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chap_LBC.tex in NEMO/trunk/doc/latex/NEMO/subfiles – NEMO

source: NEMO/trunk/doc/latex/NEMO/subfiles/chap_LBC.tex @ 11435

Last change on this file since 11435 was 11435, checked in by nicolasmartin, 5 years ago

Various corrections on chapters

Cleaning the indexes by fixing/removing wrong entries (or appending a ? to unknown items) and
improve the classification with new index definitions for CPP keys and namelist blocks:

  • from \key{...} cmd, key_ prefix no longer precedes the index entry
  • namelist block declaration moves from \ngn{nam...} to \nam{...} (i.e. \ngn{namtra\_ldf} -> \nam{tra\_ldf}) The expected prefix nam is added to the printed word but not the index entry.

Now we have indexes with a better sorting instead of all CPP keys under 'K' and namelists blocks under 'N'.

Fix missing space issues with alias commands by adding a trailing backslash (\NEMO\, \ie\, \eg\, ...).
There is no perfect solution for this, and I prefer not using a particular package to solve it.

Review the initial LaTeX code snippet for the historic changes in chapters

Finally, for readability and future diff visualisations, please avoid writing paragraphs with continuous lines.
Break the lines around 80 to 100 characters long

File size: 41.2 KB
4% ================================================================
5% Chapter — Lateral Boundary Condition (LBC)
6% ================================================================
7\chapter{Lateral Boundary Condition (LBC)}
14%gm% add here introduction to this chapter
16% ================================================================
17% Boundary Condition at the Coast
18% ================================================================
19\section[Boundary condition at the coast (\texttt{rn\_shlat})]
20{Boundary condition at the coast (\protect\np{rn\_shlat})}
27%The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt
28%(no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip).
29%They are handled automatically by the mask system (see \autoref{subsec:DOM_msk}).
31%OPA allows land and topography grid points in the computational domain due to the presence of continents or islands,
32%and includes the use of a full or partial step representation of bottom topography.
33%The computation is performed over the whole domain, \ie\ we do not try to restrict the computation to ocean-only points.
34%This choice has two motivations.
35%Firstly, working on ocean only grid points overloads the code and harms the code readability.
36%Secondly, and more importantly, it drastically reduces the vector portion of the computation,
37%leading to a dramatic increase of CPU time requirement on vector computers.
38%The current section describes how the masking affects the computation of the various terms of the equations
39%with respect to the boundary condition at solid walls.
40%The process of defining which areas are to be masked is described in \autoref{subsec:DOM_msk}.
42Options are defined through the \nam{lbc} namelist variables.
43The discrete representation of a domain with complex boundaries (coastlines and bottom topography) leads to
44arrays that include large portions where a computation is not required as the model variables remain at zero.
45Nevertheless, vectorial supercomputers are far more efficient when computing over a whole array,
46and the readability of a code is greatly improved when boundary conditions are applied in
47an automatic way rather than by a specific computation before or after each computational loop.
48An efficient way to work over the whole domain while specifying the boundary conditions,
49is to use multiplication by mask arrays in the computation.
50A mask array is a matrix whose elements are $1$ in the ocean domain and $0$ elsewhere.
51A simple multiplication of a variable by its own mask ensures that it will remain zero over land areas.
52Since most of the boundary conditions consist of a zero flux across the solid boundaries,
53they can be simply applied by multiplying variables by the correct mask arrays,
54\ie\ the mask array of the grid point where the flux is evaluated.
55For example, the heat flux in the \textbf{i}-direction is evaluated at $u$-points.
56Evaluating this quantity as,
59  % \label{eq:lbc_aaaa}
60  \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT}
61  }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u
63(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is zero inside land and
64at the boundaries, since mask$_{u}$ is zero at solid boundaries which in this case are defined at $u$-points
65(normal velocity $u$ remains zero at the coast) (\autoref{fig:LBC_uv}).
69  \begin{center}
70    \includegraphics[width=\textwidth]{Fig_LBC_uv}
71    \caption{
72      \protect\label{fig:LBC_uv}
73      Lateral boundary (thick line) at T-level.
74      The velocity normal to the boundary is set to zero.
75    }
76  \end{center}
80For momentum the situation is a bit more complex as two boundary conditions must be provided along the coast
81(one each for the normal and tangential velocities).
82The boundary of the ocean in the C-grid is defined by the velocity-faces.
83For example, at a given $T$-level,
84the lateral boundary (a coastline or an intersection with the bottom topography) is made of
85segments joining $f$-points, and normal velocity points are located between two $f-$points (\autoref{fig:LBC_uv}).
