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r9394 r9407 5 5 % ================================================================ 6 6 \chapter{Ocean Tracers (TRA)} 7 \label{ TRA}7 \label{chap:TRA} 8 8 \minitoc 9 9 … … 17 17 %$\ $\newline % force a new ligne 18 18 19 Using the representation described in Chap.~\ref{DOM}, several semi-discrete19 Using the representation described in \autoref{chap:DOM}, several semi-discrete 20 20 space forms of the tracer equations are available depending on the vertical 21 21 coordinate used and on the physics used. In all the equations presented … … 40 40 require complex inputs and complex calculations ($e.g.$ bulk formulae, estimation 41 41 of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and 42 described in chapters \S\ref{SBC}, \S\ref{LDF} and \S\ref{ZDF}, respectively.42 described in \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively. 43 43 Note that \mdl{tranpc}, the non-penetrative convection module, although 44 44 located in the NEMO/OPA/TRA directory as it directly modifies the tracer fields, … … 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}\forcode{ = .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 \autoref{chap:DIA}. 60 60 61 61 $\ $\newline % force a new ligne … … 64 64 % ================================================================ 65 65 \section{Tracer advection (\protect\mdl{traadv})} 66 \label{ TRA_adv}66 \label{sec:TRA_adv} 67 67 %------------------------------------------namtra_adv----------------------------------------------------- 68 68 \forfile{../namelists/namtra_adv} … … 72 72 the advection tendency of a tracer is expressed in flux form, 73 73 $i.e.$ as the divergence of the advective fluxes. Its discrete expression is given by : 74 \begin{equation} \label{ Eq_tra_adv}74 \begin{equation} \label{eq:tra_adv} 75 75 ADV_\tau =-\frac{1}{b_t} \left( 76 76 \;\delta _i \left[ e_{2u}\,e_{3u} \; u\; \tau _u \right] … … 79 79 \end{equation} 80 80 where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. 81 The flux form in \ eqref{Eq_tra_adv}81 The flux form in \autoref{eq:tra_adv} 82 82 implicitly requires the use of the continuity equation. Indeed, it is obtained 83 83 by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ … … 87 87 advection tendency so that it is consistent with the continuity equation in order to 88 88 enforce the conservation properties of the continuous equations. In other words, 89 by setting $\tau = 1$ in (\ ref{Eq_tra_adv}) we recover the discrete form of89 by setting $\tau = 1$ in (\autoref{eq:tra_adv}) we recover the discrete form of 90 90 the continuity equation which is used to calculate the vertical velocity. 91 91 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 92 92 \begin{figure}[!t] \begin{center} 93 93 \includegraphics[width=0.9\textwidth]{Fig_adv_scheme} 94 \caption{ \protect\label{ Fig_adv_scheme}94 \caption{ \protect\label{fig:adv_scheme} 95 95 Schematic representation of some ways used to evaluate the tracer value 96 96 at $u$-point and the amount of tracer exchanged between two neighbouring grid … … 107 107 The key difference between the advection schemes available in \NEMO is the choice 108 108 made in space and time interpolation to define the value of the tracer at the 109 velocity points ( Fig.~\ref{Fig_adv_scheme}).109 velocity points (\autoref{fig:adv_scheme}). 110 110 111 111 Along solid lateral and bottom boundaries a zero tracer flux is automatically … … 131 131 height, two quantities that are not correlated \citep{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}. 132 132 133 The velocity field that appears in (\ ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco})133 The velocity field that appears in (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_zco}) 134 134 is the centred (\textit{now}) \textit{effective} ocean velocity, $i.e.$ the \textit{eulerian} velocity 135 (see Chap.~\ref{DYN}) plus the eddy induced velocity (\textit{eiv})135 (see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) 136 136 and/or the mixed layer eddy induced velocity (\textit{eiv}) 137 when those parameterisations are used (see Chap.~\ref{LDF}).137 when those parameterisations are used (see \autoref{chap:LDF}). 138 138 139 139 Several tracer advection scheme are proposed, namely … … 174 174 % ------------------------------------------------------------------------------------------------------------- 175 175 \subsection{CEN: Centred scheme (\protect\np{ln\_traadv\_cen}\forcode{ = .true.})} 176 \label{ TRA_adv_cen}176 \label{subsec:TRA_adv_cen} 177 177 178 178 % 2nd order centred scheme … … 186 186 is evaluated as the mean of the two neighbouring $T$-point values. 187 187 For example, in the $i$-direction : 188 \begin{equation} \label{ Eq_tra_adv_cen2}188 \begin{equation} \label{eq:tra_adv_cen2} 189 189 \tau _u^{cen2} =\overline T ^{i+1/2} 190 190 \end{equation} … … 195 195 produce a sensible solution. The associated time-stepping is performed using 196 196 a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in 197 (\ ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value.197 (\autoref{eq:tra_adv_cen2}) is the \textit{now} tracer value. 198 198 199 199 Note that using the CEN2, the overall tracer advection is of second 200 order accuracy since both (\ ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2})200 order accuracy since both (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_cen2}) 201 201 have this order of accuracy. 202 202 … … 206 206 a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 207 207 For example, in the $i$-direction: 208 \begin{equation} \label{ Eq_tra_adv_cen4}208 \begin{equation} \label{eq:tra_adv_cen4} 209 209 \tau _u^{cen4} 210 210 =\overline{ T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} … … 219 219 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme 220 220 but a $4^{th}$ order evaluation of advective fluxes, since the divergence of 221 advective fluxes \ eqref{Eq_tra_adv} is kept at $2^{nd}$ order.221 advective fluxes \autoref{eq:tra_adv} is kept at $2^{nd}$ order. 222 222 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature 223 223 is usually associated with the scheme presented here. … … 232 232 to produce a sensible solution. 