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1% ================================================================
2% Chapter — 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
27Options are defined through the \ngn{namlbc} namelist variables.
28The discrete representation of a domain with complex boundaries (coastlines and
29bottom topography) leads to arrays that include large portions where a computation
30is not required as the model variables remain at zero. Nevertheless, vectorial
31supercomputers are far more efficient when computing over a whole array, and the
32readability of a code is greatly improved when boundary conditions are applied in
33an automatic way rather than by a specific computation before or after each
34computational loop. An efficient way to work over the whole domain while specifying
35the boundary conditions, is to use multiplication by mask arrays in the computation.
36A mask array is a matrix whose elements are $1$ in the ocean domain and $0$
37elsewhere. A simple multiplication of a variable by its own mask ensures that it will
38remain zero over land areas. Since most of the boundary conditions consist of a
39zero flux across the solid boundaries, they can be simply applied by multiplying
40variables by the correct mask arrays, $i.e.$ the mask array of the grid point where
41the flux is evaluated. For example, the heat flux in the \textbf{i}-direction is evaluated
42at $u$-points. Evaluating this quantity as,
43
44\begin{equation} \label{Eq_lbc_aaaa}
45\frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT}
46}{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u
47\end{equation}
48(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is
49zero inside land and at the boundaries, since mask$_{u}$ is zero at solid boundaries
50which in this case are defined at $u$-points (normal velocity $u$ remains zero at
51the coast) (Fig.~\ref{Fig_LBC_uv}).
52
53%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
54\begin{figure}[!t]     \begin{center}
55\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_uv.pdf}
56\caption{  \label{Fig_LBC_uv}
57Lateral boundary (thick line) at T-level. The velocity normal to the boundary is set to zero.}
58\end{center}   \end{figure}
59%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
60
61For momentum the situation is a bit more complex as two boundary conditions
62must be provided along the coast (one each for the normal and tangential velocities).
63The boundary of the ocean in the C-grid is defined by the velocity-faces.
64For example, at a given $T$-level, the lateral boundary (a coastline or an intersection
65with the bottom topography) is made of segments joining $f$-points, and normal
66velocity points are located between two $f-$points (Fig.~\ref{Fig_LBC_uv}).
67The boundary condition on the normal velocity (no flux through solid boundaries)
68can thus be easily implemented using the mask system. The boundary condition
69on the tangential velocity requires a more specific treatment. This boundary
70condition influences the relative vorticity and momentum diffusive trends, and is
71required in order to compute the vorticity at the coast. Four different types of
72lateral boundary condition are available, controlled by the value of the \np{rn\_shlat}
73namelist parameter. (The value of the mask$_{f}$ array along the coastline is set
74equal to this parameter.) These are:
75
76%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
77\begin{figure}[!p] \begin{center}
78\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_shlat.pdf}
79\caption{     \label{Fig_LBC_shlat}
80lateral boundary condition (a) free-slip ($rn\_shlat=0$) ; (b) no-slip ($rn\_shlat=2$)
81; (c) "partial" free-slip ($0<rn\_shlat<2$) and (d) "strong" no-slip ($2<rn\_shlat$).
82Implied "ghost" velocity inside land area is display in grey. }
83\end{center}    \end{figure}
84%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
85
86\begin{description}
87
88\item[free-slip boundary condition (\np{rn\_shlat}=0): ]  the tangential velocity at the
89coastline is equal to the offshore velocity, $i.e.$ the normal derivative of the
90tangential velocity is zero at the coast, so the vorticity: mask$_{f}$ array is set
91to zero inside the land and just at the coast (Fig.~\ref{Fig_LBC_shlat}-a).
