Changeset 11512 for NEMO/branches/2019/dev_r10984_HPC-13_IRRMANN_BDY_optimization/doc/latex/NEMO/subfiles/chap_TRA.tex
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r11263 r11512 8 8 \label{chap:TRA} 9 9 10 \ minitoc11 12 % missing/update 10 \chaptertoc 11 12 % missing/update 13 13 % traqsr: need to coordinate with SBC module 14 14 15 %STEVEN : is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? I added a comment to this effect on some instances of this below 15 %STEVEN : is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? 16 %I added a comment to this effect on some instances of this below 16 17 17 18 Using the representation described in \autoref{chap:DOM}, several semi -discrete space forms of … … 35 36 The terms QSR, BBC, BBL and DMP are optional. 36 37 The external forcings and parameterisations require complex inputs and complex calculations 37 (\eg bulk formulae, estimation of mixing coefficients) that are carried out in the SBC,38 (\eg\ bulk formulae, estimation of mixing coefficients) that are carried out in the SBC, 38 39 LDF and ZDF modules and described in \autoref{chap:SBC}, \autoref{chap:LDF} and 39 40 \autoref{chap:ZDF}, respectively. … … 47 48 associated modules \mdl{eosbn2} and \mdl{phycst}). 48 49 49 The different options available to the user are managed by namelist logicals or CPP keys.50 The different options available to the user are managed by namelist logicals. 50 51 For each equation term \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx}, 51 52 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 52 The CPP key (when it exists) is \key{traTTT}.53 53 The equivalent code can be found in the \textit{traTTT} or \textit{traTTT\_xxx} module, 54 54 in the \path{./src/OCE/TRA} directory. … … 68 68 %------------------------------------------------------------------------------------------------------------- 69 69 70 When considered (\ie when \np{ln\_traadv\_NONE} is not set to \forcode{.true.}),70 When considered (\ie\ when \np{ln\_traadv\_OFF} is not set to \forcode{.true.}), 71 71 the advection tendency of a tracer is expressed in flux form, 72 \ie as the divergence of the advective fluxes.72 \ie\ as the divergence of the advective fluxes. 73 73 Its discrete expression is given by : 74 74 \begin{equation} … … 82 82 Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which 83 83 results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$ 84 (which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, \ie \np{ln\_linssh}\forcode{ = .true.}).84 (which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, \ie\ \np{ln\_linssh}\forcode{ = .true.}). 85 85 Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that 86 86 it is consistent with the continuity equation in order to enforce the conservation properties of … … 110 110 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 111 111 112 The key difference between the advection schemes available in \NEMO is the choice made in space and112 The key difference between the advection schemes available in \NEMO\ is the choice made in space and 113 113 time interpolation to define the value of the tracer at the velocity points 114 114 (\autoref{fig:adv_scheme}). … … 125 125 the moving surface $z = \eta$. 126 126 There is a non-zero advective flux which is set for all advection schemes as 127 $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie the product of surface velocity (at $z = 0$) by127 $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie\ the product of surface velocity (at $z = 0$) by 128 128 the first level tracer value. 129 129 \item[non-linear free surface:] … … 139 139 two quantities that are not correlated \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}. 140 140 141 The velocity field that appears in (\autoref{eq:tra_adv} and \autoref{eq:tra_adv_zco?}) is142 the centred (\textit{now}) \textit{effective} ocean velocity, \ie the \textit{eulerian} velocity141 The velocity field that appears in (\autoref{eq:tra_adv} is 142 the centred (\textit{now}) \textit{effective} ocean velocity, \ie\ the \textit{eulerian} velocity 143 143 (see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or 144 144 the mixed layer eddy induced velocity (\textit{eiv}) when those parameterisations are used … … 149 149 Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), 150 150 and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST). 151 The choice is made in the \n gn{namtra\_adv} namelist, by setting to \forcode{.true.} one of151 The choice is made in the \nam{tra\_adv} namelist, by setting to \forcode{.true.} one of 152 152 the logicals \textit{ln\_traadv\_xxx}. 153 153 The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 154 154 \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 155 By default (\ie in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}.155 By default (\ie\ in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}. 