86The boundary condition on the normal velocity (no flux through solid boundaries)
87can thus be easily implemented using the mask system.
88The boundary condition on the tangential velocity requires a more specific treatment.
89This boundary condition influences the relative vorticity and momentum diffusive trends,
90and is required in order to compute the vorticity at the coast.
91Four different types of lateral boundary condition are available,
92controlled by the value of the \np{rn\_shlat} namelist parameter
93(The value of the mask$_{f}$ array along the coastline is set equal to this parameter).
94These are:
98  \begin{center}
99    \includegraphics[width=\textwidth]{Fig_LBC_shlat}
100    \caption{
101      \protect\label{fig:LBC_shlat}
102      lateral boundary condition
103      (a) free-slip ($rn\_shlat=0$);
104      (b) no-slip ($rn\_shlat=2$);
105      (c) "partial" free-slip ($0<rn\_shlat<2$) and
106      (d) "strong" no-slip ($2<rn\_shlat$).
107      Implied "ghost" velocity inside land area is display in grey.
108    }
109  \end{center}
115\item[free-slip boundary condition (\np{rn\_shlat}\forcode{ = 0}):] the tangential velocity at
116  the coastline is equal to the offshore velocity,
117  \ie\ the normal derivative of the tangential velocity is zero at the coast,
118  so the vorticity: mask$_{f}$ array is set to zero inside the land and just at the coast
119  (\autoref{fig:LBC_shlat}-a).
121\item[no-slip boundary condition (\np{rn\_shlat}\forcode{ = 2}):] the tangential velocity vanishes at the coastline.
122  Assuming that the tangential velocity decreases linearly from
123  the closest ocean velocity grid point to the coastline,
124  the normal derivative is evaluated as if the velocities at the closest land velocity gridpoint and
125  the closest ocean velocity gridpoint were of the same magnitude but in the opposite direction
126  (\autoref{fig:LBC_shlat}-b).
127  Therefore, the vorticity along the coastlines is given by:
129  \[
130    \zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ ,
131  \]
132  where $u$ and $v$ are masked fields.
133  Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with
134  the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ :
135  \[
136    % \label{eq:lbc_bbbb}
137    \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2}
138        \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]}
139    \right)\;\mbox{mask}_f
140  \]
142\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2):] the tangential velocity at
143  the coastline is smaller than the offshore velocity, \ie\ there is a lateral friction but
144  not strong enough to make the tangential velocity at the coast vanish (\autoref{fig:LBC_shlat}-c).
145  This can be selected by providing a value of mask$_{f}$ strictly inbetween $0$ and $2$.
147\item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}):] the viscous boundary layer is assumed to
148  be smaller than half the grid size (\autoref{fig:LBC_shlat}-d).
149  The friction is thus larger than in the no-slip case.
153Note that when the bottom topography is entirely represented by the $s$-coor-dinates (pure $s$-coordinate),
154the lateral boundary condition on tangential velocity is of much less importance as
155it is only applied next to the coast where the minimum water depth can be quite shallow.
158% ================================================================
159% Boundary Condition around the Model Domain
160% ================================================================
161\section[Model domain boundary condition (\texttt{jperio})]
162{Model domain boundary condition (\protect\jp{jperio})}
165At the model domain boundaries several choices are offered:
166closed, cyclic east-west, cyclic north-south, a north-fold, and combination closed-north fold or
167bi-cyclic east-west and north-fold.
168The north-fold boundary condition is associated with the 3-pole ORCA mesh.
170% -------------------------------------------------------------------------------------------------------------
171%        Closed, cyclic (\jp{jperio}\forcode{ = 0..2})
172% -------------------------------------------------------------------------------------------------------------
173\subsection[Closed, cyclic (\forcode{jperio = [0127]})]
174{Closed, cyclic (\protect\jp{jperio}\forcode{ = [0127]})}
177The choice of closed or cyclic model domain boundary condition is made by
178setting \jp{jperio} to 0, 1, 2 or 7 in namelist \nam{cfg}.
179Each time such a boundary condition is needed, it is set by a call to routine \mdl{lbclnk}.
180The computation of momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to $j=jpj-1$,
181\ie\ in the model interior.
182To choose a lateral model boundary condition is to specify the first and last rows and columns of
183the model variables.
187\item[For closed boundary (\jp{jperio}\forcode{ = 0})],
188  solid walls are imposed at all model boundaries:
189  first and last rows and columns are set to zero.
191\item[For cyclic east-west boundary (\jp{jperio}\forcode{ = 1})],
192  first and last rows are set to zero (closed) whilst the first column is set to
193  the value of the last-but-one column and the last column to the value of the second one
194  (\autoref{fig:LBC_jperio}-a).
