regional model to be constant.
645This is controlled  by the \np{ln_vol}{ln\_vol} parameter in the namelist.
646A value of \np[=.false.]{ln_vol}{ln\_vol} indicates that this option is not used.
647Two options to control the volume are available (\np{nn_volctl}{nn\_volctl}).
648If \np[=0]{nn_volctl}{nn\_volctl} then a correction is applied to the normal barotropic velocities around the boundary at
649each timestep to ensure that the integrated volume flow through the boundary is zero.
650If \np[=1]{nn_volctl}{nn\_volctl} then the calculation of the volume change on
651the timestep includes the change due to the freshwater flux across the surface and
652the correction velocity corrects for this as well.
653
654If more than one boundary set is used then volume correction is
655applied to all boundaries at once.
656
657%% =================================================================================================
658\subsection{Tidal harmonic forcing}
659\label{subsec:LBC_bdy_tides}
660
661\begin{listing}
662  \nlst{nambdy_tide}
663  \caption{\forcode{&nambdy_tide}}
664  \label{lst:nambdy_tide}
665\end{listing}
666
667Tidal forcing at open boundaries requires the activation of surface
668tides (i.e., in \nam{_tide}{\_tide}, \np[=.true.]{ln_tide}{ln\_tide} with the active tidal
669constituents listed in the \np{sn_tide_cnames}{sn\_tide\_cnames} array; see
670\autoref{sec:SBC_TDE}). The specific options related to the reading in of
671the complex harmonic amplitudes of elevation (SSH) and barotropic
672velocity components (u,v) at the open boundaries are defined through the
673\nam{bdy_tide}{bdy\_tide} namelist parameters.\par
674
675The tidal harmonic data at open boundaries can be specified in two
676different ways, either on a two-dimensional grid covering the entire
677model domain or along open boundary segments; these two variants can
678be selected by setting \np[=.true.]{ln_bdytide_2ddta}{ln\_bdytide\_2ddta} or
679\np[=.false.]{ln_bdytide_2ddta}{ln\_bdytide\_2ddta}, respectively. In either
680case, the real and imaginary parts of SSH, u, and v amplitudes associated with
681each activated tidal constituent \texttt{<constituent>} have to be provided
682separately as fields in input files with names based on
683\np[=<input>]{filtide}{filtide}: when two-dimensional data is used, variables
684\texttt{<constituent>\_z1} and \texttt{<constituent>\_z2} for the real and imaginary parts of
685SSH, respectively, are expected to be available in file
686\ifile{<input>\_grid\_T}, variables \texttt{<constituent>\_u1} and
687\texttt{<constituent>\_u2} for the real and imaginary parts of u, respectively, in file
688\ifile{<input>\_grid\_U}, and \texttt{<constituent>\_v1} and
689\texttt{<constituent>\_v2} for the real and imaginary parts of v, respectively, in file
690\ifile{<input>\_grid\_V}; when data along open boundary segments is used,
691variables \texttt{z1} and \texttt{z2} (real and imaginary part of SSH) are
692expected to be available in file \ifile{<input><constituent>\_grid\_T},
693variables \texttt{u1} and \texttt{u2} (real and imaginary part of u) in file
694\ifile{<input><constituent>\_grid\_U}, and variables \texttt{v1} and \texttt{v2}
695(real and imaginary part of v) in file
696\ifile{<input><constituent>\_grid\_V}.\par
697
698Note that the barotropic velocity components are assumed to be defined
699on the native model grid and should be rotated accordingly when they
700are converted from their definition on a different source grid. To do
701so, the u, v amplitudes and phases can be converted into tidal
702ellipses, the grid rotation added to the ellipse inclination, and then
703converted back (care should be taken regarding conventions of the
704direction of rotation). %, e.g. anticlockwise or clockwise.
705
706\subinc{\input{../../global/epilogue}}
707
708\end{document}
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