156 156 If the user does not select an advection scheme in the configuration namelist (\textit{namelist\_cfg}), 157 157 the tracers will \textit{not} be advected! … … 188 188 \label{subsec:TRA_adv_cen} 189 189 190 % 2nd order centred scheme 190 % 2nd order centred scheme 191 191 192 192 The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}\forcode{ = .true.}. … … 203 203 \end{equation} 204 204 205 CEN2 is non diffusive (\ie it conserves the tracer variance, $\tau^2$) but dispersive206 (\ie it may create false extrema).205 CEN2 is non diffusive (\ie\ it conserves the tracer variance, $\tau^2$) but dispersive 206 (\ie\ it may create false extrema). 207 207 It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to 208 208 produce a sensible solution. … … 213 213 both (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_cen2}) have this order of accuracy. 214 214 215 % 4nd order centred scheme 215 % 4nd order centred scheme 216 216 217 217 In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as … … 225 225 a $4^{th}$ COMPACT interpolation has been prefered \citep{demange_phd14}. 226 226 In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion, 227 spectral characteristics similar to schemes of higher order \citep{lele_JCP92}. 227 spectral characteristics similar to schemes of higher order \citep{lele_JCP92}. 228 228 229 229 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but … … 237 237 238 238 A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive, 239 \ie the global variance of a tracer is not preserved using CEN4.239 \ie\ the global variance of a tracer is not preserved using CEN4. 240 240 Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution. 241 241 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, … … 250 250 251 251 % ------------------------------------------------------------------------------------------------------------- 252 % FCT scheme 252 % FCT scheme 253 253 % ------------------------------------------------------------------------------------------------------------- 254 254 \subsection[FCT: Flux Corrected Transport scheme (\forcode{ln_traadv_fct = .true.})] … … 278 278 where $c_u$ is a flux limiter function taking values between 0 and 1. 279 279 The FCT order is the one of the centred scheme used 280 (\ie it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}).280 (\ie\ it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}). 281 281 There exist many ways to define $c_u$, each corresponding to a different FCT scheme. 282 The one chosen in \NEMO is described in \citet{zalesak_JCP79}.282 The one chosen in \NEMO\ is described in \citet{zalesak_JCP79}. 283 283 $c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field. 284 284 The resulting scheme is quite expensive but \textit{positive}. … … 286 286 A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{levy.estublier.ea_GRL01}. 287 287 288 An additional option has been added controlled by \np{nn\_fct\_zts}.289 By setting this integer to a value larger than zero,290 a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, but on the latter,291 a split-explicit time stepping is used, with a number of sub-timestep equals to \np{nn\_fct\_zts}.292 This option can be useful when the size of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}.293 Note that in this case, a similar split-explicit time stepping should be used on vertical advection of momentum to294 insure a better stability (see \autoref{subsec:DYN_zad}).295 288 296 289 For stability reasons (see \autoref{chap:STP}), … … 301 294 302 295 % ------------------------------------------------------------------------------------------------------------- 303 % MUSCL scheme 296 % MUSCL scheme 304 297 % ------------------------------------------------------------------------------------------------------------- 305 298 \subsection[MUSCL: Monotone Upstream Scheme for Conservative Laws (\forcode{ln_traadv_mus = .true.})] … … 310 303 MUSCL implementation can be found in the \mdl{traadv\_mus} module. 311 304 312 MUSCL has been first implemented in \NEMO by \citet{levy.estublier.ea_GRL01}.305 MUSCL has been first implemented in \NEMO\ by \citet{levy.estublier.ea_GRL01}. 313 306 In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between 314 307 two $T$-points (\autoref{fig:adv_scheme}). … … 338 331 339 332 % ------------------------------------------------------------------------------------------------------------- 340 % UBS scheme 333 % UBS scheme 341 334 % ------------------------------------------------------------------------------------------------------------- 342 335 \subsection[UBS a.k.a. UP3: Upstream-Biased Scheme (\forcode{ln_traadv_ubs = .true.})] … … 374 367 \citep{shchepetkin.mcwilliams_OM05, demange_phd14}. 375 368 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme 376 (\np{nn\_ cen\_v}\forcode{ = 2 or 4}).369 (\np{nn\_ubs\_v}\forcode{ = 2 or 4}). 