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1% ================================================================
2% Chapter Ñ Surface Boundary Condition (SBC)
3% ================================================================
4\chapter{Surface Boundary Condition (SBC) }
5\label{SBC}
6\minitoc
7
8\newpage
9$\ $\newline    % force a new ligne
10%---------------------------------------namsbc--------------------------------------------------
11\namdisplay{namsbc}
12%--------------------------------------------------------------------------------------------------------------
13$\ $\newline    % force a new ligne
14
15The ocean needs six fields as surface boundary condition:
16\begin{itemize}
17   \item the two components of the surface ocean stress $\left( {\tau _u \;,\;\tau _v} \right)$
18   \item the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$
19   \item the surface freshwater budget $\left( {\textit{emp},\;\textit{emp}_S } \right)$
20\end{itemize}
21plus an optional field:
22\begin{itemize}
23   \item the atmospheric pressure at the ocean surface $\left( p_a \right)$
24\end{itemize}
25
26Four different ways to provide the first six fields to the ocean are available which
27are controlled by namelist variables: an analytical formulation (\np{ln\_ana}~=~true),
28a flux formulation (\np{ln\_flx}~=~true), a bulk formulae formulation (CORE
29(\np{ln\_core}~=~true) or CLIO (\np{ln\_clio}~=~true) bulk formulae) and a coupled
30formulation (exchanges with a atmospheric model via the OASIS coupler)
31(\np{ln\_cpl}~=~true). The optional atmospheric pressure can be used either
32to force ocean and ice dynamics (\np{ln\_apr\_dyn}~=~true), or in the bulk
33formulae computation (\np{ln\_apr\_dyn}~=~true)
34\footnote{None of the two current bulk formulea (CLIO and CORE) uses the
35atmospheric pressure field.}.
36The frequency at which the six or seven fields have to be updated is the \np{nn\_fsbc} 
37namelist parameter.
38When the fields are supplied from data files (flux and bulk formulations), the input fields
39need not be supplied on the model grid.  Instead a file of coordinates and weights can
40be supplied which maps the data from the supplied grid to the model points
41(so called "Interpolation on the Fly").
42In addition, the resulting fields can be further modified using several namelist options.
43These options control  the rotation of vector components supplied relative to an east-north
44coordinate system onto the local grid directions in the model; the addition of a surface
45restoring term to observed SST and/or SSS (\np{ln\_ssr}~=~true); the modification of fluxes
46below ice-covered areas (using observed ice-cover or a sea-ice model)
47(\np{nn\_ice}~=~0,1, 2 or 3); the addition of river runoffs as surface freshwater
48fluxes or lateral inflow (\np{ln\_rnf}~=~true); the addition of a freshwater flux adjustment
49in order to avoid a mean sea-level drift (\np{nn\_fwb}~=~0,~1~or~2); and the
50transformation of the solar radiation (if provided as daily mean) into a diurnal
51cycle (\np{ln\_dm2dc}~=~true).
52
53In this chapter, we first discuss where the surface boundary condition appears in the
54model equations. Then we present the four ways of providing the surface boundary condition.
55Next the scheme for interpolation on the fly is described.
56Finally, the different options that further modify the fluxes applied to the ocean are discussed.
57
58
59% ================================================================
60% Surface boundary condition for the ocean
61% ================================================================
62\section{Surface boundary condition for the ocean}
63\label{SBC_general}
64
65
66The surface ocean stress is the stress exerted by the wind and the sea-ice
67on the ocean. The two components of stress are assumed to be interpolated
68onto the ocean mesh, $i.e.$ resolved onto the model (\textbf{i},\textbf{j}) direction
69at $u$- and $v$-points They are applied as a surface boundary condition of the
70computation of the momentum vertical mixing trend (\mdl{dynzdf} module) :
71\begin{equation} \label{Eq_sbc_dynzdf}
72\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
73    = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
74\end{equation}
75where $(\tau _u ,\;\tau _v )=(utau,vtau)$ are the two components of the wind
76stress vector in the $(\textbf{i},\textbf{j})$ coordinate system.
77
78The surface heat flux is decomposed into two parts, a non solar and a solar heat
79flux, $Q_{ns}$ and $Q_{sr}$, respectively. The former is the non penetrative part
80of the heat flux ($i.e.$ the sum of sensible, latent and long wave heat fluxes).
81It is applied as a surface boundary condition trend of the first level temperature
82time evolution equation (\mdl{trasbc} module).
83\begin{equation} \label{Eq_sbc_trasbc_q}
84\frac{\partial T}{\partial t}\equiv \cdots \;+\;\left. {\frac{Q_{ns} }{\rho 
85_o \;C_p \;e_{3t} }} \right|_{k=1} \quad
86\end{equation}
87$Q_{sr}$ is the penetrative part of the heat flux. It is applied as a 3D
88trends of the temperature equation (\mdl{traqsr} module) when \np{ln\_traqsr}=True.
89
90\begin{equation} \label{Eq_sbc_traqsr}
91\frac{\partial T}{\partial t}\equiv \cdots \;+\frac{Q_{sr} }{\rho_o C_p \,e_{3t} }\delta _k \left[ {I_w } \right]
92\end{equation}
93where $I_w$ is a non-dimensional function that describes the way the light
94penetrates inside the water column. It is generally a sum of decreasing
95exponentials (see \S\ref{TRA_qsr}).
96
97The surface freshwater budget is provided by fields: \textit{emp} and $\textit{emp}_S$ which
98may or may not be identical. Indeed, a surface freshwater flux has two effects:
99it changes the volume of the ocean and it changes the surface concentration of
100salt (and other tracers). Therefore it appears in the sea surface height as a volume
101flux, \textit{emp} (\textit{dynspg\_xxx} modules), and in the salinity time evolution equations
102as a concentration/dilution effect,
103$\textit{emp}_{S}$ (\mdl{trasbc} module).
