Changeset 9393 for branches/2017/dev_merge_2017/DOC/tex_sub/chap_TRA.tex
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branches/2017/dev_merge_2017/DOC/tex_sub/chap_TRA.tex
r9392 r9393 57 57 58 58 The user has the option of extracting each tendency term on the RHS of the tracer 59 equation for output (\np{ln _tra_trd} or \np{ln_tra_mxl}~=~true), as described in Chap.~\ref{DIA}.59 equation for output (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}\forcode{ = .true.}), as described in Chap.~\ref{DIA}. 60 60 61 61 $\ $\newline % force a new ligne … … 63 63 % Tracer Advection 64 64 % ================================================================ 65 \section [Tracer Advection (\textit{traadv})] 66 {Tracer Advection (\protect\mdl{traadv})} 65 \section{Tracer advection (\protect\mdl{traadv})} 67 66 \label{TRA_adv} 68 67 %------------------------------------------namtra_adv----------------------------------------------------- … … 70 69 %------------------------------------------------------------------------------------------------------------- 71 70 72 When considered ($i.e.$ when \np{ln _traadv_NONE} is not set to \textit{true}),71 When considered ($i.e.$ when \np{ln\_traadv\_NONE} is not set to \forcode{.true.}), 73 72 the advection tendency of a tracer is expressed in flux form, 74 73 $i.e.$ as the divergence of the advective fluxes. Its discrete expression is given by : … … 84 83 by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 85 84 which results from the use of the continuity equation, $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ 86 (which reduces to $\nabla \cdot \vect{U}=0$ in linear free surface, $i.e.$ \ forcode{ln_linssh= .true.}).85 (which reduces to $\nabla \cdot \vect{U}=0$ in linear free surface, $i.e.$ \np{ln\_linssh}\forcode{ = .true.}). 87 86 Therefore it is of paramount importance to design the discrete analogue of the 88 87 advection tendency so that it is consistent with the continuity equation in order to … … 114 113 boundary condition depends on the type of sea surface chosen: 115 114 \begin{description} 116 \item [linear free surface:] (\ forcode{ln_linssh= .true.}) the first level thickness is constant in time:115 \item [linear free surface:] (\np{ln\_linssh}\forcode{ = .true.}) the first level thickness is constant in time: 117 116 the vertical boundary condition is applied at the fixed surface $z=0$ 118 117 rather than on the moving surface $z=\eta$. There is a non-zero advective … … 120 119 $\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $, $i.e.$ 121 120 the product of surface velocity (at $z=0$) by the first level tracer value. 122 \item [non-linear free surface:] (\ forcode{ln_linssh= .false.})121 \item [non-linear free surface:] (\np{ln\_linssh}\forcode{ = .false.}) 123 122 convergence/divergence in the first ocean level moves the free surface 124 123 up/down. There is no tracer advection through it so that the advective … … 146 145 Estimated Streaming Terms scheme (QUICKEST). 147 146 The choice is made in the \textit{\ngn{namtra\_adv}} namelist, by 148 setting to \ textit{true} one of the logicals \textit{ln\_traadv\_xxx}.147 setting to \forcode{.true.} one of the logicals \textit{ln\_traadv\_xxx}. 149 148 The corresponding code can be found in the \textit{traadv\_xxx.F90} module, 150 149 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 151 150 By default ($i.e.$ in the reference namelist, \ngn{namelist\_ref}), all the logicals 152 are set to \ textit{false}. If the user does not select an advection scheme151 are set to \forcode{.false.}. If the user does not select an advection scheme 153 152 in the configuration namelist (\ngn{namelist\_cfg}), the tracers will \textit{not} be advected ! 154 153 … … 174 173 % 2nd and 4th order centred schemes 175 174 % ------------------------------------------------------------------------------------------------------------- 176 \subsection [Centred schemes (CEN) (\protect\np{ln_traadv_cen})] 177 {Centred schemes (CEN) (\protect\forcode{ln_traadv_cen = .true.})} 175 \subsection{CEN: Centred scheme (\protect\np{ln\_traadv\_cen}\forcode{ = .true.})} 178 176 \label{TRA_adv_cen} 179 177 180 178 % 2nd order centred scheme 181 179 182 The centred advection scheme (CEN) is used when \np{ln _traadv_cen}~=~\textit{true}.180 The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}\forcode{ = .true.}. 183 181 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) 184 and vertical direction by setting \np{nn _cen_h} and \np{nn_cen_v} to $2$ or $4$.182 and vertical direction by setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$. 185 183 CEN implementation can be found in the \mdl{traadv\_cen} module. 186 184 … … 212 210 =\overline{ T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} 213 211 \end{equation} 214 In the vertical direction (\np{nn _cen_v}=$4$), a $4^{th}$ COMPACT interpolation212 In the vertical direction (\np{nn\_cen\_v}\forcode{ = 4}), a $4^{th}$ COMPACT interpolation 215 213 has been prefered \citep{Demange_PhD2014}. 216 214 In the COMPACT scheme, both the field and its derivative are interpolated, … … 224 222 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature 225 223 is usually associated with the scheme presented here. 226 Introducing a \ textit{true} $4^{th}$ order advection scheme is feasible but,224 Introducing a \forcode{.true.} $4^{th}$ order advection scheme is feasible but, 227 225 for consistency reasons, it requires changes in the discretisation of the tracer 228 226 advection together with changes in the continuity equation, … … 246 244 % FCT scheme 247 245 % ------------------------------------------------------------------------------------------------------------- 248 \subsection [Flux Corrected Transport schemes (FCT) (\protect\np{ln_traadv_fct})] 249 {Flux Corrected Transport schemes (FCT) (\protect\forcode{ln_traadv_fct = .true.})} 246 \subsection{FCT: Flux Corrected Transport scheme (\protect\np{ln\_traadv\_fct}\forcode{ = .true.