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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   4.0} & {\em ...} & {\em ...} \\
19    {\em   3.6} & {\em ...} & {\em ...} \\
20    {\em   3.4} & {\em ...} & {\em ...} \\
21    {\em <=3.4} & {\em ...} & {\em ...}
22  \end{tabularx}
23}
24
25\clearpage
26
27\cmtgm{Add here introduction to this chapter}
28
29%% =================================================================================================
30\section[Boundary condition at the coast (\forcode{rn_shlat})]{Boundary condition at the coast (\protect\np{rn_shlat}{rn\_shlat})}
31\label{sec:LBC_coast}
32
33\begin{listing}
34  \nlst{namlbc}
35  \caption{\forcode{&namlbc}}
36  \label{lst:namlbc}
37\end{listing}
38
39%The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt
40%(no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip).
41%They are handled automatically by the mask system (see \autoref{subsec:DOM_msk}).
42
43%OPA allows land and topography grid points in the computational domain due to the presence of continents or islands,
44%and includes the use of a full or partial step representation of bottom topography.
45%The computation is performed over the whole domain, \ie\ we do not try to restrict the computation to ocean-only points.
46%This choice has two motivations.
47%Firstly, working on ocean only grid points overloads the code and harms the code readability.
48%Secondly, and more importantly, it drastically reduces the vector portion of the computation,
49%leading to a dramatic increase of CPU time requirement on vector computers.
50%The current section describes how the masking affects the computation of the various terms of the equations
51%with respect to the boundary condition at solid walls.
52%The process of defining which areas are to be masked is described in \autoref{subsec:DOM_msk}.
53
54Options are defined through the \nam{lbc}{lbc} namelist variables.
55The discrete representation of a domain with complex boundaries (coastlines and bottom topography) leads to
56arrays that include large portions where a computation is not required as the model variables remain at zero.
57Nevertheless, vectorial supercomputers are far more efficient when computing over a whole array,
58and the readability of a code is greatly improved when boundary conditions are applied in
59an automatic way rather than by a specific computation before or after each computational loop.
60An efficient way to work over the whole domain while specifying the boundary conditions,
61is to use multiplication by mask arrays in the computation.
62A mask array is a matrix whose elements are $1$ in the ocean domain and $0$ elsewhere.
63A simple multiplication of a variable by its own mask ensures that it will remain zero over land areas.
64Since most of the boundary conditions consist of a zero flux across the solid boundaries,
65they can be simply applied by multiplying variables by the correct mask arrays,
66\ie\ the mask array of the grid point where the flux is evaluated.
67For example, the heat flux in the \textbf{i}-direction is evaluated at $u$-points.
68Evaluating this quantity as,
69
70$71 % \label{eq:LBC_aaaa} 72 \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 73 }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u 74$
75(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is zero inside land and
76at the boundaries, since mask$_{u}$ is zero at solid boundaries which in this case are defined at $u$-points
77(normal velocity $u$ remains zero at the coast) (\autoref{fig:LBC_uv}).
78
79\begin{figure}[!t]
80  \centering
81  \includegraphics[width=0.66\textwidth]{LBC_uv}
82  \caption[Lateral boundary at $T$-level]{
83    Lateral boundary (thick line) at T-level.
84    The velocity normal to the boundary is set to zero.}
85  \label{fig:LBC_uv}
86\end{figure}
87
88For momentum the situation is a bit more complex as two boundary conditions must be provided along the coast
89(one each for the normal and tangential velocities).
90The boundary of the ocean in the C-grid is defined by the velocity-faces.
91For example, at a given $T$-level,
92the lateral boundary (a coastline or an intersection with the bottom topography) is made of
93segments joining $f$-points, and normal velocity points are located between two $f-$points (\autoref{fig:LBC_uv}).
94The boundary condition on the normal velocity (no flux through solid boundaries)
95can thus be easily implemented using the mask system.
96The boundary condition on the tangential velocity requires a more specific treatment.
97This boundary condition influences the relative vorticity and momentum diffusive trends,
98and is required in order to compute the vorticity at the coast.
99Four different types of lateral boundary condition are available,
100controlled by the value of the \np{rn_shlat}{rn\_shlat} namelist parameter
101(The value of the mask$_{f}$ array along the coastline is set equal to this parameter).