233 233 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction 234 with an Asselin time-filter, so $T$ in (\ ref{Eq_tra_adv_cen4}) is the \textit{now} tracer.234 with an Asselin time-filter, so $T$ in (\autoref{eq:tra_adv_cen4}) is the \textit{now} tracer. 235 235 236 236 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), … … 245 245 % ------------------------------------------------------------------------------------------------------------- 246 246 \subsection{FCT: Flux Corrected Transport scheme (\protect\np{ln\_traadv\_fct}\forcode{ = .true.})} 247 \label{ TRA_adv_tvd}247 \label{subsec:TRA_adv_tvd} 248 248 249 249 The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}\forcode{ = .true.}. … … 254 254 In FCT formulation, the tracer at velocity points is evaluated using a combination of 255 255 an upstream and a centred scheme. For example, in the $i$-direction : 256 \begin{equation} \label{ Eq_tra_adv_fct}256 \begin{equation} \label{eq:tra_adv_fct} 257 257 \begin{split} 258 258 \tau _u^{ups}&= \begin{cases} … … 280 280 by vertical advection \citep{Lemarie_OM2015}. Note that in this case, a similar split-explicit 281 281 time stepping should be used on vertical advection of momentum to insure a better stability 282 (see \ S\ref{DYN_zad}).283 284 For stability reasons (see \ S\ref{STP}), $\tau _u^{cen}$ is evaluated in (\ref{Eq_tra_adv_fct})282 (see \autoref{subsec:DYN_zad}). 283 284 For stability reasons (see \autoref{chap:STP}), $\tau _u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) 285 285 using the \textit{now} tracer while $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. In other words, 286 286 the advective part of the scheme is time stepped with a leap-frog scheme … … 291 291 % ------------------------------------------------------------------------------------------------------------- 292 292 \subsection{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln\_traadv\_mus}\forcode{ = .true.})} 293 \label{ TRA_adv_mus}293 \label{subsec:TRA_adv_mus} 294 294 295 295 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}\forcode{ = .true.}. … … 298 298 MUSCL has been first implemented in \NEMO by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points 299 299 is evaluated assuming a linear tracer variation between two $T$-points 300 ( Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction :301 \begin{equation} \label{ Eq_tra_adv_mus}300 (\autoref{fig:adv_scheme}). For example, in the $i$-direction : 301 \begin{equation} \label{eq:tra_adv_mus} 302 302 \tau _u^{mus} = \left\{ \begin{aligned} 303 303 &\tau _i &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) … … 323 323 % ------------------------------------------------------------------------------------------------------------- 324 324 \subsection{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} 325 \label{ TRA_adv_ubs}325 \label{subsec:TRA_adv_ubs} 326 326 327 327 The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}\forcode{ = .true.}. … … 332 332 third order scheme based on an upstream-biased parabolic interpolation. 333 333 For example, in the $i$-direction : 334 \begin{equation} \label{ Eq_tra_adv_ubs}334 \begin{equation} \label{eq:tra_adv_ubs} 335 335 \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ 336 336 \begin{aligned} … … 355 355 or a $4^th$ order COMPACT scheme (\np{nn\_cen\_v}\forcode{ = 2 or 4}). 356 356 357 For stability reasons (see \ S\ref{STP}),358 the first term in \ eqref{Eq_tra_adv_ubs} (which corresponds to a second order357 For stability reasons (see \autoref{chap:STP}), 358 the first term in \autoref{eq:tra_adv_ubs} (which corresponds to a second order 359 359 centred scheme) is evaluated using the \textit{now} tracer (centred in time) 360 360 while the second term (which is the diffusive part of the scheme), is … … 362 362 This choice is discussed by \citet{Webb_al_JAOT98} in the context of the 363 363 QUICK advection scheme. UBS and QUICK schemes only differ 364 by one coefficient. Replacing 1/6 with 1/8 in \ eqref{Eq_tra_adv_ubs}364 by one coefficient. Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} 365 365 leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. 366 366 This option is not available through a namelist parameter, since the … … 368 368 substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 369 369 370 Note that it is straightforward to rewrite \ eqref{Eq_tra_adv_ubs} as follows:371 \begin{equation} \label{ Eq_traadv_ubs2}370 Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows: 371 \begin{equation} \label{eq:traadv_ubs2} 372 372 \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{ 373 373 \begin{aligned} … … 377 377 \end{equation} 378 378 or equivalently 379 \begin{equation} \label{ Eq_traadv_ubs2b}379 \begin{equation} \label{eq:traadv_ubs2b} 380 380 u_{i+1/2} \ \tau _u^{ubs} 381 381 =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} … … 383 383 \end{equation} 384 384 385 \ eqref{Eq_traadv_ubs2} has several advantages. Firstly, it clearly reveals385 \autoref{eq:traadv_ubs2} has several advantages. Firstly, it clearly reveals 386 386 that the UBS scheme is based on the fourth order scheme to which an 387 387 upstream-biased diffusion term is added. Secondly, this emphasises that the 388 388 $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has 389 to be evaluated at the \emph{now} time step using \ eqref{Eq_tra_adv_ubs}.389 to be evaluated at the \emph{now} time step using \autoref{eq:tra_adv_ubs}. 390 390 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy 391 391 coefficient which is simply proportional to the velocity: 392 392 $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note the current version of NEMO uses 393 the computationally more efficient formulation \ eqref{Eq_tra_adv_ubs}.393 the computationally more efficient formulation \autoref{eq:tra_adv_ubs}. 394 394 395 395 % ------------------------------------------------------------------------------------------------------------- … … 397 397 % ------------------------------------------------------------------------------------------------------------- 398 398 \subsection{QCK: QuiCKest scheme (\protect\np{ln\_traadv\_qck}\forcode{ = .true.