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1\documentclass[../tex_main/NEMO_manual]{subfiles}
2\begin{document}
3% ================================================================
4% Chapter — Lateral Boundary Condition (LBC)
5% ================================================================
6\chapter{Lateral Boundary Condition (LBC)}
7\label{chap:LBC}
8\minitoc
9
10\newpage
11$\ $\newline    % force a new ligne
12
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 (\protect\np{rn\_shlat})}
20\label{sec:LBC_coast}
21%--------------------------------------------nam_lbc-------------------------------------------------------
22\forfile{../namelists/namlbc} 
23%--------------------------------------------------------------------------------------------------------------
24
25%The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt (no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip). They are handled automatically by the mask system (see \autoref{subsec:DOM_msk}).
26
27%OPA allows land and topography grid points in the computational domain due to the presence of continents or islands, and includes the use of a full or partial step representation of bottom topography. The computation is performed over the whole domain, i.e. we do not try to restrict the computation to ocean-only points. This choice has two motivations. Firstly, working on ocean only grid points overloads the code and harms the code readability. Secondly, and more importantly, it drastically reduces the vector portion of the computation, leading to a dramatic increase of CPU time requirement on vector computers.  The current section describes how the masking affects the computation of the various terms of the equations with respect to the boundary condition at solid walls. The process of defining which areas are to be masked is described in \autoref{subsec:DOM_msk}.
28
29Options are defined through the \ngn{namlbc} namelist variables.
30The discrete representation of a domain with complex boundaries (coastlines and
31bottom topography) leads to arrays that include large portions where a computation
32is not required as the model variables remain at zero. Nevertheless, vectorial
33supercomputers are far more efficient when computing over a whole array, and the
34readability of a code is greatly improved when boundary conditions are applied in
35an automatic way rather than by a specific computation before or after each
36computational loop. An efficient way to work over the whole domain while specifying
37the boundary conditions, is to use multiplication by mask arrays in the computation.
38A mask array is a matrix whose elements are $1$ in the ocean domain and $0$ 
39elsewhere. A simple multiplication of a variable by its own mask ensures that it will
40remain zero over land areas. Since most of the boundary conditions consist of a
41zero flux across the solid boundaries, they can be simply applied by multiplying
42variables by the correct mask arrays, $i.e.$ the mask array of the grid point where
43the flux is evaluated. For example, the heat flux in the \textbf{i}-direction is evaluated
44at $u$-points. Evaluating this quantity as,
45
46\begin{equation} \label{eq:lbc_aaaa}
47\frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 
48}{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u
49\end{equation}
50(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is
51zero inside land and at the boundaries, since mask$_{u}$ is zero at solid boundaries
52which in this case are defined at $u$-points (normal velocity $u$ remains zero at
53the coast) (\autoref{fig:LBC_uv}).
54
55%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
56\begin{figure}[!t]     \begin{center}
57\includegraphics[width=0.90\textwidth]{Fig_LBC_uv}
58\caption{  \protect\label{fig:LBC_uv}
59Lateral boundary (thick line) at T-level. The velocity normal to the boundary is set to zero.}
60\end{center}   \end{figure}
61%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
62
63For momentum the situation is a bit more complex as two boundary conditions
64must be provided along the coast (one each for the normal and tangential velocities).
65The boundary of the ocean in the C-grid is defined by the velocity-faces.
66For example, at a given $T$-level, the lateral boundary (a coastline or an intersection
67with the bottom topography) is made of segments joining $f$-points, and normal
68velocity points are located between two $f-$points (\autoref{fig:LBC_uv}).