104\begin{equation} \label{Eq_trasbc_emp}
105\begin{aligned}
106&\frac{\partial \eta }{\partial t}\equiv \cdots \;+\;\textit{emp}\quad  \\ 
107\\
108 &\frac{\partial S}{\partial t}\equiv \cdots \;+\left. {\frac{\textit{emp}_S \;S}{e_{3t} }} \right|_{k=1} \\ 
109 \end{aligned}
110\end{equation} 
111
112In the real ocean, $\textit{emp}=\textit{emp}_S$ and the ocean salt content is conserved,
113but it exist several numerical reasons why this equality should be broken.
114For example, when the ocean is coupled to a sea-ice model, the water exchanged between
115ice and ocean is slightly salty (mean sea-ice salinity is $\sim $\textit{4 psu}). In this case,
116$\textit{emp}_{S}$ take into account both concentration/dilution effect associated with
117freezing/melting and the salt flux between ice and ocean, while \textit{emp} is
118only the volume flux. In addition, in the current version of \NEMO, the sea-ice is
119assumed to be above the ocean (the so-called levitating sea-ice). Freezing/melting does
120not change the ocean volume (no impact on \textit{emp}) but it modifies the SSS.
121%gm  \colorbox{yellow}{(see {\S} on LIM sea-ice model)}.
122
123Note that SST can also be modified by a freshwater flux. Precipitation (in
124particular solid precipitation) may have a temperature significantly different from
125the SST. Due to the lack of information about the temperature of
126precipitation, we assume it is equal to the SST. Therefore, no
127concentration/dilution term appears in the temperature equation. It has to
128be emphasised that this absence does not mean that there is no heat flux
129associated with precipitation! Precipitation can change the ocean volume and thus the
130ocean heat content. It is therefore associated with a heat flux (not yet 
131diagnosed in the model) \citep{Roullet_Madec_JGR00}).
132
133%\colorbox{yellow}{Miss: }
134%
135%A extensive description of all namsbc namelist (parameter that have to be
136%created!)
137%
138%Especially the \np{nn\_fsbc}, the \mdl{sbc\_oce} module (fluxes + mean sst sss ssu
139%ssv) i.e. information required by flux computation or sea-ice
140%
141%\mdl{sbc\_oce} containt the definition in memory of the 7 fields (6+runoff), add
142%a word on runoff: included in surface bc or add as lateral obc{\ldots}.
143%
144%Sbcmod manage the ``providing'' (fourniture) to the ocean the 7 fields
145%
146%Fluxes update only each nf{\_}sbc time step (namsbc) explain relation
147%between nf{\_}sbc and nf{\_}ice, do we define nf{\_}blk??? ? only one
148%nf{\_}sbc
149%
150%Explain here all the namlist namsbc variable{\ldots}.
151%
152%\colorbox{yellow}{End Miss }
153
154The ocean model provides the surface currents, temperature and salinity
155averaged over \np{nf\_sbc} time-step (\ref{Tab_ssm}).The computation of the
156mean is done in \mdl{sbcmod} module.
157
158%-------------------------------------------------TABLE---------------------------------------------------
159\begin{table}[tb]  \label{Tab_ssm}
160\begin{center}
161\begin{tabular}{|l|l|l|l|}
162\hline
163Variable description             & Model variable  & Units  & point \\  \hline
164i-component of the surface current  & ssu\_m & $m.s^{-1}$   & U \\   \hline
165j-component of the surface current  & ssv\_m & $m.s^{-1}$   & V \\   \hline
166Sea surface temperature          & sst\_m & \r{}$K$      & T \\   \hline
167Sea surface salinty              & sss\_m & $psu$        & T \\   \hline
168\end{tabular}
169\caption{Ocean variables provided by the ocean to the surface module (SBC).
170The variable are averaged over nf{\_}sbc time step, $i.e.$ the frequency of
171computation of surface fluxes.}
172\end{center}
173\end{table}
174%--------------------------------------------------------------------------------------------------------------
175
176
177
178%\colorbox{yellow}{Penser a} mettre dans le restant l'info nf{\_}sbc ET nf{\_}sbc*rdt de sorte de reinitialiser la moyenne si on change la frequence ou le pdt
179
180
181% ================================================================
182% Analytical formulation (sbcana module)
183% ================================================================
184\section  [Analytical formulation (\textit{sbcana}) ]
185      {Analytical formulation (\mdl{sbcana} module) }
186\label{SBC_ana}
187
188%---------------------------------------namsbc_ana--------------------------------------------------
189\namdisplay{namsbc_ana}
190%--------------------------------------------------------------------------------------------------------------
191
192The analytical formulation of the surface boundary condition is the default scheme.
193In this case, all the six fluxes needed by the ocean are assumed to
194be uniform in space. They take constant values given in the namelist
195namsbc{\_}ana by the variables \np{rn\_utau0}, \np{rn\_vtau0}, \np{rn\_qns0},
196\np{rn\_qsr0}, and \np{rn\_emp0} ($\textit{emp}=\textit{emp}_S$). The runoff is set to zero.
197In addition, the wind is allowed to reach its nominal value within a given number
198of time steps (\np{nn\_tau000}).
199
200If a user wants to apply a different analytical forcing, the \mdl{sbcana} 
201module can be modified to use another scheme. As an example,
202the \mdl{sbc\_ana\_gyre} routine provides the analytical forcing for the
203GYRE configuration (see GYRE configuration manual, in preparation).
