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1\documentclass[NEMO_book]{subfiles}
2\begin{document}
3% ================================================================
4% Chapter — Lateral Boundary Condition (LBC)
5% ================================================================
6\chapter{Lateral Boundary Condition (LBC) }
7\label{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{LBC_coast}
21%--------------------------------------------nam_lbc-------------------------------------------------------
22\namdisplay{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 \S\ref{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 \S\ref{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) (Fig.~\ref{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 (Fig.~\ref{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}=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 (Fig.~\ref{Fig_LBC_shlat}-a).
94
95\item[no-slip boundary condition (\np{rn\_shlat}=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(Fig.~\ref{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(Fig.~\ref{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 (Fig.~\ref{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{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} = 0, 1 or 2)
144% -------------------------------------------------------------------------------------------------------------
145\subsection{Closed, cyclic, south symmetric (\protect\np{jperio} = 0, 1 or 2)}
146\label{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 (\textit{jperio=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 (\textit{jperio=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 (Fig.~\ref{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 (\textit{jperio=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 (Fig.~\ref{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 (\textit{jperio = 3 }to $6$) }
191\label{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(Fig.\ref{Fig_MISC_ORCA_msh}, and thus requires a specific treatment illustrated in Fig.\ref{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 (\textit{lbclnk}, \textit{lib\_mpp})]
212      {Exchange with neighbouring processors (\protect\mdl{lbclnk}, \protect\mdl{lib\_mpp})}
213\label{LBC_mpp}
214
215For massively parallel processing (mpp), a domain decomposition method is used.
216The basic idea of the method is to split the large computation domain of a numerical
217experiment into several smaller domains and solve the set of equations by addressing
218independent local problems. Each processor has its own local memory and computes
219the model equation over a subdomain of the whole model domain. The subdomain
220boundary conditions are specified through communications between processors
221which are organized by explicit statements (message passing method).
222
223A big advantage is that the method does not need many modifications of the initial
224FORTRAN code. From the modeller's point of view, each sub domain running on
225a processor is identical to the "mono-domain" code. In addition, the programmer
226manages the communications between subdomains, and the code is faster when
227the number of processors is increased. The porting of OPA code on an iPSC860
228was achieved during Guyon's PhD [Guyon et al. 1994, 1995] in collaboration with
229CETIIS and ONERA. The implementation in the operational context and the studies
230of performance on a T3D and T3E Cray computers have been made in collaboration
231with IDRIS and CNRS. The present implementation is largely inspired by Guyon's
232work  [Guyon 1995].
233
234The parallelization strategy is defined by the physical characteristics of the
235ocean model. Second order finite difference schemes lead to local discrete
236operators that depend at the very most on one neighbouring point. The only
237non-local computations concern the vertical physics (implicit diffusion,
238turbulent closure scheme, ...) (delocalization over the whole water column),
239and the solving of the elliptic equation associated with the surface pressure
240gradient computation (delocalization over the whole horizontal domain).
241Therefore, a pencil strategy is used for the data sub-structuration
242: the 3D initial domain is laid out on local processor
243memories following a 2D horizontal topological splitting. Each sub-domain
244computes its own surface and bottom boundary conditions and has a side
245wall overlapping interface which defines the lateral boundary conditions for
246computations in the inner sub-domain. The overlapping area consists of the
247two rows at each edge of the sub-domain. After a computation, a communication
248phase starts: each processor sends to its neighbouring processors the update
249values of the points corresponding to the interior overlapping area to its
250neighbouring sub-domain ($i.e.$ the innermost of the two overlapping rows).
251The communication is done through the Message Passing Interface (MPI).
252The data exchanges between processors are required at the very
253place where lateral domain boundary conditions are set in the mono-domain
254computation : the \rou{lbc\_lnk} routine (found in \mdl{lbclnk} module)
255which manages such conditions is interfaced with routines found in \mdl{lib\_mpp} module
256when running on an MPP computer ($i.e.$ when \key{mpp\_mpi} defined).
257It has to be pointed out that when using the MPP version of the model,
258the east-west cyclic boundary condition is done implicitly,
259whilst the south-symmetric boundary condition option is not available.
260
261%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
262\begin{figure}[!t]    \begin{center}
263\includegraphics[width=0.90\textwidth]{Fig_mpp}
264\caption{   \protect\label{Fig_mpp} 
265Positioning of a sub-domain when massively parallel processing is used. }
266\end{center}   \end{figure}
267%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
268
269In the standard version of \NEMO, the splitting is regular and arithmetic.