})} 250 247 \label{TRA_adv_tvd} 251 248 252 The Flux Corrected Transport schemes (FCT) is used when \np{ln _traadv_fct}~=~\textit{true}.249 The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}\forcode{ = .true.}. 253 250 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) 254 and vertical direction by setting \np{nn _fct_h} and \np{nn_fct_v} to $2$ or $4$.251 and vertical direction by setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$. 255 252 FCT implementation can be found in the \mdl{traadv\_fct} module. 256 253 … … 269 266 where $c_u$ is a flux limiter function taking values between 0 and 1. 270 267 The FCT order is the one of the centred scheme used ($i.e.$ it depends on the setting of 271 \np{nn _fct_h} and \np{nn_fct_v}.268 \np{nn\_fct\_h} and \np{nn\_fct\_v}. 272 269 There exist many ways to define $c_u$, each corresponding to a different 273 270 FCT scheme. The one chosen in \NEMO is described in \citet{Zalesak_JCP79}. … … 277 274 A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{Levy_al_GRL01}. 278 275 279 An additional option has been added controlled by \np{nn _fct_zts}. By setting this integer to276 An additional option has been added controlled by \np{nn\_fct\_zts}. By setting this integer to 280 277 a value larger than zero, a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, 281 278 but on the latter, a split-explicit time stepping is used, with a number of sub-timestep equals 282 to \np{nn _fct_zts}. This option can be useful when the size of the timestep is limited279 to \np{nn\_fct\_zts}. This option can be useful when the size of the timestep is limited 283 280 by vertical advection \citep{Lemarie_OM2015}. Note that in this case, a similar split-explicit 284 281 time stepping should be used on vertical advection of momentum to insure a better stability … … 293 290 % MUSCL scheme 294 291 % ------------------------------------------------------------------------------------------------------------- 295 \subsection[MUSCL scheme (\protect\np{ln_traadv_mus})] 296 {Monotone Upstream Scheme for Conservative Laws (MUSCL) (\protect\forcode{ln_traadv_mus = .true.})} 292 \subsection{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln\_traadv\_mus}\forcode{ = .true.})} 297 293 \label{TRA_adv_mus} 298 294 299 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln _traadv_mus}~=~\textit{true}.295 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}\forcode{ = .true.}. 300 296 MUSCL implementation can be found in the \mdl{traadv\_mus} module. 301 297 … … 321 317 the \textit{positive} character of the scheme. 322 318 In addition, fluxes round a grid-point where a runoff is applied can optionally be 323 computed using upstream fluxes (\np{ln _mus_ups}~=~\textit{true}).319 computed using upstream fluxes (\np{ln\_mus\_ups}\forcode{ = .true.}). 324 320 325 321 % ------------------------------------------------------------------------------------------------------------- 326 322 % UBS scheme 327 323 % ------------------------------------------------------------------------------------------------------------- 328 \subsection [Upstream-Biased Scheme (UBS) (\protect\np{ln_traadv_ubs})] 329 {Upstream-Biased Scheme (UBS) (\protect\forcode{ln_traadv_ubs = .true.})} 324 \subsection{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} 330 325 \label{TRA_adv_ubs} 331 326 332 The Upstream-Biased Scheme (UBS) is used when \np{ln _traadv_ubs}~=~\textit{true}.327 The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}\forcode{ = .true.}. 333 328 UBS implementation can be found in the \mdl{traadv\_mus} module. 334 329 … … 358 353 where the control of artificial diapycnal fluxes is of paramount importance \citep{Shchepetkin_McWilliams_OM05, Demange_PhD2014}. 359 354 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme 360 or a $4^th$ order COMPACT scheme (\ forcode{nn_cen_v = 2} or 4).355 or a $4^th$ order COMPACT scheme (\np{nn\_cen\_v}\forcode{ = 2 or 4}). 361 356 362 357 For stability reasons (see \S\ref{STP}), … … 401 396 % QCK scheme 402 397 % ------------------------------------------------------------------------------------------------------------- 403 \subsection [QUICKEST scheme (QCK) (\protect\np{ln_traadv_qck})] 404 {QUICKEST scheme (QCK) (\protect\forcode{ln_traadv_qck = .true.})} 398 \subsection{QCK: QuiCKest scheme (\protect\np{ln\_traadv\_qck}\forcode{ = .true.})} 405 399 \label{TRA_adv_qck} 406 400 407 401 The Quadratic Upstream Interpolation for Convective Kinematics with 408 402 Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} 409 is used when \np{ln _traadv_qck}~=~\textit{true}.403 is used when \np{ln\_traadv\_qck}\forcode{ = .true.}. 410 404 QUICKEST implementation can be found in the \mdl{traadv\_qck} module. 411 405 … … 428 422 % Tracer Lateral Diffusion 429 423 % ================================================================ 430 \section [Tracer Lateral Diffusion (\textit{traldf})] 431 {Tracer Lateral Diffusion (\protect\mdl{traldf})} 424 \section{Tracer lateral diffusion (\protect\mdl{traldf})} 432 425 \label{TRA_ldf} 433 426 %-----------------------------------------nam_traldf------------------------------------------------------ … … 449 442 except for the pure vertical component that appears when a rotation tensor is used. 450 443 This latter component is solved implicitly together with the vertical diffusion term (see \S\ref{STP}). 451 When \np{ln _traldf_msc}~=~\textit{true}, a Method of Stabilizing Correction is used in which444 When \np{ln\_traldf\_msc}\forcode{ = .true.}, a Method of Stabilizing Correction is used in which 452 445 the pure vertical component is split into an explicit and an implicit part \citep{Lemarie_OM2012}. 