195  Whatever flows out of the eastern (western) end of the basin enters the western (eastern) end.
197\item[For cyclic north-south boundary (\jp{jperio}\forcode{ = 2})],
198  first and last columns are set to zero (closed) whilst the first row is set to
199  the value of the last-but-one row and the last row to the value of the second one
200  (\autoref{fig:LBC_jperio}-a).
201  Whatever flows out of the northern (southern) end of the basin enters the southern (northern) end.
203\item[Bi-cyclic east-west and north-south boundary (\jp{jperio}\forcode{ = 7})] combines cases 1 and 2.
209  \begin{center}
210    \includegraphics[width=\textwidth]{Fig_LBC_jperio}
211    \caption{
212      \protect\label{fig:LBC_jperio}
213      setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.
214    }
215  \end{center}
219% -------------------------------------------------------------------------------------------------------------
220%        North fold (\textit{jperio = 3 }to $6)$
221% -------------------------------------------------------------------------------------------------------------
222\subsection[North-fold (\forcode{jperio = [3-6]})]
223{North-fold (\protect\jp{jperio}\forcode{ = [3-6]})}
226The north fold boundary condition has been introduced in order to handle the north boundary of
227a three-polar ORCA grid.
228Such a grid has two poles in the northern hemisphere (\autoref{fig:MISC_ORCA_msh},
229and thus requires a specific treatment illustrated in \autoref{fig:North_Fold_T}.
230Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition.
234  \begin{center}
235    \includegraphics[width=\textwidth]{Fig_North_Fold_T}
236    \caption{
237      \protect\label{fig:North_Fold_T}
238      North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$),
239      as used in ORCA 2, 1/4, and 1/12.
240      Pink shaded area corresponds to the inner domain mask (see text).
241    }
242  \end{center}
246% ====================================================================
247% Exchange with neighbouring processors
248% ====================================================================
249\section[Exchange with neighbouring processors (\textit{lbclnk.F90}, \textit{lib\_mpp.F90})]
250{Exchange with neighbouring processors (\protect\mdl{lbclnk}, \protect\mdl{lib\_mpp})}
253For massively parallel processing (mpp), a domain decomposition method is used.
254The basic idea of the method is to split the large computation domain of a numerical experiment into
255several smaller domains and solve the set of equations by addressing independent local problems.
256Each processor has its own local memory and computes the model equation over a subdomain of the whole model domain.
257The subdomain boundary conditions are specified through communications between processors which
258are organized by explicit statements (message passing method).
260A big advantage is that the method does not need many modifications of the initial \fortran code.
261From the modeller's point of view, each sub domain running on a processor is identical to the "mono-domain" code.
262In addition, the programmer manages the communications between subdomains,
263and the code is faster when the number of processors is increased.
264The porting of OPA code on an iPSC860 was achieved during Guyon's PhD [Guyon et al. 1994, 1995]
265in collaboration with CETIIS and ONERA.
266The implementation in the operational context and the studies of performance on
267a T3D and T3E Cray computers have been made in collaboration with IDRIS and CNRS.
268The present implementation is largely inspired by Guyon's work [Guyon 1995].
270The parallelization strategy is defined by the physical characteristics of the ocean model.
271Second order finite difference schemes lead to local discrete operators that
272depend at the very most on one neighbouring point.
273The only non-local computations concern the vertical physics
274(implicit diffusion, turbulent closure scheme, ...) (delocalization over the whole water column),
275and the solving of the elliptic equation associated with the surface pressure gradient computation
276(delocalization over the whole horizontal domain).
277Therefore, a pencil strategy is used for the data sub-structuration:
278the 3D initial domain is laid out on local processor memories following a 2D horizontal topological splitting.
279Each sub-domain computes its own surface and bottom boundary conditions and
280has a side wall overlapping interface which defines the lateral boundary conditions for
281computations in the inner sub-domain.
282The overlapping area consists of the two rows at each edge of the sub-domain.
283After a computation, a communication phase starts:
284each processor sends to its neighbouring processors the update values of the points corresponding to
285the interior overlapping area to its neighbouring sub-domain (\ie\ the innermost of the two overlapping rows).
286The communication is done through the Message Passing Interface (MPI).
287The data exchanges between processors are required at the very place where
288lateral domain boundary conditions are set in the mono-domain computation:
289the \rou{lbc\_lnk} routine (found in \mdl{lbclnk} module) which manages such conditions is interfaced with
290routines found in \mdl{lib\_mpp} module when running on an MPP computer (\ie\ when \key{mpp\_mpi} defined).