69The boundary condition on the normal velocity (no flux through solid boundaries)
70can thus be easily implemented using the mask system. The boundary condition
71on the tangential velocity requires a more specific treatment. This boundary
72condition influences the relative vorticity and momentum diffusive trends, and is
73required in order to compute the vorticity at the coast. Four different types of
74lateral boundary condition are available, controlled by the value of the \np{rn\_shlat} 
75namelist parameter. (The value of the mask$_{f}$ array along the coastline is set
76equal to this parameter.) These are:
77
78%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
79\begin{figure}[!p] \begin{center}
80\includegraphics[width=0.90\textwidth]{Fig_LBC_shlat}
81\caption{     \protect\label{fig:LBC_shlat} 
82lateral boundary condition (a) free-slip ($rn\_shlat=0$) ; (b) no-slip ($rn\_shlat=2$)
83; (c) "partial" free-slip ($0<rn\_shlat<2$) and (d) "strong" no-slip ($2<rn\_shlat$).
84Implied "ghost" velocity inside land area is display in grey. }
85\end{center}    \end{figure}
86%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
87
88\begin{description}
89
90\item[free-slip boundary condition (\np{rn\_shlat}\forcode{ = 0}): ]  the tangential velocity at the
91coastline is equal to the offshore velocity, $i.e.$ the normal derivative of the
92tangential velocity is zero at the coast, so the vorticity: mask$_{f}$ array is set
93to zero inside the land and just at the coast (\autoref{fig:LBC_shlat}-a).
94
95\item[no-slip boundary condition (\np{rn\_shlat}\forcode{ = 2}): ] the tangential velocity vanishes
96at the coastline. Assuming that the tangential velocity decreases linearly from
97the closest ocean velocity grid point to the coastline, the normal derivative is
98evaluated as if the velocities at the closest land velocity gridpoint and the closest
99ocean velocity gridpoint were of the same magnitude but in the opposite direction
100(\autoref{fig:LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by:
101
102\begin{equation*}
103\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) \ ,
104\end{equation*}
105where $u$ and $v$ are masked fields. Setting the mask$_{f}$ array to $2$ along
106the coastline provides a vorticity field computed with the no-slip boundary condition,
107simply by multiplying it by the mask$_{f}$ :
108\begin{equation} \label{eq:lbc_bbbb}
109\zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2} 
110\left[ {e_{2v} \,v} \right]-\delta _{j+1/2} \left[ {e_{1u} \,u} \right]} 
111\right)\;\mbox{mask}_f
112\end{equation}
113
114\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2): ] the tangential
115velocity at the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral
116friction but not strong enough to make the tangential velocity at the coast vanish
117(\autoref{fig:LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$ 
118strictly inbetween $0$ and $2$.
119
120\item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}): ] the viscous boundary
121layer is assumed to be smaller than half the grid size (\autoref{fig:LBC_shlat}-d).
122The friction is thus larger than in the no-slip case.
123
124\end{description}
125
126Note that when the bottom topography is entirely represented by the $s$-coor-dinates
127(pure $s$-coordinate), the lateral boundary condition on tangential velocity is of much
128less importance as it is only applied next to the coast where the minimum water depth
129can be quite shallow.
130
131
132% ================================================================
133% Boundary Condition around the Model Domain
134% ================================================================
135\section{Model domain boundary condition (\protect\np{jperio})}
136\label{sec:LBC_jperio}
137
138At the model domain boundaries several choices are offered: closed, cyclic east-west,
139south symmetric across the equator, a north-fold, and combination closed-north fold
140or cyclic-north-fold. The north-fold boundary condition is associated with the 3-pole ORCA mesh.
141
142% -------------------------------------------------------------------------------------------------------------
143%        Closed, cyclic, south symmetric (\np{jperio}\forcode{ = 0..2})
144% -------------------------------------------------------------------------------------------------------------
145\subsection{Closed, cyclic, south symmetric (\protect\np{jperio}\forcode{= 0..2})}
146\label{subsec:LBC_jperio012}
147
148The choice of closed, cyclic or symmetric model domain boundary condition is made
149by setting \np{jperio} to 0, 1 or 2 in namelist \ngn{namcfg}. Each time such a boundary
150condition is needed, it is set by a call to routine \mdl{lbclnk}. The computation of
151momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to
152$j=jpj-1$, $i.e.$ in the model interior. To choose a lateral model boundary condition
153is to specify the first and last rows and columns of the model variables.