})} 399 \label{ TRA_adv_qck}399 \label{subsec:TRA_adv_qck} 400 400 401 401 The Quadratic Upstream Interpolation for Convective Kinematics with … … 423 423 % ================================================================ 424 424 \section{Tracer lateral diffusion (\protect\mdl{traldf})} 425 \label{ TRA_ldf}425 \label{sec:TRA_ldf} 426 426 %-----------------------------------------nam_traldf------------------------------------------------------ 427 427 \forfile{../namelists/namtra_ldf} … … 434 434 $(iii)$ some specific options related to the rotated operators ($i.e.$ non-iso-level operator), and 435 435 $(iv)$ the specification of eddy diffusivity coefficient (either constant or variable in space and time). 436 Item $(iv)$ will be described in Chap.\ref{LDF} .436 Item $(iv)$ will be described in \autoref{chap:LDF} . 437 437 The direction along which the operators act is defined through the slope between this direction and the iso-level surfaces. 438 The slope is computed in the \mdl{ldfslp} module and will also be described in Chap.~\ref{LDF}.438 The slope is computed in the \mdl{ldfslp} module and will also be described in \autoref{chap:LDF}. 439 439 440 440 The lateral diffusion of tracers is evaluated using a forward scheme, 441 441 $i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time, 442 442 except for the pure vertical component that appears when a rotation tensor is used. 443 This latter component is solved implicitly together with the vertical diffusion term (see \ S\ref{STP}).443 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). 444 444 When \np{ln\_traldf\_msc}\forcode{ = .true.}, a Method of Stabilizing Correction is used in which 445 445 the pure vertical component is split into an explicit and an implicit part \citep{Lemarie_OM2012}. … … 450 450 \subsection[Type of operator (\protect\np{ln\_traldf}\{\_NONE,\_lap,\_blp\}\})] 451 451 {Type of operator (\protect\np{ln\_traldf\_NONE}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) } 452 \label{ TRA_ldf_op}452 \label{subsec:TRA_ldf_op} 453 453 454 454 Three operator options are proposed and, one and only one of them must be selected: … … 459 459 \item [\np{ln\_traldf\_lap}\forcode{ = .true.}]: a laplacian operator is selected. This harmonic operator 460 460 takes the following expression: $\mathpzc{L}(T)=\nabla \cdot A_{ht}\;\nabla T $, 461 where the gradient operates along the selected direction (see \ S\ref{TRA_ldf_dir}),462 and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see Chap.~\ref{LDF}).461 where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}), 462 and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}). 463 463 \item [\np{ln\_traldf\_blp}\forcode{ = .true.}]: a bilaplacian operator is selected. This biharmonic operator 464 464 takes the following expression: 465 465 $\mathpzc{B}=- \mathpzc{L}\left(\mathpzc{L}(T) \right) = -\nabla \cdot b\nabla \left( {\nabla \cdot b\nabla T} \right)$ 466 466 where the gradient operats along the selected direction, 467 and $b^2=B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see Chap.~\ref{LDF}).467 and $b^2=B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see \autoref{chap:LDF}). 468 468 In the code, the bilaplacian operator is obtained by calling the laplacian twice. 469 469 \end{description} … … 483 483 \subsection[Action direction (\protect\np{ln\_traldf}\{\_lev,\_hor,\_iso,\_triad\})] 484 484 {Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) } 485 \label{ TRA_ldf_dir}485 \label{subsec:TRA_ldf_dir} 486 486 487 487 The choice of a direction of action determines the form of operator used. … … 509 509 % ------------------------------------------------------------------------------------------------------------- 510 510 \subsection{Iso-level (bi-)laplacian operator ( \protect\np{ln\_traldf\_iso}) } 511 \label{ TRA_ldf_lev}511 \label{subsec:TRA_ldf_lev} 512 512 513 513 The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 514 \begin{equation} \label{ Eq_tra_ldf_lap}514 \begin{equation} \label{eq:tra_ldf_lap} 515 515 D_t^{lT} =\frac{1}{b_t} \left( \; 516 516 \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right] … … 533 533 Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}), tracers in horizontally 534 534 adjacent cells are located at different depths in the vicinity of the bottom. 535 In this case, horizontal derivatives in (\ ref{Eq_tra_ldf_lap}) at the bottom level535 In this case, horizontal derivatives in (\autoref{eq:tra_ldf_lap}) at the bottom level 536 536 require a specific treatment. They are calculated in the \mdl{zpshde} module, 537 described in \ S\ref{TRA_zpshde}.537 described in \autoref{sec:TRA_zpshde}. 538 538 539 539 … … 542 542 % ------------------------------------------------------------------------------------------------------------- 543 543 \subsection{Standard and triad (bi-)laplacian operator} 544 \label{ TRA_ldf_iso_triad}544 \label{subsec:TRA_ldf_iso_triad} 545 545 546 546 %&& Standard rotated (bi-)laplacian operator 547 547 %&& ---------------------------------------------- 548 548 \subsubsection{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 549 \label{ TRA_ldf_iso}549 \label{subsec:TRA_ldf_iso} 550 550 The general form of the second order lateral tracer subgrid scale physics 551 (\ ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and $s$-coordinates:552 \begin{equation} \label{ Eq_tra_ldf_iso}551 (\autoref{eq:PE_zdf}) takes the following semi-discrete space form in $z$- and $s$-coordinates: 552 \begin{equation} \label{eq:tra_ldf_iso} 553 553 \begin{split} 554 554 D_T^{lT} = \frac{1}{b_t} & \left\{ \,\;\delta_i \left[ A_u^{lT} \left( … … 576 576 in addition to \np{ln\_traldf\_lap}\forcode{ = .true.}, we have \np{ln\_traldf\_iso}\forcode{ = .true.}, 577 577 or both \np{ln\_traldf\_hor}\forcode{ = .true.} and \np{ln\_zco}\forcode{ = .true.}. The way these 578 slopes are evaluated is given in \ S\ref{LDF_slp}. At the surface, bottom578 slopes are evaluated is given in \autoref{sec:LDF_slp}. At the surface, bottom 579 579 and lateral boundaries, the turbulent fluxes of heat and salt are set to zero 580 using the mask technique (see \ S\ref{LBC_coast}).581 582 The operator in \ eqref{Eq_tra_ldf_iso} involves both lateral and vertical580 using the mask technique (see \autoref{sec:LBC_coast}). 