291It has to be pointed out that when using the MPP version of the model,
292the east-west cyclic boundary condition is done implicitly,
293whilst the south-symmetric boundary condition option is not available.
297  \begin{center}
298    \includegraphics[width=\textwidth]{Fig_mpp}
299    \caption{
300      \protect\label{fig:mpp}
301      Positioning of a sub-domain when massively parallel processing is used.
302    }
303  \end{center}
307In the standard version of \NEMO, the splitting is regular and arithmetic.
308The i-axis is divided by \jp{jpni} and
309the j-axis by \jp{jpnj} for a number of processors \jp{jpnij} most often equal to $jpni \times jpnj$
310(parameters set in  \nam{mpp} namelist).
311Each processor is independent and without message passing or synchronous process,
312programs run alone and access just its own local memory.
313For this reason, the main model dimensions are now the local dimensions of the subdomain (pencil) that
314are named \jp{jpi}, \jp{jpj}, \jp{jpk}.
315These dimensions include the internal domain and the overlapping rows.
316The number of rows to exchange (known as the halo) is usually set to one (\jp{jpreci}=1, in \mdl{par\_oce}).
317The whole domain dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}.
318The relationship between the whole domain and a sub-domain is:
320  jpi = ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci
321  jpj = ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj
323where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis.
325One also defines variables nldi and nlei which correspond to the internal domain bounds,
326and the variables nimpp and njmpp which are the position of the (1,1) grid-point in the global domain.
327An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
328a global array (whole domain) by the relationship:
330  % \label{eq:lbc_nimpp}
331  T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
333with  $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.
335Processors are numbered from 0 to $jpnij-1$, the number is saved in the variable nproc.
336In the standard version, a processor has no more than
337four neighbouring processors named nono (for north), noea (east), noso (south) and nowe (west) and
338two variables, nbondi and nbondj, indicate the relative position of the processor:
340\item       nbondi = -1    an east neighbour, no west processor,
341\item       nbondi =  0 an east neighbour, a west neighbour,
342\item       nbondi =  1    no east processor, a west neighbour,
343\item       nbondi =  2    no splitting following the i-axis.
345During the simulation, processors exchange data with their neighbours.
346If there is effectively a neighbour, the processor receives variables from this processor on its overlapping row,
347and sends the data issued from internal domain corresponding to the overlapping row of the other processor.
350The \NEMO\ model computes equation terms with the help of mask arrays (0 on land points and 1 on sea points).
351It is easily readable and very efficient in the context of a computer with vectorial architecture.
352However, in the case of a scalar processor, computations over the land regions become more expensive in
353terms of CPU time.
354It is worse when we use a complex configuration with a realistic bathymetry like the global ocean where
355more than 50 \% of points are land points.
356For this reason, a pre-processing tool can be used to choose the mpp domain decomposition with a maximum number of
357only land points processors, which can then be eliminated (\autoref{fig:mppini2})
358(For example, the mpp\_optimiz tools, available from the DRAKKAR web site).
359This optimisation is dependent on the specific bathymetry employed.
360The user then chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with $jpnij < jpni \times jpnj$,
361leading to the elimination of $jpni \times jpnj - jpnij$ land processors.
362When those parameters are specified in \nam{mpp} namelist,
363the algorithm in the \rou{inimpp2} routine sets each processor's parameters (nbound, nono, noea,...) so that
364the land-only processors are not taken into account.
366\gmcomment{Note that the inimpp2 routine is general so that the original inimpp
367routine should be suppressed from the code.}
369When land processors are eliminated,
370the value corresponding to these locations in the model output files is undefined.
371Note that this is a problem for the meshmask file which requires to be defined over the whole domain.
372Therefore, user should not eliminate land processors when creating a meshmask file
373(\ie\ when setting a non-zero value to \np{nn\_msh}).
377  \begin{center}
378    \includegraphics[width=\textwidth]{Fig_mppini2}
379    \caption {
380      \protect\label{fig:mppini2}
381      Example of Atlantic domain defined for the CLIPPER projet.
382      Initial grid is composed of 773 x 1236 horizontal points.
383      (a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
384      52 subdomains are land areas.
385      (b) 52 subdomains are eliminated (white rectangles) and
386      the resulting number of processors really used during the computation is jpnij=128.
387    }
388  \end{center}
393% ====================================================================
394% Unstructured open boundaries BDY
395% ====================================================================
396\section{Unstructured open boundary conditions (BDY)}
408Options are defined through the \nam{bdy} \nam{bdy\_dta} namelist variables.