204
205
206% ================================================================
207% Flux formulation
208% ================================================================
209\section  [Flux formulation (\textit{sbcflx}) ]
210      {Flux formulation (\mdl{sbcflx} module) }
211\label{SBC_flx}
212%------------------------------------------namsbc_flx----------------------------------------------------
213\namdisplay{namsbc_flx} 
214%-------------------------------------------------------------------------------------------------------------
215
216In the flux formulation (\np{ln\_flx}=true), the surface boundary
217condition fields are directly read from input files. The user has to define
218in the namelist namsbc{\_}flx the name of the file, the name of the variable
219read in the file, the time frequency at which it is given (in hours), and a logical
220setting whether a time interpolation to the model time step is required
221for this field). (fld\_i namelist structure).
222
223\textbf{Caution}: when the frequency is set to --12, the data are monthly
224values. These are assumed to be climatological values, so time interpolation
225between December the 15$^{th}$ and January the 15$^{th}$ is done using
226records 12 and 1
227
228When higher frequency is set and time interpolation is demanded, the model
229will try to read the last (first) record of previous (next) year in a file
230having the same name but a suffix {\_}prev{\_}year ({\_}next{\_}year) being
231added (e.g. "{\_}1989"). These files must only contain a single record. If they don't exist,
232the model assumes that the last record of the previous year is equal to the first
233record of the current year, and similarly, that the first record of the
234next year is equal to the last record of the current year. This will cause
235the forcing to remain constant over the first and last half fld\_frequ hours.
236
237Note that in general, a flux formulation is used in associated with a
238restoring term to observed SST and/or SSS. See \S\ref{SBC_ssr} for its
239specification.
240
241
242% ================================================================
243% Bulk formulation
244% ================================================================
245\section  [Bulk formulation (\textit{sbcblk\_core} or \textit{sbcblk\_clio}) ]
246      {Bulk formulation \small{(\mdl{sbcblk\_core} or \mdl{sbcblk\_clio} module)} }
247\label{SBC_blk}
248
249In the bulk formulation, the surface boundary condition fields are computed
250using bulk formulae and atmospheric fields and ocean (and ice) variables.
251
252The atmospheric fields used depend on the bulk formulae used. Two bulk formulations
253are available : the CORE and CLIO bulk formulea. The choice is made by setting to true
254one of the following namelist variable : \np{ln\_core} and \np{ln\_clio}.
255
256Note : in forced mode, when a sea-ice model is used, a bulk formulation have to be used.
257Therefore the two bulk formulea provided include the computation of the fluxes over both
258an ocean and an ice surface.
259
260% -------------------------------------------------------------------------------------------------------------
261%        CORE Bulk formulea
262% -------------------------------------------------------------------------------------------------------------
263\subsection    [CORE Bulk formulea (\np{ln\_core}=true)]
264            {CORE Bulk formulea (\np{ln\_core}=true, \mdl{sbcblk\_core})}
265\label{SBC_blk_core}
266%------------------------------------------namsbc_core----------------------------------------------------
267\namdisplay{namsbc_core} 
268%-------------------------------------------------------------------------------------------------------------
269
270The CORE bulk formulae have been developed by \citet{Large_Yeager_Rep04}.
271They have been designed to handle the CORE forcing, a mixture of NCEP
272reanalysis and satellite data. They use an inertial dissipative method to compute
273the turbulent transfer coefficients (momentum, sensible heat and evaporation)
274from the 10 metre wind speed, air temperature and specific humidity.
275This \citet{Large_Yeager_Rep04} dataset is available through the GFDL web
276site (http://nomads.gfdl.noaa.gov/nomads/forms/mom4/CORE.html).
277
278Note that substituting ERA40 to NCEP reanalysis fields
279does not require changes in the bulk formulea themself.
280This is the so-called DRAKKAR Forcing Set (DFS) \citep{Brodeau_al_OM09}.
281
282The required 8 input fields are:
283
284%--------------------------------------------------TABLE--------------------------------------------------
285\begin{table}[htbp]   \label{Tab_CORE}
286\begin{center}
287\begin{tabular}{|l|l|l|l|}
288\hline
289Variable desciption              & Model variable  & Units   & point \\    \hline
290i-component of the 10m air velocity & utau      & $m.s^{-1}$         & T  \\  \hline
291j-component of the 10m air velocity & vtau      & $m.s^{-1}$         & T  \\  \hline
29210m air temperature              & tair      & \r{}$K$            & T   \\ \hline
293Specific humidity             & humi      & \%              & T \\      \hline
294Incoming long wave radiation     & qlw    & $W.m^{-2}$         & T \\      \hline
295Incoming short wave radiation    & qsr    & $W.m^{-2}$         & T \\      \hline
296Total precipitation (liquid + solid)   & precip & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
297Solid precipitation              & snow      & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
298\end{tabular}
299\end{center}
300\end{table}
301%--------------------------------------------------------------------------------------------------------------
302
303Note that the air velocity is provided at a tracer ocean point, not at a velocity ocean point ($u$- and $v$-points). It is simpler and faster (less fields to be read), but it is not the recommended method when the ocean grid
304size is the same or larger than the one of the input atmospheric fields.
305
306% -------------------------------------------------------------------------------------------------------------
307%        CLIO Bulk formulea
308% -------------------------------------------------------------------------------------------------------------
309\subsection    [CLIO Bulk formulea (\np{ln\_clio}=true)]
310            {CLIO Bulk formulea (\np{ln\_clio}=true, \mdl{sbcblk\_clio})}
311\label{SBC_blk_clio}
312%------------------------------------------namsbc_clio----------------------------------------------------
313\namdisplay{namsbc_clio} 
314%-------------------------------------------------------------------------------------------------------------
315
316The CLIO bulk formulae were developed several years ago for the
317Louvain-la-neuve coupled ice-ocean model (CLIO, \cite{Goosse_al_JGR99}).
318They are simpler bulk formulae. They assume the stress to be known and
319compute the radiative fluxes from a climatological cloud cover.