581 582 The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical 583 583 derivatives. For numerical stability, the vertical second derivative must 584 584 be solved using the same implicit time scheme as that used in the vertical 585 physics (see \ S\ref{TRA_zdf}). For computer efficiency reasons, this term585 physics (see \autoref{sec:TRA_zdf}). For computer efficiency reasons, this term 586 586 is not computed in the \mdl{traldf\_iso} module, but in the \mdl{trazdf} module 587 587 where, if iso-neutral mixing is used, the vertical mixing coefficient is simply … … 590 590 This formulation conserves the tracer but does not ensure the decrease 591 591 of the tracer variance. Nevertheless the treatment performed on the slopes 592 (see \ S\ref{LDF}) allows the model to run safely without any additional592 (see \autoref{chap:LDF}) allows the model to run safely without any additional 593 593 background horizontal diffusion \citep{Guilyardi_al_CD01}. 594 594 595 595 Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}), the horizontal derivatives 596 at the bottom level in \ eqref{Eq_tra_ldf_iso} require a specific treatment.597 They are calculated in module zpshde, described in \ S\ref{TRA_zpshde}.596 at the bottom level in \autoref{eq:tra_ldf_iso} require a specific treatment. 597 They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}. 598 598 599 599 %&& Triad rotated (bi-)laplacian operator 600 600 %&& ------------------------------------------- 601 601 \subsubsection{Triad rotated (bi-)laplacian operator (\protect\np{ln\_traldf\_triad})} 602 \label{ TRA_ldf_triad}603 604 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}\forcode{ = .true.} ; see App.\ref{sec:triad})602 \label{subsec:TRA_ldf_triad} 603 604 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}\forcode{ = .true.} ; see \autoref{apdx:triad}) 605 605 606 606 An alternative scheme developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases 607 607 is also available in \NEMO (\np{ln\_traldf\_grif}\forcode{ = .true.}). A complete description of 608 the algorithm is given in App.\ref{sec:triad}.608 the algorithm is given in \autoref{apdx:triad}. 609 609 610 610 The lateral fourth order bilaplacian operator on tracers is obtained by 611 applying (\ ref{Eq_tra_ldf_lap}) twice. The operator requires an additional assumption611 applying (\autoref{eq:tra_ldf_lap}) twice. The operator requires an additional assumption 612 612 on boundary conditions: both first and third derivative terms normal to the 613 613 coast are set to zero. 614 614 615 615 The lateral fourth order operator formulation on tracers is obtained by 616 applying (\ ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption616 applying (\autoref{eq:tra_ldf_iso}) twice. It requires an additional assumption 617 617 on boundary conditions: first and third derivative terms normal to the 618 618 coast, normal to the bottom and normal to the surface are set to zero. … … 621 621 %&& ---------------------------------------------- 622 622 \subsubsection{Option for the rotated operators} 623 \label{ TRA_ldf_options}623 \label{subsec:TRA_ldf_options} 624 624 625 625 \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) … … 637 637 % ================================================================ 638 638 \section{Tracer vertical diffusion (\protect\mdl{trazdf})} 639 \label{ TRA_zdf}639 \label{sec:TRA_zdf} 640 640 %--------------------------------------------namzdf--------------------------------------------------------- 641 641 \forfile{../namelists/namzdf} … … 645 645 The formulation of the vertical subgrid scale tracer physics is the same 646 646 for all the vertical coordinates, and is based on a laplacian operator. 647 The vertical diffusion operator given by (\ ref{Eq_PE_zdf}) takes the647 The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the 648 648 following semi-discrete space form: 649 \begin{equation} \label{ Eq_tra_zdf}649 \begin{equation} \label{eq:tra_zdf} 650 650 \begin{split} 651 651 D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}} \delta_{k+1/2}[T] \;\right] … … 658 658 $A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is 659 659 parameterised ($i.e.$ \key{zdfddm} is defined). The way these coefficients 660 are evaluated is given in \ S\ref{ZDF} (ZDF). Furthermore, when660 are evaluated is given in \autoref{chap:ZDF} (ZDF). Furthermore, when 661 661 iso-neutral mixing is used, both mixing coefficients are increased 662 662 by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ 663 to account for the vertical second derivative of \ eqref{Eq_tra_ldf_iso}.663 to account for the vertical second derivative of \autoref{eq:tra_ldf_iso}. 664 664 665 665 At the surface and bottom boundaries, the turbulent fluxes of 666 666 heat and salt must be specified. At the surface they are prescribed 667 from the surface forcing and added in a dedicated routine (see \ S\ref{TRA_sbc}),667 from the surface forcing and added in a dedicated routine (see \autoref{subsec:TRA_sbc}), 668 668 whilst at the bottom they are set to zero for heat and salt unless 669 669 a geothermal flux forcing is prescribed as a bottom boundary 670 condition (see \ S\ref{TRA_bbc}).670 condition (see \autoref{subsec:TRA_bbc}). 671 671 672 672 The large eddy coefficient found in the mixed layer together with high … … 684 684 % ================================================================ 685 685 \section{External forcing} 686 \label{ TRA_sbc_qsr_bbc}686 \label{sec:TRA_sbc_qsr_bbc} 687 687 688 688 % ------------------------------------------------------------------------------------------------------------- … … 690 690 % ------------------------------------------------------------------------------------------------------------- 691 691 \subsection{Surface boundary condition (\protect\mdl{trasbc})} 692 \label{ TRA_sbc}692 \label{subsec:TRA_sbc} 693 693 694 694 The surface boundary condition for tracers is implemented in a separate … … 703 703 of the ocean is due both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) 704 704 and to the heat and salt content of the mass exchange. They are both included directly in $Q_{ns}$, 705 the surface heat flux, and $F_{salt}$, the surface salt flux (see \ S\ref{SBC} for further details).705 the surface heat flux, and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details). 