409The BDY module is the core implementation of open boundary conditions for regional configurations on
410temperature, salinity, barotropic and baroclinic velocities, as well as ice concentration, ice and snow thicknesses).
412The BDY module was modelled on the OBC module (see \NEMO\ 3.4) and shares many features and
413a similar coding structure \citep{chanut_rpt05}.
414The specification of the location of the open boundary is completely flexible and
415allows for example the open boundary to follow an isobath or other irregular contour.
416Boundary data files used with versions of \NEMO\ prior to Version 3.4 may need to be re-ordered to work with this version.
417See the section on the Input Boundary Data Files for details.
423The BDY module is activated by setting \np{ln\_bdy}\forcode{ = .true.} .
424It is possible to define more than one boundary ``set'' and apply different boundary conditions to each set.
425The number of boundary sets is defined by \np{nb\_bdy}.
426Each boundary set may be defined as a set of straight line segments in a namelist
427(\np{ln\_coords\_file}\forcode{ = .false.}) or read in from a file (\np{ln\_coords\_file}\forcode{ = .true.}).
428If the set is defined in a namelist, then the namelists \nam{bdy\_index} must be included separately, one for each set.
429If the set is defined by a file, then a ``\ifile{coordinates.bdy}'' file must be provided.
430The coordinates.bdy file is analagous to the usual \NEMO\ ``\ifile{coordinates}'' file.
431In the example above, there are two boundary sets, the first of which is defined via a file and
432the second is defined in a namelist.
433For more details of the definition of the boundary geometry see section \autoref{subsec:BDY_geometry}.
435For each boundary set a boundary condition has to be chosen for the barotropic solution
436(``u2d'':sea-surface height and barotropic velocities), for the baroclinic velocities (``u3d''),
437for the active tracers \footnote{The BDY module does not deal with passive tracers at this version} (``tra''), and sea-ice (``ice'').
438For each set of variables there is a choice of algorithm and a choice for the data,
439eg. for the active tracers the algorithm is set by \np{cn\_tra} and the choice of data is set by \np{nn\_tra\_dta}.\\
441The choice of algorithm is currently as follows:
444\item[\forcode{'none'}:] No boundary condition applied.
445  So the solution will ``see'' the land points around the edge of the edge of the domain.
446\item[\forcode{'specified'}:] Specified boundary condition applied (only available for baroclinic velocity and tracer variables).
447\item[\forcode{'neumann'}:] Value at the boundary are duplicated (No gradient). Only available for baroclinic velocity and tracer variables.
448\item[\forcode{'frs'}:] Flow Relaxation Scheme (FRS) available for all variables.
449\item[\forcode{'Orlanski'}:] Orlanski radiation scheme (fully oblique) for barotropic, baroclinic and tracer variables.
450\item[\forcode{'Orlanski_npo'}:] Orlanski radiation scheme for barotropic, baroclinic and tracer variables.
451\item[\forcode{'flather'}:] Flather radiation scheme for the barotropic variables only.
454The main choice for the boundary data is to use initial conditions as boundary data
455(\np{nn\_tra\_dta}\forcode{ = 0}) or to use external data from a file (\np{nn\_tra\_dta}\forcode{ = 1}).
456In case the 3d velocity data contain the total velocity (ie, baroclinic and barotropic velocity),
457the bdy code can derived baroclinic and barotropic velocities by setting \np{ln\_full\_vel}\forcode{ = .true. }
458For the barotropic solution there is also the option to use tidal harmonic forcing either by
459itself (\np{nn\_dyn2d\_dta}\forcode{ = 2}) or in addition to other external data (\np{nn\_dyn2d\_dta}\forcode{ = 3}).\\
460Sea-ice salinity, temperature and age data at the boundary are constant and defined repectively by \np{rn\_ice\_sal}, \np{rn\_ice\_tem} and \np{rn\_ice\_age}.
462If external boundary data is required then the \nam{bdy\_dta} namelist must be defined.
463One \nam{bdy\_dta} namelist is required for each boundary set in the order in which
464the boundary sets are defined in nambdy.
465In the example given, two boundary sets have been defined. The first one is reading data file in the \nam{bdy\_dta} namelist shown above
466and the second one is using data from intial condition (no namelist bloc needed).
467The boundary data is read in using the fldread module,
468so the \nam{bdy\_dta} namelist is in the format required for fldread.
469For each variable required, the filename, the frequency of the files and
470the frequency of the data in the files is given.
471Also whether or not time-interpolation is required and whether the data is climatological (time-cyclic) data.\\
473There is currently an option to vertically interpolate the open boundary data onto the native grid at run-time.
