Changeset 2268 for branches/DEV_r1826_DOC
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branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_TRA.tex
r2195 r2268 48 48 In the present chapter we also describe the diagnostic equations used to compute 49 49 the sea-water properties (density, Brunt-Vais\"{a}l\"{a} frequency, specific heat and 50 freezing point) although the associated modules ($i.e.$ \mdl{eosbn2}, \mdl{ocfzpt} 51 and \mdl{phycst}) are (temporarily) located in the NEMO/OPA directory. 52 53 The different options available to the user are managed by namelist logical or 50 freezing point with associated modules \mdl{eosbn2}, \mdl{ocfzpt} and \mdl{phycst}). 51 52 The different options available to the user are managed by namelist logicals or 54 53 CPP keys. For each equation term \textit{ttt}, the namelist logicals are \textit{ln\_trattt\_xxx}, 55 where \textit{xxx} is a 3 or 4 letter acronym accounting foreach optional scheme.56 The CPP key (when it exists) is \textbf{key\_trattt}. The correspondingcode can be54 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 55 The CPP key (when it exists) is \textbf{key\_trattt}. The equivalent code can be 57 56 found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory. 58 57 … … 88 87 since the vertical scale factors are functions of $k$ only, and thus 89 88 $e_{3u} =e_{3v} =e_{3t} $. The flux form in \eqref{Eq_tra_adv} 90 requires implicitlythe use of the continuity equation. Indeed, it is obtained89 implicitly requires the use of the continuity equation. Indeed, it is obtained 91 90 by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 92 91 which results from the use of the continuity equation, $\nabla \cdot \vect{U}=0$ or 93 $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant (default option)94 or variable (\key{vvl} defined) volumecase, respectively.92 $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant volume (default option) 93 or variable volume (\key{vvl} defined) case, respectively. 95 94 Therefore it is of paramount importance to design the discrete analogue of the 96 95 advection tendency so that it is consistent with the continuity equation in order to 97 96 enforce the conservation properties of the continuous equations. In other words, 98 by substituting $\tau$ by1 in (\ref{Eq_tra_adv}) we recover the discrete form of97 by replacing $\tau$ by the number 1 in (\ref{Eq_tra_adv}) we recover the discrete form of 99 98 the continuity equation which is used to calculate the vertical velocity. 100 99 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 113 112 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 114 113 115 The key difference between the advection schemes usedin \NEMO is the choice114 The key difference between the advection schemes available in \NEMO is the choice 116 115 made in space and time interpolation to define the value of the tracer at the 117 116 velocity points (Fig.~\ref{Fig_adv_scheme}). 118 117 119 Along solid lateral and bottom boundaries a zero tracer flux is naturally118 Along solid lateral and bottom boundaries a zero tracer flux is automatically 120 119 specified, since the normal velocity is zero there. At the sea surface the 121 120 boundary condition depends on the type of sea surface chosen: … … 167 166 that are pure numerical artefacts. 168 167 168 \sgacomment{not sure whether 3 is still relevant after TRA-TRC branch is merged in?} 169 169 % ------------------------------------------------------------------------------------------------------------- 170 170 % 2nd order centred scheme … … 188 188 (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. The centered second 189 189 order advection is computed in the \mdl{traadv\_cen2} module. In this module, 190 it is a lso proposedto combine the \textit{cen2} scheme with an upstream scheme191 in specific areas which require sa strong diffusion in order to avoid the generation190 it is advantageous to combine the \textit{cen2} scheme with an upstream scheme 191 in specific areas which require a strong diffusion in order to avoid the generation 192 192 of false extrema. These areas are the vicinity of large river mouths, some straits 193 193 with coarse resolution, and the vicinity of ice cover area ($i.e.$ when the ocean 194 194 temperature is close to the freezing point). 