102These are:
103
104\begin{figure}[!p]
105  \centering
106  \includegraphics[width=0.66\textwidth]{LBC_shlat}
107  \caption[Lateral boundary conditions]{
108    Lateral boundary conditions
109    (a) free-slip                       (\protect\np[=0]{rn_shlat}{rn\_shlat});
110    (b) no-slip                         (\protect\np[=2]{rn_shlat}{rn\_shlat});
111    (c) "partial" free-slip (\forcode{0<}\protect\np[<2]{rn_shlat}{rn\_shlat}) and
112    (d) "strong" no-slip    (\forcode{2<}\protect\np{rn_shlat}{rn\_shlat}).
113    Implied "ghost" velocity inside land area is display in grey.}
114  \label{fig:LBC_shlat}
115\end{figure}
116
117\begin{description}
118
119\item [free-slip boundary condition ({\np[=0]{rn_shlat}{rn\_shlat}})] the tangential velocity at
120  the coastline is equal to the offshore velocity,
121  \ie\ the normal derivative of the tangential velocity is zero at the coast,
122  so the vorticity: mask$_{f}$ array is set to zero inside the land and just at the coast
123  (\autoref{fig:LBC_shlat}-a).
124
125\item [no-slip boundary condition ({\np[=2]{rn_shlat}{rn\_shlat}})] the tangential velocity vanishes at the coastline.
126  Assuming that the tangential velocity decreases linearly from
127  the closest ocean velocity grid point to the coastline,
128  the normal derivative is evaluated as if the velocities at the closest land velocity gridpoint and
129  the closest ocean velocity gridpoint were of the same magnitude but in the opposite direction
130  (\autoref{fig:LBC_shlat}-b).
131  Therefore, the vorticity along the coastlines is given by:
132
133  $134 \zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ , 135$
136  where $u$ and $v$ are masked fields.
137  Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with
138  the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ :
139  $140 % \label{eq:LBC_bbbb} 141 \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2} 142 \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]} 143 \right)\;\mbox{mask}_f 144$
145
146\item ["partial" free-slip boundary condition (0$<$\np{rn_shlat}{rn\_shlat}$<$2)] the tangential velocity at
147  the coastline is smaller than the offshore velocity, \ie\ there is a lateral friction but
148  not strong enough to make the tangential velocity at the coast vanish (\autoref{fig:LBC_shlat}-c).
149  This can be selected by providing a value of mask$_{f}$ strictly inbetween $0$ and $2$.
150
151\item ["strong" no-slip boundary condition (2$<$\np{rn_shlat}{rn\_shlat})] the viscous boundary layer is assumed to
152  be smaller than half the grid size (\autoref{fig:LBC_shlat}-d).
153  The friction is thus larger than in the no-slip case.
154
155\end{description}
156
157Note that when the bottom topography is entirely represented by the $s$-coordinates (pure $s$-coordinate),
158the lateral boundary condition on tangential velocity is of much less importance as
159it is only applied next to the coast where the minimum water depth can be quite shallow.
160
161%% =================================================================================================
162\section[Model domain boundary condition (\forcode{jperio})]{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\subsection[Closed, cyclic (\forcode{=0,1,2,7})]{Closed, cyclic (\protect\jp{jperio}\forcode{=0,1,2,7})}
172\label{subsec:LBC_jperio012}
173
174The choice of closed or cyclic model domain boundary condition is made by
175setting \jp{jperio} to 0, 1, 2 or 7 in namelist \nam{cfg}{cfg}.
176Each time such a boundary condition is needed, it is set by a call to routine \mdl{lbclnk}.
177The computation of momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to $j=jpj-1$,
178\ie\ in the model interior.
179To choose a lateral model boundary condition is to specify the first and last rows and columns of
180the model variables.
181
182\begin{description}
183
184\item [For closed boundary (\jp{jperio}\forcode{=0})], solid walls are imposed at all model boundaries:
185  first and last rows and columns are set to zero.
186
187\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
188  the value of the last-but-one column and the last column to the value of the second one
189  (\autoref{fig:LBC_jperio}-a).
190  Whatever flows out of the eastern (western) end of the basin enters the western (eastern) end.
191
192\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
193  the value of the last-but-one row and the last row to the value of the second one
194  (\autoref{fig:LBC_jperio}-a).
195  Whatever flows out of the northern (southern) end of the basin enters the southern (northern) end.
196
197\item [Bi-cyclic east-west and north-south boundary (\jp{jperio}\forcode{=7})] combines cases 1 and 2.