92
93\item[no-slip boundary condition (\np{rn\_shlat}=2): ] the tangential velocity vanishes
94at the coastline. Assuming that the tangential velocity decreases linearly from
95the closest ocean velocity grid point to the coastline, the normal derivative is
96evaluated as if the velocities at the closest land velocity gridpoint and the closest
97ocean velocity gridpoint were of the same magnitude but in the opposite direction
98(Fig.~\ref{Fig_LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by:
99
100\begin{equation*}
101\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) \ ,
102\end{equation*}
103where $u$ and $v$ are masked fields. Setting the mask$_{f}$ array to $2$ along
104the coastline provides a vorticity field computed with the no-slip boundary condition,
105simply by multiplying it by the mask$_{f}$ :
106\begin{equation} \label{Eq_lbc_bbbb}
107\zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2}
108\left[ {e_{2v} \,v} \right]-\delta _{j+1/2} \left[ {e_{1u} \,u} \right]}
110\end{equation}
111
112\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2): ] the tangential
113velocity at the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral
114friction but not strong enough to make the tangential velocity at the coast vanish
115(Fig.~\ref{Fig_LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$
116strictly inbetween $0$ and $2$.
117
118\item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}): ] the viscous boundary
119layer is assumed to be smaller than half the grid size (Fig.~\ref{Fig_LBC_shlat}-d).
120The friction is thus larger than in the no-slip case.
121
122\end{description}
123
124Note that when the bottom topography is entirely represented by the $s$-coor-dinates
125(pure $s$-coordinate), the lateral boundary condition on tangential velocity is of much
126less importance as it is only applied next to the coast where the minimum water depth
127can be quite shallow.
128
129
130% ================================================================
131% Boundary Condition around the Model Domain
132% ================================================================
133\section{Model Domain Boundary Condition (\np{jperio})}
134\label{LBC_jperio}
135
136At the model domain boundaries several choices are offered: closed, cyclic east-west,
137south symmetric across the equator, a north-fold, and combination closed-north fold
138or cyclic-north-fold. The north-fold boundary condition is associated with the 3-pole ORCA mesh.
139
140% -------------------------------------------------------------------------------------------------------------
141%        Closed, cyclic, south symmetric (\np{jperio} = 0, 1 or 2)
142% -------------------------------------------------------------------------------------------------------------
143\subsection{Closed, cyclic, south symmetric (\np{jperio} = 0, 1 or 2)}
144\label{LBC_jperio012}
145
146The choice of closed, cyclic or symmetric model domain boundary condition is made
147by setting \np{jperio} to 0, 1 or 2 in namelist \ngn{namcfg}. Each time such a boundary
148condition is needed, it is set by a call to routine \mdl{lbclnk}. The computation of
149momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to
150$j=jpj-1$, $i.e.$ in the model interior. To choose a lateral model boundary condition
151is to specify the first and last rows and columns of the model variables.
152
153\begin{description}
154
155\item[For closed boundary (\textit{jperio=0})], solid walls are imposed at all model
156boundaries: first and last rows and columns are set to zero.
157
158\item[For cyclic east-west boundary (\textit{jperio=1})], first and last rows are set
159to zero (closed) whilst the first column is set to the value of the last-but-one column
160and the last column to the value of the second one (Fig.~\ref{Fig_LBC_jperio}-a).
161Whatever flows out of the eastern (western) end of the basin enters the western
162(eastern) end. Note that there is no option for north-south cyclic or for doubly
163cyclic cases.
164
165\item[For symmetric boundary condition across the equator (\textit{jperio=2})],
166last rows, and first and last columns are set to zero (closed). The row of symmetry
167is chosen to be the $u$- and $T-$points equator line ($j=2$, i.e. at the southern
168end of the domain). For arrays defined at $u-$ or $T-$points, the first row is set
169to the value of the third row while for most of $v$- and $f$-point arrays ($v$, $\zeta$,
170$j\psi$, but \gmcomment{not sure why this is "but"} scalar arrays such as eddy coefficients)
171the first row is set to minus the value of the second row (Fig.~\ref{Fig_LBC_jperio}-b).
172Note that this boundary condition is not yet available for the case of a massively
173parallel computer (\textbf{key{\_}mpp} defined).
174
175\end{description}
176
177%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
178\begin{figure}[!t]     \begin{center}
179\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_jperio.pdf}
180\caption{    \label{Fig_LBC_jperio}
181setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.}
182\end{center}   \end{figure}
183%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
184
185% -------------------------------------------------------------------------------------------------------------
186%        North fold (\textit{jperio = 3 }to $6)$
187% -------------------------------------------------------------------------------------------------------------
188\subsection{North-fold (\textit{jperio = 3 }to $6$) }
189\label{LBC_north_fold}
190
191The north fold boundary condition has been introduced in order to handle the north
192boundary of a three-polar ORCA grid. Such a grid has two poles in the northern hemisphere
193(Fig.\ref{Fig_MISC_ORCA_msh}, and thus requires a specific treatment illustrated in Fig.\ref{Fig_North_Fold_T}.
194Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition.
195
196%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
197\begin{figure}[!t]    \begin{center}
198\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_North_Fold_T.pdf}
199\caption{    \label{Fig_North_Fold_T}
200North fold boundary with a $T$-point pivot and cyclic east-west boundary condition
201($jperio=4$), as used in ORCA 2, 1/4, and 1/12. Pink shaded area corresponds
202to the inner domain mask (see text). }
203\end{center}   \end{figure}
204%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
205
206% ====================================================================
207% Exchange with neighbouring processors
208% ====================================================================
209\section  [Exchange with neighbouring processors (\textit{lbclnk}, \textit{lib\_mpp})]
210      {Exchange with neighbouring processors (\mdl{lbclnk}, \mdl{lib\_mpp})}
211\label{LBC_mpp}
212
213For massively parallel processing (mpp), a domain decomposition method is used.
214The basic idea of the method is to split the large computation domain of a numerical
215experiment into several smaller domains and solve the set of equations by addressing
216independent local problems. Each processor has its own local memory and computes
217the model equation over a subdomain of the whole model domain. The subdomain
218boundary conditions are specified through communications between processors
219which are organized by explicit statements (message passing method).
220
221A big advantage is that the method does not need many modifications of the initial
222FORTRAN code. From the modeller's point of view, each sub domain running on
223a processor is identical to the "mono-domain" code. In addition, the programmer
224manages the communications between subdomains, and the code is faster when
225the number of processors is increased. The porting of OPA code on an iPSC860
226was achieved during Guyon's PhD [Guyon et al. 1994, 1995] in collaboration with
227CETIIS and ONERA. The implementation in the operational context and the studies
228of performance on a T3D and T3E Cray computers have been made in collaboration
229with IDRIS and CNRS. The present implementation is largely inspired by Guyon's
230work  [Guyon 1995].
231
232The parallelization strategy is defined by the physical characteristics of the
233ocean model. Second order finite difference schemes lead to local discrete
234operators that depend at the very most on one neighbouring point. The only
235non-local computations concern the vertical physics (implicit diffusion,
236turbulent closure scheme, ...) (delocalization over the whole water column),
237and the solving of the elliptic equation associated with the surface pressure
238gradient computation (delocalization over the whole horizontal domain).
239Therefore, a pencil strategy is used for the data sub-structuration
240: the 3D initial domain is laid out on local processor
241memories following a 2D horizontal topological splitting. Each sub-domain
242computes its own surface and bottom boundary conditions and has a side
243wall overlapping interface which defines the lateral boundary conditions for
244computations in the inner sub-domain. The overlapping area consists of the
245two rows at each edge of the sub-domain. After a computation, a communication
246phase starts: each processor sends to its neighbouring processors the update
247values of the points corresponding to the interior overlapping area to its
248neighbouring sub-domain ($i.e.$ the innermost of the two overlapping rows).
249The communication is done through the Message Passing Interface (MPI).
250The data exchanges between processors are required at the very
251place where lateral domain boundary conditions are set in the mono-domain
252computation : the \rou{lbc\_lnk} routine (found in \mdl{lbclnk} module)
253which manages such conditions is interfaced with routines found in \mdl{lib\_mpp} module
254when running on an MPP computer ($i.e.$ when \key{mpp\_mpi} defined).
255It has to be pointed out that when using the MPP version of the model,
256the east-west cyclic boundary condition is done implicitly,
257whilst the south-symmetric boundary condition option is not available.
258
259%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
260\begin{figure}[!t]    \begin{center}
261\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_mpp.pdf}
262\caption{   \label{Fig_mpp}
263Positioning of a sub-domain when massively parallel processing is used. }
264\end{center}   \end{figure}
265%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
266
267In the standard version of \NEMO, the splitting is regular and arithmetic.