453 446 … … 455 448 % Type of operator 456 449 % ------------------------------------------------------------------------------------------------------------- 457 \subsection [Type of operator (\protect\np{ln\_traldf\{\_NONE, \_lap, \_blp\}})]458 {Type of operator (\protect\np{ln _traldf_NONE}, \protect\np{ln_traldf_lap}, or \protect\np{ln_traldf_blp} = true) }450 \subsection[Type of operator (\protect\np{ln\_traldf}\{\_NONE,\_lap,\_blp\}\})] 451 {Type of operator (\protect\np{ln\_traldf\_NONE}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) } 459 452 \label{TRA_ldf_op} 460 453 461 454 Three operator options are proposed and, one and only one of them must be selected: 462 455 \begin{description} 463 \item [\np{ln _traldf_NONE}] = true: no operator selected, the lateral diffusive tendency will not be456 \item [\np{ln\_traldf\_NONE}\forcode{ = .true.}]: no operator selected, the lateral diffusive tendency will not be 464 457 applied to the tracer equation. This option can be used when the selected advection scheme 465 458 is diffusive enough (MUSCL scheme for example). 466 \item [ \np{ln_traldf_lap}] = true: a laplacian operator is selected. This harmonic operator459 \item [\np{ln\_traldf\_lap}\forcode{ = .true.}]: a laplacian operator is selected. This harmonic operator 467 460 takes the following expression: $\mathpzc{L}(T)=\nabla \cdot A_{ht}\;\nabla T $, 468 461 where the gradient operates along the selected direction (see \S\ref{TRA_ldf_dir}), 469 462 and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see Chap.~\ref{LDF}). 470 \item [\np{ln _traldf_blp}] = true: a bilaplacian operator is selected. This biharmonic operator463 \item [\np{ln\_traldf\_blp}\forcode{ = .true.}]: a bilaplacian operator is selected. This biharmonic operator 471 464 takes the following expression: 472 465 $\mathpzc{B}=- \mathpzc{L}\left(\mathpzc{L}(T) \right) = -\nabla \cdot b\nabla \left( {\nabla \cdot b\nabla T} \right)$ … … 488 481 % Direction of action 489 482 % ------------------------------------------------------------------------------------------------------------- 490 \subsection [Direction of action (\protect\np{ln\_traldf\{\_lev, \_hor, \_iso, \_triad\}})]491 {Direction of action (\protect\np{ln _traldf_lev}, \textit{...\_hor}, \textit{...\_iso}, or \textit{...\_triad} = true) }483 \subsection[Action direction (\protect\np{ln\_traldf}\{\_lev,\_hor,\_iso,\_triad\})] 484 {Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) } 492 485 \label{TRA_ldf_dir} 493 486 494 487 The choice of a direction of action determines the form of operator used. 495 488 The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane 496 when iso-level option is used (\np{ln _traldf_lev}~=~\textit{true})489 when iso-level option is used (\np{ln\_traldf\_lev}\forcode{ = .true.}) 497 490 or when a horizontal ($i.e.$ geopotential) operator is demanded in \textit{z}-coordinate 498 (\np{ln _traldf_hor} and \np{ln_zco} equal \textit{true}).491 (\np{ln\_traldf\_hor} and \np{ln\_zco} equal \forcode{.true.}). 499 492 The associated code can be found in the \mdl{traldf\_lap\_blp} module. 500 493 The operator is a rotated (re-entrant) laplacian when the direction along which it acts 501 494 does not coincide with the iso-level surfaces, 502 that is when standard or triad iso-neutral option is used (\np{ln _traldf_iso} or503 \np{ln _traldf_triad} equals \textit{true}, see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.),495 that is when standard or triad iso-neutral option is used (\np{ln\_traldf\_iso} or 496 \np{ln\_traldf\_triad} equals \forcode{.true.}, see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), 504 497 or when a horizontal ($i.e.$ geopotential) operator is demanded in \textit{s}-coordinate 505 (\np{ln _traldf_hor} and \np{ln_sco} equal \textit{true})498 (\np{ln\_traldf\_hor} and \np{ln\_sco} equal \forcode{.true.}) 506 499 \footnote{In this case, the standard iso-neutral operator will be automatically selected}. 507 500 In that case, a rotation is applied to the gradient(s) that appears in the operator … … 515 508 % iso-level operator 516 509 % ------------------------------------------------------------------------------------------------------------- 517 \subsection [Iso-level (bi-)laplacian operator ( \protect\np{ln_traldf_iso})] 518 {Iso-level (bi-)laplacian operator ( \protect\np{ln_traldf_iso}) } 510 \subsection{Iso-level (bi-)laplacian operator ( \protect\np{ln\_traldf\_iso}) } 519 511 \label{TRA_ldf_lev} 520 512 … … 534 526 It is a \emph{horizontal} operator ($i.e.$ acting along geopotential surfaces) in the $z$-coordinate 535 527 with or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 536 It is thus used when, in addition to \np{ln _traldf_lap} or \np{ln_traldf_blp}~=~\textit{true},537 we have \np{ln _traldf_lev}~=~\textit{true} or \np{ln_traldf_hor}~=~\np{ln_zco}~=~\textit{true}.528 It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}\forcode{ = .true.}, 529 we have \np{ln\_traldf\_lev}\forcode{ = .true.} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}\forcode{ = .true.}. 538 530 In both cases, it significantly contributes to diapycnal mixing. 539 531 It is therefore never recommended, even when using it in the bilaplacian case. 540 532 541 Note that in the partial step $z$-coordinate (\ forcode{ln_zps= .true.}), tracers in horizontally533 Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}), tracers in horizontally 542 534 adjacent cells are located at different depths in the vicinity of the bottom. 543 535 In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level … … 549 541 % Rotated laplacian operator 550 542 % ------------------------------------------------------------------------------------------------------------- 551 \subsection [Standard and triad rotated (bi-)laplacian operator] 552 {Standard and triad (bi-)laplacian operator} 543 \subsection{Standard and triad (bi-)laplacian operator} 553 544 \label{TRA_ldf_iso_triad} 554 545 555 546 %&& Standard rotated (bi-)laplacian operator 556 547 %&& ---------------------------------------------- 557 \subsubsection [Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})] 558 {Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 548 \subsubsection{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 559 549 \label{TRA_ldf_iso} 560 550 The general form of the second order lateral tracer subgrid scale physics … … 584 574 ($z$- or $s$-surfaces) and the surface along which the diffusion operator 585 575 acts ($i.e.