154
155\begin{description}
156
157\item[For closed boundary (\np{jperio}\forcode{ = 0})], solid walls are imposed at all model
158boundaries: first and last rows and columns are set to zero.
159
160\item[For cyclic east-west boundary (\np{jperio}\forcode{ = 1})], first and last rows are set
161to zero (closed) whilst the first column is set to the value of the last-but-one column
162and the last column to the value of the second one (\autoref{fig:LBC_jperio}-a).
163Whatever flows out of the eastern (western) end of the basin enters the western
164(eastern) end. Note that there is no option for north-south cyclic or for doubly
165cyclic cases.
166
167\item[For symmetric boundary condition across the equator (\np{jperio}\forcode{ = 2})],
168last rows, and first and last columns are set to zero (closed). The row of symmetry
169is chosen to be the $u$- and $T-$points equator line ($j=2$, i.e. at the southern
170end of the domain). For arrays defined at $u-$ or $T-$points, the first row is set
171to the value of the third row while for most of $v$- and $f$-point arrays ($v$, $\zeta$,
172$j\psi$, but \gmcomment{not sure why this is "but"} scalar arrays such as eddy coefficients)
173the first row is set to minus the value of the second row (\autoref{fig:LBC_jperio}-b).
174Note that this boundary condition is not yet available for the case of a massively
175parallel computer (\textbf{key{\_}mpp} defined).
176
177\end{description}
178
179%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
180\begin{figure}[!t]     \begin{center}
181\includegraphics[width=1.0\textwidth]{Fig_LBC_jperio}
182\caption{    \protect\label{fig:LBC_jperio}
183setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.}
184\end{center}   \end{figure}
185%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
186
187% -------------------------------------------------------------------------------------------------------------
188%        North fold (\textit{jperio = 3 }to $6)$
189% -------------------------------------------------------------------------------------------------------------
190\subsection{North-fold (\protect\np{jperio}\forcode{ = 3..6})}
191\label{subsec:LBC_north_fold}
192
193The north fold boundary condition has been introduced in order to handle the north
194boundary of a three-polar ORCA grid. Such a grid has two poles in the northern hemisphere
195(\autoref{fig:MISC_ORCA_msh}, and thus requires a specific treatment illustrated in \autoref{fig:North_Fold_T}.
196Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition.
197
198%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
199\begin{figure}[!t]    \begin{center}
200\includegraphics[width=0.90\textwidth]{Fig_North_Fold_T}
201\caption{    \protect\label{fig:North_Fold_T} 
202North fold boundary with a $T$-point pivot and cyclic east-west boundary condition
203($jperio=4$), as used in ORCA 2, 1/4, and 1/12. Pink shaded area corresponds
204to the inner domain mask (see text). }
205\end{center}   \end{figure}
206%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
207
208% ====================================================================
209% Exchange with neighbouring processors
210% ====================================================================
211\section{Exchange with neighbouring processors (\protect\mdl{lbclnk}, \protect\mdl{lib\_mpp})}
212\label{sec:LBC_mpp}
213
214For massively parallel processing (mpp), a domain decomposition method is used.
215The basic idea of the method is to split the large computation domain of a numerical
216experiment into several smaller domains and solve the set of equations by addressing
217independent local problems. Each processor has its own local memory and computes
218the model equation over a subdomain of the whole model domain. The subdomain
219boundary conditions are specified through communications between processors
220which are organized by explicit statements (message passing method).
221
222A big advantage is that the method does not need many modifications of the initial
223FORTRAN code. From the modeller's point of view, each sub domain running on
224a processor is identical to the "mono-domain" code. In addition, the programmer
225manages the communications between subdomains, and the code is faster when
226the number of processors is increased. The porting of OPA code on an iPSC860
227was achieved during Guyon's PhD [Guyon et al. 1994, 1995] in collaboration with
228CETIIS and ONERA. The implementation in the operational context and the studies
229of performance on a T3D and T3E Cray computers have been made in collaboration
230with IDRIS and CNRS. The present implementation is largely inspired by Guyon's
231work  [Guyon 1995].