268The i-axis is divided by \jp{jpni} and the j-axis by \jp{jpnj} for a number of processors
269\jp{jpnij} most often equal to $jpni \times jpnj$ (parameters set in
270 \ngn{nammpp} namelist). Each processor is independent and without message passing
271 or synchronous process, programs run alone and access just its own local memory.
272 For this reason, the main model dimensions are now the local dimensions of the subdomain (pencil)
273 that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. These dimensions include the internal
274 domain and the overlapping rows. The number of rows to exchange (known as
275 the halo) is usually set to one (\jp{jpreci}=1, in \mdl{par\_oce}). The whole domain
276 dimensions are named \np{jpiglo}, \np{jpjglo} and \jp{jpk}. The relationship between
277 the whole domain and a sub-domain is:
278\begin{eqnarray}
279      jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\
280      jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj  \label{Eq_lbc_jpi}
281\end{eqnarray}
282where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis.
283
284One also defines variables nldi and nlei which correspond to the internal domain bounds,
285and the variables nimpp and njmpp which are the position of the (1,1) grid-point in the global domain.
286An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
287a global array (whole domain) by the relationship:
288\begin{equation} \label{Eq_lbc_nimpp}
289T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
290\end{equation}
291with  $1 \leq i \leq jpi$, $1 \leq j \leq jpj$ , and  $1 \leq k \leq jpk$.
292
293Processors are numbered from 0 to $jpnij-1$, the number is saved in the variable
294nproc. In the standard version, a processor has no more than four neighbouring
295processors named nono (for north), noea (east), noso (south) and nowe (west)
296and two variables, nbondi and nbondj, indicate the relative position of the processor :
297\begin{itemize}
298\item       nbondi = -1    an east neighbour, no west processor,
299\item       nbondi =  0 an east neighbour, a west neighbour,
300\item       nbondi =  1    no east processor, a west neighbour,
301\item       nbondi =  2    no splitting following the i-axis.
302\end{itemize}
303During the simulation, processors exchange data with their neighbours.
304If there is effectively a neighbour, the processor receives variables from this
305processor on its overlapping row, and sends the data issued from internal
306domain corresponding to the overlapping row of the other processor.
307
308
309The \NEMO model computes equation terms with the help of mask arrays (0 on land
310points and 1 on sea points). It is easily readable and very efficient in the context of
311a computer with vectorial architecture. However, in the case of a scalar processor,
312computations over the land regions become more expensive in terms of CPU time.
313It is worse when we use a complex configuration with a realistic bathymetry like the
314global ocean where more than 50 \% of points are land points. For this reason, a
315pre-processing tool can be used to choose the mpp domain decomposition with a
316maximum number of only land points processors, which can then be eliminated (Fig. \ref{Fig_mppini2})
317(For example, the mpp\_optimiz tools, available from the DRAKKAR web site).
318This optimisation is dependent on the specific bathymetry employed. The user
319then chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with
320$jpnij < jpni \times jpnj$, leading to the elimination of $jpni \times jpnj - jpnij$
321land processors. When those parameters are specified in \ngn{nammpp} namelist,
322the algorithm in the \rou{inimpp2} routine sets each processor's parameters (nbound,
323nono, noea,...) so that the land-only processors are not taken into account.
324
325\gmcomment{Note that the inimpp2 routine is general so that the original inimpp
326routine should be suppressed from the code.}
327
328When land processors are eliminated, the value corresponding to these locations in
329the model output files is undefined. Note that this is a problem for the meshmask file
330which requires to be defined over the whole domain. Therefore, user should not eliminate
331land processors when creating a meshmask file ($i.e.$ when setting a non-zero value to \np{nn\_msh}).
332
333%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
334\begin{figure}[!ht]     \begin{center}
335\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_mppini2.pdf}
336\caption {    \label{Fig_mppini2}
337Example of Atlantic domain defined for the CLIPPER projet. Initial grid is
338composed of 773 x 1236 horizontal points.
339(a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
34052 subdomains are land areas.