474If \np{nn\_bdy\_jpk} $< -1$, it is assumed that the lateral boundary data are already on the native grid.
475However, if \np{nn\_bdy\_jpk} is set to the number of vertical levels present in the boundary data,
476a bilinear interpolation onto the native grid will be triggered at runtime.
477For this to be successful the additional variables: $gdept$, $gdepu$, $gdepv$, $e3t$, $e3u$ and $e3v$, are required to be present in the lateral boundary files.
478These correspond to the depths and scale factors of the input data,
479the latter used to make any adjustment to the velocity fields due to differences in the total water depths between the two vertical grids.\\
481In the example namelists given, two boundary sets are defined.
482The first set is defined via a file and applies FRS conditions to temperature and salinity and
483Flather conditions to the barotropic variables. No condition specified for the baroclinic velocity and sea-ice.
484External data is provided in daily files (from a large-scale model).
485Tidal harmonic forcing is also used.
486The second set is defined in a namelist.
487FRS conditions are applied on temperature and salinity and climatological data is read from initial condition files.
490\subsection{Flow relaxation scheme}
493The Flow Relaxation Scheme (FRS) \citep{davies_QJRMS76,engedahl_T95},
494applies a simple relaxation of the model fields to externally-specified values over
495a zone next to the edge of the model domain.
496Given a model prognostic variable $\Phi$
498  % \label{eq:bdy_frs1}
499  \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
501where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified external field,
502$d$ gives the discrete distance from the model boundary and
503$\alpha$ is a parameter that varies from $1$ at $d=1$ to a small value at $d=N$.
504It can be shown that this scheme is equivalent to adding a relaxation term to
505the prognostic equation for $\Phi$ of the form:
507  % \label{eq:bdy_frs2}
508  -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
510where the relaxation time scale $\tau$ is given by a function of $\alpha$ and the model time step $\Delta t$:
512  % \label{eq:bdy_frs3}
513  \tau = \frac{1-\alpha}{\alpha\,\rdt
515Thus the model solution is completely prescribed by the external conditions at the edge of the model domain and
516is relaxed towards the external conditions over the rest of the FRS zone.
517The application of a relaxation zone helps to prevent spurious reflection of
518outgoing signals from the model boundary.
520The function $\alpha$ is specified as a $tanh$ function:
522  % \label{eq:bdy_frs4}
523  \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
525The width of the FRS zone is specified in the namelist as \np{nn\_rimwidth}.
526This is typically set to a value between 8 and 10.
529\subsection{Flather radiation scheme}
532The \citet{flather_JPO94} scheme is a radiation condition on the normal,
533depth-mean transport across the open boundary.
534It takes the form
535\begin{equation}  \label{eq:bdy_fla1}
536U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
538where $U$ is the depth-mean velocity normal to the boundary and $\eta$ is the sea surface height,
539both from the model.
540The subscript $e$ indicates the same fields from external sources.
541The speed of external gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the water column.
542The depth-mean normal velocity along the edge of the model domain is set equal to
543the external depth-mean normal velocity,
544plus a correction term that allows gravity waves generated internally to exit the model boundary.
545Note that the sea-surface height gradient in \autoref{eq:bdy_fla1} is a spatial gradient across the model boundary,
546so that $\eta_{e}$ is defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the $T$ points with $nbr=2$.
547$U$ and $U_{e}$ are defined on the $U$ or $V$ points with $nbr=1$, \ie\ between the two $T$ grid points.
550\subsection{Orlanski radiation scheme}
553The Orlanski scheme is based on the algorithm described by \citep{marchesiello.mcwilliams.ea_OM01}, hereafter MMS.
555The adaptive Orlanski condition solves a wave plus relaxation equation at the boundary:
557\frac{\partial\phi}{\partial t} + c_x \frac{\partial\phi}{\partial x} + c_y \frac{\partial\phi}{\partial y} =
558                                                -\frac{1}{\tau}(\phi - \phi^{ext})
562where $\phi$ is the model field, $x$ and $y$ refer to the normal and tangential directions to the boundary respectively, and the phase
563velocities are diagnosed from the model fields as:
565\begin{equation} \label{eq:cx}
566c_x = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial x}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}
570c_y = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial y}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}
573(As noted by MMS, this is a circular diagnosis of the phase speeds which only makes sense on a discrete grid).
574Equation (\autoref{eq:wave_continuous}) is defined adaptively depending on the sign of the phase velocity normal to the boundary $c_x$.