232
233The parallelization strategy is defined by the physical characteristics of the
234ocean model. Second order finite difference schemes lead to local discrete
235operators that depend at the very most on one neighbouring point. The only
236non-local computations concern the vertical physics (implicit diffusion,
237turbulent closure scheme, ...) (delocalization over the whole water column),
238and the solving of the elliptic equation associated with the surface pressure
239gradient computation (delocalization over the whole horizontal domain).
240Therefore, a pencil strategy is used for the data sub-structuration
241: the 3D initial domain is laid out on local processor
242memories following a 2D horizontal topological splitting. Each sub-domain
243computes its own surface and bottom boundary conditions and has a side
244wall overlapping interface which defines the lateral boundary conditions for
245computations in the inner sub-domain. The overlapping area consists of the
246two rows at each edge of the sub-domain. After a computation, a communication
247phase starts: each processor sends to its neighbouring processors the update
248values of the points corresponding to the interior overlapping area to its
249neighbouring sub-domain ($i.e.$ the innermost of the two overlapping rows).
250The communication is done through the Message Passing Interface (MPI).
251The data exchanges between processors are required at the very
252place where lateral domain boundary conditions are set in the mono-domain
253computation : the \rou{lbc\_lnk} routine (found in \mdl{lbclnk} module)
254which manages such conditions is interfaced with routines found in \mdl{lib\_mpp} module
255when running on an MPP computer ($i.e.$ when \key{mpp\_mpi} defined).
256It has to be pointed out that when using the MPP version of the model,
257the east-west cyclic boundary condition is done implicitly,
258whilst the south-symmetric boundary condition option is not available.
259
260%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
261\begin{figure}[!t]    \begin{center}
262\includegraphics[width=0.90\textwidth]{Fig_mpp}
263\caption{   \protect\label{fig:mpp} 
264Positioning of a sub-domain when massively parallel processing is used. }
265\end{center}   \end{figure}
266%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
267
268In the standard version of \NEMO, the splitting is regular and arithmetic.
269The i-axis is divided by \jp{jpni} and the j-axis by \jp{jpnj} for a number of processors
270\jp{jpnij} most often equal to $jpni \times jpnj$ (parameters set in
271 \ngn{nammpp} namelist). Each processor is independent and without message passing
272 or synchronous process, programs run alone and access just its own local memory.
273 For this reason, the main model dimensions are now the local dimensions of the subdomain (pencil)
274 that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. These dimensions include the internal
275 domain and the overlapping rows. The number of rows to exchange (known as
276 the halo) is usually set to one (\jp{jpreci}=1, in \mdl{par\_oce}). The whole domain
277 dimensions are named \np{jpiglo}, \np{jpjglo} and \jp{jpk}. The relationship between
278 the whole domain and a sub-domain is:
279\begin{eqnarray} 
280      jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\
281      jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj  \label{eq:lbc_jpi}
282\end{eqnarray}
283where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis.
284
285One also defines variables nldi and nlei which correspond to the internal domain bounds,
286and the variables nimpp and njmpp which are the position of the (1,1) grid-point in the global domain.
287An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
288a global array (whole domain) by the relationship:
289\begin{equation} \label{eq:lbc_nimpp}
290T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
291\end{equation}
292with  $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.
293
294Processors are numbered from 0 to $jpnij-1$, the number is saved in the variable
295nproc. In the standard version, a processor has no more than four neighbouring
296processors named nono (for north), noea (east), noso (south) and nowe (west)
297and two variables, nbondi and nbondj, indicate the relative position of the processor :
298\begin{itemize}
299\item       nbondi = -1    an east neighbour, no west processor,
300\item       nbondi =  0 an east neighbour, a west neighbour,
301\item       nbondi =  1    no east processor, a west neighbour,
302\item       nbondi =  2    no splitting following the i-axis.