198
199\end{description}
200
201\begin{figure}[!t]
202  \centering
203  \includegraphics[width=0.66\textwidth]{LBC_jperio}
204  \caption[Setting of east-west cyclic and symmetric across the Equator boundary conditions]{
205    Setting of (a) east-west cyclic (b) symmetric across the Equator boundary conditions}
206  \label{fig:LBC_jperio}
207\end{figure}
208
209%% =================================================================================================
210\subsection[North-fold (\forcode{=3,6})]{North-fold (\protect\jp{jperio}\forcode{=3,6})}
211\label{subsec:LBC_north_fold}
212
213The north fold boundary condition has been introduced in order to handle the north boundary of
214a three-polar ORCA grid.
215Such a grid has two poles in the northern hemisphere (\autoref{fig:CFGS_ORCA_msh},
216and thus requires a specific treatment illustrated in \autoref{fig:LBC_North_Fold_T}.
217Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition.
218
219\begin{figure}[!t]
220  \centering
221  \includegraphics[width=0.66\textwidth]{LBC_North_Fold_T}
222  \caption[North fold boundary in ORCA 2\deg, 1/4\deg and 1/12\deg]{
223    North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$),
224    as used in ORCA 2\deg, 1/4\deg and 1/12\deg.
225    Pink shaded area corresponds to the inner domain mask (see text).}
226  \label{fig:LBC_North_Fold_T}
227\end{figure}
228
229%% =================================================================================================
230\section[Exchange with neighbouring processors (\textit{lbclnk.F90}, \textit{lib\_mpp.F90})]{Exchange with neighbouring processors (\protect\mdl{lbclnk}, \protect\mdl{lib\_mpp})}
231\label{sec:LBC_mpp}
232
233\begin{listing}
234  \nlst{nammpp}
235  \caption{\forcode{&nammpp}}
236  \label{lst:nammpp}
237\end{listing}
238
239For massively parallel processing (mpp), a domain decomposition method is used.
240The basic idea of the method is to split the large computation domain of a numerical experiment into several smaller domains and
241solve the set of equations by addressing independent local problems.
242Each processor has its own local memory and computes the model equation over a subdomain of the whole model domain.
243The subdomain boundary conditions are specified through communications between processors which are organized by
244explicit statements (message passing method).
245The present implementation is largely inspired by Guyon's work [Guyon 1995].
246
247The parallelization strategy is defined by the physical characteristics of the ocean model.
248Second order finite difference schemes lead to local discrete operators that
249depend at the very most on one neighbouring point.
250The only non-local computations concern the vertical physics
251(implicit diffusion, turbulent closure scheme, ...).
252Therefore, a pencil strategy is used for the data sub-structuration:
253the 3D initial domain is laid out on local processor memories following a 2D horizontal topological splitting.
254Each sub-domain computes its own surface and bottom boundary conditions and
255has a side wall overlapping interface which defines the lateral boundary conditions for
256computations in the inner sub-domain.
257The overlapping area consists of the two rows at each edge of the sub-domain.
258After a computation, a communication phase starts:
259each processor sends to its neighbouring processors the update values of the points corresponding to
260the interior overlapping area to its neighbouring sub-domain (\ie\ the innermost of the two overlapping rows).
261Communications are first done according to the east-west direction and next according to the north-south direction.
262There is no specific communications for the corners.
263The communication is done through the Message Passing Interface (MPI) and requires \key{mpp\_mpi}.
264Use also \key{mpi2} if MPI3 is not available on your computer.
265The data exchanges between processors are required at the very place where
266lateral domain boundary conditions are set in the mono-domain computation:
267the \rou{lbc\_lnk} routine (found in \mdl{lbclnk} module) which manages such conditions is interfaced with
268routines found in \mdl{lib\_mpp} module.
269The output file \textit{communication\_report.txt} provides the list of which routines do how
270many communications during 1 time step of the model.\\
271
272\begin{figure}[!t]
273  \centering
274  \includegraphics[width=0.66\textwidth]{LBC_mpp}
275  \caption{Positioning of a sub-domain when massively parallel processing is used}
276  \label{fig:LBC_mpp}
277\end{figure}
278
279In \NEMO, the splitting is regular and arithmetic.
280The total number of subdomains corresponds to the number of MPI processes allocated to \NEMO\ when the model is launched
281(\ie\ mpirun -np x ./nemo will automatically give x subdomains).
282The i-axis is divided by \np{jpni}{jpni} and the j-axis by \np{jpnj}{jpnj}.