270The i-axis is divided by \jp{jpni} and the j-axis by \jp{jpnj} for a number of processors
271\jp{jpnij} most often equal to $jpni \times jpnj$ (parameters set in
272 \ngn{nammpp} namelist). Each processor is independent and without message passing
273 or synchronous process, programs run alone and access just its own local memory.
274 For this reason, the main model dimensions are now the local dimensions of the subdomain (pencil)
275 that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. These dimensions include the internal
276 domain and the overlapping rows. The number of rows to exchange (known as
277 the halo) is usually set to one (\jp{jpreci}=1, in \mdl{par\_oce}). The whole domain
278 dimensions are named \np{jpiglo}, \np{jpjglo} and \jp{jpk}. The relationship between
279 the whole domain and a sub-domain is:
280\begin{eqnarray} 
281      jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\
282      jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj  \label{Eq_lbc_jpi}
283\end{eqnarray}
284where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis.
285
286One also defines variables nldi and nlei which correspond to the internal domain bounds,
287and the variables nimpp and njmpp which are the position of the (1,1) grid-point in the global domain.
288An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
289a global array (whole domain) by the relationship:
290\begin{equation} \label{Eq_lbc_nimpp}
291T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
292\end{equation}
293with  $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.
294
295Processors are numbered from 0 to $jpnij-1$, the number is saved in the variable
296nproc. In the standard version, a processor has no more than four neighbouring
297processors named nono (for north), noea (east), noso (south) and nowe (west)
298and two variables, nbondi and nbondj, indicate the relative position of the processor :
299\begin{itemize}
300\item       nbondi = -1    an east neighbour, no west processor,
301\item       nbondi =  0 an east neighbour, a west neighbour,
302\item       nbondi =  1    no east processor, a west neighbour,
303\item       nbondi =  2    no splitting following the i-axis.
304\end{itemize}
305During the simulation, processors exchange data with their neighbours.
306If there is effectively a neighbour, the processor receives variables from this
307processor on its overlapping row, and sends the data issued from internal
308domain corresponding to the overlapping row of the other processor.
309
310
311The \NEMO model computes equation terms with the help of mask arrays (0 on land
312points and 1 on sea points). It is easily readable and very efficient in the context of
313a computer with vectorial architecture. However, in the case of a scalar processor,
314computations over the land regions become more expensive in terms of CPU time.
315It is worse when we use a complex configuration with a realistic bathymetry like the
316global ocean where more than 50 \% of points are land points. For this reason, a
317pre-processing tool can be used to choose the mpp domain decomposition with a
318maximum number of only land points processors, which can then be eliminated (Fig. \ref{Fig_mppini2})
319(For example, the mpp\_optimiz tools, available from the DRAKKAR web site).
320This optimisation is dependent on the specific bathymetry employed. The user
321then chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with
322$jpnij < jpni \times jpnj$, leading to the elimination of $jpni \times jpnj - jpnij$ 
323land processors. When those parameters are specified in \ngn{nammpp} namelist,
324the algorithm in the \rou{inimpp2} routine sets each processor's parameters (nbound,
325nono, noea,...) so that the land-only processors are not taken into account.
326
327\gmcomment{Note that the inimpp2 routine is general so that the original inimpp
328routine should be suppressed from the code.}
329
330When land processors are eliminated, the value corresponding to these locations in
331the model output files is undefined. Note that this is a problem for the meshmask file
332which requires to be defined over the whole domain. Therefore, user should not eliminate
333land processors when creating a meshmask file ($i.e.$ when setting a non-zero value to \np{nn\_msh}).
334
335%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
336\begin{figure}[!ht]     \begin{center}
337\includegraphics[width=0.90\textwidth]{Fig_mppini2}
338\caption {    \protect\label{Fig_mppini2}
339Example of Atlantic domain defined for the CLIPPER projet. Initial grid is
340composed of 773 x 1236 horizontal points.
341(a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
34252 subdomains are land areas.