377 370 378 371 For stability reasons (see \autoref{chap:STP}), the first term in \autoref{eq:tra_adv_ubs} … … 408 401 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which 409 402 is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$. 410 Note the current version of NEMOuses the computationally more efficient formulation \autoref{eq:tra_adv_ubs}.411 412 % ------------------------------------------------------------------------------------------------------------- 413 % QCK scheme 403 Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:tra_adv_ubs}. 404 405 % ------------------------------------------------------------------------------------------------------------- 406 % QCK scheme 414 407 % ------------------------------------------------------------------------------------------------------------- 415 408 \subsection[QCK: QuiCKest scheme (\forcode{ln_traadv_qck = .true.})] … … 423 416 QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter 424 417 \citep{leonard_CMAME91}. 425 It has been implemented in NEMOby G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module.418 It has been implemented in \NEMO\ by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. 426 419 The resulting scheme is quite expensive but \textit{positive}. 427 420 It can be used on both active and passive tracers. … … 444 437 \nlst{namtra_ldf} 445 438 %------------------------------------------------------------------------------------------------------------- 446 447 Options are defined through the \n gn{namtra\_ldf} namelist variables.448 They are regrouped in four items, allowing to specify 439 440 Options are defined through the \nam{tra\_ldf} namelist variables. 441 They are regrouped in four items, allowing to specify 449 442 $(i)$ the type of operator used (none, laplacian, bilaplacian), 450 443 $(ii)$ the direction along which the operator acts (iso-level, horizontal, iso-neutral), 451 $(iii)$ some specific options related to the rotated operators (\ie non-iso-level operator), and444 $(iii)$ some specific options related to the rotated operators (\ie\ non-iso-level operator), and 452 445 $(iv)$ the specification of eddy diffusivity coefficient (either constant or variable in space and time). 453 446 Item $(iv)$ will be described in \autoref{chap:LDF}. … … 457 450 458 451 The lateral diffusion of tracers is evaluated using a forward scheme, 459 \ie the tracers appearing in its expression are the \textit{before} tracers in time,452 \ie\ the tracers appearing in its expression are the \textit{before} tracers in time, 460 453 except for the pure vertical component that appears when a rotation tensor is used. 461 454 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). … … 466 459 % Type of operator 467 460 % ------------------------------------------------------------------------------------------------------------- 468 \subsection[Type of operator (\texttt{ln\_traldf}\{\texttt{\_ NONE,\_lap,\_blp}\})]469 {Type of operator (\protect\np{ln\_traldf\_ NONE}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) }461 \subsection[Type of operator (\texttt{ln\_traldf}\{\texttt{\_OFF,\_lap,\_blp}\})] 462 {Type of operator (\protect\np{ln\_traldf\_OFF}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) } 470 463 \label{subsec:TRA_ldf_op} 471 464 … … 473 466 474 467 \begin{description} 475 \item[\np{ln\_traldf\_ NONE}\forcode{ = .true.}:]468 \item[\np{ln\_traldf\_OFF}\forcode{ = .true.}:] 476 469 no operator selected, the lateral diffusive tendency will not be applied to the tracer equation. 477 470 This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example). … … 494 487 minimizing the impact on the larger scale features. 495 488 The main difference between the two operators is the scale selectiveness. 496 The bilaplacian damping time (\ie its spin down time) scales like $\lambda^{-4}$ for489 The bilaplacian damping time (\ie\ its spin down time) scales like $\lambda^{-4}$ for 497 490 disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones), 498 491 whereas the laplacian damping time scales only like $\lambda^{-2}$. … … 502 495 % ------------------------------------------------------------------------------------------------------------- 503 496 \subsection[Action direction (\texttt{ln\_traldf}\{\texttt{\_lev,\_hor,\_iso,\_triad}\})] 504 {Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) } 497 {Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) } 505 498 \label{subsec:TRA_ldf_dir} 506 499 … … 508 501 The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when 509 502 iso-level option is used (\np{ln\_traldf\_lev}\forcode{ = .true.}) or 510 when a horizontal (\ie geopotential) operator is demanded in \textit{z}-coordinate503 when a horizontal (\ie\ geopotential) operator is demanded in \textit{z}-coordinate 511 504 (\np{ln\_traldf\_hor} and \np{ln\_zco} equal \forcode{.true.}). 512 505 The associated code can be found in the \mdl{traldf\_lap\_blp} module. … … 516 509 (\np{ln\_traldf\_iso} or \np{ln\_traldf\_triad} equals \forcode{.true.