283These parameters are defined in \nam{mpp}{mpp} namelist.
284If \np{jpni}{jpni} and \np{jpnj}{jpnj} are < 1, they will be automatically redefined in the code to give the best domain decomposition
285(see bellow).
286
287Each processor is independent and without message passing or synchronous process, programs run alone and access just its own local memory.
288For this reason,
289the main model dimensions are now the local dimensions of the subdomain (pencil) that are named \jp{jpi}, \jp{jpj}, \jp{jpk}.
290These dimensions include the internal domain and the overlapping rows.
291The number of rows to exchange (known as the halo) is usually set to one (nn\_hls=1, in \mdl{par\_oce},
292and must be kept to one until further notice).
293The whole domain dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}.
294The relationship between the whole domain and a sub-domain is:
295\begin{gather*}
296  jpi = ( jpiglo-2\times nn\_hls + (jpni-1) ) / jpni + 2\times nn\_hls \\
297  jpj = ( jpjglo-2\times nn\_hls + (jpnj-1) ) / jpnj + 2\times nn\_hls
298\end{gather*}
299
300One 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.
301
302An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
303a global array (whole domain) by the relationship:
304$305 % \label{eq:LBC_nimpp} 306 T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k), 307$
308with $1 \leq i \leq jpi$, $1 \leq j \leq jpj$ , and  $1 \leq k \leq jpk$.
309
310The 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).\\
311
312The \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:
313$314 N_{mpi} = jpni \times jpnj - N_{land} + N_{useless} 315$
316$N_{land}$ 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.
317
318If 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.
319In 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.
320
321Processors 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 $N_{useless}$.
322
323When 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}).
324
325\begin{figure}[!ht]
326  \centering
327  \includegraphics[width=0.66\textwidth]{LBC_mppini2}
328  \caption[Atlantic domain defined for the CLIPPER projet]{
329    Example of Atlantic domain defined for the CLIPPER projet.
330    Initial grid is composed of 773 x 1236 horizontal points.
331    (a) the domain is split onto 9 $times$ 20 subdomains (jpni=9, jpnj=20).
332    52 subdomains are land areas.
333    (b) 52 subdomains are eliminated (white rectangles) and
334    the resulting number of processors really used during the computation is jpnij=128.}
335  \label{fig:LBC_mppini2}
336\end{figure}
337
338%% =================================================================================================
339\section{Unstructured open boundary conditions (BDY)}
340\label{sec:LBC_bdy}
341
342\begin{listing}
343  \nlst{nambdy}
344  \caption{\forcode{&nambdy}}
345  \label{lst:nambdy}
346\end{listing}
347
348\begin{listing}
349  \nlst{nambdy_dta}
350  \caption{\forcode{&nambdy_dta}}
351  \label{lst:nambdy_dta}
352\end{listing}
353
354Options are defined through the \nam{bdy}{bdy} and \nam{bdy_dta}{bdy\_dta} namelist variables.
355The BDY module is the core implementation of open boundary conditions for regional configurations on
356ocean temperature, salinity, barotropic-baroclinic velocities, ice-snow concentration, thicknesses, temperatures, salinity and melt ponds concentration and thickness.
357
358The BDY module was modelled on the OBC module (see \NEMO\ 3.4) and shares many features and
359a similar coding structure \citep{chanut_rpt05}.
360The specification of the location of the open boundary is completely flexible and
361allows any type of setup, from regular boundaries to irregular contour (it includes the possibility to set an open boundary able to follow an isobath).
362Boundary data files used with versions of \NEMO\ prior to Version 3.4 may need to be re-ordered to work with this version.
363See the section on the Input Boundary Data Files for details.
364
365%% =================================================================================================
366\subsection{Namelists}
367\label{subsec:LBC_bdy_namelist}
368
369The BDY module is activated by setting \np[=.true.]{ln_bdy}{ln\_bdy} .
370It is possible to define more than one boundary set'' and apply different boundary conditions to each set.
371The number of boundary sets is defined by \np{nb_bdy}{nb\_bdy}.
372Each boundary set can be either defined as a series of straight line segments directly in the namelist
373(\np[=.false.]{ln_coords_file}{ln\_coords\_file}, and a namelist block \nam{bdy_index}{bdy\_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).
374The coordinates.bdy file is analagous to the usual \NEMO\ \ifile{coordinates}'' file.
375In the example above, there are two boundary sets, the first of which is defined via a file and
376the second is defined in the namelist.
