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1% ================================================================
2% Chapter Ñ Lateral Boundary Condition (LBC)
3% ================================================================
4\chapter{Lateral Boundary Condition (LBC) }
5\label{LBC}
6\minitoc
7
8\newpage
9$\ $\newline    % force a new ligne
10
11
12%gm% add here introduction to this chapter
13
14% ================================================================
15% Boundary Condition at the Coast
16% ================================================================
17\section{Boundary Condition at the Coast (\np{rn\_shlat})}
18\label{LBC_coast}
19%--------------------------------------------nam_lbc-------------------------------------------------------
20\namdisplay{namlbc} 
21%--------------------------------------------------------------------------------------------------------------
22
23%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}).
24
25%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}.
26
27The discrete representation of a domain with complex boundaries (coastlines and
28bottom topography) leads to arrays that include large portions where a computation
29is not required as the model variables remain at zero. Nevertheless, vectorial
30supercomputers are far more efficient when computing over a whole array, and the
31readability of a code is greatly improved when boundary conditions are applied in
32an automatic way rather than by a specific computation before or after each
33computational loop. An efficient way to work over the whole domain while specifying
34the boundary conditions, is to use multiplication by mask arrays in the computation.
35A mask array is a matrix whose elements are $1$ in the ocean domain and $0$ 
36elsewhere. A simple multiplication of a variable by its own mask ensures that it will
37remain zero over land areas. Since most of the boundary conditions consist of a
38zero flux across the solid boundaries, they can be simply applied by multiplying
39variables by the correct mask arrays, $i.e.$ the mask array of the grid point where
40the flux is evaluated. For example, the heat flux in the \textbf{i}-direction is evaluated
41at $u$-points. Evaluating this quantity as,
42
43\begin{equation} \label{Eq_lbc_aaaa}
44\frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 
45}{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u
46\end{equation}
47(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is
48zero inside land and at the boundaries, since mask$_{u}$ is zero at solid boundaries
49which in this case are defined at $u$-points (normal velocity $u$ remains zero at
50the coast) (Fig.~\ref{Fig_LBC_uv}).
51
52%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
53\begin{figure}[!t]     \begin{center}
54\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_uv.pdf}
55\caption{  \label{Fig_LBC_uv}
56Lateral boundary (thick line) at T-level. The velocity normal to the boundary is set to zero.}
57\end{center}   \end{figure}
58%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
59
60For momentum the situation is a bit more complex as two boundary conditions
61must be provided along the coast (one each for the normal and tangential velocities).
62The boundary of the ocean in the C-grid is defined by the velocity-faces.
63For example, at a given $T$-level, the lateral boundary (a coastline or an intersection
64with the bottom topography) is made of segments joining $f$-points, and normal
65velocity points are located between two $f-$points (Fig.~\ref{Fig_LBC_uv}).
66The boundary condition on the normal velocity (no flux through solid boundaries)
67can thus be easily implemented using the mask system. The boundary condition
68on the tangential velocity requires a more specific treatment. This boundary
69condition influences the relative vorticity and momentum diffusive trends, and is
70required in order to compute the vorticity at the coast. Four different types of
71lateral boundary condition are available, controlled by the value of the \np{rn\_shlat} 
72namelist parameter. (The value of the mask$_{f}$ array along the coastline is set
73equal to this parameter.) These are:
74
75%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
76\begin{figure}[!p] \begin{center}
77\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_shlat.pdf}
78\caption{     \label{Fig_LBC_shlat} 
79lateral boundary condition (a) free-slip ($rn\_shlat=0$) ; (b) no-slip ($rn\_shlat=2$)
80; (c) "partial" free-slip ($0<rn\_shlat<2$) and (d) "strong" no-slip ($2<rn\_shlat$).
81Implied "ghost" velocity inside land area is display in grey. }
82\end{center}    \end{figure}
83%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
84
85\begin{description}
86
87\item[free-slip boundary condition (\np{rn\_shlat}=0): ]  the tangential velocity at the
88coastline is equal to the offshore velocity, $i.e.$ the normal derivative of the
89tangential velocity is zero at the coast, so the vorticity: mask$_{f}$ array is set
90to zero inside the land and just at the coast (Fig.~\ref{Fig_LBC_shlat}-a).
91
92\item[no-slip boundary condition (\np{rn\_shlat}=2): ] the tangential velocity vanishes
93at the coastline. Assuming that the tangential velocity decreases linearly from
94the closest ocean velocity grid point to the coastline, the normal derivative is
95evaluated as if the velocities at the closest land velocity gridpoint and the closest
96ocean velocity gridpoint were of the same magnitude but in the opposite direction
97(Fig.~\ref{Fig_LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by:
98
99\begin{equation*}
100\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) \ ,
101\end{equation*}
102where $u$ and $v$ are masked fields. Setting the mask$_{f}$ array to $2$ along
103the coastline provides a vorticity field computed with the no-slip boundary condition,
104simply by multiplying it by the mask$_{f}$ :
105\begin{equation} \label{Eq_lbc_bbbb}
106\zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2} 
107\left[ {e_{2v} \,v} \right]-\delta _{j+1/2} \left[ {e_{1u} \,u} \right]} 
108\right)\;\mbox{mask}_f
109\end{equation}
110
111\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2): ] the tangential
112velocity at the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral
113friction but not strong enough to make the tangential velocity at the coast vanish
114(Fig.~\ref{Fig_LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$ 
115strictly inbetween $0$ and $2$.
116
117\item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}): ] the viscous boundary
118layer is assumed to be smaller than half the grid size (Fig.~\ref{Fig_LBC_shlat}-d).
119The friction is thus larger than in the no-slip case.
120
121\end{description}
122
123Note that when the bottom topography is entirely represented by the $s$-coor-dinates
124(pure $s$-coordinate), the lateral boundary condition on tangential velocity is of much
125less importance as it is only applied next to the coast where the minimum water depth
126can be quite shallow.
127
128The alternative numerical implementation of the no-slip boundary conditions for an
129arbitrary coast line of \citet{Shchepetkin1996} is also available through the
130\key{noslip\_accurate} CPP key. It is based on a fourth order evaluation of the shear at the
131coast which, in turn, allows a true second order scheme in the interior of the domain
132($i.e.$ the numerical boundary scheme simulates the truncation error of the numerical
133scheme used in the interior of the domain). \citet{Shchepetkin1996} found that such a
134technique considerably improves the quality of the numerical solution. In \NEMO, such
135spectacular improvements have not been found in the half-degree global ocean
136(ORCA05), but significant reductions of numerically induced coastal upwellings were
137found in an eddy resolving simulation of the Alboran Sea \citep{Olivier_PhD01}.