}, 517 510 see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or 518 when a horizontal (\ie geopotential) operator is demanded in \textit{s}-coordinate511 when a horizontal (\ie\ geopotential) operator is demanded in \textit{s}-coordinate 519 512 (\np{ln\_traldf\_hor} and \np{ln\_sco} equal \forcode{.true.}) 520 513 \footnote{In this case, the standard iso-neutral operator will be automatically selected}. … … 532 525 \label{subsec:TRA_ldf_lev} 533 526 534 The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 527 The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 535 528 \begin{equation} 536 529 \label{eq:tra_ldf_lap} … … 541 534 where zero diffusive fluxes is assumed across solid boundaries, 542 535 first (and third in bilaplacian case) horizontal tracer derivative are masked. 543 It is implemented in the \rou{tra ldf\_lap} subroutine found in the \mdl{traldf\_lap} module.544 The module also contains \rou{tra ldf\_blp}, the subroutine calling twice \rou{traldf\_lap} in order to536 It is implemented in the \rou{tra\_ldf\_lap} subroutine found in the \mdl{traldf\_lap\_blp}} module. 537 The module also contains \rou{tra\_ldf\_blp}, the subroutine calling twice \rou{tra\_ldf\_lap} in order to 545 538 compute the iso-level bilaplacian operator. 546 539 … … 584 577 where $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells, 585 578 $r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and 586 the surface along which the diffusion operator acts (\ie horizontal or iso-neutral surfaces).579 the surface along which the diffusion operator acts (\ie\ horizontal or iso-neutral surfaces). 587 580 It is thus used when, in addition to \np{ln\_traldf\_lap}\forcode{ = .true.}, 588 581 we have \np{ln\_traldf\_iso}\forcode{ = .true.}, … … 613 606 \label{subsec:TRA_ldf_triad} 614 607 615 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:triad})616 617 608 An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases 618 is also available in \NEMO (\np{ln\_traldf\_grif}\forcode{ = .true.}).609 is also available in \NEMO\ (\np{ln\_traldf\_triad}\forcode{ = .true.}). 619 610 A complete description of the algorithm is given in \autoref{apdx:triad}. 620 611 … … 637 628 \item \np{rn\_slpmax} = slope limit (both operators) 638 629 \item \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only) 639 \item \np{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only) 630 \item \np{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only) 640 631 \item \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only) 641 632 \end{itemize} … … 652 643 %-------------------------------------------------------------------------------------------------------------- 653 644 654 Options are defined through the \n gn{namzdf} namelist variables.645 Options are defined through the \nam{zdf} namelist variables. 655 646 The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates, 656 647 and is based on a laplacian operator. … … 664 655 respectively. 665 656 Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised 666 (\ie \key{zdfddm} is defined).657 (\ie\ \np{ln\_zdfddm} equals \forcode{.true.},). 667 658 The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF). 668 659 Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by … … 676 667 677 668 The large eddy coefficient found in the mixed layer together with high vertical resolution implies that 678 in the case of explicit time stepping (\np{ln\_zdfexp}\forcode{ = .true.}) 679 there would be too restrictive a constraint on the time step. 680 Therefore, the default implicit time stepping is preferred for the vertical diffusion since 669 there would be too restrictive constraint on the time step if we use explicit time stepping. 670 Therefore an implicit time stepping is preferred for the vertical diffusion since 681 671 it overcomes the stability constraint. 682 A forward time differencing scheme (\np{ln\_zdfexp}\forcode{ = .true.}) using683 a time splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative.684 Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.685 672 686 673 % ================================================================ … … 704 691 705 692 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components 706 (\ie atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due693 (\ie\ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due 707 694 both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and 708 695 to the heat and salt content of the mass exchange. … … 716 703 \item 717 704 $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 718 (\ie the difference between the total surface heat flux and the fraction of the short wave flux that705 (\ie\ the difference between the total surface heat flux and the fraction of the short wave flux that 719 706 penetrates into the water column, see \autoref{subsec:TRA_qsr}) 720 707 plus the heat content associated with of the mass exchange with the atmosphere and lands. … … 758 745 &\overline{(\textit{sfx} - \textit{emp} \lt. S \rt|_{k = 1})}^t 759 746 \end{alignedat} 760 \end{equation} 747 \end{equation} 761 748 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. 