195 This combined scheme has been included for specific grid points in the ORCA2 and ORCA4 configurations only. 195 196 196 197 Note that using the cen2 scheme, the overall tracer advection is of second 197 198 order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) 198 have this order of accuracy. Note also that199 have this order of accuracy. \gmcomment{Note also that ... blah, blah} 199 200 200 201 % ------------------------------------------------------------------------------------------------------------- … … 206 207 207 208 In the $4^{th}$ order formulation (to be implemented), tracer values are 208 evaluated at velocity points as a $4^{th}$ order interpolation, and thus uses209 evaluated at velocity points as a $4^{th}$ order interpolation, and thus depend on 209 210 the four neighbouring $T$-points. For example, in the $i$-direction: 210 211 \begin{equation} \label{Eq_tra_adv_cen4} … … 257 258 \end{equation} 258 259 where $c_u$ is a flux limiter function taking values between 0 and 1. 259 There exist many ways to define $c_u$, each corre cponding to a different260 There exist many ways to define $c_u$, each corresponding to a different 260 261 total variance decreasing scheme. The one chosen in \NEMO is described in 261 262 \citet{Zalesak_JCP79}. $c_u$ only departs from $1$ when the advective term … … 264 265 This scheme is tested and compared with MUSCL and the MPDATA scheme in 265 266 \citet{Levy_al_GRL01}; note that in this paper it is referred to as "FCT" (Flux corrected 266 transport) rather than TVD. The TVD scheme is computed in the \mdl{traadv\_tvd} module.267 268 For stability reasons (see \S\ref{DOM_nxt}), in (\ref{Eq_tra_adv_tvd})269 $\tau _u^{cen2}$ is evaluated using the \textit{now} tracer while $\tau _u^{ups}$267 transport) rather than TVD. The TVD scheme is implemented in the \mdl{traadv\_tvd} module. 268 269 For stability reasons (see \S\ref{DOM_nxt}), 270 $\tau _u^{cen2}$ is evaluated in (\ref{Eq_tra_adv_tvd}) using the \textit{now} tracer while $\tau _u^{ups}$ 270 271 is evaluated using the \textit{before} tracer. In other words, the advective part of 271 272 the scheme is time stepped with a leap-frog scheme while a forward scheme is … … 302 303 (\np{ln\_traadv\_muscl2}=true). Note that the latter choice does not ensure 303 304 the \textit{positive} character of the scheme. Only the former can be used 304 on both active and passive tracers. The two MUSCL schemes are computed305 on both active and passive tracers. The two MUSCL schemes are implemented 305 306 in the \mdl{traadv\_tvd} and \mdl{traadv\_tvd2} modules. 306 307 … … 339 340 \np{ln\_traadv\_ubs}=true. 340 341 341 For stability reasons (see \S\ref{DOM_nxt}), in \eqref{Eq_tra_adv_ubs},342 the first term (which corresponds to a second order centred scheme)342 For stability reasons (see \S\ref{DOM_nxt}), 343 the first term in \eqref{Eq_tra_adv_ubs} (which corresponds to a second order centred scheme) 343 344 is evaluated using the \textit{now} tracer (centred in time) while the 344 345 second term (which is the diffusive part of the scheme), is … … 352 353 substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 353 354 355 Four different options are possible for the vertical 356 component used in the UBS scheme. $\tau _w^{ubs}$ can be evaluated 357 using either \textit{(a)} a centred $2^{nd}$ order scheme, or \textit{(b)} 358 a TVD scheme, or \textit{(c)} an interpolation based on conservative 359 parabolic splines following the \citet{Shchepetkin_McWilliams_OM05} 360 implementation of UBS in ROMS, or \textit{(d)} a UBS. The $3^{rd}$ case 361 has dispersion properties similar to an eighth-order accurate conventional scheme. 362 The current reference version uses method b) 363 354 364 Note that : 355 365 … … 361 371 in the current reference version. 362 372 363 (2) In a forthcoming release four options will be available for the vertical 364 component used in the UBS scheme. $\tau _w^{ubs}$ will be evaluated 365 using either \textit{(a)} a centred $2^{nd}$ order scheme, or \textit{(b)} 366 a TVD scheme, or \textit{(c)} an interpolation based on conservative 367 parabolic splines following the \citet{Shchepetkin_McWilliams_OM05} 368 implementation of UBS in ROMS, or \textit{(d)} a UBS. The $3^{rd}$ case 369 has dispersion properties similar to an eighth-order accurate conventional scheme. 370 371 (3) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: 373 (2) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: 372 374 \begin{equation} \label{Eq_traadv_ubs2} 373 375 \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{ … … 391 393 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy 392 394 coefficient which is simply proportional to the velocity: 393 $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v2.3 still uses 394 \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}. This should be 395 changed in forthcoming release. 