138Nevertheless, since a no-slip boundary condition is not recommended in an eddy
139permitting or resolving simulation \citep{Penduff_al_OS07}, the use of this option is also
140not recommended.
141
142In practice, the no-slip accurate option changes the way the curl is evaluated at the
143coast (see \mdl{divcur} module), and requires the nature of each coastline grid point
144(convex or concave corners, straight north-south or east-west coast) to be specified. 
145This is performed in routine \rou{dom\_msk\_nsa} in the \mdl{domask} module.
146
147% ================================================================
148% Boundary Condition around the Model Domain
149% ================================================================
150\section{Model Domain Boundary Condition (\jp{jperio})}
151\label{LBC_jperio}
152
153At the model domain boundaries several choices are offered: closed, cyclic east-west,
154south symmetric across the equator, a north-fold, and combination closed-north fold
155or cyclic-north-fold. The north-fold boundary condition is associated with the 3-pole ORCA mesh.
156
157% -------------------------------------------------------------------------------------------------------------
158%        Closed, cyclic, south symmetric (\jp{jperio} = 0, 1 or 2)
159% -------------------------------------------------------------------------------------------------------------
160\subsection{Closed, cyclic, south symmetric (\jp{jperio} = 0, 1 or 2)}
161\label{LBC_jperio012}
162
163The choice of closed, cyclic or symmetric model domain boundary condition is made
164by setting \jp{jperio} to 0, 1 or 2 in file \mdl{par\_oce}. Each time such a boundary
165condition is needed, it is set by a call to routine \mdl{lbclnk}. The computation of
166momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to
167$j=jpj-1$, $i.e.$ in the model interior. To choose a lateral model boundary condition
168is to specify the first and last rows and columns of the model variables.
169
170\begin{description}
171
172\item[For closed boundary (\textit{jperio=0})], solid walls are imposed at all model
173boundaries: first and last rows and columns are set to zero.
174
175\item[For cyclic east-west boundary (\textit{jperio=1})], first and last rows are set
176to zero (closed) whilst the first column is set to the value of the last-but-one column
177and the last column to the value of the second one (Fig.~\ref{Fig_LBC_jperio}-a).
178Whatever flows out of the eastern (western) end of the basin enters the western
179(eastern) end. Note that there is no option for north-south cyclic or for doubly
180cyclic cases.
181
182\item[For symmetric boundary condition across the equator (\textit{jperio=2})],
183last rows, and first and last columns are set to zero (closed). The row of symmetry
184is chosen to be the $u$- and $T-$points equator line ($j=2$, i.e. at the southern
185end of the domain). For arrays defined at $u-$ or $T-$points, the first row is set
186to the value of the third row while for most of $v$- and $f$-point arrays ($v$, $\zeta$,
187$j\psi$, but \gmcomment{not sure why this is "but"} scalar arrays such as eddy coefficients)
188the first row is set to minus the value of the second row (Fig.~\ref{Fig_LBC_jperio}-b).
189Note that this boundary condition is not yet available for the case of a massively
190parallel computer (\textbf{key{\_}mpp} defined).
191
192\end{description}
193
194%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
195\begin{figure}[!t]     \begin{center}
196\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_jperio.pdf}
197\caption{    \label{Fig_LBC_jperio}
198setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.}
199\end{center}   \end{figure}
200%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
201
202% -------------------------------------------------------------------------------------------------------------
203%        North fold (\textit{jperio = 3 }to $6)$
204% -------------------------------------------------------------------------------------------------------------
205\subsection{North-fold (\textit{jperio = 3 }to $6)$ }
206\label{LBC_north_fold}
207
208The north fold boundary condition has been introduced in order to handle the north
209boundary of a three-polar ORCA grid. Such a grid has two poles in the northern hemisphere.
210\colorbox{yellow}{to be completed...}
211
212%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
213\begin{figure}[!t]    \begin{center}
214\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_North_Fold_T.pdf}
215\caption{    \label{Fig_North_Fold_T} 
216North fold boundary with a $T$-point pivot and cyclic east-west boundary condition
217($jperio=4$), as used in ORCA 2, 1/4, and 1/12. Pink shaded area corresponds
218to the inner domain mask (see text). }
219\end{center}   \end{figure}
220%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
221
222% ====================================================================
223% Exchange with neighbouring processors
224% ====================================================================
225\section  [Exchange with neighbouring processors (\textit{lbclnk}, \textit{lib\_mpp})]
226      {Exchange with neighbouring processors (\mdl{lbclnk}, \mdl{lib\_mpp})}
227\label{LBC_mpp}
228
229For massively parallel processing (mpp), a domain decomposition method is used.
230The basic idea of the method is to split the large computation domain of a numerical
231experiment into several smaller domains and solve the set of equations by addressing
232independent local problems. Each processor has its own local memory and computes
233the model equation over a subdomain of the whole model domain. The subdomain
234boundary conditions are specified through communications between processors
235which are organized by explicit statements (message passing method).
236
237A big advantage is that the method does not need many modifications of the initial
238FORTRAN code. From the modeller's point of view, each sub domain running on
239a processor is identical to the "mono-domain" code. In addition, the programmer
240manages the communications between subdomains, and the code is faster when
241the number of processors is increased. The porting of OPA code on an iPSC860
242was achieved during Guyon's PhD [Guyon et al. 1994, 1995] in collaboration with
243CETIIS and ONERA. The implementation in the operational context and the studies
244of performance on a T3D and T3E Cray computers have been made in collaboration
245with IDRIS and CNRS. The present implementation is largely inspired by Guyon's
246work  [Guyon 1995].
247
248The parallelization strategy is defined by the physical characteristics of the
249ocean model. Second order finite difference schemes lead to local discrete
250operators that depend at the very most on one neighbouring point. The only
251non-local computations concern the vertical physics (implicit diffusion, 1.5
252turbulent closure scheme, ...) (delocalization over the whole water column),
253and the solving of the elliptic equation associated with the surface pressure
254gradient computation (delocalization over the whole horizontal domain).
255Therefore, a pencil strategy is used for the data sub-structuration
256\gmcomment{no idea what this means!}
257: the 3D initial domain is laid out on local processor
258memories following a 2D horizontal topological splitting. Each sub-domain
259computes its own surface and bottom boundary conditions and has a side
260wall overlapping interface which defines the lateral boundary conditions for
261computations in the inner sub-domain. The overlapping area consists of the
262two rows at each edge of the sub-domain. After a computation, a communication
263phase starts: each processor sends to its neighbouring processors the update
264values of the points corresponding to the interior overlapping area to its
265neighbouring sub-domain (i.e. the innermost of the two overlapping rows).