377For more details of the definition of the boundary geometry see section \autoref{subsec:LBC_bdy_geometry}.
378
379For each boundary set a boundary condition has to be chosen for the barotropic solution
380(u2d'':sea-surface height and barotropic velocities), for the baroclinic velocities (u3d''),
381for the active tracers \footnote{The BDY module does not deal with passive tracers at this version} (tra''), and for sea-ice (ice'').
382For 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).\\
383
384The choice of algorithm is currently as follows:
385
386\begin{description}
387\item [\forcode{'none'}:] No boundary condition applied.
388  So the solution will see'' the land points around the edge of the edge of the domain.
389\item [\forcode{'specified'}:] Specified boundary condition applied (only available for baroclinic velocity and tracer variables).
390\item [\forcode{'neumann'}:] Value at the boundary are duplicated (No gradient). Only available for baroclinic velocity and tracer variables.
391\item [\forcode{'frs'}:] Flow Relaxation Scheme (FRS) available for all variables.
392\item [\forcode{'Orlanski'}:] Orlanski radiation scheme (fully oblique) for barotropic, baroclinic and tracer variables.
393\item [\forcode{'Orlanski_npo'}:] Orlanski radiation scheme for barotropic, baroclinic and tracer variables.
394\item [\forcode{'flather'}:] Flather radiation scheme for the barotropic variables only.
395\end{description}
396
397The boundary data is either set to initial conditions
398(\np[=0]{nn_tra_dta}{nn\_tra\_dta}) or forced with external data from a file (\np[=1]{nn_tra_dta}{nn\_tra\_dta}).
399In case the 3d velocity data contain the total velocity (ie, baroclinic and barotropic velocity),
400the bdy code can derived baroclinic and barotropic velocities by setting \np[=.true.]{ln_full_vel}{ln\_full\_vel}
401For the barotropic solution there is also the option to use tidal harmonic forcing either by
402itself (\np[=2]{nn_dyn2d_dta}{nn\_dyn2d\_dta}) or in addition to other external data (\np[=3]{nn_dyn2d_dta}{nn\_dyn2d\_dta}).\\
403If 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}.
404
405If external boundary data is required then the \nam{bdy_dta}{bdy\_dta} namelist must be defined.
406One \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.
407In 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
408and the second one is using data from intial condition (no namelist block needed).
410so the \nam{bdy_dta}{bdy\_dta} namelist is in the format required for fldread.
411For each required variable, the filename, the frequency of the files and
412the frequency of the data in the files are given.
413Also whether or not time-interpolation is required and whether the data is climatological (time-cyclic) data.
414For sea-ice salinity, temperatures and melt ponds, reading the files are skipped and constant values are used if filenames are defined as {'NOT USED'}.\\
415
416There is currently an option to vertically interpolate the open boundary data onto the native grid at run-time.
417If \np{nn_bdy_jpk}{nn\_bdy\_jpk}$<-1$, it is assumed that the lateral boundary data are already on the native grid.
418However, if \np{nn_bdy_jpk}{nn\_bdy\_jpk} is set to the number of vertical levels present in the boundary data,
419a bilinear interpolation onto the native grid will be triggered at runtime.
420For this to be successful the additional variables: $gdept$, $gdepu$, $gdepv$, $e3t$, $e3u$ and $e3v$, are required to be present in the lateral boundary files.
421These correspond to the depths and scale factors of the input data,
422the latter used to make any adjustment to the velocity fields due to differences in the total water depths between the two vertical grids.\\
423
424In the example of given namelists, two boundary sets are defined.
425The first set is defined via a file and applies FRS conditions to temperature and salinity and
426Flather conditions to the barotropic variables. No condition specified for the baroclinic velocity and sea-ice.
427External data is provided in daily files (from a large-scale model).
428Tidal harmonic forcing is also used.
429The second set is defined in a namelist.
430FRS conditions are applied on temperature and salinity and climatological data is read from initial condition files.
431
432%% =================================================================================================
433\subsection{Flow relaxation scheme}
434\label{subsec:LBC_bdy_FRS_scheme}
435
436The Flow Relaxation Scheme (FRS) \citep{davies_QJRMS76,engedahl_T95},
437applies a simple relaxation of the model fields to externally-specified values over
438a zone next to the edge of the model domain.