303\end{itemize}
304During the simulation, processors exchange data with their neighbours.
305If there is effectively a neighbour, the processor receives variables from this
306processor on its overlapping row, and sends the data issued from internal
307domain corresponding to the overlapping row of the other processor.
308
309
310The \NEMO model computes equation terms with the help of mask arrays (0 on land
311points and 1 on sea points). It is easily readable and very efficient in the context of
312a computer with vectorial architecture. However, in the case of a scalar processor,
313computations over the land regions become more expensive in terms of CPU time.
314It is worse when we use a complex configuration with a realistic bathymetry like the
315global ocean where more than 50 \% of points are land points. For this reason, a
316pre-processing tool can be used to choose the mpp domain decomposition with a
317maximum number of only land points processors, which can then be eliminated (\autoref{fig:mppini2})
318(For example, the mpp\_optimiz tools, available from the DRAKKAR web site).
319This optimisation is dependent on the specific bathymetry employed. The user
320then chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with
321$jpnij < jpni \times jpnj$, leading to the elimination of $jpni \times jpnj - jpnij$ 
322land processors. When those parameters are specified in \ngn{nammpp} namelist,
323the algorithm in the \rou{inimpp2} routine sets each processor's parameters (nbound,
324nono, noea,...) so that the land-only processors are not taken into account.
325
326\gmcomment{Note that the inimpp2 routine is general so that the original inimpp
327routine should be suppressed from the code.}
328
329When land processors are eliminated, the value corresponding to these locations in
330the model output files is undefined. Note that this is a problem for the meshmask file
331which requires to be defined over the whole domain. Therefore, user should not eliminate
332land processors when creating a meshmask file ($i.e.$ when setting a non-zero value to \np{nn\_msh}).
333
334%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
335\begin{figure}[!ht]     \begin{center}
336\includegraphics[width=0.90\textwidth]{Fig_mppini2}
337\caption {    \protect\label{fig:mppini2}
338Example of Atlantic domain defined for the CLIPPER projet. Initial grid is
339composed of 773 x 1236 horizontal points.
340(a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
34152 subdomains are land areas.
342(b) 52 subdomains are eliminated (white rectangles) and the resulting number
343of processors really used during the computation is jpnij=128.}
344\end{center}   \end{figure}
345%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
346
347
348% ====================================================================
349% Unstructured open boundaries BDY
350% ====================================================================
351\section{Unstructured open boundary conditions (BDY)}
352\label{sec:LBC_bdy}
353
354%-----------------------------------------nambdy--------------------------------------------
355\forfile{../namelists/nambdy} 
356%-----------------------------------------------------------------------------------------------
357%-----------------------------------------nambdy_index--------------------------------------------
358%\forfile{../namelists/nambdy_index}
359%-----------------------------------------------------------------------------------------------
360%-----------------------------------------nambdy_dta--------------------------------------------
361\forfile{../namelists/nambdy_dta} 
362%-----------------------------------------------------------------------------------------------
363%-----------------------------------------nambdy_dta--------------------------------------------
364%\forfile{../namelists/nambdy_dta2}
365%-----------------------------------------------------------------------------------------------
366
367Options are defined through the \ngn{nambdy} \ngn{nambdy\_index} 
368\ngn{nambdy\_dta} \ngn{nambdy\_dta2} namelist variables.
369The BDY module is the core implementation of open boundary
370conditions for regional configurations. It implements the Flow
371Relaxation Scheme algorithm for temperature, salinity, velocities and
372ice fields, and the Flather radiation condition for the depth-mean
373transports. The specification of the location of the open boundary is
374completely flexible and allows for example the open boundary to follow
375an isobath or other irregular contour.
376
377The BDY module was modelled on the OBC module (see NEMO 3.4) and shares many
378features and a similar coding structure \citep{Chanut2005}.