762 749 In the linear free surface case, there is a small imbalance. … … 765 752 766 753 % ------------------------------------------------------------------------------------------------------------- 767 % Solar Radiation Penetration 754 % Solar Radiation Penetration 768 755 % ------------------------------------------------------------------------------------------------------------- 769 756 \subsection[Solar radiation penetration (\textit{traqsr.F90})] … … 775 762 %-------------------------------------------------------------------------------------------------------------- 776 763 777 Options are defined through the \n gn{namtra\_qsr} namelist variables.778 When the penetrative solar radiation option is used (\np{ln\_ flxqsr}\forcode{ = .true.}),764 Options are defined through the \nam{tra\_qsr} namelist variables. 765 When the penetrative solar radiation option is used (\np{ln\_traqsr}\forcode{ = .true.}), 779 766 the solar radiation penetrates the top few tens of meters of the ocean. 780 If it is not used (\np{ln\_ flxqsr}\forcode{ = .false.}) all the heat flux is absorbed in the first ocean level.767 If it is not used (\np{ln\_traqsr}\forcode{ = .false.}) all the heat flux is absorbed in the first ocean level. 781 768 Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:PE_tra_T} and 782 the surface boundary condition is modified to take into account only the non-penetrative part of the surface 769 the surface boundary condition is modified to take into account only the non-penetrative part of the surface 783 770 heat flux: 784 771 \begin{equation} … … 789 776 \end{gathered} 790 777 \end{equation} 791 where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie the shortwave radiation) and778 where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie\ the shortwave radiation) and 792 779 $I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$). 793 780 The additional term in \autoref{eq:PE_qsr} is discretized as follows: … … 803 790 (specified through namelist parameter \np{rn\_abs}). 804 791 It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 805 of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn\_si0} in the \n gn{namtra\_qsr} namelist).792 of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn\_si0} in the \nam{tra\_qsr} namelist). 806 793 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to 807 794 larger depths where it contributes to local heating. … … 836 823 837 824 The RGB formulation is used when \np{ln\_qsr\_rgb}\forcode{ = .true.}. 838 The RGB attenuation coefficients (\ie the inverses of the extinction length scales) are tabulated over825 The RGB attenuation coefficients (\ie\ the inverses of the extinction length scales) are tabulated over 839 826 61 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L 840 827 (see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module). … … 842 829 843 830 \begin{description} 844 \item[\np{nn\_ch dta}\forcode{ = 0}]845 a constant 0.05 g.Chl/L value everywhere ; 846 \item[\np{nn\_ch dta}\forcode{ = 1}]831 \item[\np{nn\_chldta}\forcode{ = 0}] 832 a constant 0.05 g.Chl/L value everywhere ; 833 \item[\np{nn\_chldta}\forcode{ = 1}] 847 834 an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in 848 835 the vertical direction; 849 \item[\np{nn\_ch dta}\forcode{ = 2}]836 \item[\np{nn\_chldta}\forcode{ = 2}] 850 837 same as previous case except that a vertical profile of chlorophyl is used. 851 838 Following \cite{morel.berthon_LO89}, the profile is computed from the local surface chlorophyll value; … … 853 840 simulated time varying chlorophyll by TOP biogeochemical model. 854 841 In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in 855 PISCES or LOBSTERand the oceanic heating rate.856 \end{description} 842 PISCES and the oceanic heating rate. 843 \end{description} 857 844 858 845 The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation is added to … … 862 849 the depth of $w-$levels does not significantly vary with location. 863 850 The level at which the light has been totally absorbed 864 (\ie it is less than the computer precision) is computed once,851 (\ie\ it is less than the computer precision) is computed once, 865 852 and the trend associated with the penetration of the solar radiation is only added down to that level. 866 853 Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor. 867 854 In this case, we have chosen that all remaining radiation is absorbed in the last ocean level 868 (\ie $I$ is masked).855 (\ie\ $I$ is masked). 869 856 870 857 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 889 876 % Bottom Boundary Condition 890 877 % ------------------------------------------------------------------------------------------------------------- 891 \subsection[Bottom boundary condition (\textit{trabbc.F90}) ]878 \subsection[Bottom boundary condition (\textit{trabbc.F90}) - \forcode{ln_trabbc = .true.