706 706 By doing this, the forcing formulation is the same for any tracer (including temperature and salinity). 707 707 708 The surface module (\mdl{sbcmod}, see \ S\ref{SBC}) provides the following708 The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following 709 709 forcing fields (used on tracers): 710 710 711 711 $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 712 712 (i.e. the difference between the total surface heat flux and the fraction of the short wave flux that 713 penetrates into the water column, see \ S\ref{TRA_qsr}) plus the heat content associated with713 penetrates into the water column, see \autoref{subsec:TRA_qsr}) plus the heat content associated with 714 714 of the mass exchange with the atmosphere and lands. 715 715 … … 720 720 721 721 $\bullet$ \textit{rnf}, the mass flux associated with runoff 722 (see \ S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)722 (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 723 723 724 724 $\bullet$ \textit{fwfisf}, the mass flux associated with ice shelf melt, 725 (see \ S\ref{SBC_isf} for further details on how the ice shelf melt is computed and applied).725 (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied). 726 726 727 727 The surface boundary condition on temperature and salinity is applied as follows: 728 \begin{equation} \label{ Eq_tra_sbc}728 \begin{equation} \label{eq:tra_sbc} 729 729 \begin{aligned} 730 730 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ … … 734 734 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps 735 735 ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the 736 divergence of odd and even time step (see \ S\ref{STP}).736 divergence of odd and even time step (see \autoref{chap:STP}). 737 737 738 738 In the linear free surface case (\np{ln\_linssh}\forcode{ = .true.}), … … 742 742 would have resulted from a change in the volume of the first level. 743 743 The resulting surface boundary condition is applied as follows: 744 \begin{equation} \label{ Eq_tra_sbc_lin}744 \begin{equation} \label{eq:tra_sbc_lin} 745 745 \begin{aligned} 746 746 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } … … 754 754 In the linear free surface case, there is a small imbalance. The imbalance is larger 755 755 than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. 756 This is the reason why the modified filter is not applied in the linear free surface case (see \ S\ref{STP}).756 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:STP}). 757 757 758 758 % ------------------------------------------------------------------------------------------------------------- … … 760 760 % ------------------------------------------------------------------------------------------------------------- 761 761 \subsection{Solar radiation penetration (\protect\mdl{traqsr})} 762 \label{ TRA_qsr}762 \label{subsec:TRA_qsr} 763 763 %--------------------------------------------namqsr-------------------------------------------------------- 764 764 \forfile{../namelists/namtra_qsr} 765 765 %-------------------------------------------------------------------------------------------------------------- 766 766 767 Options are defined through the 767 Options are defined through the \ngn{namtra\_qsr} namelist variables. 768 768 When the penetrative solar radiation option is used (\np{ln\_flxqsr}\forcode{ = .true.}), 769 769 the solar radiation penetrates the top few tens of meters of the ocean. If it is not used 770 770 (\np{ln\_flxqsr}\forcode{ = .false.}) all the heat flux is absorbed in the first ocean level. 771 771 Thus, in the former case a term is added to the time evolution equation of 772 temperature \ eqref{Eq_PE_tra_T} and the surface boundary condition is772 temperature \autoref{eq:PE_tra_T} and the surface boundary condition is 773 773 modified to take into account only the non-penetrative part of the surface 774 774 heat flux: 775 \begin{equation} \label{ Eq_PE_qsr}775 \begin{equation} \label{eq:PE_qsr} 776 776 \begin{split} 777 777 \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \\ … … 781 781 where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) 782 782 and $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). 783 The additional term in \ eqref{Eq_PE_qsr} is discretized as follows:784 \begin{equation} \label{ Eq_tra_qsr}783 The additional term in \autoref{eq:PE_qsr} is discretized as follows: 784 \begin{equation} \label{eq:tra_qsr} 785 785 \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3t}} \delta_k \left[ I_w \right] 786 786 \end{equation} 787 787 788 The shortwave radiation, 788 The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. 789 789 The ocean is strongly absorbing for wavelengths longer than 700~nm and these 790 790 wavelengths contribute to heating the upper few tens of centimetres. The fraction of $Q_{sr}$ 791 791 that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified 792 through namelist parameter \np{rn\_abs}). 792 through namelist parameter \np{rn\_abs}). It is assumed to penetrate the ocean 793 793 with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 794 of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsrnamelist).794 of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the \ngn{namtra\_qsr} namelist). 795 795 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy 796 796 propagates to larger depths where it contributes to 797 797 local heating. 798 798 The way this second part of the solar energy penetrates into the ocean depends on 799 which formulation is chosen. In the simple 2-waveband light penetration scheme 799 which formulation is chosen. In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}\forcode{ = .true.