575For $c_x$ outward, we have
578\tau = \tau_{out}
581For $c_x$ inward, the radiation equation is not applied:
584\tau = \tau_{in}\,\,\,;\,\,\, c_x = c_y = 0
588Generally the relaxation time scale at inward propagation points (\np{rn\_time\_dmp}) is set much shorter than the time scale at outward propagation
589points (\np{rn\_time\_dmp\_out}) so that the solution is constrained more strongly by the external data at inward propagation points.
590See \autoref{subsec:BDY_relaxation} for detailed on the spatial shape of the scaling.\\
591The ``normal propagation of oblique radiation'' or NPO approximation (called \forcode{'orlanski_npo'}) involves assuming
592that $c_y$ is zero in equation (\autoref{eq:wave_continuous}), but including
593this term in the denominator of equation (\autoref{eq:cx}). Both versions of the scheme are options in BDY. Equations
594(\autoref{eq:wave_continuous}) - (\autoref{eq:tau_in}) correspond to equations (13) - (15) and (2) - (3) in MMS.\\
597\subsection{Relaxation at the boundary}
600In addition to a specific boundary condition specified as \np{cn\_tra} and \np{cn\_dyn3d}, relaxation on baroclinic velocities and tracers variables are available.
601It is control by the namelist parameter \np{ln\_tra\_dmp} and \np{ln\_dyn3d\_dmp} for each boundary set.
603The relaxation time scale value (\np{rn\_time\_dmp} and \np{rn\_time\_dmp\_out}, $\tau$) are defined at the boundaries itself.
604This time scale ($\alpha$) is weighted by the distance ($d$) from the boundary over \np{nn\_rimwidth} cells ($N$):
607  \alpha = \frac{1}{\tau}(\frac{N+1-d}{N})^2,       \quad d=1,N
610The same scaling is applied in the Orlanski damping.
613\subsection{Boundary geometry}
616Each open boundary set is defined as a list of points.
617The information is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$ structure.
618The $nbi$ and $nbj$ arrays define the local $(i,j)$ indices of each point in the boundary zone and
619the $nbr$ array defines the discrete distance from the boundary with $nbr=1$ meaning that
620the point is next to the edge of the model domain and $nbr>1$ showing that
621the point is increasingly further away from the edge of the model domain.
622A set of $nbi$, $nbj$, and $nbr$ arrays is defined for each of the $T$, $U$ and $V$ grids.
623Figure \autoref{fig:LBC_bdy_geom} shows an example of an irregular boundary.
625The boundary geometry for each set may be defined in a namelist nambdy\_index or
626by reading in a ``\ifile{coordinates.bdy}'' file.
627The nambdy\_index namelist defines a series of straight-line segments for north, east, south and west boundaries.
628One nambdy\_index namelist bloc is needed for each boundary condition defined by indexes.
629For the northern boundary, \texttt{nbdysegn} gives the number of segments,
630\jp{jpjnob} gives the $j$ index for each segment and \jp{jpindt} and
631\jp{jpinft} give the start and end $i$ indices for each segment with similar for the other boundaries.
632These segments define a list of $T$ grid points along the outermost row of the boundary ($nbr\,=\, 1$).
633The code deduces the $U$ and $V$ points and also the points for $nbr\,>\, 1$ if \np{nn\_rimwidth}\forcode{ > 1}.
635The boundary geometry may also be defined from a ``\ifile{coordinates.bdy}'' file.
636Figure \autoref{fig:LBC_nc_header} gives an example of the header information from such a file.
637The file should contain the index arrays for each of the $T$, $U$ and $V$ grids.
638The arrays must be in order of increasing $nbr$.
639Note that the $nbi$, $nbj$ values in the file are global values and are converted to local values in the code.
640Typically this file will be used to generate external boundary data via interpolation and so
641will also contain the latitudes and longitudes of each point as shown.
642However, this is not necessary to run the model.
644For some choices of irregular boundary the model domain may contain areas of ocean which
645are not part of the computational domain.
646For example if an open boundary is defined along an isobath, say at the shelf break,
647then the areas of ocean outside of this boundary will need to be masked out.
648This can be done by reading a mask file defined as \np{cn\_mask\_file} in the nam\_bdy namelist.
649Only one mask file is used even if multiple boundary sets are defined.
653  \begin{center}
654    \includegraphics[width=\textwidth]{Fig_LBC_bdy_geom}
655    \caption {
656      \protect\label{fig:LBC_bdy_geom}
657      Example of geometry of unstructured open boundary
658    }
659  \end{center}
664\subsection{Input boundary data files}
667The data files contain the data arrays in the order in which the points are defined in the $nbi$ and $nbj$ arrays.