$ horizontal or iso-neutral surfaces). It is thus used when, 586 in addition to \np{ln _traldf_lap}= true, we have \forcode{ln_traldf_iso= .true.},587 or both \ forcode{ln_traldf_hor = .true.} and \forcode{ln_zco= .true.}. The way these576 in addition to \np{ln\_traldf\_lap}\forcode{ = .true.}, we have \np{ln\_traldf\_iso}\forcode{ = .true.}, 577 or both \np{ln\_traldf\_hor}\forcode{ = .true.} and \np{ln\_zco}\forcode{ = .true.}. The way these 588 578 slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom 589 579 and lateral boundaries, the turbulent fluxes of heat and salt are set to zero … … 603 593 background horizontal diffusion \citep{Guilyardi_al_CD01}. 604 594 605 Note that in the partial step $z$-coordinate (\ forcode{ln_zps= .true.}), the horizontal derivatives595 Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}), the horizontal derivatives 606 596 at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific treatment. 607 597 They are calculated in module zpshde, described in \S\ref{TRA_zpshde}. … … 609 599 %&& Triad rotated (bi-)laplacian operator 610 600 %&& ------------------------------------------- 611 \subsubsection [Triad rotated (bi-)laplacian operator (\protect\np{ln_traldf_triad})] 612 {Triad rotated (bi-)laplacian operator (\protect\np{ln_traldf_triad})} 601 \subsubsection{Triad rotated (bi-)laplacian operator (\protect\np{ln\_traldf\_triad})} 613 602 \label{TRA_ldf_triad} 614 603 615 If the Griffies triad scheme is employed (\ forcode{ln_traldf_triad= .true.} ; see App.\ref{sec:triad})604 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}\forcode{ = .true.} ; see App.\ref{sec:triad}) 616 605 617 606 An alternative scheme developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases 618 is also available in \NEMO (\ forcode{ln_traldf_grif= .true.}). A complete description of607 is also available in \NEMO (\np{ln\_traldf\_grif}\forcode{ = .true.}). A complete description of 619 608 the algorithm is given in App.\ref{sec:triad}. 620 609 … … 631 620 %&& Option for the rotated operators 632 621 %&& ---------------------------------------------- 633 \subsubsection [Option for the rotated operators] 634 {Option for the rotated operators} 622 \subsubsection{Option for the rotated operators} 635 623 \label{TRA_ldf_options} 636 624 637 \np{ln _traldf_msc} = Method of Stabilizing Correction (both operators)638 639 \np{rn _slpmax} = slope limit (both operators)640 641 \np{ln _triad_iso} = pure horizontal mixing in ML (triad only)642 643 \np{rn _sw_triad} =1 switching triad ; =0 all 4 triads used (triad only)644 645 \np{ln _botmix_triad} = lateral mixing on bottom (triad only)625 \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) 626 627 \np{rn\_slpmax} = slope limit (both operators) 628 629 \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only) 630 631 \np{rn\_sw\_triad} =1 switching triad ; =0 all 4 triads used (triad only) 632 633 \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only) 646 634 647 635 % ================================================================ 648 636 % Tracer Vertical Diffusion 649 637 % ================================================================ 650 \section [Tracer Vertical Diffusion (\textit{trazdf})] 651 {Tracer Vertical Diffusion (\protect\mdl{trazdf})} 638 \section{Tracer vertical diffusion (\protect\mdl{trazdf})} 652 639 \label{TRA_zdf} 653 640 %--------------------------------------------namzdf--------------------------------------------------------- … … 685 672 The large eddy coefficient found in the mixed layer together with high 686 673 vertical resolution implies that in the case of explicit time stepping 687 (\ forcode{ln_zdfexp= .true.}) there would be too restrictive a constraint on674 (\np{ln\_zdfexp}\forcode{ = .true.}) there would be too restrictive a constraint on 688 675 the time step. Therefore, the default implicit time stepping is preferred 689 676 for the vertical diffusion since it overcomes the stability constraint. 690 A forward time differencing scheme (\ forcode{ln_zdfexp= .true.}) using a time691 splitting technique (\np{nn _zdfexp} $> 1$) is provided as an alternative.692 Namelist variables \np{ln _zdfexp} and \np{nn_zdfexp} apply to both677 A forward time differencing scheme (\np{ln\_zdfexp}\forcode{ = .true.}) using a time 678 splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative. 679 Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both 693 680 tracers and dynamics. 694 681 … … 696 683 % External Forcing 697 684 % ================================================================ 698 \section{External Forcing}685 \section{External forcing} 699 686 \label{TRA_sbc_qsr_bbc} 700 687 … … 702 689 % surface boundary condition 703 690 % ------------------------------------------------------------------------------------------------------------- 704 \subsection [Surface boundary condition (\textit{trasbc})] 705 {Surface boundary condition (\protect\mdl{trasbc})} 691 \subsection{Surface boundary condition (\protect\mdl{trasbc})} 706 692 \label{TRA_sbc} 707 693 … … 750 736 divergence of odd and even time step (see \S\ref{STP}). 751 737 752 In the linear free surface case (\np{ln _linssh}~=~\textit{true}),738 In the linear free surface case (\np{ln\_linssh}\forcode{ = .true.}), 753 739 an additional term has to be added on both temperature and salinity. 754 740 On temperature, this term remove the heat content associated with mass exchange … … 773 759 % Solar Radiation Penetration 774 760 % ------------------------------------------------------------------------------------------------------------- 775 \subsection [Solar Radiation Penetration (\textit{traqsr})] 776 {Solar Radiation Penetration (\protect\mdl{traqsr})} 761 \subsection{Solar radiation penetration (\protect\mdl{traqsr})} 777 762 \label{TRA_qsr} 778 763 %--------------------------------------------namqsr-------------------------------------------------------- … … 781 766 782 767 Options are defined through the \ngn{namtra\_qsr} namelist variables. 