})] 892 879 {Bottom boundary condition (\protect\mdl{trabbc})} 893 880 \label{subsec:TRA_bbc} … … 910 897 911 898 Usually it is assumed that there is no exchange of heat or salt through the ocean bottom, 912 \ie a no flux boundary condition is applied on active tracers at the bottom.899 \ie\ a no flux boundary condition is applied on active tracers at the bottom. 913 900 This is the default option in \NEMO, and it is implemented using the masking technique. 914 901 However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling. … … 916 903 but it warms systematically the ocean and acts on the densest water masses. 917 904 Taking this flux into account in a global ocean model increases the deepest overturning cell 918 (\ie the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{emile-geay.madec_OS09}.919 920 Options are defined through the \ngn{namtra\_bbc} namelist variables.905 (\ie\ the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{emile-geay.madec_OS09}. 906 907 Options are defined through the \nam{bbc} namelist variables. 921 908 The presence of geothermal heating is controlled by setting the namelist parameter \np{ln\_trabbc} to true. 922 909 Then, when \np{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by 923 the \np{ nn\_geoflx\_cst}, which is also a namelist parameter.910 the \np{rn\_geoflx\_cst}, which is also a namelist parameter. 924 911 When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in 925 912 the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:geothermal}) \citep{emile-geay.madec_OS09}. … … 928 915 % Bottom Boundary Layer 929 916 % ================================================================ 930 \section[Bottom boundary layer (\textit{trabbl.F90} - \ texttt{\textbf{key\_trabbl}})]931 {Bottom boundary layer (\protect\mdl{trabbl} - \protect\ key{trabbl})}917 \section[Bottom boundary layer (\textit{trabbl.F90} - \forcode{ln_trabbl = .true.})] 918 {Bottom boundary layer (\protect\mdl{trabbl} - \protect\np{ln\_trabbl}\forcode{ = .true.})} 932 919 \label{sec:TRA_bbl} 933 920 %--------------------------------------------nambbl--------------------------------------------------------- … … 936 923 %-------------------------------------------------------------------------------------------------------------- 937 924 938 Options are defined through the \n gn{nambbl} namelist variables.925 Options are defined through the \nam{bbl} namelist variables. 939 926 In a $z$-coordinate configuration, the bottom topography is represented by a series of discrete steps. 940 927 This is not adequate to represent gravity driven downslope flows. … … 965 952 \label{subsec:TRA_bbl_diff} 966 953 967 When applying sigma-diffusion (\ key{trabbl} definedand \np{nn\_bbl\_ldf} set to 1),968 the diffusive flux between two adjacent cells at the ocean floor is given by 954 When applying sigma-diffusion (\np{ln\_trabbl}\forcode{ = .true.} and \np{nn\_bbl\_ldf} set to 1), 955 the diffusive flux between two adjacent cells at the ocean floor is given by 969 956 \[ 970 957 % \label{eq:tra_bbl_diff} … … 974 961 $A_l^\sigma$ the lateral diffusivity in the BBL. 975 962 Following \citet{beckmann.doscher_JPO97}, the latter is prescribed with a spatial dependence, 976 \ie in the conditional form963 \ie\ in the conditional form 977 964 \begin{equation} 978 965 \label{eq:tra_bbl_coef} … … 990 977 (see green arrow in \autoref{fig:bbl}). 991 978 In practice, this constraint is applied separately in the two horizontal directions, 992 and the density gradient in \autoref{eq:tra_bbl_coef} is evaluated with the log gradient formulation: 979 and the density gradient in \autoref{eq:tra_bbl_coef} is evaluated with the log gradient formulation: 993 980 \[ 994 981 % \label{eq:tra_bbl_Drho} … … 1041 1028 (see black arrow in \autoref{fig:bbl}) \citep{beckmann.doscher_JPO97}. 1042 1029 It is a \textit{conditional advection}, that is, advection is allowed only 1043 if dense water overlies less dense water on the slope (\ie $\nabla_\sigma \rho \cdot \nabla H < 0$) and1044 if the velocity is directed towards greater depth (\ie $\vect U \cdot \nabla H > 0$).1030 if dense water overlies less dense water on the slope (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and 1031 if the velocity is directed towards greater depth (\ie\ $\vect U \cdot \nabla H > 0$). 1045 1032 1046 1033 \np{nn\_bbl\_adv}\forcode{ = 2}: … … 1048 1035 the density difference between the higher cell and lower cell densities \citep{campin.goosse_T99}. 1049 1036 The advection is allowed only if dense water overlies less dense water on the slope 1050 (\ie $\nabla_\sigma \rho \cdot \nabla H < 0$).1037 (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$). 