341(b) 52 subdomains are eliminated (white rectangles) and the resulting number
342of processors really used during the computation is jpnij=128.}
343\end{center}   \end{figure}
344%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
345
346
347% ====================================================================
348% Unstructured open boundaries BDY
349% ====================================================================
350\section{Unstructured Open Boundary Conditions (\key{bdy}) (BDY)}
351\label{LBC_bdy}
352
353%-----------------------------------------nambdy--------------------------------------------
354\namdisplay{nambdy}
355%-----------------------------------------------------------------------------------------------
356%-----------------------------------------nambdy_index--------------------------------------------
357\namdisplay{nambdy_index}
358%-----------------------------------------------------------------------------------------------
359%-----------------------------------------nambdy_dta--------------------------------------------
360\namdisplay{nambdy_dta}
361%-----------------------------------------------------------------------------------------------
362%-----------------------------------------nambdy_dta--------------------------------------------
363\namdisplay{nambdy_dta2}
364%-----------------------------------------------------------------------------------------------
365
366Options are defined through the \ngn{nambdy} \ngn{nambdy\_index}
367\ngn{nambdy\_dta} \ngn{nambdy\_dta2} namelist variables.
368The BDY module is an alternative implementation of open boundary
369conditions for regional configurations. It implements the Flow
370Relaxation Scheme algorithm for temperature, salinity, velocities and
371ice fields, and the Flather radiation condition for the depth-mean
372transports. The specification of the location of the open boundary is
373completely flexible and allows for example the open boundary to follow
374an isobath or other irregular contour.
375
376The BDY module was modelled on the OBC module and shares many features
377and a similar coding structure \citep{Chanut2005}.
378
379The BDY module is completely rewritten at NEMO 3.4 and there is a new
380set of namelists. Boundary data files used with earlier versions of
381NEMO may need to be re-ordered to work with this version. See the
382section on the Input Boundary Data Files for details.
383
384%----------------------------------------------
385\subsection{The namelists}
386\label{BDY_namelist}
387
388It is possible to define more than one boundary set'' and apply
389different boundary conditions to each set. The number of boundary
390sets is defined by \np{nb\_bdy}.  Each boundary set may be defined
391as a set of straight line segments in a namelist
392(\np{ln\_coords\_file}=.false.) or read in from a file
393(\np{ln\_coords\_file}=.true.). If the set is defined in a namelist,
394then the namelists nambdy\_index must be included separately, one for
395each set. If the set is defined by a file, then a
396coordinates.bdy.nc'' file must be provided. The coordinates.bdy file
397is analagous to the usual NEMO coordinates.nc'' file. In the example
398above, there are two boundary sets, the first of which is defined via
399a file and the second is defined in a namelist. For more details of
400the definition of the boundary geometry see section
401\ref{BDY_geometry}.
402
403For each boundary set a boundary
404condition has to be chosen for the barotropic solution (u2d'':
405sea-surface height and barotropic velocities), for the baroclinic
406velocities (u3d''), and for the active tracers\footnote{The BDY
407  module does not deal with passive tracers at this version}
408(tra''). For each set of variables there is a choice of algorithm
409and a choice for the data, eg. for the active tracers the algorithm is
410set by \np{nn\_tra} and the choice of data is set by
411\np{nn\_tra\_dta}.
412
413The choice of algorithm is currently as follows:
414
415\mbox{}
416
417\begin{itemize}
418\item[0.] No boundary condition applied. So the solution will see''
419  the land points around the edge of the edge of the domain.
420\item[1.] Flow Relaxation Scheme (FRS) available for all variables.
421\item[2.] Flather radiation scheme for the barotropic variables. The
422  Flather scheme is not compatible with the filtered free surface
423  ({\it dynspg\_ts}).
424\end{itemize}
425
426\mbox{}
427
428The main choice for the boundary data is
429to use initial conditions as boundary data (\np{nn\_tra\_dta}=0) or to
430use external data from a file (\np{nn\_tra\_dta}=1). For the
431barotropic solution there is also the option to use tidal
432harmonic forcing either by itself or in addition to other external
433data.
434
435If external boundary data is required then the nambdy\_dta namelist
436must be defined. One nambdy\_dta namelist is required for each boundary
437set in the order in which the boundary sets are defined in nambdy. In
438the example given, two boundary sets have been defined and so there
439are two nambdy\_dta namelists. The boundary data is read in using the
440fldread module, so the nambdy\_dta namelist is in the format required
441for fldread. For each variable required, the filename, the frequency
442of the files and the frequency of the data in the files is given. Also
443whether or not time-interpolation is required and whether the data is
444climatological (time-cyclic) data. Note that on-the-fly spatial
445interpolation of boundary data is not available at this version.