783 When the penetrative solar radiation option is used (\ forcode{ln_flxqsr= .true.}),768 When the penetrative solar radiation option is used (\np{ln\_flxqsr}\forcode{ = .true.}), 784 769 the solar radiation penetrates the top few tens of meters of the ocean. If it is not used 785 (\ forcode{ln_flxqsr= .false.}) all the heat flux is absorbed in the first ocean level.770 (\np{ln\_flxqsr}\forcode{ = .false.}) all the heat flux is absorbed in the first ocean level. 786 771 Thus, in the former case a term is added to the time evolution equation of 787 772 temperature \eqref{Eq_PE_tra_T} and the surface boundary condition is … … 805 790 wavelengths contribute to heating the upper few tens of centimetres. The fraction of $Q_{sr}$ 806 791 that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified 807 through namelist parameter \np{rn _abs}). It is assumed to penetrate the ocean792 through namelist parameter \np{rn\_abs}). It is assumed to penetrate the ocean 808 793 with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 809 of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn _si0} in the namtra\_qsr namelist).794 of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsr namelist). 810 795 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy 811 796 propagates to larger depths where it contributes to 812 797 local heating. 813 798 The way this second part of the solar energy penetrates into the ocean depends on 814 which formulation is chosen. In the simple 2-waveband light penetration scheme (\ forcode{ln_qsr_2bd= .true.})799 which formulation is chosen. In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}\forcode{ = .true.}) 815 800 a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 816 801 leading to the following expression \citep{Paulson1977}: … … 819 804 \end{equation} 820 805 where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. 821 It is usually chosen to be 23~m by setting the \np{rn _si0} namelist parameter.806 It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter. 822 807 The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in 823 808 Jerlov's (1968) classification (oligotrophic waters). … … 839 824 computational efficiency. The 2-bands formulation does not reproduce the full model very well. 840 825 841 The RGB formulation is used when \ forcode{ln_qsr_rgb= .true.}. The RGB attenuation coefficients826 The RGB formulation is used when \np{ln\_qsr\_rgb}\forcode{ = .true.}. The RGB attenuation coefficients 842 827 ($i.e.$ the inverses of the extinction length scales) are tabulated over 61 nonuniform 843 828 chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} 844 829 in \mdl{trc\_oce} module). Four types of chlorophyll can be chosen in the RGB formulation: 845 830 \begin{description} 846 \item[\ forcode{nn_chdta= 0}]831 \item[\np{nn\_chdta}\forcode{ = 0}] 847 832 a constant 0.05 g.Chl/L value everywhere ; 848 \item[\ forcode{nn_chdta= 1}]833 \item[\np{nn\_chdta}\forcode{ = 1}] 849 834 an observed time varying chlorophyll deduced from satellite surface ocean color measurement 850 835 spread uniformly in the vertical direction ; 851 \item[\ forcode{nn_chdta= 2}]836 \item[\np{nn\_chdta}\forcode{ = 2}] 852 837 same as previous case except that a vertical profile of chlorophyl is used. 853 838 Following \cite{Morel_Berthon_LO89}, the profile is computed from the local surface chlorophyll value ; 854 \item[\ forcode{ln_qsr_bio= .true.}]839 \item[\np{ln\_qsr\_bio}\forcode{ = .true.}] 855 840 simulated time varying chlorophyll by TOP biogeochemical model. 856 841 In this case, the RGB formulation is used to calculate both the phytoplankton … … 884 869 % Bottom Boundary Condition 885 870 % ------------------------------------------------------------------------------------------------------------- 886 \subsection [Bottom Boundary Condition (\textit{trabbc})] 887 {Bottom Boundary Condition (\protect\mdl{trabbc})} 871 \subsection{Bottom boundary condition (\protect\mdl{trabbc})} 888 872 \label{TRA_bbc} 889 873 %--------------------------------------------nambbc-------------------------------------------------------- … … 913 897 Options are defined through the \ngn{namtra\_bbc} namelist variables. 914 898 The presence of geothermal heating is controlled by setting the namelist 915 parameter \np{ln _trabbc} to true. Then, when \np{nn_geoflx} is set to 1,899 parameter \np{ln\_trabbc} to true. Then, when \np{nn\_geoflx} is set to 1, 916 900 a constant geothermal heating is introduced whose value is given by the 917 \np{nn _geoflx_cst}, which is also a namelist parameter.918 When \np{nn _geoflx} is set to 2, a spatially varying geothermal heat flux is901 \np{nn\_geoflx\_cst}, which is also a namelist parameter. 902 When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is 919 903 introduced which is provided in the \ifile{geothermal\_heating} NetCDF file 920 904 (Fig.