266The communication is done through message passing. Usually the parallel virtual
267language, PVM, is used as it is a standard language available on  nearly  all
268MPP computers. More specific languages (i.e. computer dependant languages)
269can be easily used to speed up the communication, such as SHEM on a T3E
270computer. The data exchanges between processors are required at the very
271place where lateral domain boundary conditions are set in the mono-domain
272computation (\S III.10-c): the lbc\_lnk routine which manages such conditions
273is substituted by mpplnk.F or mpplnk2.F routine when running on an MPP
274computer (\key{mpp\_mpi} defined). It has to be pointed out that when using
275the MPP version of the model, the east-west cyclic boundary condition is done
276implicitly, whilst the south-symmetric boundary condition option is not available.
277
278%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
279\begin{figure}[!t]    \begin{center}
280\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_mpp.pdf}
281\caption{   \label{Fig_mpp} 
282Positioning of a sub-domain when massively parallel processing is used. }
283\end{center}   \end{figure}
284%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
285
286In the standard version of the OPA model, the splitting is regular and arithmetic.
287 the i-axis is divided by \jp{jpni} and the j-axis by \jp{jpnj} for a number of processors
288 \jp{jpnij} most often equal to $jpni \times jpnj$ (model parameters set in
289 \mdl{par\_oce}). Each processor is independent and without message passing
290 or synchronous process
291 \gmcomment{how does a synchronous process relate to this?},
292 programs run alone and access just its own local memory. For this reason, the
293 main model dimensions are now the local dimensions of the subdomain (pencil)
294 that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. These dimensions include the internal
295 domain and the overlapping rows. The number of rows to exchange (known as
296 the halo) is usually set to one (\jp{jpreci}=1, in \mdl{par\_oce}). The whole domain
297 dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}. The relationship between
298 the whole domain and a sub-domain is:
299\begin{eqnarray} 
300      jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\
301      jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj  \label{Eq_lbc_jpi}
302\end{eqnarray}
303where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis.
304
305\colorbox{yellow}{Figure IV.3: example of a domain splitting with 9 processors and
306no east-west cyclic boundary conditions.}
307
308One also defines variables nldi and nlei which correspond to the internal
309domain bounds, and the variables nimpp and njmpp which are the position
310of the (1,1) grid-point in the global domain. An element of $T_{l}$, a local array
311(subdomain) corresponds to an element of $T_{g}$, a global array
312(whole domain) by the relationship:
313\begin{equation} \label{Eq_lbc_nimpp}
314T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
315\end{equation}
316with  $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.
317
318Processors are numbered from 0 to $jpnij-1$, the number is saved in the variable
319nproc. In the standard version, a processor has no more than four neighbouring
320processors named nono (for north), noea (east), noso (south) and nowe (west)
321and two variables, nbondi and nbondj, indicate the relative position of the processor
322\colorbox{yellow}{(see Fig.IV.3)}:
323\begin{itemize}
324\item       nbondi = -1    an east neighbour, no west processor,
325\item       nbondi =  0 an east neighbour, a west neighbour,
326\item       nbondi =  1    no east processor, a west neighbour,
327\item       nbondi =  2    no splitting following the i-axis.
328\end{itemize}
329During the simulation, processors exchange data with their neighbours.
330If there is effectively a neighbour, the processor receives variables from this
331processor on its overlapping row, and sends the data issued from internal
332domain corresponding to the overlapping row of the other processor.
333       
334\colorbox{yellow}{Figure IV.4: pencil splitting with the additional outer halos }
335
336
337The \NEMO model computes equation terms with the help of mask arrays (0 on land
338points and 1 on sea points). It is easily readable and very efficient in the context of
339a computer with vectorial architecture. However, in the case of a scalar processor,
340computations over the land regions become more expensive in terms of CPU time.
341It is worse when we use a complex configuration with a realistic bathymetry like the
342global ocean where more than 50 \% of points are land points. For this reason, a
343pre-processing tool can be used to choose the mpp domain decomposition with a
344maximum number of only land points processors, which can then be eliminated.
345(For example, the mpp\_optimiz tools, available from the DRAKKAR web site.)
346This optimisation is dependent on the specific bathymetry employed. The user
347then chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with
348$jpnij < jpni \times jpnj$, leading to the elimination of $jpni \times jpnj - jpnij$ 
349land processors. When those parameters are specified in module \mdl{par\_oce},
350the algorithm in the \rou{inimpp2} routine sets each processor's parameters (nbound,
351nono, noea,...) so that the land-only processors are not taken into account.
352
353\colorbox{yellow}{Note that the inimpp2 routine is general so that the original inimpp
354routine should be suppressed from the code.}
355
356When land processors are eliminated, the value corresponding to these locations in
357the model output files is zero. Note that this is a problem for a mesh output file written
358by such a model configuration, because model users often divide by the scale factors
359($e1t$, $e2t$, etc) and do not expect the grid size to be zero, even on land. It may be
360best not to eliminate land processors when running the model especially to write the
361mesh files as outputs (when \np{nn\_msh} namelist parameter differs from 0).
362%%
363\gmcomment{Steven : dont understand this, no land processor means no output file
364covering this part of globe; its only when files are stitched together into one that you
365can leave a hole}
366%%
367
368%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
369\begin{figure}[!ht]     \begin{center}
370\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_mppini2.pdf}
371\caption {    \label{Fig_mppini2}
372Example of Atlantic domain defined for the CLIPPER projet. Initial grid is
373composed of 773 x 1236 horizontal points.
374(a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
37552 subdomains are land areas.
376(b) 52 subdomains are eliminated (white rectangles) and the resulting number
377of processors really used during the computation is jpnij=128.}
378\end{center}   \end{figure}
379%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
380
381
382% ================================================================
383% Open Boundary Conditions
384% ================================================================
385\section{Open Boundary Conditions (\key{obc}) (OBC)}
386\label{LBC_obc}
387%-----------------------------------------nam_obc  -------------------------------------------
388%-    nobc_dta    =    0     !  = 0 the obc data are equal to the initial state
389%-                           !  = 1 the obc data are read in 'obc   .dta' files
390%-    rn_dpein      =    1.    !  ???
391%-    rn_dpwin      =    1.    !  ???
392%-    rn_dpnin      =   30.    !  ???
393%-    rn_dpsin      =    1.    !  ???
394%-    rn_dpeob      = 1500.    !  time relaxation (days) for the east  open boundary
395%-    rn_dpwob      =   15.    !    "        "           for the west  open boundary
396%-    rn_dpnob      =  150.    !    "        "           for the north open boundary
397%-    rn_dpsob      =   15.    !    "        "           for the south open boundary
398%-    ln_obc_clim = .true.   !  climatological obc data files (default T)
399%-    ln_vol_cst  = .true.   !  total volume conserved
400\namdisplay{namobc} 
401
402It is often necessary to implement a model configuration limited to an oceanic
403region or a basin, which communicates with the rest of the global ocean through
404''open boundaries''. As stated by \citet{Roed1986}, an open boundary is a
405computational border where the aim of the calculations is to allow the perturbations
406generated inside the computational domain to leave it without deterioration of the
407inner model solution. However, an open boundary also has to let information from
408the outer ocean enter the model and should support inflow and outflow conditions.