320
321The required 7 input fields are:
322
323%--------------------------------------------------TABLE--------------------------------------------------
324\begin{table}[htbp]   \label{Tab_CLIO}
325\begin{center}
326\begin{tabular}{|l|l|l|l|}
327\hline
328Variable desciption           & Model variable  & Units           & point \\  \hline
329i-component of the ocean stress     & utau         & $N.m^{-2}$         & U \\   \hline
330j-component of the ocean stress     & vtau         & $N.m^{-2}$         & V \\   \hline
331Wind speed module             & vatm         & $m.s^{-1}$         & T \\   \hline
33210m air temperature              & tair         & \r{}$K$            & T \\   \hline
333Specific humidity                & humi         & \%              & T \\   \hline
334Cloud cover                   &           & \%              & T \\   \hline
335Total precipitation (liquid + solid)   & precip    & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
336Solid precipitation              & snow         & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
337\end{tabular}
338\end{center}
339\end{table}
340%--------------------------------------------------------------------------------------------------------------
341
342As for the flux formulation, information about the input data required by the
343model is provided in the namsbc\_blk\_core or namsbc\_blk\_clio
344namelist (via the structure fld\_i). The first and last record assumption is also made
345(see \S\ref{SBC_flx})
346
347% ================================================================
348% Coupled formulation
349% ================================================================
350\section  [Coupled formulation (\textit{sbccpl}) ]
351      {Coupled formulation (\mdl{sbccpl} module)}
352\label{SBC_cpl}
353%------------------------------------------namsbc_cpl----------------------------------------------------
354\namdisplay{namsbc_cpl} 
355%-------------------------------------------------------------------------------------------------------------
356
357In the coupled formulation of the surface boundary condition, the fluxes are
358provided by the OASIS coupler at a frequency which is defined in the OASIS coupler,
359while sea and ice surface temperature, ocean and ice albedo, and ocean currents
360are sent to the atmospheric component.
361
362A generalised coupled interface has been developed. It is currently interfaced with OASIS 3
363(\key{oasis3}) and does not support OASIS 4
364\footnote{The \key{oasis4} exist. It activates portion of the code that are still under development.}.
365It has been successfully used to interface \NEMO to most of the European atmospheric
366GCM (ARPEGE, ECHAM, ECMWF, HadAM, LMDz),
367as well as to WRF (Weather Research and Forecasting Model) (http://wrf-model.org/).
368
369Note that in addition to the setting of \np{ln\_cpl} to true, the \key{coupled} have to be defined.
370The CPP key is mainly used in sea-ice to ensure that the atmospheric fluxes are
371actually recieved by the ice-ocean system (no calculation of ice sublimation in coupled mode).
372When PISCES biogeochemical model (\key{top} and \key{pisces}) is also used in the coupled system,
373the whole carbon cycle is computed by defining \key{cpl\_carbon\_cycle}. In this case,
374CO$_2$ fluxes are exchanged between the atmosphere and the ice-ocean system.
375
376
377% ================================================================
378%        Atmospheric pressure
379% ================================================================
380\section   [Atmospheric pressure (\textit{sbcapr})]
381         {Atmospheric pressure (\mdl{sbcapr})}
382\label{SBC_apr}
383%------------------------------------------namsbc_apr----------------------------------------------------
384\namdisplay{namsbc_apr} 
385%-------------------------------------------------------------------------------------------------------------
386
387The optional atmospheric pressure can be used either to force ocean and ice dynamics
388(\np{ln\_apr\_dyn}~=~true), or in the bulk formulae computation (\np{ln\_apr\_dyn}~=~true).
389The input atmospheric forcing is interpolated in time to the model time step, and optionally
390in space when interpolation on-the-fly is used. When used to force the dynamics, it is further
391transformed into an equivalent inverse barometer sea surface height, $\eta_{ib}$, using:
392\begin{equation} \label{SBC_ssh_ib}
393   \eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right)
394\end{equation}
395where $P_{atm}$ is the atmospheric pressure and $P_o$ a reference atmospheric pressure.
396A value of $101,000~N/m^2$ is used unless \np{ln\_ref\_apr} is set to true. In this case $P_o$ 
397is set to the value of $P_{atm}$ averaged over the ocean domain, $i.e.$ the mean value of
398$\eta_{ib}$ is kept to zero at all time step.
399
400A gradient of $\eta_{ib}$ is added to the RHS of the ocean momentum equation
401(see \mdl{dynspg} for the ocean). For sea-ice, the sea surface height, $\eta_m$,
402which is provided to the sea ice model is set to $\eta - \eta_{ib}$ (see \mdl{sbcssr} module).
403Furthermore, $\eta_{ib}$ can be set in the output. This simplifies the altirmetry data
404and model comparison as inverse barometer sea surface height is usually removed
405from thise date prior to their distribution.
406
407% ================================================================
408%        River runoffs
409% ================================================================
410\section   [River runoffs (\textit{sbcrnf})]
411         {River runoffs (\mdl{sbcrnf})}
412\label{SBC_rnf}
413%------------------------------------------namsbc_rnf----------------------------------------------------
414\namdisplay{namsbc_rnf} 
415%-------------------------------------------------------------------------------------------------------------
416
417%River runoff generally enters the ocean at a nonzero depth rather than through the surface.
418%Many models, however, have traditionally inserted river runoff to the top model cell.
419%This was the case in \NEMO prior to the version 3.3. The switch toward a input of runoff
420%throughout a nonzero depth has been motivated by the numerical and physical problems
421%that arise when the top grid cells are of the order of one meter. This situation is common in
422%coastal modelling and becomes more and more often open ocean and climate modelling
423%\footnote{At least a top cells thickness of 1~meter and a 3 hours forcing frequency are
424%required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \S\ref{SBC_dcy}.}.