\ref{Fig_geothermal}) \citep{Emile-Geay_Madec_OS09}. … … 923 907 % Bottom Boundary Layer 924 908 % ================================================================ 925 \section [Bottom Boundary Layer (\protect\mdl{trabbl} - \protect\key{trabbl})] 926 {Bottom Boundary Layer (\protect\mdl{trabbl} - \protect\key{trabbl})} 909 \section{Bottom boundary layer (\protect\mdl{trabbl} - \protect\key{trabbl})} 927 910 \label{TRA_bbl} 928 911 %--------------------------------------------nambbl--------------------------------------------------------- … … 959 942 % Diffusive BBL 960 943 % ------------------------------------------------------------------------------------------------------------- 961 \subsection{Diffusive Bottom Boundary layer (\protect\forcode{nn_bbl_ldf= 1})}944 \subsection{Diffusive bottom boundary layer (\protect\np{nn\_bbl\_ldf}\forcode{ = 1})} 962 945 \label{TRA_bbl_diff} 963 946 964 When applying sigma-diffusion (\key{trabbl} defined and \np{nn _bbl_ldf} set to 1),947 When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), 965 948 the diffusive flux between two adjacent cells at the ocean floor is given by 966 949 \begin{equation} \label{Eq_tra_bbl_diff} … … 978 961 \end{equation} 979 962 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist 980 parameter \np{rn _ahtbbl} and usually set to a value much larger963 parameter \np{rn\_ahtbbl} and usually set to a value much larger 981 964 than the one used for lateral mixing in the open ocean. The constraint in \eqref{Eq_tra_bbl_coef} 982 965 implies that sigma-like diffusion only occurs when the density above the sea floor, at the top of … … 994 977 % Advective BBL 995 978 % ------------------------------------------------------------------------------------------------------------- 996 \subsection {Advective Bottom Boundary Layer (\protect\np{nn_bbl_adv}= 1 or 2)}979 \subsection{Advective bottom boundary layer (\protect\np{nn\_bbl\_adv}\forcode{ = 1..2})} 997 980 \label{TRA_bbl_adv} 998 981 … … 1022 1005 %%%gmcomment : this section has to be really written 1023 1006 1024 When applying an advective BBL (\np{nn _bbl_adv} = 1 or 2), an overturning1007 When applying an advective BBL (\np{nn\_bbl\_adv}\forcode{ = 1..2}), an overturning 1025 1008 circulation is added which connects two adjacent bottom grid-points only if dense 1026 1009 water overlies less dense water on the slope. The density difference causes dense 1027 1010 water to move down the slope. 1028 1011 1029 \np{nn _bbl_adv} = 1: the downslope velocity is chosen to be the Eulerian1012 \np{nn\_bbl\_adv}\forcode{ = 1} : the downslope velocity is chosen to be the Eulerian 1030 1013 ocean velocity just above the topographic step (see black arrow in Fig.\ref{Fig_bbl}) 1031 1014 \citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection … … 1034 1017 greater depth ($i.e.$ $\vect{U} \cdot \nabla H>0$). 1035 1018 1036 \np{nn _bbl_adv} = 2: the downslope velocity is chosen to be proportional to $\Delta \rho$,1019 \np{nn\_bbl\_adv}\forcode{ = 2} : the downslope velocity is chosen to be proportional to $\Delta \rho$, 1037 1020 the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}. 1038 1021 The advection is allowed only if dense water overlies less dense water on the slope ($i.e.$ … … 1044 1027 \end{equation} 1045 1028 where $\gamma$, expressed in seconds, is the coefficient of proportionality 1046 provided as \np{rn _gambbl}, a namelist parameter, and \textit{kup} and \textit{kdwn}1029 provided as \np{rn\_gambbl}, a namelist parameter, and \textit{kup} and \textit{kdwn} 1047 1030 are the vertical index of the higher and lower cells, respectively. 1048 1031 The parameter $\gamma$ should take a different value for each bathymetric … … 1083 1066 % Tracer damping 1084 1067 % ================================================================ 1085 \section [Tracer damping (\textit{tradmp})] 1086 {Tracer damping (\protect\mdl{tradmp})} 1068 \section{Tracer damping (\protect\mdl{tradmp})} 1087 1069 \label{TRA_dmp} 1088 1070 %--------------------------------------------namtra_dmp------------------------------------------------- … … 1101 1083 are given temperature and salinity fields (usually a climatology). 1102 1084 Options are defined through the \ngn{namtra\_dmp} namelist variables. 1103 The restoring term is added when the namelist parameter \np{ln _tradmp} is set to true.1104 It also requires that both \np{ln _tsd_init} and \np{ln_tsd_tradmp} are set to true1105 in \textit{namtsd} namelist as well as \np{sn _tem} and \np{sn_sal} structures are1085 The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true. 1086 It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true 1087 in \textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are 1106 1088 correctly set ($i.e.$ that $T_o$ and $S_o$ are provided in input files and read 1107 1089 using \mdl{fldread}, see \S\ref{SBC_fldread}). 1108 The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. The file name is specified by the namelist variable \np{cn _resto}. The DMP\_TOOLS tool is provided to allow users to generate the netcdf file.1090 The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. The file name is specified by the namelist variable \np{cn\_resto}. The DMP\_TOOLS tool is provided to allow users to generate the netcdf file. 1109 1091 1110 1092 The two main cases in which \eqref{Eq_tra_dmp} is used are \textit{(a)} … … 1128 1110 by stabilising the water column too much. 1129 1111 1130 The namelist parameter \np{nn _zdmp} sets whether the damping should be applied in the whole water column or only below the mixed layer (defined either on a density or $S_o$ criterion). It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here \citep{Madec_al_JPO96}.1131 1132 \subsection [DMP\_TOOLS]{Generating \ifile{resto} using DMP\_TOOLS}1112 The namelist parameter \np{nn\_zdmp} sets whether the damping should be applied in the whole water column or only below the mixed layer (defined either on a density or $S_o$ criterion). It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here \citep{Madec_al_JPO96}. 1113 1114 \subsection{Generating \ifile{resto} using DMP\_TOOLS} 1133 1115 1134 1116 DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$. 1135 1117 Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled 1136 1118 and run on the same machine as the NEMO model. A \ifile{mesh\_mask} file for the model configuration is required as an input. 1137 This can be generated by carrying out a short model run with the namelist parameter \np{nn _msh} set to 1.1138 The namelist parameter \np{ln _tradmp} will also need to be set to .false. for this to work.1119 This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1. 