439Given a model prognostic variable $\Phi$
440$441 % \label{eq:LBC_bdy_frs1} 442 \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N 443$
444where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified external field,
445$d$ gives the discrete distance from the model boundary and
446$\alpha$ is a parameter that varies from $1$ at $d=1$ to a small value at $d=N$.
447It can be shown that this scheme is equivalent to adding a relaxation term to
448the prognostic equation for $\Phi$ of the form:
449$450 % \label{eq:LBC_bdy_frs2} 451 -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right) 452$
453where the relaxation time scale $\tau$ is given by a function of $\alpha$ and the model time step $\Delta t$:
454$455 % \label{eq:LBC_bdy_frs3} 456 \tau = \frac{1-\alpha}{\alpha\,\rdt 457$
458Thus the model solution is completely prescribed by the external conditions at the edge of the model domain and
459is relaxed towards the external conditions over the rest of the FRS zone.
460The application of a relaxation zone helps to prevent spurious reflection of
461outgoing signals from the model boundary.
462
463The function $\alpha$ is specified as a $tanh$ function:
464$465 % \label{eq:LBC_bdy_frs4} 466 \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right), \quad d=1,N 467$
468The width of the FRS zone is specified in the namelist as \np{nn_rimwidth}{nn\_rimwidth}.
469This is typically set to a value between 8 and 10.
470
471%% =================================================================================================
473\label{subsec:LBC_bdy_flather_scheme}
474
475The \citet{flather_JPO94} scheme is a radiation condition on the normal,
476depth-mean transport across the open boundary.
477It takes the form
478\begin{equation}
479  \label{eq:LBC_bdy_fla1}
480  U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
481\end{equation}
482where $U$ is the depth-mean velocity normal to the boundary and $\eta$ is the sea surface height,
483both from the model.
484The subscript $e$ indicates the same fields from external sources.
485The speed of external gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the water column.
486The depth-mean normal velocity along the edge of the model domain is set equal to
487the external depth-mean normal velocity,
488plus a correction term that allows gravity waves generated internally to exit the model boundary.
489Note that the sea-surface height gradient in \autoref{eq:LBC_bdy_fla1} is a spatial gradient across the model boundary,
490so that $\eta_{e}$ is defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the $T$ points with $nbr=2$.
491$U$ and $U_{e}$ are defined on the $U$ or $V$ points with $nbr=1$, \ie\ between the two $T$ grid points.
492
493%% =================================================================================================
495\label{subsec:LBC_bdy_orlanski_scheme}
496
497The Orlanski scheme is based on the algorithm described by \citep{marchesiello.mcwilliams.ea_OM01}, hereafter MMS.
498
499The adaptive Orlanski condition solves a wave plus relaxation equation at the boundary:
500\begin{equation}
501  \label{eq:LBC_wave_continuous}
502  \frac{\partial\phi}{\partial t} + c_x \frac{\partial\phi}{\partial x} + c_y \frac{\partial\phi}{\partial y} =
503  -\frac{1}{\tau}(\phi - \phi^{ext})
504\end{equation}
505
506where $\phi$ is the model field, $x$ and $y$ refer to the normal and tangential directions to the boundary respectively, and the phase
507velocities are diagnosed from the model fields as:
508
509\begin{equation}
510  \label{eq:LBC_cx}
511  c_x = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial x}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}
512\end{equation}
513\begin{equation}
514  \label{eq:LBC_cy}
515  c_y = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial y}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}
516\end{equation}
517
518(As noted by MMS, this is a circular diagnosis of the phase speeds which only makes sense on a discrete grid).
519Equation (\autoref{eq:LBC_wave_continuous}) is defined adaptively depending on the sign of the phase velocity normal to the boundary $c_x$.
520For $c_x$ outward, we have
521
522\begin{equation}
523\tau = \tau_{out}
524\end{equation}
525
526For $c_x$ inward, the radiation equation is not applied:
527
528\begin{equation}
529  \label{eq:LBC_tau_in}
530  \tau = \tau_{in}\,\,\,;\,\,\, c_x = c_y = 0
531\end{equation}
532
533Generally 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
534points (\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.
535See \autoref{subsec:LBC_bdy_relaxation} for detailed on the spatial shape of the scaling.\\
536The normal propagation of oblique radiation'' or NPO approximation (called \forcode{'orlanski_npo'}) involves assuming
537that $c_y$ is zero in equation (\autoref{eq:LBC_wave_continuous}), but including
538this term in the denominator of equation (\autoref{eq:LBC_cx}). Both versions of the scheme are options in BDY. Equations
539(\autoref{eq:LBC_wave_continuous}) - (\autoref{eq:LBC_tau_in}) correspond to equations (13) - (15) and (2) - (3) in MMS.\\
540
541%% =================================================================================================
542\subsection{Relaxation at the boundary}
543\label{subsec:LBC_bdy_relaxation}
544
545In 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.