343(b) 52 subdomains are eliminated (white rectangles) and the resulting number
344of processors really used during the computation is jpnij=128.}
345\end{center}   \end{figure}
346%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
347
348
349% ====================================================================
350% Unstructured open boundaries BDY
351% ====================================================================
352\section{Unstructured Open Boundary Conditions (BDY)}
353\label{LBC_bdy}
354
355%-----------------------------------------nambdy--------------------------------------------
356\namdisplay{nambdy} 
357%-----------------------------------------------------------------------------------------------
358%-----------------------------------------nambdy_index--------------------------------------------
359\namdisplay{nambdy_index} 
360%-----------------------------------------------------------------------------------------------
361%-----------------------------------------nambdy_dta--------------------------------------------
362\namdisplay{nambdy_dta} 
363%-----------------------------------------------------------------------------------------------
364%-----------------------------------------nambdy_dta--------------------------------------------
365\namdisplay{nambdy_dta2} 
366%-----------------------------------------------------------------------------------------------
367
368Options are defined through the \ngn{nambdy} \ngn{nambdy\_index} 
369\ngn{nambdy\_dta} \ngn{nambdy\_dta2} namelist variables.
370The BDY module is the core implementation of open boundary
371conditions for regional configurations. It implements the Flow
372Relaxation Scheme algorithm for temperature, salinity, velocities and
373ice fields, and the Flather radiation condition for the depth-mean
374transports. The specification of the location of the open boundary is
375completely flexible and allows for example the open boundary to follow
376an isobath or other irregular contour.
377
378The BDY module was modelled on the OBC module (see NEMO 3.4) and shares many
379features and a similar coding structure \citep{Chanut2005}.
380
381Boundary data files used with earlier versions of NEMO may need
382to be re-ordered to work with this version. See the
383section on the Input Boundary Data Files for details.
384
385%----------------------------------------------
386\subsection{The namelists}
387\label{BDY_namelist}
388
389The BDY module is activated by setting \np{ln\_bdy} to true.
390It is possible to define more than one boundary ``set'' and apply
391different boundary conditions to each set. The number of boundary
392sets is defined by \np{nb\_bdy}.  Each boundary set may be defined
393as a set of straight line segments in a namelist
394(\np{ln\_coords\_file}=.false.) or read in from a file
395(\np{ln\_coords\_file}=.true.). If the set is defined in a namelist,
396then the namelists nambdy\_index must be included separately, one for
397each set. If the set is defined by a file, then a
398``coordinates.bdy.nc'' file must be provided. The coordinates.bdy file
399is analagous to the usual NEMO ``coordinates.nc'' file. In the example
400above, there are two boundary sets, the first of which is defined via
401a file and the second is defined in a namelist. For more details of
402the definition of the boundary geometry see section
403\ref{BDY_geometry}.
404
405For each boundary set a boundary
406condition has to be chosen for the barotropic solution (``u2d'':
407sea-surface height and barotropic velocities), for the baroclinic
408velocities (``u3d''), and for the active tracers\footnote{The BDY
409  module does not deal with passive tracers at this version}
410(``tra''). For each set of variables there is a choice of algorithm
411and a choice for the data, eg. for the active tracers the algorithm is
412set by \np{nn\_tra} and the choice of data is set by
413\np{nn\_tra\_dta}.
414
415The choice of algorithm is currently as follows:
416
417\mbox{}
418
419\begin{itemize}
420\item[0.] No boundary condition applied. So the solution will ``see''
421  the land points around the edge of the edge of the domain.
422\item[1.] Flow Relaxation Scheme (FRS) available for all variables.
423\item[2.] Flather radiation scheme for the barotropic variables. The
424  Flather scheme is not compatible with the filtered free surface
425  ({\it dynspg\_ts}).
426\end{itemize}
427
428\mbox{}
429
430The main choice for the boundary data is
431to use initial conditions as boundary data (\np{nn\_tra\_dta}=0) or to
432use external data from a file (\np{nn\_tra\_dta}=1). For the
433barotropic solution there is also the option to use tidal
434harmonic forcing either by itself or in addition to other external
435data.
436
437If external boundary data is required then the nambdy\_dta namelist
438must be defined. One nambdy\_dta namelist is required for each boundary
439set in the order in which the boundary sets are defined in nambdy. In
440the example given, two boundary sets have been defined and so there
441are two nambdy\_dta namelists. The boundary data is read in using the
442fldread module, so the nambdy\_dta namelist is in the format required
443for fldread. For each variable required, the filename, the frequency
444of the files and the frequency of the data in the files is given. Also
445whether or not time-interpolation is required and whether the data is
446climatological (time-cyclic) data. Note that on-the-fly spatial
447interpolation of boundary data is not available at this version.