425
426
427%To do this we need to treat evaporation/precipitation fluxes and river runoff differently in the
428%\mdl{tra\_sbc} module.  We decided to separate them throughout the code, so that the variable
429%\textit{emp} represented solely evaporation minus precipitation fluxes, and a new 2d variable
430%rnf was added which represents the volume flux of river runoff (in kg/m2s to remain consistent with
431%emp).  This meant many uses of emp and emps needed to be changed, a list of all modules which use
432%emp or emps and the changes made are below:
433
434
435%Rachel:
436River runoff generally enters the ocean at a nonzero depth rather than through the surface.
437Many models, however, have traditionally inserted river runoff to the top model cell.
438This was the case in \NEMO prior to the version 3.3, and was combined with an option to increase vertical mixing near the river mouth.
439
440However, with this method numerical and physical problems arise when the top grid cells are
441of the order of one meter. This situation is common in coastal modelling and is becoming
442more common in open ocean and climate modelling
443\footnote{At least a top cells thickness of 1~meter and a 3 hours forcing frequency are
444required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \S\ref{SBC_dcy}.}.
445
446As such from V~3.3 onwards it is possible to add river runoff through a non-zero depth, and for the
447temperature and salinity of the river to effect the surrounding ocean.
448The user is able to specify, in a NetCDF input file, the temperature and salinity of the river, along with the   
449depth (in metres) which the river should be added to.
450
451Namelist options, \np{ln\_rnf\_depth}, \np{ln\_rnf\_sal} and \np{ln\_rnf\_temp} control whether
452the river attributes (depth, salinity and temperature) are read in and used.  If these are set
453as false the river is added to the surface box only, assumed to be fresh (0~psu), and/or
454taken as surface temperature respectively.
455
456The runoff value and attributes are read in in sbcrnf. 
457For temperature -999 is taken as missing data and the river temperature is taken to be the
458surface temperatue at the river point.
459For the depth parameter a value of -1 means the river is added to the surface box only,
460and a value of -999 means the river is added through the entire water column.
461After being read in the temperature and salinity variables are multiplied by the amount of runoff (converted into m/s)
462to give the heat and salt content of the river runoff.
463After the user specified depth is read ini, the number of grid boxes this corresponds to is
464calculated and stored in the variable \np{nz\_rnf}.
465The variable \textit{h\_dep} is then calculated to be the depth (in metres) of the bottom of the
466lowest box the river water is being added to (i.e. the total depth that river water is being added to in the model).
467
468The mass/volume addition due to the river runoff is, at each relevant depth level, added to the horizontal divergence
469(\textit{hdivn}) in the subroutine \rou{sbc\_rnf\_div} (called from \mdl{divcur}).
470This increases the diffusion term in the vicinity of the river, thereby simulating a momentum flux.
471The sea surface height is calculated using the sum of the horizontal divergence terms, and so the
472river runoff indirectly forces an increase in sea surface height.
473
474The \textit{hdivn} terms are used in the tracer advection modules to force vertical velocities.
475This causes a mass of water, equal to the amount of runoff, to be moved into the box above.
476The heat and salt content of the river runoff is not included in this step, and so the tracer
477concentrations are diluted as water of ocean temperature and salinity is moved upward out of the box
478and replaced by the same volume of river water with no corresponding heat and salt addition.
479
480For the linear free surface case, at the surface box the tracer advection causes a flux of water
481(of equal volume to the runoff) through the sea surface out of the domain, which causes a salt and heat flux out of the model.
482As such the volume of water does not change, but the water is diluted.
483
484For the non-linear free surface case (\key{vvl}), no flux is allowed through the surface.
485Instead in the surface box (as well as water moving up from the boxes below) a volume of runoff water
486is added with no corresponding heat and salt addition and so as happens in the lower boxes there is a dilution effect.
487(The runoff addition to the top box along with the water being moved up through boxes below means the surface box has a large
488increase in volume, whilst all other boxes remain the same size)
489
490In trasbc the addition of heat and salt due to the river runoff is added.
491This is done in the same way for both vvl and non-vvl.
492The temperature and salinity are increased through the specified depth according to the heat and salt content of the river.
493
494In the non-linear free surface case (vvl), near the end of the time step the change in sea surface height is redistrubuted
495through the grid boxes, so that the original ratios of grid box heights are restored.
496In doing this water is moved into boxes below, throughout the water column, so the large volume addition to the surface box is spread between all the grid boxes.
497
498It is also possible for runnoff to be specified as a negative value for modelling flow through straits, i.e. modelling the Baltic flow in and out of the North Sea.
499When the flow is out of the domain there is no change in temperature and salinity, regardless of the namelist options used, as the ocean water leaving the domain removes heat and salt (at the same concentration) with it.
500
501
502%\colorbox{yellow}{Nevertheless, Pb of vertical resolution and 3D input : increase vertical mixing near river mouths to mimic a 3D river
503
504%All river runoff and emp fluxes are assumed to be fresh water (zero salinity) and at the same temperature as the sea surface.}
505
506%\colorbox{yellow}{river mouths{\ldots}}
507
508%IF( ln_rnf ) THEN                                     ! increase diffusivity at rivers mouths
509%        DO jk = 2, nkrnf   ;   avt(:,:,jk) = avt(:,:,jk) + rn_avt_rnf * rnfmsk(:,:)   ;   END DO
510%ENDIF
511
512%\gmcomment{  word doc of runoffs:
513%
514%In the current \NEMO setup river runoff is added to emp fluxes, these are then applied at just the sea surface as a volume change (in the variable volume case this is a literal volume change, and in the linear free surface case the free surface is moved) and a salt flux due to the concentration/dilution effect.  There is also an option to increase vertical mixing near river mouths; this gives the effect of having a 3d river.  All river runoff and emp fluxes are assumed to be fresh water (zero salinity) and at the same temperature as the sea surface.
515%Our aim was to code the option to specify the temperature and salinity of river runoff, (as well as the amount), along with the depth that the river water will affect.  This would make it possible to model low salinity outflow, such as the Baltic, and would allow the ocean temperature to be affected by river runoff. 