546It 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.
547
548The 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.
549This time scale ($\alpha$) is weighted by the distance ($d$) from the boundary over \np{nn_rimwidth}{nn\_rimwidth} cells ($N$):
550
551$552 \alpha = \frac{1}{\tau}(\frac{N+1-d}{N})^2, \quad d=1,N 553$
554
555The same scaling is applied in the Orlanski damping.
556
557%% =================================================================================================
558\subsection{Boundary geometry}
559\label{subsec:LBC_bdy_geometry}
560
561Each open boundary set is defined as a list of points.
562The information is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$ structure.
563The $nbi$ and $nbj$ arrays define the local $(i,j)$ indexes of each point in the boundary zone and
564the $nbr$ array defines the discrete distance from the boundary: $nbr=1$ means that
565the boundary point is next to the edge of the model domain, while $nbr>1$ means that
566the boundary point is increasingly further away from the edge of the model domain.
567A set of $nbi$, $nbj$, and $nbr$ arrays is defined for each of the $T$, $U$ and $V$ grids.
568\autoref{fig:LBC_bdy_geom} shows an example of an irregular boundary.
569
570The boundary geometry for each set may be defined in a namelist nambdy\_index or
571by reading in a \ifile{coordinates.bdy}'' file.
572The nambdy\_index namelist defines a series of straight-line segments for north, east, south and west boundaries.
573One nambdy\_index namelist block is needed for each boundary condition defined by indexes.
574For the northern boundary, \texttt{nbdysegn} gives the number of segments,
575\jp{jpjnob} gives the $j$ index for each segment and \jp{jpindt} and
576\jp{jpinft} give the start and end $i$ indices for each segment with similar for the other boundaries.
577These segments define a list of $T$ grid points along the outermost row of the boundary ($nbr\,=\, 1$).
578The code deduces the $U$ and $V$ points and also the points for $nbr\,>\, 1$ if \np[>1]{nn_rimwidth}{nn\_rimwidth}.
579
580The boundary geometry may also be defined from a \ifile{coordinates.bdy}'' file.
581\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.
582The file should contain the index arrays for each of the $T$, $U$ and $V$ grids.
583The arrays must be in order of increasing $nbr$.
584Note that the $nbi$, $nbj$ values in the file are global values and are converted to local values in the code.
585Typically this file will be used to generate external boundary data via interpolation and so
586will also contain the latitudes and longitudes of each point as shown.
587However, this is not necessary to run the model.
588
589For some choices of irregular boundary the model domain may contain areas of ocean which
590are not part of the computational domain.
591For example, if an open boundary is defined along an isobath, say at the shelf break,
592then the areas of ocean outside of this boundary will need to be masked out.
594Only one mask file is used even if multiple boundary sets are defined.
595
596\begin{figure}[!t]
597  \centering
598  \includegraphics[width=0.66\textwidth]{LBC_bdy_geom}
599  \caption[Geometry of unstructured open boundary]{Example of geometry of unstructured open boundary}
600  \label{fig:LBC_bdy_geom}
601\end{figure}
602
603%% =================================================================================================
604\subsection{Input boundary data files}
605\label{subsec:LBC_bdy_data}
606
607The data files contain the data arrays in the order in which the points are defined in the $nbi$ and $nbj$ arrays.
608The data arrays are dimensioned on:
609a time dimension;
610$xb$ which is the index of the boundary data point in the horizontal;
611and $yb$ which is a degenerate dimension of 1 to enable the file to be read by the standard \NEMO\ I/O routines.
612The 3D fields also have a depth dimension.
613
614From Version 3.4 there are new restrictions on the order in which the boundary points are defined
615(and therefore restrictions on the order of the data in the file).
616In particular:
617
618\begin{enumerate}
619\item The data points must be in order of increasing $nbr$,
620  ie. all the $nbr=1$ points, then all the $nbr=2$ points etc.
621\item All the data for a particular boundary set must be in the same order.
622  (Prior to 3.4 it was possible to define barotropic data in a different order to
623  the data for tracers and baroclinic velocities).