1051 1038 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}), 1052 1039 is simply given by the following expression: … … 1070 1057 The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by 1071 1058 the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and 1072 the upward \autoref{eq:bbl_up} return flows as follows: 1059 the upward \autoref{eq:bbl_up} return flows as follows: 1073 1060 \begin{alignat}{3} 1074 1061 \label{eq:bbl_dw} … … 1108 1095 \pd[S]{t} = \cdots - \gamma (S - S_o) 1109 1096 \end{gathered} 1110 \end{equation} 1097 \end{equation} 1111 1098 where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields 1112 1099 (usually a climatology). 1113 Options are defined through the \n gn{namtra\_dmp} namelist variables.1100 Options are defined through the \nam{tra\_dmp} namelist variables. 1114 1101 The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true. 1115 It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_ tradmp} are set to true in1116 \n gn{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set1117 (\ie that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread},1102 It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_dmp} are set to true in 1103 \nam{tsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set 1104 (\ie\ that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread}, 1118 1105 see \autoref{subsec:SBC_fldread}). 1119 1106 The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. … … 1157 1144 %-------------------------------------------------------------------------------------------------------------- 1158 1145 1159 Options are defined through the \n gn{namdom} namelist variables.1146 Options are defined through the \nam{dom} namelist variables. 1160 1147 The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09}, 1161 \ie a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}):1148 \ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 1162 1149 \begin{equation} 1163 1150 \label{eq:tra_nxt} … … 1165 1152 &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\ 1166 1153 &(e_{3t}T)_f^t &&= (e_{3t}T)^t &&+ \, \gamma \, \lt[ (e_{3t}T)_f^{t - \rdt} - 2(e_{3t}T)^t + (e_{3t}T)^{t + \rdt} \rt] \\ 1167 & && &&- \, \gamma \, \rdt \, \lt[ Q^{t + \rdt/2} - Q^{t - \rdt/2} \rt] 1154 & && &&- \, \gamma \, \rdt \, \lt[ Q^{t + \rdt/2} - Q^{t - \rdt/2} \rt] 1168 1155 \end{alignedat} 1169 \end{equation} 1156 \end{equation} 1170 1157 where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values, 1171 1158 $\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$ 1172 (\ie fluxes plus content in mass exchanges).1159 (\ie\ fluxes plus content in mass exchanges). 1173 1160 $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 1174 1161 Its default value is \np{rn\_atfp}\forcode{ = 10.e-3}. 1175 1162 Note that the forcing correction term in the filter is not applied in linear free surface 1176 (\jp{l k\_vvl}\forcode{ = .false.}) (see \autoref{subsec:TRA_sbc}).1163 (\jp{ln\_linssh}\forcode{ = .true.}) (see \autoref{subsec:TRA_sbc}). 1177 1164 Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$. 1178 1165 … … 1185 1172 1186 1173 % ================================================================ 1187 % Equation of State (eosbn2) 1174 % Equation of State (eosbn2) 1188 1175 % ================================================================ 1189 1176 \section[Equation of state (\textit{eosbn2.F90})] … … 1198 1185 % Equation of State 1199 1186 % ------------------------------------------------------------------------------------------------------------- 1200 \subsection[Equation of seawater (\ forcode{nn_eos = {-1,1}})]1201 {Equation of seawater (\protect\np{ nn\_eos}\forcode{ = {-1,1}})}1187 \subsection[Equation of seawater (\texttt{ln}\{\texttt{\_teso10,\_eos80,\_seos}\})] 1188 {Equation of seawater (\protect\np{ln\_teos10}, \protect\np{ln\_teos80}, or \protect\np{ln\_seos}) } 1202 1189 \label{subsec:TRA_eos} 1190 1203 1191 1204 1192 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density, … … 1217 1205 \textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and 1218 1206 \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and 1219 practical salinity for EOS- 980, both variables being more suitable for use as model variables1207 practical salinity for EOS-80, both variables being more suitable for use as model variables 1220 1208 \citep{ioc.iapso_bk10, graham.mcdougall_JPO13}. 1221 EOS-80 is an obsolescent feature of the NEMOsystem, kept only for backward compatibility.1209 EOS-80 is an obsolescent feature of the \NEMO\ system, kept only for backward compatibility. 1222 1210 For process studies, it is often convenient to use an approximation of the EOS. 1223 1211 To that purposed, a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is also available. … … 1229 1217 density in the World Ocean varies by no more than 2$\%$ from that value \citep{gill_bk82}. 1230 1218 1231 Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} which 1232 controls the EOS used (\forcode{= -1} for TEOS10 ; \forcode{= 0} for EOS-80 ; \forcode{= 1} for S-EOS). 