379
380Boundary data files used with earlier versions of NEMO may need
381to be re-ordered to work with this version. See the
382section on the Input Boundary Data Files for details.
383
384%----------------------------------------------
385\subsection{Namelists}
386\label{subsec:BDY_namelist}
387
388The BDY module is activated by setting \np{ln\_bdy} to true.
389It is possible to define more than one boundary ``set'' and apply
390different boundary conditions to each set. The number of boundary
391sets is defined by \np{nb\_bdy}.  Each boundary set may be defined
392as a set of straight line segments in a namelist
393(\np{ln\_coords\_file}\forcode{ = .false.}) or read in from a file
394(\np{ln\_coords\_file}\forcode{ = .true.}). If the set is defined in a namelist,
395then the namelists nambdy\_index must be included separately, one for
396each set. If the set is defined by a file, then a
397``\ifile{coordinates.bdy}'' file must be provided. The coordinates.bdy file
398is analagous to the usual NEMO ``\ifile{coordinates}'' file. In the example
399above, there are two boundary sets, the first of which is defined via
400a file and the second is defined in a namelist. For more details of
401the definition of the boundary geometry see section
402\autoref{subsec:BDY_geometry}.
403
404For each boundary set a boundary
405condition has to be chosen for the barotropic solution (``u2d'':
406sea-surface height and barotropic velocities), for the baroclinic
407velocities (``u3d''), and for the active tracers\footnote{The BDY
408  module does not deal with passive tracers at this version}
409(``tra''). For each set of variables there is a choice of algorithm
410and a choice for the data, eg. for the active tracers the algorithm is
411set by \np{nn\_tra} and the choice of data is set by
412\np{nn\_tra\_dta}.
413
414The choice of algorithm is currently as follows:
415
416\mbox{}
417
418\begin{itemize}
419\item[0.] No boundary condition applied. So the solution will ``see''
420  the land points around the edge of the edge of the domain.
421\item[1.] Flow Relaxation Scheme (FRS) available for all variables.
422\item[2.] Flather radiation scheme for the barotropic variables. The
423  Flather scheme is not compatible with the filtered free surface
424  ({\it dynspg\_ts}).
425\end{itemize}
426
427\mbox{}
428
429The main choice for the boundary data is
430to use initial conditions as boundary data (\np{nn\_tra\_dta}\forcode{ = 0}) or to
431use external data from a file (\np{nn\_tra\_dta}\forcode{ = 1}). For the
432barotropic solution there is also the option to use tidal
433harmonic forcing either by itself or in addition to other external
434data.
435
436If external boundary data is required then the nambdy\_dta namelist
437must be defined. One nambdy\_dta namelist is required for each boundary
438set in the order in which the boundary sets are defined in nambdy. In
439the example given, two boundary sets have been defined and so there
440are two nambdy\_dta namelists. The boundary data is read in using the
441fldread module, so the nambdy\_dta namelist is in the format required
442for fldread. For each variable required, the filename, the frequency
443of the files and the frequency of the data in the files is given. Also
444whether or not time-interpolation is required and whether the data is
445climatological (time-cyclic) data. Note that on-the-fly spatial
446interpolation of boundary data is not available at this version.
447
448In the example namelists given, two boundary sets are defined. The
449first set is defined via a file and applies FRS conditions to
450temperature and salinity and Flather conditions to the barotropic
451variables. External data is provided in daily files (from a
452large-scale model). Tidal harmonic forcing is also used. The second
453set is defined in a namelist. FRS conditions are applied on
454temperature and salinity and climatological data is read from external
455files.