}) 800 800 a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 801 leading to the following expression 802 \begin{equation} \label{ Eq_traqsr_iradiance}801 leading to the following expression \citep{Paulson1977}: 802 \begin{equation} \label{eq:traqsr_iradiance} 803 803 I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] 804 804 \end{equation} … … 810 810 Such assumptions have been shown to provide a very crude and simplistic 811 811 representation of observed light penetration profiles (\cite{Morel_JGR88}, see also 812 Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on812 \autoref{fig:traqsr_irradiance}). Light absorption in the ocean depends on 813 813 particle concentration and is spectrally selective. \cite{Morel_JGR88} has shown 814 814 that an accurate representation of light penetration can be provided by a 61 waveband … … 819 819 attenuation coefficient is fitted to the coefficients computed from the full spectral model 820 820 of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}), assuming 821 the same power-law relationship. As shown in Fig.\ref{Fig_traqsr_irradiance},821 the same power-law relationship. As shown in \autoref{fig:traqsr_irradiance}, 822 822 this formulation, called RGB (Red-Green-Blue), reproduces quite closely 823 823 the light penetration profiles predicted by the full spectal model, but with much greater … … 842 842 light limitation in PISCES or LOBSTER and the oceanic heating rate. 843 843 \end{description} 844 The trend in \ eqref{Eq_tra_qsr} associated with the penetration of the solar radiation844 The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation 845 845 is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. 846 846 … … 857 857 \begin{figure}[!t] \begin{center} 858 858 \includegraphics[width=1.0\textwidth]{Fig_TRA_Irradiance} 859 \caption{ \protect\label{ Fig_traqsr_irradiance}859 \caption{ \protect\label{fig:traqsr_irradiance} 860 860 Penetration profile of the downward solar irradiance calculated by four models. 861 861 Two waveband chlorophyll-independent formulation (blue), a chlorophyll-dependent … … 870 870 % ------------------------------------------------------------------------------------------------------------- 871 871 \subsection{Bottom boundary condition (\protect\mdl{trabbc})} 872 \label{ TRA_bbc}872 \label{subsec:TRA_bbc} 873 873 %--------------------------------------------nambbc-------------------------------------------------------- 874 874 \forfile{../namelists/nambbc} … … 877 877 \begin{figure}[!t] \begin{center} 878 878 \includegraphics[width=1.0\textwidth]{Fig_TRA_geoth} 879 \caption{ \protect\label{ Fig_geothermal}879 \caption{ \protect\label{fig:geothermal} 880 880 Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}. 881 881 It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.} … … 902 902 When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is 903 903 introduced which is provided in the \ifile{geothermal\_heating} NetCDF file 904 ( Fig.\ref{Fig_geothermal}) \citep{Emile-Geay_Madec_OS09}.904 (\autoref{fig:geothermal}) \citep{Emile-Geay_Madec_OS09}. 905 905 906 906 % ================================================================ … … 908 908 % ================================================================ 909 909 \section{Bottom boundary layer (\protect\mdl{trabbl} - \protect\key{trabbl})} 910 \label{ TRA_bbl}910 \label{sec:TRA_bbl} 911 911 %--------------------------------------------nambbl--------------------------------------------------------- 912 912 \forfile{../namelists/nambbl} … … 943 943 % ------------------------------------------------------------------------------------------------------------- 944 944 \subsection{Diffusive bottom boundary layer (\protect\np{nn\_bbl\_ldf}\forcode{ = 1})} 945 \label{ TRA_bbl_diff}945 \label{subsec:TRA_bbl_diff} 946 946 947 947 When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), 948 948 the diffusive flux between two adjacent cells at the ocean floor is given by 949 \begin{equation} \label{ Eq_tra_bbl_diff}949 \begin{equation} \label{eq:tra_bbl_diff} 950 950 {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T 951 951 \end{equation} … … 953 953 and $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{Beckmann_Doscher1997}, 954 954 the latter is prescribed with a spatial dependence, $i.e.$ in the conditional form 955 \begin{equation} \label{ Eq_tra_bbl_coef}955 \begin{equation} \label{eq:tra_bbl_coef} 956 956 A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l} 957 957 A_{bbl} \quad \quad \mbox{if} \quad \nabla_\sigma \rho \cdot \nabla H<0 \\ … … 962 962 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist 963 963 parameter \np{rn\_ahtbbl} and usually set to a value much larger 964 than the one used for lateral mixing in the open ocean. The constraint in \ eqref{Eq_tra_bbl_coef}964 than the one used for lateral mixing in the open ocean. The constraint in \autoref{eq:tra_bbl_coef} 965 965 implies that sigma-like diffusion only occurs when the density above the sea floor, at the top of 966 the slope, is larger than in the deeper ocean (see green arrow in Fig.\ref{Fig_bbl}).966 the slope, is larger than in the deeper ocean (see green arrow in \autoref{fig:bbl}). 967 967 In practice, this constraint is applied separately in the two horizontal directions, 968 and the density gradient in \ eqref{Eq_tra_bbl_coef} is evaluated with the log gradient formulation:969 \begin{equation} \label{ Eq_tra_bbl_Drho}968 and the density gradient in \autoref{eq:tra_bbl_coef} is evaluated with the log gradient formulation: 969 \begin{equation} \label{eq:tra_bbl_Drho} 970 970 \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta \,\nabla_\sigma S 971 971 \end{equation} … … 978 978 % ------------------------------------------------------------------------------------------------------------- 979 979 \subsection{Advective bottom boundary layer (\protect\np{nn\_bbl\_adv}\forcode{ = 1..2})} 980 \label{ TRA_bbl_adv}980 \label{subsec:TRA_bbl_adv} 981 981 982 982 \sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following … … 986 986 \begin{figure}[!t] \begin{center} 987 987 \includegraphics[width=0.7\textwidth]{Fig_BBL_adv} 988 \caption{ \protect\label{ Fig_bbl}988 \caption{ \protect\label{fig:bbl} 989 989 Advective/diffusive Bottom Boundary Layer. The BBL parameterisation is 990 990 activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$. … … 1011 1011 1012 1012 \np{nn\_bbl\_adv}\forcode{ = 1} : the downslope velocity is chosen to be the Eulerian 1013 ocean velocity just above the topographic step (see black arrow in Fig.