446
447In the example namelists given, two boundary sets are defined. The
448first set is defined via a file and applies FRS conditions to
449temperature and salinity and Flather conditions to the barotropic
450variables. External data is provided in daily files (from a
451large-scale model). Tidal harmonic forcing is also used. The second
452set is defined in a namelist. FRS conditions are applied on
453temperature and salinity and climatological data is read from external
454files.
455
456%----------------------------------------------
457\subsection{The Flow Relaxation Scheme}
458\label{BDY_FRS_scheme}
459
460The Flow Relaxation Scheme (FRS) \citep{Davies_QJRMS76,Engerdahl_Tel95},
461applies a simple relaxation of the model fields to
462externally-specified values over a zone next to the edge of the model
463domain. Given a model prognostic variable $\Phi$
464\begin{equation}  \label{Eq_bdy_frs1}
465\Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
466\end{equation}
467where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified
468external field, $d$ gives the discrete distance from the model
469boundary  and $\alpha$ is a parameter that varies from $1$ at $d=1$ to
470a small value at $d=N$. It can be shown that this scheme is equivalent
471to adding a relaxation term to the prognostic equation for $\Phi$ of
472the form:
473\begin{equation}  \label{Eq_bdy_frs2}
474-\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
475\end{equation}
476where the relaxation time scale $\tau$ is given by a function of
477$\alpha$ and the model time step $\Delta t$:
478\begin{equation}  \label{Eq_bdy_frs3}
479\tau = \frac{1-\alpha}{\alpha}  \,\rdt
480\end{equation}
481Thus the model solution is completely prescribed by the external
482conditions at the edge of the model domain and is relaxed towards the
483external conditions over the rest of the FRS zone. The application of
484a relaxation zone helps to prevent spurious reflection of outgoing
485signals from the model boundary.
486
487The function $\alpha$ is specified as a $tanh$ function:
488\begin{equation}  \label{Eq_bdy_frs4}
489\alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
490\end{equation}
491The width of the FRS zone is specified in the namelist as
492\np{nn\_rimwidth}. This is typically set to a value between 8 and 10.
493
494%----------------------------------------------
496\label{BDY_flather_scheme}
497
498The \citet{Flather_JPO94} scheme is a radiation condition on the normal, depth-mean
499transport across the open boundary. It takes the form
500\begin{equation}  \label{Eq_bdy_fla1}
501U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
502\end{equation}
503where $U$ is the depth-mean velocity normal to the boundary and $\eta$
504is the sea surface height, both from the model. The subscript $e$
505indicates the same fields from external sources. The speed of external
506gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the
507water column. The depth-mean normal velocity along the edge of the
508model domain is set equal to the
509external depth-mean normal velocity, plus a correction term that
510allows gravity waves generated internally to exit the model boundary.
511Note that the sea-surface height gradient in \eqref{Eq_bdy_fla1}
512is a spatial gradient across the model boundary, so that $\eta_{e}$ is
513defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the
514$T$ points with $nbr=2$. $U$ and $U_{e}$ are defined on the $U$ or
515$V$ points with $nbr=1$, $i.e.$ between the two $T$ grid points.
516
517%----------------------------------------------
518\subsection{Boundary geometry}
519\label{BDY_geometry}
520
521Each open boundary set is defined as a list of points. The information
522is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$
523structure.  The $nbi$ and $nbj$ arrays
524define the local $(i,j)$ indices of each point in the boundary zone
525and the $nbr$ array defines the discrete distance from the boundary
526with $nbr=1$ meaning that the point is next to the edge of the
527model domain and $nbr>1$ showing that the point is increasingly
528further away from the edge of the model domain. A set of $nbi$, $nbj$,
529and $nbr$ arrays is defined for each of the $T$, $U$ and $V$
530grids. Figure \ref{Fig_LBC_bdy_geom} shows an example of an irregular
531boundary.