409
410The open boundary package OBC is the first open boundary option developed in
411NEMO (originally in OPA8.2). It allows the user to
412\begin{itemize}
413\item tell the model that a boundary is ''open'' and not closed by a wall, for example
414by modifying the calculation of the divergence of velocity there;
415\item impose values of tracers and velocities at that boundary (values which may
416be taken from a climatology): this is the``fixed OBC'' option.
417\item calculate boundary values by a sophisticated algorithm combining radiation
418and relaxation (``radiative OBC'' option)
419\end{itemize}
420
421The package resides in the OBC directory. It is described here in four parts: the
422boundary geometry (parameters to be set in \mdl{obc\_par}), the forcing data at
423the boundaries (module \mdl{obcdta}),  the radiation algorithm involving the
424namelist and module \mdl{obcrad}, and a brief presentation of boundary update
425and restart files.
426
427%----------------------------------------------
428\subsection{Boundary geometry}
429\label{OBC_geom}
430%
431First one has to realize that open boundaries may not necessarily be located
432at the extremities of the computational domain. They may exist in the middle
433of the domain, for example at Gibraltar Straits if one wants to avoid including
434the Mediterranean in an Atlantic domain. This flexibility has been found necessary
435for the CLIPPER project \citep{Treguier_al_JGR01}. Because of the complexity of the
436geometry of ocean basins, it may even be necessary to have more than one
437''west'' open boundary, more than one ''north'', etc. This is not possible with
438the OBC option: only one open boundary of each kind, west, east, south and
439north is allowed; these names refer to the grid geometry (not to the direction
440of the geographical ''west'', ''east'', etc).
441
442The open boundary geometry is set by a series of parameters in the module
443\mdl{obc\_par}. For an eastern open boundary, parameters are \jp{lp\_obc\_east} 
444(true if an east open boundary exists), \jp{jpieob} the $i$-index along which
445the eastern open boundary (eob) is located, \jp{jpjed} the $j$-index at which
446it starts, and \jp{jpjef} the $j$-index where it ends (note $d$ is for ''d\'{e}but''
447and $f$ for ''fin'' in French). Similar parameters exist for the west, south and
448north cases (Table~\ref{Tab_obc_param}).
449
450
451%--------------------------------------------------TABLE--------------------------------------------------
452\begin{table}[htbp]     \begin{center}    \begin{tabular}{|l|c|c|c|}
453\hline
454Boundary and  & Constant index  & Starting index (d\'{e}but) & Ending index (fin) \\
455Logical flag  &                 &                            &                     \\
456\hline
457West          & \jp{jpiwob} $>= 2$         &  \jp{jpjwd}$>= 2$          &  \jp{jpjwf}<= \jp{jpjglo}-1 \\
458lp\_obc\_west & $i$-index of a $u$ point   & $j$ of a $T$ point   &$j$ of a $T$ point \\
459\hline
460East            & \jp{jpieob}$<=$\jp{jpiglo}-2&\jp{jpjed} $>= 2$         & \jp{jpjef}$<=$ \jp{jpjglo}-1 \\
461 lp\_obc\_east  & $i$-index of a $u$ point    & $j$ of a $T$ point & $j$ of a $T$ point \\
462\hline
463South           & \jp{jpjsob} $>= 2$         & \jp{jpisd} $>= 2$          & \jp{jpisf}$<=$\jp{jpiglo}-1 \\
464lp\_obc\_south  & $j$-index of a $v$ point   & $i$ of a $T$ point   & $i$ of a $T$ point \\
465\hline
466North           & \jp{jpjnob} $<=$ \jp{jpjglo}-2& \jp{jpind} $>= 2$        & \jp{jpinf}$<=$\jp{jpiglo}-1 \\
467lp\_obc\_north  & $j$-index of a $v$ point      & $i$  of a $T$ point & $i$ of a $T$ point \\
468\hline
469\end{tabular}    \end{center}
470\caption{     \label{Tab_obc_param}
471Names of different indices relating to the open boundaries. In the case
472of a completely open ocean domain with four ocean boundaries, the parameters
473take exactly the values indicated.}
474\end{table}
475%------------------------------------------------------------------------------------------------------------
476
477The open boundaries must be along coordinate lines. On the C-grid, the boundary
478itself is along a line of normal velocity points: $v$ points for a zonal open boundary
479(the south or north one), and $u$ points for a meridional open boundary (the west
480or east one). Another constraint is that there still must be a row of masked points
481all around the domain, as if the domain were a closed basin (unless periodic conditions
482are used together with open boundary conditions). Therefore, an open boundary
483cannot be located at the first/last index, namely, 1, \jp{jpiglo} or \jp{jpjglo}. Also,
484the open boundary algorithm involves calculating the normal velocity points situated
485just on the boundary, as well as the tangential velocity and temperature and salinity
486just outside the boundary. This means that for a west/south boundary, normal
487velocities and temperature are calculated at the same index \jp{jpiwob} and
488\jp{jpjsob}, respectively. For an east/north boundary, the normal velocity is
489calculated at index \jp{jpieob} and \jp{jpjnob}, but the ``outside'' temperature is
490at index \jp{jpieob}+1 and \jp{jpjnob}+1. This means that \jp{jpieob}, \jp{jpjnob} 
491cannot be bigger than \jp{jpiglo}-2, \jp{jpjglo}-2.
492
493
494The starting and ending indices are to be thought of as $T$ point indices: in many
495cases they indicate the first land $T$-point, at the extremity of an open boundary
496(the coast line follows the $f$ grid points, see Fig.~\ref{Fig_obc_north} for an example
497of a northern open boundary). All indices are relative to the global domain. In the
498free surface case it is possible to have ``ocean corners'', that is, an open boundary
499starting and ending in the ocean.
500
501%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
502\begin{figure}[!t]     \begin{center}
503\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_obc_north.pdf}
504\caption{    \label{Fig_obc_north}
505Localization of the North open boundary points.}
506\end{center}     \end{figure}
507%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
508
509Although not compulsory, it is highly recommended that the bathymetry in the
510vicinity of an open boundary follows the following rule: in the direction perpendicular
511to the open line, the water depth should be constant for 4 grid points. This is in
512order to ensure that the radiation condition, which involves model variables next
513to the boundary, is calculated in a consistent way. On Fig.\ref{Fig_obc_north} we
514indicate by an $=$ symbol, the points which should have the same depth. It means
515that at the 4 points near the boundary, the bathymetry is cylindrical \gmcomment{not sure
516why cylindrical}. The line behind the open $T$-line must be 0 in the bathymetry file
517(as shown on Fig.\ref{Fig_obc_north} for example).