668The data arrays are dimensioned on:
669a time dimension;
670$xb$ which is the index of the boundary data point in the horizontal;
671and $yb$ which is a degenerate dimension of 1 to enable the file to be read by the standard \NEMO\ I/O routines.
672The 3D fields also have a depth dimension.
674From Version 3.4 there are new restrictions on the order in which the boundary points are defined
675(and therefore restrictions on the order of the data in the file).
676In particular:
679\item The data points must be in order of increasing $nbr$,
680  ie. all the $nbr=1$ points, then all the $nbr=2$ points etc.
681\item All the data for a particular boundary set must be in the same order.
682  (Prior to 3.4 it was possible to define barotropic data in a different order to
683  the data for tracers and baroclinic velocities).
686These restrictions mean that data files used with versions of the
687model prior to Version 3.4 may not work with Version 3.4 onwards.
688A \fortran utility {\itshape bdy\_reorder} exists in the TOOLS directory which
689will re-order the data in old BDY data files.
693  \begin{center}
694    \includegraphics[width=\textwidth]{Fig_LBC_nc_header}
695    \caption {
696      \protect\label{fig:LBC_nc_header}
697      Example of the header for a \protect\ifile{coordinates.bdy} file
698    }
699  \end{center}
704\subsection{Volume correction}
707There is an option to force the total volume in the regional model to be constant.
708This is controlled  by the \np{ln\_vol} parameter in the namelist.
709A value of \np{ln\_vol}\forcode{ = .false.} indicates that this option is not used.
710Two options to control the volume are available (\np{nn\_volctl}).
711If \np{nn\_volctl}\forcode{ = 0} then a correction is applied to the normal barotropic velocities around the boundary at
712each timestep to ensure that the integrated volume flow through the boundary is zero.
713If \np{nn\_volctl}\forcode{ = 1} then the calculation of the volume change on
714the timestep includes the change due to the freshwater flux across the surface and
715the correction velocity corrects for this as well.
717If more than one boundary set is used then volume correction is
718applied to all boundaries at once.
721\subsection{Tidal harmonic forcing}
729Tidal forcing at open boundaries requires the activation of surface
730tides (i.e., in \nam{\_tide}, \np{ln\_tide} needs to be set to
731\forcode{.true.} and the required constituents need to be activated by
732including their names in the \np{clname} array; see
733\autoref{sec:SBC_tide}). Specific options related to the reading in of
734the complex harmonic amplitudes of elevation (SSH) and barotropic
735velocity (u,v) at open boundaries are defined through the
736\nam{bdy\_tide} namelist parameters.\\
738The tidal harmonic data at open boundaries can be specified in two
739different ways, either on a two-dimensional grid covering the entire
740model domain or along open boundary segments; these two variants can
741be selected by setting \np{ln\_bdytide\_2ddta } to \forcode{.true.} or
742\forcode{.false.}, respectively. In either case, the real and
743imaginary parts of SSH and the two barotropic velocity components for
744each activated tidal constituent \textit{tcname} have to be provided
745separately: when two-dimensional data is used, variables
746\textit{tcname\_z1} and \textit{tcname\_z2} for real and imaginary SSH,
747respectively, are expected in input file \np{filtide} with suffix
748\ifile{\_grid\_T}, variables \textit{tcname\_u1} and
749\textit{tcname\_u2} for real and imaginary u, respectively, are
750expected in input file \np{filtide} with suffix \ifile{\_grid\_U}, and
751\textit{tcname\_v1} and \textit{tcname\_v2} for real and imaginary v,
752respectively, are expected in input file \np{filtide} with suffix
753\ifile{\_grid\_V}; when data along open boundary segments is used,
754variables \textit{z1} and \textit{z2} (real and imaginary part of SSH)
755are expected to be available from file \np{filtide} with suffix
756\ifile{tcname\_grid\_T}, variables \textit{u1} and \textit{u2} (real
757and imaginary part of u) are expected to be available from file
758\np{filtide} with suffix \ifile{tcname\_grid\_U}, and variables
759\textit{v1} and \textit{v2} (real and imaginary part of v) are
760expected to be available from file \np{filtide} with suffix
761\ifile{tcname\_grid\_V}. If \np{ln\_bdytide\_conj} is set to
762\forcode{.true.}, the data is expected to be in complex conjugate
765Note that the barotropic velocity components are assumed to be defined
766on the native model grid and should be rotated accordingly when they
767are converted from their definition on a different source grid. To do
768so, the u, v amplitudes and phases can be converted into tidal
769ellipses, the grid rotation added to the ellipse inclination, and then
770converted back (care should be taken regarding conventions of the
771direction of rotation). %, e.g. anticlockwise or clockwise.
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