448
449In the example namelists given, two boundary sets are defined. The
450first set is defined via a file and applies FRS conditions to
451temperature and salinity and Flather conditions to the barotropic
452variables. External data is provided in daily files (from a
453large-scale model). Tidal harmonic forcing is also used. The second
454set is defined in a namelist. FRS conditions are applied on
455temperature and salinity and climatological data is read from external
456files.
457
458%----------------------------------------------
459\subsection{The Flow Relaxation Scheme}
460\label{BDY_FRS_scheme}
461
462The Flow Relaxation Scheme (FRS) \citep{Davies_QJRMS76,Engerdahl_Tel95},
463applies a simple relaxation of the model fields to
464externally-specified values over a zone next to the edge of the model
465domain. Given a model prognostic variable $\Phi$ 
466\begin{equation}  \label{Eq_bdy_frs1}
467\Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
468\end{equation}
469where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified
470external field, $d$ gives the discrete distance from the model
471boundary  and $\alpha$ is a parameter that varies from $1$ at $d=1$ to
472a small value at $d=N$. It can be shown that this scheme is equivalent
473to adding a relaxation term to the prognostic equation for $\Phi$ of
474the form:
475\begin{equation}  \label{Eq_bdy_frs2}
476-\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
477\end{equation}
478where the relaxation time scale $\tau$ is given by a function of
479$\alpha$ and the model time step $\Delta t$:
480\begin{equation}  \label{Eq_bdy_frs3}
481\tau = \frac{1-\alpha}{\alpha}  \,\rdt
482\end{equation}
483Thus the model solution is completely prescribed by the external
484conditions at the edge of the model domain and is relaxed towards the
485external conditions over the rest of the FRS zone. The application of
486a relaxation zone helps to prevent spurious reflection of outgoing
487signals from the model boundary.
488
489The function $\alpha$ is specified as a $tanh$ function:
490\begin{equation}  \label{Eq_bdy_frs4}
491\alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
492\end{equation}
493The width of the FRS zone is specified in the namelist as
494\np{nn\_rimwidth}. This is typically set to a value between 8 and 10.
495
496%----------------------------------------------
497\subsection{The Flather radiation scheme}
498\label{BDY_flather_scheme}
499
500The \citet{Flather_JPO94} scheme is a radiation condition on the normal, depth-mean
501transport across the open boundary. It takes the form
502\begin{equation}  \label{Eq_bdy_fla1}
503U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
504\end{equation}
505where $U$ is the depth-mean velocity normal to the boundary and $\eta$
506is the sea surface height, both from the model. The subscript $e$
507indicates the same fields from external sources. The speed of external
508gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the
509water column. The depth-mean normal velocity along the edge of the
510model domain is set equal to the
511external depth-mean normal velocity, plus a correction term that
512allows gravity waves generated internally to exit the model boundary.
513Note that the sea-surface height gradient in \eqref{Eq_bdy_fla1}
514is a spatial gradient across the model boundary, so that $\eta_{e}$ is
515defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the
516$T$ points with $nbr=2$. $U$ and $U_{e}$ are defined on the $U$ or
517$V$ points with $nbr=1$, $i.e.$ between the two $T$ grid points.
518
519%----------------------------------------------
520\subsection{Boundary geometry}
521\label{BDY_geometry}
522
523Each open boundary set is defined as a list of points. The information
524is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$
525structure.  The $nbi$ and $nbj$ arrays
526define the local $(i,j)$ indices of each point in the boundary zone
527and the $nbr$ array defines the discrete distance from the boundary
528with $nbr=1$ meaning that the point is next to the edge of the
529model domain and $nbr>1$ showing that the point is increasingly
530further away from the edge of the model domain. A set of $nbi$, $nbj$,
531and $nbr$ arrays is defined for each of the $T$, $U$ and $V$
532grids. Figure \ref{Fig_LBC_bdy_geom} shows an example of an irregular
533boundary.