518
519%----------------------------------------------
520\subsection{Boundary data}
521\label{OBC_data}
522
523It is necessary to provide information at the boundaries. The simplest case is
524when this information does not change in time and is equal to the initial conditions
525(namelist variable \np{nn\_obcdta}=0). This is the case for the standard configuration
526EEL5 with open boundaries. When (\np{nn\_obcdta}=1), open boundary information
527is read from netcdf files. For convenience the input files are supposed to be similar
528to the ''history'' NEMO output files, for dimension names and variable names.
529Open boundary arrays must be dimensioned according to the parameters of table~
530\ref{Tab_obc_param}: for example, at the western boundary, arrays have a
531dimension of \jp{jpwf}-\jp{jpwd}+1 in the horizontal and \jp{jpk} in the vertical.
532
533When ocean observations are used to generate the boundary data (a hydrographic
534section for example, as in \citet{Treguier_al_JGR01}) it happens often that only the velocity
535normal to the boundary is known, which is the reason why the initial OBC code
536assumes that only $T$, $S$, and the normal velocity ($u$ or $v$) needs to be
537specified. As more and more global model solutions and ocean analysis products
538become available, it will be possible to provide information about all the variables
539(including the tangential velocity) so that the specification of four variables at each
540boundaries will become standard. For the sea surface height, one must distinguish
541between the filtered free surface case and the time-splitting or explicit treatment of
542the free surface.
543 In the first case, it is assumed that the user does not wish to represent high
544 frequency motions such as tides. The boundary condition is thus one of zero
545 normal gradient of sea surface height at the open boundaries, following \citet{Marchesiello2001}.
546No information other than the total velocity needs to be provided at the open
547boundaries in that case. In the other two cases (time splitting or explicit free surface),
548the user must provide barotropic information (sea surface height and barotropic
549velocities) and the use of the Flather algorithm for barotropic variables is
550recommanded. However, this algorithm has not yet been fully tested and bugs
551remain in NEMO v2.3. Users should read the code carefully before using it. Finally,
552in the case of the rigid lid approximation the barotropic streamfunction must be
553provided, as documented in \citet{Treguier_al_JGR01}). This option is no longer
554recommended but remains in NEMO V2.3.
555
556One frequently encountered case is when an open boundary domain is constructed
557from a global or larger scale NEMO configuration. Assuming the domain corresponds
558to indices $ib:ie$, $jb:je$ of the global domain, the bathymetry and forcing of the
559small domain can be created by using the following netcdf utility on the global files:
560ncks -F $-d\;x,ib,ie$ $-d\;y,jb,je$ (part of the nco series of utilities,
561see their \href{http://nco.sourceforge.net}{website}).
562The open boundary files can be constructed using ncks
563commands, following table~\ref{Tab_obc_ind}.
564
565%--------------------------------------------------TABLE--------------------------------------------------
566\begin{table}[htbp]     \begin{center}      \begin{tabular}{|l|c|c|c|c|c|}
567\hline
568OBC  & Variable   & file name      & Index  & Start  & end  \\
569West &  T,S       &   obcwest\_TS.nc &  $ib$+1     &   $jb$+1 &  $je-1$  \\
570     &    U       &   obcwest\_U.nc  &  $ib$+1     &   $jb$+1 &  $je-1$  \\ 
571     &    V       &   obcwest\_V.nc  &  $ib$+1     &   $jb$+1 &  $je-1$  \\       
572\hline
573East &  T,S       &   obceast\_TS.nc &  $ie$-1     &   $jb$+1 &  $je-1$  \\
574     &    U       &   obceast\_U.nc  &  $ie$-2     &   $jb$+1 &  $je-1$  \\ 
575     &    V       &   obceast\_V.nc  &  $ie$-1     &   $jb$+1 &  $je-1$  \\       
576\hline         
577South &  T,S      &   obcsouth\_TS.nc &  $jb$+1     &  $ib$+1 &  $ie-1$  \\
578      &    U      &   obcsouth\_U.nc  &  $jb$+1     &  $ib$+1 &  $ie-1$  \\ 
579      &    V      &   obcsouth\_V.nc  &  $jb$+1     &  $ib$+1 &  $ie-1$  \\   
580\hline
581North &  T,S      &   obcnorth\_TS.nc &  $je$-1     &  $ib$+1 &  $ie-1$  \\
582      &    U      &   obcnorth\_U.nc  &  $je$-1     &  $ib$+1 &  $ie-1$  \\ 
583      &    V      &   obcnorth\_V.nc  &  $je$-2     &  $ib$+1 &  $ie-1$  \\ 
584\hline
585\end{tabular}     \end{center}
586\caption{    \label{Tab_obc_ind}
587Requirements for creating open boundary files from a global configuration,
588appropriate for the subdomain of indices $ib:ie$, $jb:je$. ``Index'' designates the
589$i$ or $j$ index along which the $u$ of $v$ boundary point is situated in the global
590configuration, starting and ending with the $j$ or $i$ indices indicated.
591For example, to generate file obcnorth\_V.nc, use the command ncks
592$-F$ $-d\;y,je-2$  $-d\;x,ib+1,ie-1$ } 
593\end{table}
594%-----------------------------------------------------------------------------------------------------------
595
596It is assumed that the open boundary files contain the variables for the period of
597the model integration. If the boundary files contain one time frame, the boundary
598data is held fixed in time. If the files contain 12 values, it is assumed that the input
599is a climatology for a repeated annual cycle (corresponding to the case \np{ln\_obc\_clim} 
600=true). The case of an arbitrary number of time frames is not yet implemented
601correctly; the user is required to write his own code in the module \mdl{obc\_dta} 
602to deal with this situation.
603
604\subsection{Radiation algorithm}
605\label{OBC_rad}
606
607The art of open boundary management consists in applying a constraint strong
608enough that the inner domain "feels" the rest of the ocean, but weak enough
609that perturbations are allowed to leave the domain with minimum false reflections
610of energy. The constraints are specified separately at each boundary as time
611scales for ''inflow'' and ''outflow'' as defined below. The time scales are set (in days)
612by namelist parameters such as \np{rn\_dpein}, \np{rn\_dpeob} for the eastern open
613boundary for example. When both time scales are zero for a given boundary
614($e.g.$ for the western boundary, \jp{lp\_obc\_west}=true, \np{rn\_dpwob}=0 and
615\np{rn\_dpwin}=0) this means that the boundary in question is a ''fixed '' boundary
616where the solution is set exactly by the boundary data. This is not recommended,
617except in combination with increased viscosity in a ''sponge'' layer next to the
618boundary in order to avoid spurious reflections. 