\ref{Fig_bbl})1013 ocean velocity just above the topographic step (see black arrow in \autoref{fig:bbl}) 1014 1014 \citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection 1015 1015 is allowed only if dense water overlies less dense water on the slope ($i.e.$ … … 1021 1021 The advection is allowed only if dense water overlies less dense water on the slope ($i.e.$ 1022 1022 $\nabla_\sigma \rho \cdot \nabla H<0$). For example, the resulting transport of the 1023 downslope flow, here in the $i$-direction ( Fig.\ref{Fig_bbl}), is simply given by the1023 downslope flow, here in the $i$-direction (\autoref{fig:bbl}), is simply given by the 1024 1024 following expression: 1025 \begin{equation} \label{ Eq_bbl_Utr}1025 \begin{equation} \label{eq:bbl_Utr} 1026 1026 u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o} e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right) 1027 1027 \end{equation} … … 1039 1039 water at intermediate depths. The entrainment is replaced by the vertical mixing 1040 1040 implicit in the advection scheme. Let us consider as an example the 1041 case displayed in Fig.\ref{Fig_bbl} where the density at level $(i,kup)$ is1041 case displayed in \autoref{fig:bbl} where the density at level $(i,kup)$ is 1042 1042 larger than the one at level $(i,kdwn)$. The advective BBL scheme 1043 1043 modifies the tracer time tendency of the ocean cells near the 1044 topographic step by the downslope flow \ eqref{Eq_bbl_dw},1045 the horizontal \ eqref{Eq_bbl_hor} and the upward \eqref{Eq_bbl_up}1044 topographic step by the downslope flow \autoref{eq:bbl_dw}, 1045 the horizontal \autoref{eq:bbl_hor} and the upward \autoref{eq:bbl_up} 1046 1046 return flows as follows: 1047 1047 \begin{align} 1048 1048 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 1049 + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} \left( T^{sh}_{kup} - T^{do}_{kdw} \right) \label{ Eq_bbl_dw} \\1049 + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} \left( T^{sh}_{kup} - T^{do}_{kdw} \right) \label{eq:bbl_dw} \\ 1050 1050 % 1051 1051 \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 1052 + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} \left( T^{do}_{kup} - T^{sh}_{kup} \right) \label{ Eq_bbl_hor} \\1052 + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} \left( T^{do}_{kup} - T^{sh}_{kup} \right) \label{eq:bbl_hor} \\ 1053 1053 % 1054 1054 \intertext{and for $k =kdw-1,\;..., \; kup$ :} 1055 1055 % 1056 1056 \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 1057 + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} \left( T^{do}_{k+1} - T^{sh}_{k} \right) \label{ Eq_bbl_up}1057 + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} \left( T^{do}_{k+1} - T^{sh}_{k} \right) \label{eq:bbl_up} 1058 1058 \end{align} 1059 1059 where $b_t$ is the $T$-cell volume. … … 1067 1067 % ================================================================ 1068 1068 \section{Tracer damping (\protect\mdl{tradmp})} 1069 \label{ TRA_dmp}1069 \label{sec:TRA_dmp} 1070 1070 %--------------------------------------------namtra_dmp------------------------------------------------- 1071 1071 \forfile{../namelists/namtra_dmp} … … 1074 1074 In some applications it can be useful to add a Newtonian damping term 1075 1075 into the temperature and salinity equations: 1076 \begin{equation} \label{ Eq_tra_dmp}1076 \begin{equation} \label{eq:tra_dmp} 1077 1077 \begin{split} 1078 1078 \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right) \\ … … 1087 1087 in \textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are 1088 1088 correctly set ($i.e.$ that $T_o$ and $S_o$ are provided in input files and read 1089 using \mdl{fldread}, see \ S\ref{SBC_fldread}).1089 using \mdl{fldread}, see \autoref{subsec:SBC_fldread}). 1090 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. 1091 1091 1092 The two main cases in which \ eqref{Eq_tra_dmp} is used are \textit{(a)}1092 The two main cases in which \autoref{eq:tra_dmp} is used are \textit{(a)} 1093 1093 the specification of the boundary conditions along artificial walls of a 1094 1094 limited domain basin and \textit{(b)} the computation of the velocity … … 1151 1151 % ================================================================ 1152 1152 \section{Tracer time evolution (\protect\mdl{tranxt})} 1153 \label{ TRA_nxt}1153 \label{sec:TRA_nxt} 1154 1154 %--------------------------------------------namdom----------------------------------------------------- 1155 1155 \forfile{../namelists/namdom} … … 1159 1159 The general framework for tracer time stepping is a modified leap-frog scheme 1160 1160 \citep{Leclair_Madec_OM09}, $i.e.$ a three level centred time scheme associated 1161 with a Asselin time filter (cf. \ S\ref{STP_mLF}):1162 \begin{equation} \label{ Eq_tra_nxt}1161 with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 1162 \begin{equation} \label{eq:tra_nxt} 1163 1163 \begin{aligned} 1164 1164 (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt \,e_{3t}^t\ \text{RHS}^t & \\ … … 1174 1174 $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 1175 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}.1176 is not applied in linear free surface (\jp{lk\_vvl}\forcode{ = .false.}) (see \autoref{subsec:TRA_sbc}. 1177 1177 Not also that in constant volume case, the time stepping is performed on $T$, 1178 1178 not on its content, $e_{3t}T$. … … 1189 1189 % ================================================================ 1190 1190 \section{Equation of state (\protect\mdl{eosbn2}) } 1191 \label{ TRA_eosbn2}1191 \label{sec:TRA_eosbn2} 1192 1192 %--------------------------------------------nameos----------------------------------------------------- 1193 1193 \forfile{../namelists/nameos} … … 1198 1198 % ------------------------------------------------------------------------------------------------------------- 1199 1199 \subsection{Equation of seawater (\protect\np{nn\_eos}\forcode{ = -1..1})} 1200 \label{ TRA_eos}1200 \label{subsec:TRA_eos} 1201 1201 1202 1202 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship … … 1272 1272 and \textit{practical} salinity. 