395 $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v3.3 still uses 396 \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}. 396 397 %%% 397 398 \gmcomment{the change in UBS scheme has to be done} … … 412 413 The resulting scheme is quite expensive but \emph{positive}. 413 414 It can be used on both active and passive tracers. 414 Nevertheless, the intrinsic diffusion of QCK makes its use risky in the vertical415 However, the intrinsic diffusion of QCK makes its use risky in the vertical 415 416 direction where the control of artificial diapycnal fluxes is of paramount importance. 416 417 Therefore the vertical flux is evaluated using the CEN2 scheme. 417 This no more ensurethe positivity of the scheme. The use of TVD in the vertical418 direction as for the UBS case should be implemented to maintain theproperty.418 This no longer guarantees the positivity of the scheme. The use of TVD in the vertical 419 direction (as for the UBS case) should be implemented to restore this property. 419 420 420 421 … … 427 428 428 429 The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984) 429 is based on a quadradic piecewise rebuilding. Like the QCK scheme, it is associated 430 \sgacomment{reference?} 431 is based on a quadradic piecewise construction. Like the QCK scheme, it is associated 430 432 with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented 431 433 in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference 432 version 3. 0.434 version 3.3. 433 435 434 436 % ================================================================ … … 469 471 \end{equation} 470 472 where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. 471 It can be found in the \mdl{traadv\_lap} module.473 It is implemented in the \mdl{traadv\_lap} module. 472 474 473 475 This lateral operator is computed in \mdl{traldf\_lap}. It is a \emph{horizontal} 474 476 operator ($i.e.$ acting along geopotential surfaces) in the $z$-coordinate with 475 or without partial step , but is simply an iso-level operator in the $s$-coordinate.477 or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 476 478 It is thus used when, in addition to \np{ln\_traldf\_lap}=true, we have 477 \np{ln\_traldf\_level}=true ,or \np{ln\_traldf\_hor}=\np{ln\_zco}=true.479 \np{ln\_traldf\_level}=true or \np{ln\_traldf\_hor}=\np{ln\_zco}=true. 478 480 In both cases, it significantly contributes to diapycnal mixing. 479 481 It is therefore not recommended. 480 482 481 483 Note that 482 (a) In pure $z$-coordinate (\key{zco} is defined), $e_{3u}$=$e_{3v}$=$e_{3t}$,484 (a) In the pure $z$-coordinate (\key{zco} is defined), $e_{3u}$=$e_{3v}$=$e_{3t}$, 483 485 so that the vertical scale factors disappear from (\ref{Eq_tra_ldf_lap}) ; 484 (b) In partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally486 (b) In the partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally 485 487 adjacent cells are located at different depths in the vicinity of the bottom. 486 488 In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level … … 541 543 background horizontal diffusion \citep{Guilyardi_al_CD01}. An alternative scheme 542 544 developed by \cite{Griffies_al_JPO98} which preserves both tracer and its variance 543 is currently been tested in \NEMO. It should be available in a forthcoming 544 release. 545 is also available in \NEMO (\np{ln\_traldf_grif}=true). 545 546 546 547 Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), the horizontal … … 572 573 where $a=\sqrt{|A|}$ and $A<0$. This was a mistake: both formulations 573 574 ensure the total variance decrease, but the former requires a larger 574 number of code-lines. It will be corrected in a forthcoming release.575 number of code-lines. 575 576 576 577 % ------------------------------------------------------------------------------------------------------------- … … 584 585 applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption 585 586 on boundary conditions: first and third derivative terms normal to the 586 coast, the bottom andthe surface are set to zero. It can be found in the587 coast, normal to the bottom and normal to the surface are set to zero. It can be found in the 587 588 \mdl{traldf\_bilapg}. 