619
620
621The radiation\/relaxation \gmcomment{the / doesnt seem to appear in the output} 
622algorithm is applied when either relaxation time (for ''inflow'' or ''outflow'') is
623non-zero. It has been developed and tested in the SPEM model and its
624successor ROMS \citep{Barnier1996, Marchesiello2001}, which is an
625$s$-coordinate model on an Arakawa C-grid. Although the algorithm has
626been numerically successful in the CLIPPER Atlantic models, the physics
627do not work as expected \citep{Treguier_al_JGR01}. Users are invited to consider
628open boundary conditions (OBC hereafter) with some scepticism
629\citep{Durran2001, Blayo2005}.
630
631The first part of the algorithm calculates a phase velocity to determine
632whether perturbations tend to propagate toward, or away from, the
633boundary. Let us consider a model variable $\phi$.
634The phase velocities ($C_{\phi x}$,$C_{\phi y}$) for the variable $\phi$,
635in the directions normal and tangential to the boundary are
636\begin{equation} \label{Eq_obc_cphi}
637C_{\phi x} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{x} 
638\;\;\;\;\; \;\;\; 
639C_{\phi y} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{y}.
640\end{equation}
641Following \citet{Treguier_al_JGR01} and \citet{Marchesiello2001} we retain only
642the normal component of the velocity, $C_{\phi x}$, setting $C_{\phi y} =0$ 
643(but unlike the original Orlanski radiation algorithm we retain $\phi_{y}$ in
644the expression for $C_{\phi x}$). 
645
646The discrete form of (\ref{Eq_obc_cphi}), described by \citet{Barnier1998},
647takes into account the two rows of grid points situated inside the domain
648next to the boundary, and the three previous time steps ($n$, $n-1$,
649and $n-2$). The same equation can then be discretized at the boundary at
650time steps $n-1$, $n$ and $n+1$ \gmcomment{since the original was three time-level} 
651in order to extrapolate for the new boundary value $\phi^{n+1}$.
652
653In the open boundary algorithm as implemented in NEMO v2.3, the new boundary
654values are updated differently depending on the sign of $C_{\phi x}$. Let us take
655an eastern boundary as an example. The solution for variable $\phi$ at the
656boundary is given by a generalized wave equation with phase velocity $C_{\phi}$,
657with the addition of a relaxation term, as:
658\begin{eqnarray}
659\phi_{t} &  =  & -C_{\phi x} \phi_{x} + \frac{1}{\tau_{o}} (\phi_{c}-\phi)
660                        \;\;\; \;\;\; \;\;\; (C_{\phi x} > 0), \label{Eq_obc_rado} \\
661\phi_{t} &  =  & \frac{1}{\tau_{i}} (\phi_{c}-\phi)
662\;\;\; \;\;\; \;\;\;\;\;\; (C_{\phi x} < 0), \label{Eq_obc_radi}
663\end{eqnarray}
664where $\phi_{c}$ is the estimate of $\phi$ at the boundary, provided as boundary
665data. Note that in (\ref{Eq_obc_rado}), $C_{\phi x}$ is bounded by the ratio
666$\delta x/\delta t$ for stability reasons. When $C_{\phi x}$ is eastward (outward
667propagation), the radiation condition (\ref{Eq_obc_rado}) is used.
668When  $C_{\phi x}$ is westward (inward propagation), (\ref{Eq_obc_radi}) is
669used with a strong relaxation to climatology (usually $\tau_{i}=\np{rn\_dpein}=$1~day).
670Equation (\ref{Eq_obc_radi}) is solved with a Euler time-stepping scheme. As a
671consequence, setting $\tau_{i}$ smaller than, or equal to the time step is equivalent
672to a fixed boundary condition. A time scale of one day is usually a good compromise
673which guarantees that the inflow conditions remain close to climatology while ensuring
674numerical stability.
675
676In  the case of a western boundary located in the Eastern Atlantic, \citet{Penduff_al_JGR00} 
677have been able to implement the radiation algorithm without any boundary data,
678using persistence from the previous time step instead. This solution has not worked
679in other cases \citep{Treguier_al_JGR01}, so that the use of boundary data is recommended.
680Even in the outflow condition (\ref{Eq_obc_rado}), we have found it desirable to
681maintain a weak relaxation to climatology. The time step is usually chosen so as to
682be larger than typical turbulent scales (of order 1000~days \gmcomment{or maybe seconds?}).
683
684The radiation condition is applied to the model variables: temperature, salinity,
685tangential and normal velocities. For normal and tangential velocities, $u$ and $v$,
686radiation is applied with phase velocities calculated from $u$ and $v$ respectively. 
687For the radiation of tracers, we use the phase velocity calculated from the tangential
688velocity in order to avoid calculating too many independent radiation velocities and
689because tangential velocities and tracers have the same position along the boundary
690on a C-grid. 
691
692\subsection{Domain decomposition (\key{mpp\_mpi})}
693\label{OBC_mpp}
694When \key{mpp\_mpi} is active in the code, the computational domain is divided
695into rectangles that are attributed each to a different processor. The open boundary
696code is ``mpp-compatible'' up to a certain point. The radiation algorithm will not
697work if there is an mpp subdomain boundary parallel to the open boundary at the
698index of the boundary, or the grid point after (outside), or three grid points before
699(inside). On the other hand, there is no problem if an mpp subdomain boundary
700cuts the open boundary perpendicularly. These geometrical limitations must be
701checked for by the user (there is no safeguard in the code). 
702The general principle for the open boundary mpp code is that loops over the open
703boundaries {not sure what this means} are performed on local indices (nie0,
704nie1, nje0, nje1 for an eastern boundary for instance) that are initialized in module
705\mdl{obc\_ini}. Those indices have relevant values on the processors that contain
706a segment of an open boundary. For processors that do not include an open
707boundary segment, the indices are such that the calculations within the loops are
708not performed.
709\gmcomment{I dont understand most of the last few sentences}
710 
711Arrays of climatological data that are read from files are seen by all processors
712and have the same dimensions for all (for instance, for the eastern boundary,
713uedta(jpjglo,jpk,2)). On the other hand, the arrays for the calculation of radiation
714are local to each processor (uebnd(jpj,jpk,3,3) for instance).  This allowed the
715CLIPPER model for example, to save on memory where the eastern boundary
716crossed 8 processors so that \jp{jpj} was much smaller than (\jp{jpjef}-\jp{jpjed}+1).
717
718\subsection{Volume conservation}
719\label{OBC_vol}
720
721It is necessary to control the volume inside a domain when using open boundaries.