532
533The boundary geometry for each set may be defined in a namelist
534nambdy\_index or by reading in a coordinates.bdy.nc'' file. The
535nambdy\_index namelist defines a series of straight-line segments for
536north, east, south and west boundaries. For the northern boundary,
537\np{nbdysegn} gives the number of segments, \np{jpjnob} gives the $j$
538index for each segment and \np{jpindt} and \np{jpinft} give the start
539and end $i$ indices for each segment with similar for the other
540boundaries. These segments define a list of $T$ grid points along the
541outermost row of the boundary ($nbr\,=\, 1$). The code deduces the $U$ and
542$V$ points and also the points for $nbr\,>\, 1$ if
543$nn\_rimwidth\,>\,1$.
544
545The boundary geometry may also be defined from a
546coordinates.bdy.nc'' file. Figure \ref{Fig_LBC_nc_header}
547gives an example of the header information from such a file. The file
548should contain the index arrays for each of the $T$, $U$ and $V$
549grids. The arrays must be in order of increasing $nbr$. Note that the
550$nbi$, $nbj$ values in the file are global values and are converted to
551local values in the code. Typically this file will be used to generate
552external boundary data via interpolation and so will also contain the
553latitudes and longitudes of each point as shown. However, this is not
554necessary to run the model.
555
556For some choices of irregular boundary the model domain may contain
557areas of ocean which are not part of the computational domain. For
558example if an open boundary is defined along an isobath, say at the
559shelf break, then the areas of ocean outside of this boundary will
562used even if multiple boundary sets are defined.
563
564%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
565\begin{figure}[!t]      \begin{center}
566\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_bdy_geom.pdf}
567\caption {      \label{Fig_LBC_bdy_geom}
568Example of geometry of unstructured open boundary}
569\end{center}   \end{figure}
570%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
571
572%----------------------------------------------
573\subsection{Input boundary data files}
574\label{BDY_data}
575
576The data files contain the data arrays
577in the order in which the points are defined in the $nbi$ and $nbj$
578arrays. The data arrays are dimensioned on: a time dimension;
579$xb$ which is the index of the boundary data point in the horizontal;
580and $yb$ which is a degenerate dimension of 1 to enable the file to be
581read by the standard NEMO I/O routines. The 3D fields also have a
582depth dimension.
583
584At Version 3.4 there are new restrictions on the order in which the
585boundary points are defined (and therefore restrictions on the order
586of the data in the file). In particular:
587
588\mbox{}
589
590\begin{enumerate}
591\item The data points must be in order of increasing $nbr$, ie. all
592  the $nbr=1$ points, then all the $nbr=2$ points etc.
593\item All the data for a particular boundary set must be in the same
594  order. (Prior to 3.4 it was possible to define barotropic data in a
595  different order to the data for tracers and baroclinic velocities).
596\end{enumerate}
597
598\mbox{}
599
600These restrictions mean that data files used with previous versions of
601the model may not work with version 3.4. A fortran utility
602{\it bdy\_reorder} exists in the TOOLS directory which will re-order the
603data in old BDY data files.
604
605%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
606\begin{figure}[!t]     \begin{center}
609Example of the header for a coordinates.bdy.nc file}
610\end{center}   \end{figure}
611%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
612
613%----------------------------------------------
614\subsection{Volume correction}
615\label{BDY_vol_corr}
616
617There is an option to force the total volume in the regional model to be constant,
618similar to the option in the OBC module. This is controlled  by the \np{nn\_volctl}
619parameter in the namelist. A value of \np{nn\_volctl}~=~0 indicates that this option is not used.
620If  \np{nn\_volctl}~=~1 then a correction is applied to the normal velocities
621around the boundary at each timestep to ensure that the integrated volume flow
622through the boundary is zero. If \np{nn\_volctl}~=~2 then the calculation of
623the volume change on the timestep includes the change due to the freshwater
624flux across the surface and the correction velocity corrects for this as well.
625
626If more than one boundary set is used then volume correction is
627applied to all boundaries at once.
628
629\newpage
630%----------------------------------------------
631\subsection{Tidal harmonic forcing}
632\label{BDY_tides}
633
634%-----------------------------------------nambdy_tide--------------------------------------------
635\namdisplay{nambdy_tide}
636%-----------------------------------------------------------------------------------------------
637
638Options are defined through the  \ngn{nambdy\_tide} namelist variables.
639 To be written....
640
641
642
643
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