624\end{enumerate}
625
626These restrictions mean that data files used with versions of the
627model prior to Version 3.4 may not work with Version 3.4 onwards.
628A \fortran\ utility {\itshape bdy\_reorder} exists in the TOOLS directory which
629will re-order the data in old BDY data files.
630
631\begin{figure}[!t]
632  \centering
634  \caption[Header for a \protect\ifile{coordinates.bdy} file]{
635    Example of the header for a \protect\ifile{coordinates.bdy} file}
637\end{figure}
638
639%% =================================================================================================
640\subsection{Volume correction}
641\label{subsec:LBC_bdy_vol_corr}
642
643There is an option to force the total volume in the regional model to be constant.
644This is controlled  by the \np{ln_vol}{ln\_vol} parameter in the namelist.
645A value of \np[=.false.]{ln_vol}{ln\_vol} indicates that this option is not used.
646Two options to control the volume are available (\np{nn_volctl}{nn\_volctl}).
647If \np[=0]{nn_volctl}{nn\_volctl} then a correction is applied to the normal barotropic velocities around the boundary at
648each timestep to ensure that the integrated volume flow through the boundary is zero.
649If \np[=1]{nn_volctl}{nn\_volctl} then the calculation of the volume change on
650the timestep includes the change due to the freshwater flux across the surface and
651the correction velocity corrects for this as well.
652
653If more than one boundary set is used then volume correction is
654applied to all boundaries at once.
655
656%% =================================================================================================
657\subsection{Tidal harmonic forcing}
658\label{subsec:LBC_bdy_tides}
659
660\begin{listing}
661  \nlst{nambdy_tide}
662  \caption{\forcode{&nambdy_tide}}
663  \label{lst:nambdy_tide}
664\end{listing}
665
666Tidal forcing at open boundaries requires the activation of surface
667tides (i.e., in \nam{_tide}{\_tide}, \np{ln_tide}{ln\_tide} needs to be set to
668\forcode{.true.} and the required constituents need to be activated by
669including their names in the \np{clname}{clname} array; see
670\autoref{sec:SBC_tide}). Specific options related to the reading in of
671the complex harmonic amplitudes of elevation (SSH) and barotropic
672velocity (u,v) at open boundaries are defined through the
673\nam{bdy_tide}{bdy\_tide} namelist parameters.\\
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{ln_bdytide_2ddta }{ln\_bdytide\_2ddta } to \forcode{.true.} or
679\forcode{.false.}, respectively. In either case, the real and
680imaginary parts of SSH and the two barotropic velocity components for
681each activated tidal constituent \textit{tcname} have to be provided
682separately: when two-dimensional data is used, variables
683\textit{tcname\_z1} and \textit{tcname\_z2} for real and imaginary SSH,
684respectively, are expected in input file \np{filtide}{filtide} with suffix
685\ifile{\_grid\_T}, variables \textit{tcname\_u1} and
686\textit{tcname\_u2} for real and imaginary u, respectively, are
687expected in input file \np{filtide}{filtide} with suffix \ifile{\_grid\_U}, and
688\textit{tcname\_v1} and \textit{tcname\_v2} for real and imaginary v,
689respectively, are expected in input file \np{filtide}{filtide} with suffix
690\ifile{\_grid\_V}; when data along open boundary segments is used,
691variables \textit{z1} and \textit{z2} (real and imaginary part of SSH)
692are expected to be available from file \np{filtide}{filtide} with suffix
693\ifile{tcname\_grid\_T}, variables \textit{u1} and \textit{u2} (real
694and imaginary part of u) are expected to be available from file
695\np{filtide}{filtide} with suffix \ifile{tcname\_grid\_U}, and variables
696\textit{v1} and \textit{v2} (real and imaginary part of v) are
697expected to be available from file \np{filtide}{filtide} with suffix
698\ifile{tcname\_grid\_V}. If \np{ln_bdytide_conj}{ln\_bdytide\_conj} is set to
699\forcode{.true.}, the data is expected to be in complex conjugate
700form.
701
702Note that the barotropic velocity components are assumed to be defined
703on the native model grid and should be rotated accordingly when they
704are converted from their definition on a different source grid. To do
705so, the u, v amplitudes and phases can be converted into tidal
706ellipses, the grid rotation added to the ellipse inclination, and then
707converted back (care should be taken regarding conventions of the
708direction of rotation). %, e.g. anticlockwise or clockwise.
709
710\subinc{\input{../../global/epilogue}}
711
712\end{document}
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