456
457%----------------------------------------------
458\subsection{Flow relaxation scheme}
459\label{subsec:BDY_FRS_scheme}
460
461The Flow Relaxation Scheme (FRS) \citep{Davies_QJRMS76,Engerdahl_Tel95},
462applies a simple relaxation of the model fields to
463externally-specified values over a zone next to the edge of the model
464domain. Given a model prognostic variable $\Phi$ 
465\begin{equation}  \label{eq:bdy_frs1}
466\Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
467\end{equation}
468where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified
469external field, $d$ gives the discrete distance from the model
470boundary  and $\alpha$ is a parameter that varies from $1$ at $d=1$ to
471a small value at $d=N$. It can be shown that this scheme is equivalent
472to adding a relaxation term to the prognostic equation for $\Phi$ of
473the form:
474\begin{equation}  \label{eq:bdy_frs2}
475-\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
476\end{equation}
477where the relaxation time scale $\tau$ is given by a function of
478$\alpha$ and the model time step $\Delta t$:
479\begin{equation}  \label{eq:bdy_frs3}
480\tau = \frac{1-\alpha}{\alpha}  \,\rdt
481\end{equation}
482Thus the model solution is completely prescribed by the external
483conditions at the edge of the model domain and is relaxed towards the
484external conditions over the rest of the FRS zone. The application of
485a relaxation zone helps to prevent spurious reflection of outgoing
486signals from the model boundary.
487
488The function $\alpha$ is specified as a $tanh$ function:
489\begin{equation}  \label{eq:bdy_frs4}
490\alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
491\end{equation}
492The width of the FRS zone is specified in the namelist as
493\np{nn\_rimwidth}. This is typically set to a value between 8 and 10.
494
495%----------------------------------------------
496\subsection{Flather radiation scheme}
497\label{subsec:BDY_flather_scheme}
498
499The \citet{Flather_JPO94} scheme is a radiation condition on the normal, depth-mean
500transport across the open boundary. It takes the form
501\begin{equation}  \label{eq:bdy_fla1}
502U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
503\end{equation}
504where $U$ is the depth-mean velocity normal to the boundary and $\eta$
505is the sea surface height, both from the model. The subscript $e$
506indicates the same fields from external sources. The speed of external
507gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the
508water column. The depth-mean normal velocity along the edge of the
509model domain is set equal to the
510external depth-mean normal velocity, plus a correction term that
511allows gravity waves generated internally to exit the model boundary.
512Note that the sea-surface height gradient in \autoref{eq:bdy_fla1}
513is a spatial gradient across the model boundary, so that $\eta_{e}$ is
514defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the
515$T$ points with $nbr=2$. $U$ and $U_{e}$ are defined on the $U$ or
516$V$ points with $nbr=1$, $i.e.$ between the two $T$ grid points.
517
518%----------------------------------------------
519\subsection{Boundary geometry}
520\label{subsec:BDY_geometry}
521
522Each open boundary set is defined as a list of points. The information
523is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$
524structure.  The $nbi$ and $nbj$ arrays
525define the local $(i,j)$ indices of each point in the boundary zone
526and the $nbr$ array defines the discrete distance from the boundary
527with $nbr=1$ meaning that the point is next to the edge of the
528model domain and $nbr>1$ showing that the point is increasingly
529further away from the edge of the model domain. A set of $nbi$, $nbj$,
530and $nbr$ arrays is defined for each of the $T$, $U$ and $V$
531grids. Figure \autoref{fig:LBC_bdy_geom} shows an example of an irregular
532boundary.
533
534The boundary geometry for each set may be defined in a namelist
535nambdy\_index or by reading in a ``\ifile{coordinates.bdy}'' file. The
536nambdy\_index namelist defines a series of straight-line segments for
537north, east, south and west boundaries. For the northern boundary,
538\np{nbdysegn} gives the number of segments, \np{jpjnob} gives the $j$
539index for each segment and \np{jpindt} and \np{jpinft} give the start
540and end $i$ indices for each segment with similar for the other
541boundaries. These segments define a list of $T$ grid points along the
542outermost row of the boundary ($nbr\,=\, 1$). The code deduces the $U$ and
543$V$ points and also the points for $nbr\,>\, 1$ if
544$nn\_rimwidth\,>\,1$.