516
517%The depth option makes it possible to have the river water affecting just the surface layer, throughout depth, or some specified point in between.
518
519%To do this we need to treat evaporation/precipitation fluxes and river runoff differently in the tra_sbc module.  We decided to separate them throughout the code, so that the variable emp represented solely evaporation minus precipitation fluxes, and a new 2d variable rnf was added which represents the volume flux of river runoff (in kg/m2s to remain consistent with emp).  This meant many uses of emp and emps needed to be changed, a list of all modules which use emp or emps and the changes made are below:
520
521}
522
523
524%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
525\begin{figure}[!t] \label{Fig_SBC_diurnal}  \begin{center}
526\includegraphics[width=0.8\textwidth]{./TexFiles/Figures/Fig_SBC_diurnal.pdf}
527\caption{Example of recontruction of the diurnal cycle variation of short wave flux 
528from daily mean values. The reconstructed diurnal cycle (black line) is chosen
529as the mean value of the analytical cycle (blue line) over a time step, not
530as the mid time step value of the analytically cycle (red square). From \citet{Bernie_al_CD07}.}
531\end{center}   \end{figure}
532%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
533
534% ================================================================
535%        Diurnal cycle
536% ================================================================
537\section   [Diurnal  cycle (\textit{sbcdcy})]
538         {Diurnal cycle (\mdl{sbcdcy})}
539\label{SBC_dcy}
540%------------------------------------------namsbc_rnf----------------------------------------------------
541%\namdisplay{namsbc}
542%-------------------------------------------------------------------------------------------------------------
543
544\cite{Bernie_al_JC05} have shown that to capture 90$\%$ of the diurnal variability of
545SST requires a vertical resolution in upper ocean of 1~m or better and a temporal resolution
546of the surface fluxes of 3~h or less. Unfortunately high frequency forcing fields are rare,
547not to say inexistent. Nevertheless, it is possible to obtain a reasonable diurnal cycle
548of the SST knowning only short wave flux (SWF) at high frequency \citep{Bernie_al_CD07}.
549Furthermore, only the knowledge of daily mean value of SWF is needed,
550as higher frequency variations can be reconstructed from them, assuming that
551the diurnal cycle of SWF is a scaling of the top of the atmosphere diurnal cycle
552of incident SWF. The \cite{Bernie_al_CD07} reconstruction algorithm is available
553in \NEMO by setting \np{ln\_dm2dc}~=~true (a \textit{namsbc} namelist parameter) when using
554CORE bulk formulea (\np{ln\_blk\_core}~=~true) or the flux formulation (\np{ln\_flx}~=~true).
555The reconstruction is performed in the \mdl{sbcdcy} module. The detail of the algoritm used
556can be found in the appendix~A of \cite{Bernie_al_CD07}. The algorithm preserve the daily
557mean incomming SWF as the reconstructed SWF at a given time step is the mean value
558of the analytical cycle over this time step (Fig.\ref{Fig_SBC_diurnal}).
559The use of diurnal cycle reconstruction requires the input SWF to be daily
560($i.e.$ a frequency of 24 and a time interpolation set to true in \np{sn\_qsr} namelist parameter).
561Furthermore, it is recommended to have a least 8 surface module time step per day,
562that is  $\rdt \ \np{nn\_fsbc} < 10,800~s = 3~h$. An example of recontructed SWF
563is given in Fig.\ref{Fig_SBC_dcy} for a 12 reconstructed diurnal cycle, one every 2~hours
564(from 1am to 11pm).
565
566%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
567\begin{figure}[!t] \label{Fig_SBC_dcy}  \begin{center}
568\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_SBC_dcy.pdf}
569\caption{Example of recontruction of the diurnal cycle variation of short wave flux 
570from daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm).
571The display is on (i,j) plane. }
572\end{center}   \end{figure}
573%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
574
575Note also that the setting a diurnal cycle in SWF is highly recommended  when
576the top layer thickness approach 1~m or less, otherwise large error in SST can
577appear due to an inconsistency between the scale of the vertical resolution
578and the forcing acting on that scale.
579
580% ================================================================
581% Interpolation on the Fly
582% ================================================================
583
584\section [Interpolation on the Fly] {Interpolation on the Fly}
585\label{SBC_iof}
586
587Interpolation on the Fly allows the user to supply input files required
588for the surface forcing on grids other than the model grid.
589To do this he or she must supply, in addition to the source data file,
590a file of weights to be used to interpolate from the data grid to the model
591grid.
592The original development of this code used the SCRIP package (freely available
593under a copyright agreement from http://climate.lanl.gov/Software/SCRIP).
594In principle, any package can be used to generate the weights, but the
595variables in the input weights file must have the same names and meanings as
596assumed by the model.
597Two methods are currently available: bilinear and bicubic interpolation.
598
599\subsection{Bilinear Interpolation}
600\label{SBC_iof_bilinear}
601
602The input weights file in this case has two sets of variables: src01, src02,
603src03, src04 and wgt01, wgt02, wgt03, wgt04.
604The "src" variables correspond to the point in the input grid to which the weight
605"wgt" is to be applied. Each src value is an integer corresponding to the index of a
606point in the input grid when written as a one dimensional array.  For example, for an input grid
607of size 5x10, point (3,2) is referenced as point 8, since (2-1)*5+3=8.
608There are four of each variable because bilinear interpolation uses the four points defining
609the grid box containing the point to be interpolated.
610All of these arrays are on the model grid, so that values src01(i,j) and
611wgt01(i,j) are used to generate a value for point (i,j) in the model.
612
613Symbolically, the algorithm used is:
614
615\begin{equation}
616f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
617\end{equation}
618where function idx() transforms a one dimensional index src(k) into a two dimensional index,
619and wgt(1) corresponds to variable "wgt01" for example.