722With fixed boundaries, it is enough to ensure that the total inflow/outflow has
723reasonable values (either zero or a value compatible with an observed volume
724balance). When using radiative boundary conditions it is necessary to have a
725volume constraint because each open boundary works independently from the
726others. The methodology used to control this volume is identical to the one
727coded in the ROMS model \citep{Marchesiello2001}.
728
729
730%---------------------------------------- EXTRAS
731\colorbox{yellow}{Explain obc\_vol{\ldots}}
732
733\colorbox{yellow}{OBC algorithm for update, OBC restart, list of routines where obc key appears{\ldots}}
734
735\colorbox{yellow}{OBC rigid lid? {\ldots}}
736
737% ====================================================================
738% Unstructured open boundaries BDY
739% ====================================================================
740\section{Unstructured Open Boundary Conditions (\key{bdy}) (BDY)}
741\label{LBC_bdy}
742
743%-----------------------------------------nambdy--------------------------------------------
744\namdisplay{nambdy} 
745%-----------------------------------------------------------------------------------------------
746%-----------------------------------------nambdy_index--------------------------------------------
747\namdisplay{nambdy_index} 
748%-----------------------------------------------------------------------------------------------
749%-----------------------------------------nambdy_dta--------------------------------------------
750\namdisplay{nambdy_dta} 
751%-----------------------------------------------------------------------------------------------
752%-----------------------------------------nambdy_dta--------------------------------------------
753\namdisplay{nambdy_dta2} 
754%-----------------------------------------------------------------------------------------------
755
756The BDY module is an alternative implementation of open boundary
757conditions for regional configurations. It implements the Flow
758Relaxation Scheme algorithm for temperature, salinity, velocities and
759ice fields, and the Flather radiation condition for the depth-mean
760transports. The specification of the location of the open boundary is
761completely flexible and allows for example the open boundary to follow
762an isobath or other irregular contour.
763
764The BDY module was modelled on the OBC module and shares many features
765and a similar coding structure \citep{Chanut2005}.
766
767The BDY module is completely rewritten at NEMO 3.4 and there is a new
768set of namelists. Boundary data files used with earlier versions of
769NEMO may need to be re-ordered to work with this version. See the
770section on the Input Boundary Data Files for details.
771
772%----------------------------------------------
773\subsection{The namelists}
774\label{BDY_namelist}
775
776It is possible to define more than one boundary ``set'' and apply
777different boundary conditions to each set. The number of boundary
778sets is defined by \np{nb\_bdy}.  Each boundary set may be defined
779as a set of straight line segments in a namelist
780(\np{ln\_coords\_file}=.false.) or read in from a file
781(\np{ln\_coords\_file}=.true.). If the set is defined in a namelist,
782then the namelists nambdy\_index must be included separately, one for
783each set. If the set is defined by a file, then a
784``coordinates.bdy.nc'' file must be provided. The coordinates.bdy file
785is analagous to the usual NEMO ``coordinates.nc'' file. In the example
786above, there are two boundary sets, the first of which is defined via
787a file and the second is defined in a namelist. For more details of
788the definition of the boundary geometry see section
789\ref{BDY_geometry}.
790
791For each boundary set a boundary
792condition has to be chosen for the barotropic solution (``u2d'':
793sea-surface height and barotropic velocities), for the baroclinic
794velocities (``u3d''), and for the active tracers\footnote{The BDY
795  module does not deal with passive tracers at this version}
796(``tra''). For each set of variables there is a choice of algorithm
797and a choice for the data, eg. for the active tracers the algorithm is
798set by \np{nn\_tra} and the choice of data is set by
799\np{nn\_tra\_dta}.
800
801The choice of algorithm is currently as follows:
802
803\mbox{}
804
805\begin{itemize}
806\item[0.] No boundary condition applied. So the solution will ``see''
807  the land points around the edge of the edge of the domain.
808\item[1.] Flow Relaxation Scheme (FRS) available for all variables.
809\item[2.] Flather radiation scheme for the barotropic variables. The
810  Flather scheme is not compatible with the filtered free surface
811  ({\it dynspg\_ts}).
812\end{itemize}
813
814\mbox{}
815
816The main choice for the boundary data is
817to use initial conditions as boundary data (\np{nn\_tra\_dta}=0) or to
818use external data from a file (\np{nn\_tra\_dta}=1). For the
819barotropic solution there is also the option to use tidal
820harmonic forcing either by itself or in addition to other external
821data.
822
823If external boundary data is required then the nambdy\_dta namelist
824must be defined. One nambdy\_dta namelist is required for each boundary
825set in the order in which the boundary sets are defined in nambdy. In
826the example given, two boundary sets have been defined and so there
827are two nambdy\_dta namelists. The boundary data is read in using the
828fldread module, so the nambdy\_dta namelist is in the format required
829for fldread. For each variable required, the filename, the frequency
830of the files and the frequency of the data in the files is given. Also
831whether or not time-interpolation is required and whether the data is
832climatological (time-cyclic) data. Note that on-the-fly spatial
833interpolation of boundary data is not available at this version.
834
835In the example namelists given, two boundary sets are defined. The
836first set is defined via a file and applies FRS conditions to
837temperature and salinity and Flather conditions to the barotropic
838variables. External data is provided in daily files (from a
839large-scale model). Tidal harmonic forcing is also used. The second
840set is defined in a namelist. FRS conditions are applied on
841temperature and salinity and climatological data is read from external
842files.
843
844%----------------------------------------------
845\subsection{The Flow Relaxation Scheme}
846\label{BDY_FRS_scheme}
847
848The Flow Relaxation Scheme (FRS) \citep{Davies_QJRMS76,Engerdahl_Tel95},
849applies a simple relaxation of the model fields to
850externally-specified values over a zone next to the edge of the model
851domain. Given a model prognostic variable $\Phi$ 
852\begin{equation}  \label{Eq_bdy_frs1}
853\Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
854\end{equation}
855where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified
856external field, $d$ gives the discrete distance from the model
857boundary  and $\alpha$ is a parameter that varies from $1$ at $d=1$ to
858a small value at $d=N$. It can be shown that this scheme is equivalent
859to adding a relaxation term to the prognostic equation for $\Phi$ of
860the form:
861\begin{equation}  \label{Eq_bdy_frs2}
862-\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
863\end{equation}
864where the relaxation time scale $\tau$ is given by a function of
865$\alpha$ and the model time step $\Delta t$:
866\begin{equation}  \label{Eq_bdy_frs3}
867\tau = \frac{1-\alpha}{\alpha}  \,\rdt
868\end{equation}
869Thus the model solution is completely prescribed by the external
870conditions at the edge of the model domain and is relaxed towards the
871external conditions over the rest of the FRS zone. The application of
872a relaxation zone helps to prevent spurious reflection of outgoing
873signals from the model boundary.