545
546The boundary geometry may also be defined from a
547``\ifile{coordinates.bdy}'' file. Figure \autoref{fig:LBC_nc_header}
548gives an example of the header information from such a file. The file
549should contain the index arrays for each of the $T$, $U$ and $V$
550grids. The arrays must be in order of increasing $nbr$. Note that the
551$nbi$, $nbj$ values in the file are global values and are converted to
552local values in the code. Typically this file will be used to generate
553external boundary data via interpolation and so will also contain the
554latitudes and longitudes of each point as shown. However, this is not
555necessary to run the model.
556
557For some choices of irregular boundary the model domain may contain
558areas of ocean which are not part of the computational domain. For
559example if an open boundary is defined along an isobath, say at the
560shelf break, then the areas of ocean outside of this boundary will
561need to be masked out. This can be done by reading a mask file defined
562as \np{cn\_mask\_file} in the nam\_bdy namelist. Only one mask file is
563used even if multiple boundary sets are defined.
564
565%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
566\begin{figure}[!t]      \begin{center}
567\includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom}
568\caption {      \protect\label{fig:LBC_bdy_geom}
569Example of geometry of unstructured open boundary}
570\end{center}   \end{figure}
571%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
572
573%----------------------------------------------
574\subsection{Input boundary data files}
575\label{subsec:BDY_data}
576
577The data files contain the data arrays
578in the order in which the points are defined in the $nbi$ and $nbj$
579arrays. The data arrays are dimensioned on: a time dimension;
580$xb$ which is the index of the boundary data point in the horizontal;
581and $yb$ which is a degenerate dimension of 1 to enable the file to be
582read by the standard NEMO I/O routines. The 3D fields also have a
583depth dimension.
584
585At Version 3.4 there are new restrictions on the order in which the
586boundary points are defined (and therefore restrictions on the order
587of the data in the file). In particular:
588
589\mbox{}
590
591\begin{enumerate}
592\item The data points must be in order of increasing $nbr$, ie. all
593  the $nbr=1$ points, then all the $nbr=2$ points etc.
594\item All the data for a particular boundary set must be in the same
595  order. (Prior to 3.4 it was possible to define barotropic data in a
596  different order to the data for tracers and baroclinic velocities).
597\end{enumerate}
598
599\mbox{}
600
601These restrictions mean that data files used with previous versions of
602the model may not work with version 3.4. A fortran utility
603{\it bdy\_reorder} exists in the TOOLS directory which will re-order the
604data in old BDY data files.
605
606%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
607\begin{figure}[!t]     \begin{center}
608\includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header}
609\caption {     \protect\label{fig:LBC_nc_header} 
610Example of the header for a \protect\ifile{coordinates.bdy} file}
611\end{center}   \end{figure}
612%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
613
614%----------------------------------------------
615\subsection{Volume correction}
616\label{subsec:BDY_vol_corr}
617
618There is an option to force the total volume in the regional model to be constant,
619similar to the option in the OBC module. This is controlled  by the \np{nn\_volctl} 
620parameter in the namelist. A value of \np{nn\_volctl}\forcode{ = 0} indicates that this option is not used.
621If  \np{nn\_volctl}\forcode{ = 1} then a correction is applied to the normal velocities
622around the boundary at each timestep to ensure that the integrated volume flow
623through the boundary is zero. If \np{nn\_volctl}\forcode{ = 2} then the calculation of
624the volume change on the timestep includes the change due to the freshwater
625flux across the surface and the correction velocity corrects for this as well.
626
627If more than one boundary set is used then volume correction is
628applied to all boundaries at once.
629
630\newpage
631%----------------------------------------------
632\subsection{Tidal harmonic forcing}
633\label{subsec:BDY_tides}
634
635%-----------------------------------------nambdy_tide--------------------------------------------
636\forfile{../namelists/nambdy_tide} 
637%-----------------------------------------------------------------------------------------------
638
639Options are defined through the  \ngn{nambdy\_tide} namelist variables.
640 To be written....
641
642
643
644
645\end{document}
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