1273 1273 S-EOS takes the following expression: 1274 \begin{equation} \label{ Eq_tra_S-EOS}1274 \begin{equation} \label{eq:tra_S-EOS} 1275 1275 \begin{split} 1276 1276 d_a(T,S,z) = ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a \\ … … 1280 1280 \end{split} 1281 1281 \end{equation} 1282 where the computer name of the coefficients as well as their standard value are given in \ ref{Tab_SEOS}.1282 where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}. 1283 1283 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing 1284 1284 the associated coefficients. … … 1303 1303 $\mu_2$ & \np{rn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline 1304 1304 \end{tabular} 1305 \caption{ \protect\label{ Tab_SEOS}1305 \caption{ \protect\label{tab:SEOS} 1306 1306 Standard value of S-EOS coefficients. } 1307 1307 \end{center} … … 1314 1314 % ------------------------------------------------------------------------------------------------------------- 1315 1315 \subsection{Brunt-V\"{a}is\"{a}l\"{a} frequency (\protect\np{nn\_eos}\forcode{ = 0..2})} 1316 \label{ TRA_bn2}1316 \label{subsec:TRA_bn2} 1317 1317 1318 1318 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} … … 1323 1323 (pressure in decibar being approximated by the depth in meters). The expression for $N^2$ 1324 1324 is given by: 1325 \begin{equation} \label{ Eq_tra_bn2}1325 \begin{equation} \label{eq:tra_bn2} 1326 1326 N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T] \right) 1327 1327 \end{equation} … … 1336 1336 % ------------------------------------------------------------------------------------------------------------- 1337 1337 \subsection{Freezing point of seawater} 1338 \label{ TRA_fzp}1338 \label{subsec:TRA_fzp} 1339 1339 1340 1340 The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}: 1341 \begin{equation} \label{ Eq_tra_eos_fzp}1341 \begin{equation} \label{eq:tra_eos_fzp} 1342 1342 \begin{split} 1343 1343 T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} … … 1347 1347 \end{equation} 1348 1348 1349 \ eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of1349 \autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of 1350 1350 sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent 1351 terms in \ eqref{Eq_tra_eos_fzp} (last term) have been dropped. The freezing1351 terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped. The freezing 1352 1352 point is computed through \textit{eos\_fzp}, a \textsc{Fortran} function that can be found 1353 1353 in \mdl{eosbn2}. … … 1358 1358 % ------------------------------------------------------------------------------------------------------------- 1359 1359 %\subsection{Potential Energy anomalies} 1360 %\label{ TRA_bn2}1360 %\label{subsec:TRA_bn2} 1361 1361 1362 1362 % =====>>>>> TO BE written … … 1368 1368 % ================================================================ 1369 1369 \section{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})} 1370 \label{ TRA_zpshde}1370 \label{sec:TRA_zpshde} 1371 1371 1372 1372 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, … … 1382 1382 Before taking horizontal gradients between the tracers next to the bottom, a linear 1383 1383 interpolation in the vertical is used to approximate the deeper tracer as if it actually 1384 lived at the depth of the shallower tracer point ( Fig.~\ref{Fig_Partial_step_scheme}).1384 lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}). 1385 1385 For example, for temperature in the $i$-direction the needed interpolated 1386 1386 temperature, $\widetilde{T}$, is: … … 1389 1389 \begin{figure}[!p] \begin{center} 1390 1390 \includegraphics[width=0.9\textwidth]{Partial_step_scheme} 1391 \caption{ \protect\label{ Fig_Partial_step_scheme}1391 \caption{ \protect\label{fig:Partial_step_scheme} 1392 1392 Discretisation of the horizontal difference and average of tracers in the $z$-partial 1393 step coordinate (\ np{ln\_zps}\forcode{ = .true.}) in the case $( e3w_k^{i+1} - e3w_k^i )>0$.1393 step coordinate (\protect\np{ln\_zps}\forcode{ = .true.}) in the case $( e3w_k^{i+1} - e3w_k^i )>0$. 1394 1394 A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value 1395 1395 at the depth of the shallower tracer point of the two adjacent bottom $T$-points. … … 1409 1409 and the resulting forms for the horizontal difference and the horizontal average 1410 1410 value of $T$ at a $U$-point are: 1411 \begin{equation} \label{ Eq_zps_hde}1411 \begin{equation} \label{eq:zps_hde} 1412 1412 \begin{aligned} 1413 1413 \delta _{i+1/2} T= \begin{cases} … … 1432 1432 of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$ 1433 1433 and $S$, and the pressure at a $u$-point (in the equation of state pressure is 1434 approximated by depth, see \ S\ref{TRA_eos} ) :1435 \begin{equation} \label{ Eq_zps_hde_rho}1434 approximated by depth, see \autoref{subsec:TRA_eos} ) : 1435 \begin{equation} \label{eq:zps_hde_rho} 1436 1436 \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) 1437 1437 \quad \text{where }\ z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) … … 1441 1441 thus pressure) is highly non-linear with a true equation of state and thus is badly 1442 1442 approximated with a linear interpolation. This approximation is used to compute 1443 both the horizontal pressure gradient (\ S\ref{DYN_hpg}) and the slopes of neutral1444 surfaces (\ S\ref{LDF_slp})1443 both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and the slopes of neutral 1444 surfaces (\autoref{sec:LDF_slp}) 1445 1445 1446 1446 Note that in almost all the advection schemes presented in this Chapter, both 1447 averaging and differencing operators appear. Yet \ eqref{Eq_zps_hde} has not1447 averaging and differencing operators appear. Yet \autoref{eq:zps_hde} has not 1448 1448 been used in these schemes: in contrast to diffusion and pressure gradient 1449 1449 computations, no correction for partial steps is applied for advection. The main
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