1120 The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work. 1139 1121 The \nl{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for the restoration coefficient. 1140 1122 … … 1143 1125 %------------------------------------------------------------------------------------------------------- 1144 1126 1145 \np{cp _cfg}, \np{cp_cpz}, \np{jp_cfg} and \np{jperio} specify the model configuration being used and should be the same as specified in \nl{namcfg}. The variable \nl{lzoom} is used to specify that the damping is being used as in case \textit{a} above to provide boundary conditions to a zoom configuration. In the case of the arctic or antarctic zoom configurations this includes some specific treatment. Otherwise damping is applied to the 6 grid points along the ocean boundaries. The open boundaries are specified by the variables \np{lzoom_n}, \np{lzoom_e}, \np{lzoom_s}, \np{lzoom_w} in the \nl{nam\_zoom\_dmp} name list.1127 \np{cp\_cfg}, \np{cp\_cpz}, \np{jp\_cfg} and \np{jperio} specify the model configuration being used and should be the same as specified in \nl{namcfg}. The variable \nl{lzoom} is used to specify that the damping is being used as in case \textit{a} above to provide boundary conditions to a zoom configuration. In the case of the arctic or antarctic zoom configurations this includes some specific treatment. Otherwise damping is applied to the 6 grid points along the ocean boundaries. The open boundaries are specified by the variables \np{lzoom\_n}, \np{lzoom\_e}, \np{lzoom\_s}, \np{lzoom\_w} in the \nl{nam\_zoom\_dmp} name list. 1146 1128 1147 1129 The remaining switch namelist variables determine the spatial variation of the restoration coefficient in non-zoom configurations. 1148 \np{ln _full_field} specifies that newtonian damping should be applied to the whole model domain.1149 \np{ln _med_red_seas} specifies grid specific restoration coefficients in the Mediterranean Sea1130 \np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain. 1131 \np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea 1150 1132 for the ORCA4, ORCA2 and ORCA05 configurations. 1151 If \np{ln _old_31_lev_code} is set then the depth variation of the coeffients will be specified as1133 If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as 1152 1134 a function of the model number. This option is included to allow backwards compatability of the ORCA2 reference 1153 1135 configurations with previous model versions. 1154 \np{ln _coast} specifies that the restoration coefficient should be reduced near to coastlines.1155 This option only has an effect if \np{ln _full_field} is true.1156 \np{ln _zero_top_layer} specifies that the restoration coefficient should be zero in the surface layer.1157 Finally \np{ln _custom} specifies that the custom module will be called.1136 \np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines. 1137 This option only has an effect if \np{ln\_full\_field} is true. 1138 \np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer. 1139 Finally \np{ln\_custom} specifies that the custom module will be called. 1158 1140 This module is contained in the file custom.F90 and can be edited by users. For example damping could be applied in a specific region. 1159 1141 1160 The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \np{nn _hdmp}.1142 The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \np{nn\_hdmp}. 1161 1143 Equatorward of this latitude the restoration coefficient will be zero with a smooth transition to 1162 1144 the full values of a 10\deg latitud band. 1163 1145 This is often used because of the short adjustment time scale in the equatorial region 1164 1146 \citep{Reverdin1991, Fujio1991, Marti_PhD92}. The time scale associated with the damping depends on the depth as a 1165 hyperbolic tangent, with \np{rn _surf} as surface value, \np{rn_bot} as bottom value and a transition depth of \np{rn_dep}.1147 hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}. 1166 1148 1167 1149 % ================================================================ 1168 1150 % Tracer time evolution 1169 1151 % ================================================================ 1170 \section [Tracer time evolution (\textit{tranxt})] 1171 {Tracer time evolution (\protect\mdl{tranxt})} 1152 \section{Tracer time evolution (\protect\mdl{tranxt})} 1172 1153 \label{TRA_nxt} 1173 1154 %--------------------------------------------namdom----------------------------------------------------- … … 1191 1172 the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient, 1192 1173 and $S$ is the total forcing applied on $T$ ($i.e.$ fluxes plus content in mass exchanges). 1193 $\gamma$ is initialized as \np{rn _atfp} (\textbf{namelist} parameter).1194 Its default value is \np{rn _atfp}=$10^{-3}$. Note that the forcing correction term in the filter1195 is not applied in linear free surface (\jp{lk\_vvl} =false) (see \S\ref{TRA_sbc}.1174 $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 1175 Its default value is \np{rn\_atfp}\forcode{ = 10.e-3}. Note that the forcing correction term in the filter 1176 is not applied in linear free surface (\jp{lk\_vvl}\forcode{ = .false.}) (see \S\ref{TRA_sbc}. 1196 1177 Not also that in constant volume case, the time stepping is performed on $T$, 1197 1178 not on its content, $e_{3t}T$. … … 1207 1188 % Equation of State (eosbn2) 1208 1189 % ================================================================ 1209 \section [Equation of State (\textit{eosbn2}) ] 1210 {Equation of State (\protect\mdl{eosbn2}) } 1190 \section{Equation of state (\protect\mdl{eosbn2}) } 1211 1191 \label{TRA_eosbn2} 1212 1192 %--------------------------------------------nameos----------------------------------------------------- … … 1217 1197 % Equation of State 1218 1198 % ------------------------------------------------------------------------------------------------------------- 1219 \subsection{Equation Of Seawater (\protect\np{nn_eos} = -1, 0, or 1)}1199 \subsection{Equation of seawater (\protect\np{nn\_eos}\forcode{ = -1..1})} 1220 1200 \label{TRA_eos} 1221 1201 … … 1248 1228 density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}. 