1219 Options which control the EOS used are defined through the \ngn{nameos} namelist variables. 1233 1220 1234 1221 \begin{description} 1235 \item[\np{ nn\_eos}\forcode{ = -1}]1222 \item[\np{ln\_teos10}\forcode{ = .true.}] 1236 1223 the polyTEOS10-bsq equation of seawater \citep{roquet.madec.ea_OM15} is used. 1237 1224 The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, … … 1249 1236 In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and 1250 1237 \textit{Absolute} Salinity. 1251 In addition, setting \np{ln\_useCT} to \forcode{.true.} convert the Conservative SSTto potential SST prior to1238 In addition, when using TEOS10, the Conservative SST is converted to potential SST prior to 1252 1239 either computing the air-sea and ice-sea fluxes (forced mode) or 1253 1240 sending the SST field to the atmosphere (coupled mode). 1254 \item[\np{ nn\_eos}\forcode{ = 0}]1241 \item[\np{ln\_eos80}\forcode{ = .true.}] 1255 1242 the polyEOS80-bsq equation of seawater is used. 1256 1243 It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to … … 1264 1251 Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which 1265 1252 is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value. 1266 \item[\np{ nn\_eos}\forcode{ = 1}]1253 \item[\np{ln\_seos}\forcode{ = .true.}] 1267 1254 a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is chosen, 1268 1255 the coefficients of which has been optimized to fit the behavior of TEOS10 … … 1274 1261 as well as between \textit{absolute} and \textit{practical} salinity. 1275 1262 S-EOS takes the following expression: 1263 1276 1264 \begin{gather*} 1277 1265 % \label{eq:tra_S-EOS} 1278 1266 \begin{alignedat}{2} 1279 1267 &d_a(T,S,z) = \frac{1}{\rho_o} \big[ &- a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * &T_a \big. \\ 1280 & &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a \\ 1268 & &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a \\ 1281 1269 & \big. &- \nu \; T_a &S_a \big] \\ 1282 1270 \end{alignedat} … … 1326 1314 % Brunt-V\"{a}is\"{a}l\"{a} Frequency 1327 1315 % ------------------------------------------------------------------------------------------------------------- 1328 \subsection[Brunt-V\"{a}is\"{a}l\"{a} frequency (\forcode{nn_eos = [0-2]})]1329 {Brunt-V\"{a}is\"{a}l\"{a} frequency (\protect\np{nn\_eos}\forcode{ = [0-2]})}1316 \subsection[Brunt-V\"{a}is\"{a}l\"{a} frequency] 1317 {Brunt-V\"{a}is\"{a}l\"{a} frequency} 1330 1318 \label{subsec:TRA_bn2} 1331 1319 … … 1336 1324 In particular, $N^2$ has to be computed at the local pressure 1337 1325 (pressure in decibar being approximated by the depth in meters). 1338 The expression for $N^2$ is given by: 1326 The expression for $N^2$ is given by: 1339 1327 \[ 1340 1328 % \label{eq:tra_bn2} … … 1358 1346 \begin{split} 1359 1347 &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\ 1360 &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\ 1348 &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\ 1361 1349 &\text{and~} d = -7.53~10^{-3} 1362 1350 \end{split} … … 1364 1352 1365 1353 \autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of sea water 1366 (\ie referenced to the surface $p = 0$),1354 (\ie\ referenced to the surface $p = 0$), 1367 1355 thus the pressure dependent terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped. 1368 1356 The freezing point is computed through \textit{eos\_fzp}, … … 1370 1358 1371 1359 % ------------------------------------------------------------------------------------------------------------- 1372 % Potential Energy 1360 % Potential Energy 1373 1361 % ------------------------------------------------------------------------------------------------------------- 1374 1362 %\subsection{Potential Energy anomalies} … … 1379 1367 1380 1368 % ================================================================ 1381 % Horizontal Derivative in zps-coordinate 1369 % Horizontal Derivative in zps-coordinate 1382 1370 % ================================================================ 1383 1371 \section[Horizontal derivative in \textit{zps}-coordinate (\textit{zpshde.F90})] … … 1385 1373 \label{sec:TRA_zpshde} 1386 1374 1387 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, 1375 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, 1388 1376 I've changed "derivative" to "difference" and "mean" to "average"} 1389 1377 … … 1427 1415 \rt. 1428 1416 \] 1429 and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are: 1417 and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are: 1430 1418 \begin{equation} 1431 1419 \label{eq:zps_hde} … … 1453 1441 Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of 1454 1442 $T$ and $S$, and the pressure at a $u$-point 1455 (in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}): 1443 (in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}): 1456 1444 \[ 1457 1445 % \label{eq:zps_hde_rho}
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