534
535The boundary geometry for each set may be defined in a namelist
536nambdy\_index or by reading in a ``coordinates.bdy.nc'' file. The
537nambdy\_index namelist defines a series of straight-line segments for
538north, east, south and west boundaries. For the northern boundary,
539\np{nbdysegn} gives the number of segments, \np{jpjnob} gives the $j$
540index for each segment and \np{jpindt} and \np{jpinft} give the start
541and end $i$ indices for each segment with similar for the other
542boundaries. These segments define a list of $T$ grid points along the
543outermost row of the boundary ($nbr\,=\, 1$). The code deduces the $U$ and
544$V$ points and also the points for $nbr\,>\, 1$ if
545$nn\_rimwidth\,>\,1$.
546
547The boundary geometry may also be defined from a
548``coordinates.bdy.nc'' file. Figure \ref{Fig_LBC_nc_header}
549gives an example of the header information from such a file. The file
550should contain the index arrays for each of the $T$, $U$ and $V$
551grids. The arrays must be in order of increasing $nbr$. Note that the
552$nbi$, $nbj$ values in the file are global values and are converted to
553local values in the code. Typically this file will be used to generate
554external boundary data via interpolation and so will also contain the
555latitudes and longitudes of each point as shown. However, this is not
556necessary to run the model.
557
558For some choices of irregular boundary the model domain may contain
559areas of ocean which are not part of the computational domain. For
560example if an open boundary is defined along an isobath, say at the
561shelf break, then the areas of ocean outside of this boundary will
562need to be masked out. This can be done by reading a mask file defined
563as \np{cn\_mask\_file} in the nam\_bdy namelist. Only one mask file is
564used even if multiple boundary sets are defined.
565
566%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
567\begin{figure}[!t]      \begin{center}
568\includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom}
569\caption {      \protect\label{Fig_LBC_bdy_geom}
570Example of geometry of unstructured open boundary}
571\end{center}   \end{figure}
572%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
573
574%----------------------------------------------
575\subsection{Input boundary data files}
576\label{BDY_data}
577
578The data files contain the data arrays
579in the order in which the points are defined in the $nbi$ and $nbj$
580arrays. The data arrays are dimensioned on: a time dimension;
581$xb$ which is the index of the boundary data point in the horizontal;
582and $yb$ which is a degenerate dimension of 1 to enable the file to be
583read by the standard NEMO I/O routines. The 3D fields also have a
584depth dimension.
585
586At Version 3.4 there are new restrictions on the order in which the
587boundary points are defined (and therefore restrictions on the order
588of the data in the file). In particular:
589
590\mbox{}
591
592\begin{enumerate}
593\item The data points must be in order of increasing $nbr$, ie. all
594  the $nbr=1$ points, then all the $nbr=2$ points etc.
595\item All the data for a particular boundary set must be in the same
596  order. (Prior to 3.4 it was possible to define barotropic data in a
597  different order to the data for tracers and baroclinic velocities).
598\end{enumerate}
599
600\mbox{}
601
602These restrictions mean that data files used with previous versions of
603the model may not work with version 3.4. A fortran utility
604{\it bdy\_reorder} exists in the TOOLS directory which will re-order the
605data in old BDY data files.
606
607%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
608\begin{figure}[!t]     \begin{center}
609\includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header}
610\caption {     \protect\label{Fig_LBC_nc_header} 
611Example of the header for a coordinates.bdy.nc file}
612\end{center}   \end{figure}
613%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
614
615%----------------------------------------------
616\subsection{Volume correction}
617\label{BDY_vol_corr}
618
619There is an option to force the total volume in the regional model to be constant,
620similar to the option in the OBC module. This is controlled  by the \np{nn\_volctl} 
621parameter in the namelist. A value of \np{nn\_volctl}~=~0 indicates that this option is not used.
622If  \np{nn\_volctl}~=~1 then a correction is applied to the normal velocities
623around the boundary at each timestep to ensure that the integrated volume flow
624through the boundary is zero. If \np{nn\_volctl}~=~2 then the calculation of
625the volume change on the timestep includes the change due to the freshwater
626flux across the surface and the correction velocity corrects for this as well.
627
628If more than one boundary set is used then volume correction is
629applied to all boundaries at once.
630
631\newpage
632%----------------------------------------------
633\subsection{Tidal harmonic forcing}
634\label{BDY_tides}
635
636%-----------------------------------------nambdy_tide--------------------------------------------
637\namdisplay{nambdy_tide} 
638%-----------------------------------------------------------------------------------------------
639
640Options are defined through the  \ngn{nambdy\_tide} namelist variables.
641 To be written....
642
643
644
645
646\end{document}
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