1249 1229 1250 Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn _eos}1251 which controls the EOS used ( =-1 for TEOS10 ; =0 for EOS-80 ; =1for S-EOS).1230 Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} 1231 which controls the EOS used (\forcode{= -1} for TEOS10 ; \forcode{= 0} for EOS-80 ; \forcode{= 1} for S-EOS). 1252 1232 \begin{description} 1253 1233 1254 \item[\np{nn _eos}$=-1$] the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used.1234 \item[\np{nn\_eos}\forcode{ = -1}] the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used. 1255 1235 The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, 1256 1236 but it is optimized for a boussinesq fluid and the polynomial expressions have simpler … … 1268 1248 $\Theta$ and $S_A$. In particular, the initial state deined by the user have to be given as 1269 1249 \textit{Conservative} Temperature and \textit{Absolute} Salinity. 1270 In addition, setting \np{ln _useCT} to \textit{true} convert the Conservative SST to potential SST1250 In addition, setting \np{ln\_useCT} to \forcode{.true.} convert the Conservative SST to potential SST 1271 1251 prior to either computing the air-sea and ice-sea fluxes (forced mode) 1272 1252 or sending the SST field to the atmosphere (coupled mode). 1273 1253 1274 \item[\np{nn _eos}$=0$] the polyEOS80-bsq equation of seawater is used.1254 \item[\np{nn\_eos}\forcode{ = 0}] the polyEOS80-bsq equation of seawater is used. 1275 1255 It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized 1276 1256 to accurately fit EOS80 (Roquet, personal comm.). The state variables used in both the EOS80 … … 1283 1263 value, the TEOS10 value. 1284 1264 1285 \item[\np{nn _eos}$=1$] a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen,1265 \item[\np{nn\_eos}\forcode{ = 1}] a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen, 1286 1266 the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.) 1287 1267 (see also \citet{Roquet_JPO2015}). It provides a simplistic linear representation of both … … 1315 1295 \hline 1316 1296 coeff. & computer name & S-EOS & description \\ \hline 1317 $a_0$ & \np{rn _a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline1318 $b_0$ & \np{rn _b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline1319 $\lambda_1$ & \np{rn _lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline1320 $\lambda_2$ & \np{rn _lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline1321 $\nu$ & \np{rn _nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline1322 $\mu_1$ & \np{rn _mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline1323 $\mu_2$ & \np{rn _mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline1297 $a_0$ & \np{rn\_a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline 1298 $b_0$ & \np{rn\_b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline 1299 $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline 1300 $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline 1301 $\nu$ & \np{rn\_nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline 1302 $\mu_1$ & \np{rn\_mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline 1303 $\mu_2$ & \np{rn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline 1324 1304 \end{tabular} 1325 1305 \caption{ \protect\label{Tab_SEOS} … … 1333 1313 % Brunt-V\"{a}is\"{a}l\"{a} Frequency 1334 1314 % ------------------------------------------------------------------------------------------------------------- 1335 \subsection{Brunt-V\"{a}is\"{a}l\"{a} Frequency (\protect\np{nn_eos} = 0, 1 or 2)}1315 \subsection{Brunt-V\"{a}is\"{a}l\"{a} frequency (\protect\np{nn\_eos}\forcode{ = 0..2})} 1336 1316 \label{TRA_bn2} 1337 1317 … … 1355 1335 % Freezing Point of Seawater 1356 1336 % ------------------------------------------------------------------------------------------------------------- 1357 \subsection [Freezing Point of Seawater] 1358 {Freezing Point of Seawater} 1337 \subsection{Freezing point of seawater} 1359 1338 \label{TRA_fzp} 1360 1339 … … 1388 1367 % Horizontal Derivative in zps-coordinate 1389 1368 % ================================================================ 1390 \section [Horizontal Derivative in \textit{zps}-coordinate (\textit{zpshde})] 1391 {Horizontal Derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})} 1369 \section{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})} 1392 1370 \label{TRA_zpshde} 1393 1371 … … 1395 1373 I've changed "derivative" to "difference" and "mean" to "average"} 1396 1374 1397 With partial cells (\ forcode{ln_zps = .true.}) at bottom and top (\forcode{ln_isfcav= .true.}), in general,1375 With partial cells (\np{ln\_zps}\forcode{ = .true.}) at bottom and top (\np{ln\_isfcav}\forcode{ = .true.}), in general, 1398 1376 tracers in horizontally adjacent cells live at different depths. 1399 1377 Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) 1400 1378 and the hydrostatic pressure gradient calculations (\mdl{dynhpg} module). 1401 The partial cell properties at the top (\ forcode{ln_isfcav= .true.}) are computed in the same way as for the bottom.1379 The partial cell properties at the top (\np{ln\_isfcav}\forcode{ = .true.}) are computed in the same way as for the bottom. 1402 1380 So, only the bottom interpolation is explained below. 1403 1381 … … 1413 1391 \caption{ \protect\label{Fig_Partial_step_scheme} 1414 1392 Discretisation of the horizontal difference and average of tracers in the $z$-partial 1415 step coordinate (\ protect\forcode{ln_zps= .true.}) in the case $( e3w_k^{i+1} - e3w_k^i )>0$.1393 step coordinate (\np{ln\_zps}\forcode{ = .true.}) in the case $( e3w_k^{i+1} - e3w_k^i )>0$. 1416 1394 A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value 1417 1395 at the depth of the shallower tracer point of the two adjacent bottom $T$-points.
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