620
621\subsection{Bicubic Interpolation}
622\label{SBC_iof_bicubic}
623
624Again there are two sets of variables: "src" and "wgt".
625But in this case there are 16 of each.
626The symbolic algorithm used to calculate values on the model grid is now:
627
628\begin{equation*} \begin{split}
629f_{m}(i,j) =  f_{m}(i,j) +& \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}     
630              +   \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }    \\
631              +& \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} }   
632              +   \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} }
633\end{split}
634\end{equation*}
635The gradients here are taken with respect to the horizontal indices and not distances since the spatial dependency has been absorbed into the weights.
636
637\subsection{Implementation}
638\label{SBC_iof_imp}
639
640To activate this option, a non-empty string should be supplied in the weights filename column of the relevant namelist;
641if this is left as an empty string no action is taken.
642In the model, weights files are read in and stored in a structured type (WGT) in the fldread module, as and when they are first required.
643This initialisation procedure tries to determine whether the input data grid should be treated as cyclical or not.
644(In fact this only matters when bicubic interpolation is required.)
645To do this the model looks in the input data file (i.e. the data to which the weights are to be applied) for a variable with name "nav\_lon" or "lon".
646If found, it checks the difference between the first and last values of longitude along a single row.
647If the absolute value of this difference is close to 360 degrees or less than twice the maximum spacing from 360 degrees, the grid is assumed to be cyclical, and the difference determines whether the first column is a repeat of the last one or not.
648If neither "nav\_lon" or "lon" can be found, the model resorts to looking at the first and last columns of data.
649If the sum of the absolute values of the differences between the columns is very small, then the grid is assumed to be cyclical with coincident first and last columns.
650If both of these tests fail, the grid is assumed not to be cyclical.
651
652Next the routine reads in the weights.
653Bicubic interpolation is assumed if it finds a variable with name "src05", otherwise bilinear interpolation is used.
654The WGT structure includes dynamic arrays both for the storage of the weights (on the model grid), and when required, for reading in the variable to be interpolated (on the input data grid).
655The size of the input data array is determined by examining the values in the "src" arrays to find the minimum and maximum i and j values required.
656Since bicubic interpolation requires the calculation of gradients at each point on the grid, the corresponding arrays are dimensioned with a halo of width one grid point all the way around.
657When the array of points from the data file is adjacent to an edge of the data grid, the halo is either a copy of the row/column next to it (non-cyclical case), or is a copy of one from the first two rows/columns on the opposite side of the grid (cyclical case with coincident end rows/columns, or cyclical case with non-coincident end rows/columns).
658
659\subsection{Limitations}
660\label{SBC_iof_lim}
661
662\begin{description}
663\item
664Input data grids must be logically rectangular.
665\item
666This code is not guaranteed to produce positive definite answers from positive definite inputs.
667\item
668The cyclic condition is only applied on left and right columns, and not to top and bottom rows.
669\item
670The gradients across the ends of a cyclical grid assume that the grid spacing between the two columns involved are consistent with the weights used.
671\item
672Neither interpolation scheme is conservative.
673(There is a conservative scheme available in SCRIP, but this has not been implemented.)
674\end{description}
675
676\subsection{Utilities}
677\label{SBC_iof_util}
678
679% to be completed
680A set of utilities to create a weights file for a rectilinear input grid is available.
681
682% ================================================================
683% Miscellanea options
684% ================================================================
685\section{Miscellaneous options}
686\label{SBC_misc}
687
688% -------------------------------------------------------------------------------------------------------------
689%        Rotation of vector pairs onto the model grid directions
690% -------------------------------------------------------------------------------------------------------------
691\subsection{Rotation of vector pairs onto the model grid directions}
692\label{SBC_rotation}
693
694When using a flux (\np{ln\_flx}=true) or bulk (\np{ln\_clio}=true or \np{ln\_core}=true) formulation,
695pairs of vector components can be rotated from east-north directions onto the local grid directions. 
696This is particularly useful when interpolation on the fly is used since here any vectors are likely to be defined
697relative to a rectilinear grid.
698To activate this option a non-empty string is supplied in the rotation pair column of the relevant namelist.
699The eastward component must start with "U" and the northward component with "V". 
700The remaining characters in the strings are used to identify which pair of components go together.
701So for example, strings "U1" and "V1" next to "utau" and "vtau" would pair the wind stress components together
702and rotate them on to the model grid directions; "U2" and "V2" could be used against a second pair of components,
703and so on.
704The extra characters used in the strings are arbitrary.
705The rot\_rep routine from the \mdl{geo2ocean} module is used to perform the rotation.
706
707% -------------------------------------------------------------------------------------------------------------
708%        Surface restoring to observed SST and/or SSS
709% -------------------------------------------------------------------------------------------------------------
710\subsection    [Surface restoring to observed SST and/or SSS (\textit{sbcssr})]
711         {Surface restoring to observed SST and/or SSS (\mdl{sbcssr})}
712\label{SBC_ssr}
713%------------------------------------------namsbc_ssr----------------------------------------------------
714\namdisplay{namsbc_ssr} 
715%-------------------------------------------------------------------------------------------------------------
716
717In forced mode using a flux formulation (\np{ln\_flx}~=~true), a
718feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$:
719\begin{equation} \label{Eq_sbc_dmp_q}
720Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right)
721\end{equation}
722where SST is a sea surface temperature field (observed or climatological), $T$ is
723the model surface layer temperature and $\frac{dQ}{dT}$ is a negative feedback
724coefficient usually taken equal to $-40~W/m^2/K$. For a $50~m$ 
725mixed-layer depth, this value corresponds to a relaxation time scale of two months.
726This term ensures that if $T$ perfectly matches the supplied SST, then $Q$ is
727equal to $Q_o$.