874
875The function $\alpha$ is specified as a $tanh$ function:
876\begin{equation}  \label{Eq_bdy_frs4}
877\alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
878\end{equation}
879The width of the FRS zone is specified in the namelist as
880\np{nn\_rimwidth}. This is typically set to a value between 8 and 10.
881
882%----------------------------------------------
883\subsection{The Flather radiation scheme}
884\label{BDY_flather_scheme}
885
886The \citet{Flather_JPO94} scheme is a radiation condition on the normal, depth-mean
887transport across the open boundary. It takes the form
888\begin{equation}  \label{Eq_bdy_fla1}
889U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
890\end{equation}
891where $U$ is the depth-mean velocity normal to the boundary and $\eta$
892is the sea surface height, both from the model. The subscript $e$
893indicates the same fields from external sources. The speed of external
894gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the
895water column. The depth-mean normal velocity along the edge of the
896model domain is set equal to the
897external depth-mean normal velocity, plus a correction term that
898allows gravity waves generated internally to exit the model boundary.
899Note that the sea-surface height gradient in \eqref{Eq_bdy_fla1}
900is a spatial gradient across the model boundary, so that $\eta_{e}$ is
901defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the
902$T$ points with $nbr=2$. $U$ and $U_{e}$ are defined on the $U$ or
903$V$ points with $nbr=1$, $i.e.$ between the two $T$ grid points.
904
905%----------------------------------------------
906\subsection{Boundary geometry}
907\label{BDY_geometry}
908
909Each open boundary set is defined as a list of points. The information
910is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$
911structure.  The $nbi$ and $nbj$ arrays
912define the local $(i,j)$ indices of each point in the boundary zone
913and the $nbr$ array defines the discrete distance from the boundary
914with $nbr=1$ meaning that the point is next to the edge of the
915model domain and $nbr>1$ showing that the point is increasingly
916further away from the edge of the model domain. A set of $nbi$, $nbj$,
917and $nbr$ arrays is defined for each of the $T$, $U$ and $V$
918grids. Figure \ref{Fig_LBC_bdy_geom} shows an example of an irregular
919boundary.
920
921The boundary geometry for each set may be defined in a namelist
922nambdy\_index or by reading in a ``coordinates.bdy.nc'' file. The
923nambdy\_index namelist defines a series of straight-line segments for
924north, east, south and west boundaries. For the northern boundary,
925\np{nbdysegn} gives the number of segments, \np{jpjnob} gives the $j$
926index for each segment and \np{jpindt} and \np{jpinft} give the start
927and end $i$ indices for each segment with similar for the other
928boundaries. These segments define a list of $T$ grid points along the
929outermost row of the boundary ($nbr\,=\, 1$). The code deduces the $U$ and
930$V$ points and also the points for $nbr\,>\, 1$ if
931$nn\_rimwidth\,>\,1$.
932
933The boundary geometry may also be defined from a
934``coordinates.bdy.nc'' file. Figure \ref{Fig_LBC_nc_header}
935gives an example of the header information from such a file. The file
936should contain the index arrays for each of the $T$, $U$ and $V$
937grids. The arrays must be in order of increasing $nbr$. Note that the
938$nbi$, $nbj$ values in the file are global values and are converted to
939local values in the code. Typically this file will be used to generate
940external boundary data via interpolation and so will also contain the
941latitudes and longitudes of each point as shown. However, this is not
942necessary to run the model.
943
944For some choices of irregular boundary the model domain may contain
945areas of ocean which are not part of the computational domain. For
946example if an open boundary is defined along an isobath, say at the
947shelf break, then the areas of ocean outside of this boundary will
948need to be masked out. This can be done by reading a mask file defined
949as \np{cn\_mask\_file} in the nam\_bdy namelist. Only one mask file is
950used even if multiple boundary sets are defined.
951
952%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
953\begin{figure}[!t]      \begin{center}
954\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_bdy_geom.pdf}
955\caption {      \label{Fig_LBC_bdy_geom}
956Example of geometry of unstructured open boundary}
957\end{center}   \end{figure}
958%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
959
960%----------------------------------------------
961\subsection{Input boundary data files}
962\label{BDY_data}
963
964The data files contain the data arrays
965in the order in which the points are defined in the $nbi$ and $nbj$
966arrays. The data arrays are dimensioned on: a time dimension;
967$xb$ which is the index of the boundary data point in the horizontal;
968and $yb$ which is a degenerate dimension of 1 to enable the file to be
969read by the standard NEMO I/O routines. The 3D fields also have a
970depth dimension.
971
972At Version 3.4 there are new restrictions on the order in which the
973boundary points are defined (and therefore restrictions on the order
974of the data in the file). In particular:
975
976\mbox{}
977
978\begin{enumerate}
979\item The data points must be in order of increasing $nbr$, ie. all
980  the $nbr=1$ points, then all the $nbr=2$ points etc.
981\item All the data for a particular boundary set must be in the same
982  order. (Prior to 3.4 it was possible to define barotropic data in a
983  different order to the data for tracers and baroclinic velocities).
984\end{enumerate}
985
986\mbox{}
987
988These restrictions mean that data files used with previous versions of
989the model may not work with version 3.4. A fortran utility
990{\it bdy\_reorder} exists in the TOOLS directory which will re-order the
991data in old BDY data files.
992
993%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
994\begin{figure}[!t]     \begin{center}
995\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_nc_header.pdf}
996\caption {     \label{Fig_LBC_nc_header} 
997Example of the header for a coordinates.bdy.nc file}
998\end{center}   \end{figure}
999%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1000
1001%----------------------------------------------
1002\subsection{Volume correction}
1003\label{BDY_vol_corr}
1004
1005There is an option to force the total volume in the regional model to be constant,
1006similar to the option in the OBC module. This is controlled  by the \np{nn\_volctl} 
1007parameter in the namelist. A value of \np{nn\_volctl}~=~0 indicates that this option is not used.
1008If  \np{nn\_volctl}~=~1 then a correction is applied to the normal velocities
1009around the boundary at each timestep to ensure that the integrated volume flow
1010through the boundary is zero. If \np{nn\_volctl}~=~2 then the calculation of
1011the volume change on the timestep includes the change due to the freshwater
1012flux across the surface and the correction velocity corrects for this as well.
1013
1014If more than one boundary set is used then volume correction is
1015applied to all boundaries at once.
1016
1017\newpage
1018%----------------------------------------------
1019\subsection{Tidal harmonic forcing}
1020\label{BDY_tides}
1021
1022%-----------------------------------------nambdy_tide--------------------------------------------
1023\namdisplay{nambdy_tide} 
1024%-----------------------------------------------------------------------------------------------
1025
1026To be written....
1027
1028
1029
1030
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