728
729In the fresh water budget, a feedback term can also be added. Converted into an
730equivalent freshwater flux, it takes the following expression :
731
732\begin{equation} \label{Eq_sbc_dmp_emp}
733\textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)}
734                                             {\left.S\right|_{k=1}}
735\end{equation}
736
737where $\textit{emp}_{o }$ is a net surface fresh water flux (observed, climatological or an
738atmospheric model product), \textit{SSS}$_{Obs}$ is a sea surface salinity (usually a time
739interpolation of the monthly mean Polar Hydrographic Climatology \citep{Steele2001}),
740$\left.S\right|_{k=1}$ is the model surface layer salinity and $\gamma_s$ is a negative
741feedback coefficient which is provided as a namelist parameter. Unlike heat flux, there is no
742physical justification for the feedback term in \ref{Eq_sbc_dmp_emp} as the atmosphere
743does not care about ocean surface salinity \citep{Madec1997}. The SSS restoring
744term should be viewed as a flux correction on freshwater fluxes to reduce the
745uncertainties we have on the observed freshwater budget.
746
747% -------------------------------------------------------------------------------------------------------------
748%        Handling of ice-covered area
749% -------------------------------------------------------------------------------------------------------------
750\subsection{Handling of ice-covered area  (\textit{sbcice\_...})}
751\label{SBC_ice-cover}
752
753The presence at the sea surface of an ice covered area modifies all the fluxes
754transmitted to the ocean. There are several way to handle sea-ice in the system
755depending on the value of the \np{nn{\_}ice} namelist parameter. 
756\begin{description}
757\item[nn{\_}ice = 0]  there will never be sea-ice in the computational domain.
758This is a typical namelist value used for tropical ocean domain. The surface fluxes
759are simply specified for an ice-free ocean. No specific things is done for sea-ice.
760\item[nn{\_}ice = 1]  sea-ice can exist in the computational domain, but no sea-ice model
761is used. An observed ice covered area is read in a file. Below this area, the SST is
762restored to the freezing point and the heat fluxes are set to $-4~W/m^2$ ($-2~W/m^2$)
763in the northern (southern) hemisphere. The associated modification of the freshwater
764fluxes are done in such a way that the change in buoyancy fluxes remains zero.
765This prevents deep convection to occur when trying to reach the freezing point
766(and so ice covered area condition) while the SSS is too large. This manner of
767managing sea-ice area, just by using si IF case, is usually referred as the \textit{ice-if} 
768model. It can be found in the \mdl{sbcice{\_}if} module.
769\item[nn{\_}ice = 2 or more]  A full sea ice model is used. This model computes the
770ice-ocean fluxes, that are combined with the air-sea fluxes using the ice fraction of
771each model cell to provide the surface ocean fluxes. Note that the activation of a
772sea-ice model is is done by defining a CPP key (\key{lim2} or \key{lim3}).
773The activation automatically ovewrite the read value of nn{\_}ice to its appropriate
774value ($i.e.$ $2$ for LIM-2 and $3$ for LIM-3).
775\end{description}
776
777% {Description of Ice-ocean interface to be added here or in LIM 2 and 3 doc ?}
778
779% -------------------------------------------------------------------------------------------------------------
780%        Freshwater budget control
781% -------------------------------------------------------------------------------------------------------------
782\subsection   [Freshwater budget control (\textit{sbcfwb})]
783         {Freshwater budget control (\mdl{sbcfwb})}
784\label{SBC_fwb}
785
786For global ocean simulation it can be useful to introduce a control of the mean sea
787level in order to prevent unrealistic drift of the sea surface height due to inaccuracy
788in the freshwater fluxes. In \NEMO, two way of controlling the the freshwater budget.
789\begin{description}
790\item[\np{nn\_fwb}=0]  no control at all. The mean sea level is free to drift, and will
791certainly do so.
792\item[\np{nn\_fwb}=1]  global mean \textit{emp} set to zero at each model time step.
793%Note that with a sea-ice model, this technique only control the mean sea level with linear free surface (\key{vvl} not defined) and no mass flux between ocean and ice (as it is implemented in the current ice-ocean coupling).
794\item[\np{nn\_fwb}=2]  freshwater budget is adjusted from the previous year annual
795mean budget which is read in the \textit{EMPave\_old.dat} file. As the model uses the
796Boussinesq approximation, the annual mean fresh water budget is simply evaluated
797from the change in the mean sea level at January the first and saved in the
798\textit{EMPav.dat} file.
799\end{description}
800
801% Griffies doc:
802% When running ocean-ice simulations, we are not explicitly representing land processes, such as rivers, catchment areas, snow accumulation, etc. However, to reduce model drift, it is important to balance the hydrological cycle in ocean-ice models. We thus need to prescribe some form of global normalization to the precipitation minus evaporation plus river runoff. The result of the normalization should be a global integrated zero net water input to the ocean-ice system over a chosen time scale.
803%How often the normalization is done is a matter of choice. In mom4p1, we choose to do so at each model time step, so that there is always a zero net input of water to the ocean-ice system. Others choose to normalize over an annual cycle, in which case the net imbalance over an annual cycle is used to alter the subsequent yearÕs water budget in an attempt to damp the annual water imbalance. Note that the annual budget approach may be inappropriate with interannually varying precipitation forcing.
804%When running ocean-ice coupled models, it is incorrect to include the water transport between the ocean and ice models when aiming to balance the hydrological cycle. The reason is that it is the sum of the water in the ocean plus ice that should be balanced when running ocean-ice models, not the water in any one sub-component. As an extreme example to illustrate the issue, consider an ocean-ice model with zero initial sea ice. As the ocean-ice model spins up, there should be a net accumulation of water in the growing sea ice, and thus a net loss of water from the ocean. The total water contained in the ocean plus ice system is constant, but there is an exchange of water between the subcomponents. This exchange should not be part of the normalization used to balance the hydrological cycle in ocean-ice models.
805
806
807
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