588 589 589 590 It is used when, in addition to \np{ln\_traldf\_bilap}=true, we have 590 591 \np{ln\_traldf\_iso}= .true, or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. 591 Nevertheless, this rotated bilaplacian operator has never been seriously592 tested. No warranties that it is neither free of bugs or correctly formulated.592 This rotated bilaplacian operator has never been seriously 593 tested. There are no guarantees that it is either free of bugs or correctly formulated. 593 594 Moreover, the stability range of such an operator will be probably quite 594 narrow, requiring a significantly smaller time-step than the one used on595 narrow, requiring a significantly smaller time-step than the one used with an 595 596 unrotated operator. 596 597 … … 662 663 the thickness of the top model layer. 663 664 664 Due to interactions and mass exchange with other media($i.e.$ atmosphere, sea-ice, land),665 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components ($i.e.$ atmosphere, sea-ice, land), 665 666 the change in the heat and salt content of the surface layer of the ocean is due both 666 to the heat and salt fluxes crossing the sea surface and not linked with $F_{mass}$, the water 667 exchange with the other media, and to the heat and salt content of this water exchange. 668 In a forcoming release, these two parts, computed in the surface module (SBC), will included directly 667 to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) 668 and to the heat and salt content of the mass exchange. 669 \sgacomment{ the following does not apply to the release to which this documentation is 670 attached and so should not be included .... 671 In a forthcoming release, these two parts, computed in the surface module (SBC), will be included directly 669 672 in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux. 670 \gmcomment{ The specification of those fluxes is further detail in the SBC chapter (see \S\ref{SBC}). } 671 This change will provide a same forcing formulation for any tracers(including temperature and salinity).673 The specification of these fluxes is further detailed in the SBC chapter (see \S\ref{SBC}). 674 This change will provide a forcing formulation which is the same for any tracer (including temperature and salinity). 672 675 673 In the current version, the situation is a little bit more complicated. 676 In the current version, the situation is a little bit more complicated. } 674 677 The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following 675 678 forcing fields (used on tracers): 676 679 677 $\bullet$ $Q_{ns}$, the non solar part of the net surface heat flux that cross the sea surface678 ( difference between the total surface heat flux and the fraction of the short wave flux that680 $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 681 (i.e. the difference between the total surface heat flux and the fraction of the short wave flux that 679 682 penetrates into the water column, see \S\ref{TRA_qsr}) 680 683 681 684 $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) 682 685 683 $\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchange d686 $\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchange 684 687 685 688 $\bullet$ \textit{rnf}, the mass flux associated with runoff (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 686 689 687 The $\textit{emp}_S$ field is not simply the budget evaporation-precipitation+freezing-melting because688 the sea-ice is not currently embedded in the ocean but levitates above it. There is no tmass690 The $\textit{emp}_S$ field is not simply the budget of evaporation-precipitation+freezing-melting because 691 the sea-ice is not currently embedded in the ocean but levitates above it. There is no mass 689 692 exchanged between the sea-ice and the ocean. Instead we only take into account the salt 690 flux link to the fact that sea-ice has a non-sero salinity, and the concentration/dilution effect693 flux associated with the non-zero salinity of sea-ice, and the concentration/dilution effect 691 694 due to the freezing/melting (F/M) process. These two parts of the forcing are then converted into 692 a equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess,695 an equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess, 693 696 the surface boundary condition on temperature and salinity is applied as follows: 694 697 695 In the nonlinear free surface case (\key{vvl} is defined , \jp{lk\_vvl}=true):698 In the nonlinear free surface case (\key{vvl} is defined): 696 699 \begin{equation} \label{Eq_tra_sbc} 697 700 \begin{aligned} … … 704 707 \end{equation} 705 708 706 In the linear free surface case (\key{vvl} not defined , , \jp{lk\_vvl}=false):709 In the linear free surface case (\key{vvl} not defined): 707 710 \begin{equation} \label{Eq_tra_sbc_lin} 708 711 \begin{aligned} … … 713 716 \end{aligned} 714 717 \end{equation} 715 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time step 716 ($t-\rdt/2$ and $t+\rdt/2$). Such a time averaged prevents the excitation ofthe718 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps 719 ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the 717 720 divergence of odd and even time step (see \S\ref{STP}). 718 721 719 722 The two set of equations, \eqref{Eq_tra_sbc} and \eqref{Eq_tra_sbc_lin}, are obtained 720 723 by assuming that the temperature of precipitation and evaporation are equal to 721 the ocean surface temperature while their salinity is zero. Therefore, the heat content 722 of \textit{emp} budget must be added to the temperature equation in variable volume case, 723 while it does not appear in constant volume. Similarly, the \textit{emp} budget affects 724 the ocean surface salinity in constant volume case (through the concentration dilution effect) 725 while it does not appears explicitly in variable volume as salinity change will be 726 induced by volume change. In both constant and variable volume, surface salinity 727 will change with ice-ocean salt flux and F/M flux without mass exchanges 728 ($\textit{emp}_S - \textit{emp}$). 729 730 Note that concentration/dilution effect due to F/M is computed using 724 the ocean surface temperature and that their salinity is zero. Therefore, the heat content 725 of the \textit{emp} budget must be added to the temperature equation in the variable volume case, 726 while it does not appear in the constant volume case. Similarly, the \textit{emp} budget affects 727 the ocean surface salinity in the constant volume case (through the concentration dilution effect) 728 while it does not appears explicitly in the variable volume case since salinity change will be 729 induced by volume change. In both constant and variable volume cases, surface salinity 730 will change with ice-ocean salt flux and F/M flux (both contained in $\textit{emp}_S - \textit{emp}$) without mass exchanges. 731 732 Note that the concentration/dilution effect due to F/M is computed using 731 733 a constant ice salinity as well as a constant ocean salinity. 732 734 This approximation suppresses the correlation between \textit{SSS} … … 734 736 Indeed, if this approximation is not made, even if the F/M budget is zero 735 737 on average over the whole ocean domain and over the seasonal cycle, 736 the associated salt flux is not , since sea-surface salinity and F/M flux are738 the associated salt flux is not zero, since sea-surface salinity and F/M flux are 737 739 intrinsically correlated (high \textit{SSS} are found where freezing is 738 strong whilst low \textit{SSS} is usually associated with high melting areas .740 strong whilst low \textit{SSS} is usually associated with high melting areas). 739 741 740 742 Even using this approximation, an exact conservation of heat and salt content 741 743 is only achieved in the variable volume case. In the constant volume case, 742 there is a small unbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$.744 there is a small imbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$. 743 745 Nevertheless, the salt content variation is quite small and will not induce 744 a long term drift as there is no physical reason that$(\partial_t\eta - \textit{emp})$745 and \textit{SSS} are correlated \citep{Roullet_Madec_JGR00}.746 Note that, while quite small, the unbalance inconstant volume case is larger747 than the unbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}.748 This is the reason why the modified filter is not applied in constant volume case.746 a long term drift as there is no physical reason for $(\partial_t\eta - \textit{emp})$ 747 and \textit{SSS} to be correlated \citep{Roullet_Madec_JGR00}. 748 Note that, while quite small, the imbalance in the constant volume case is larger 749 than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. 750 This is the reason why the modified filter is not applied in the constant volume case. 749 751 750 752 % ------------------------------------------------------------------------------------------------------------- … … 759 761 760 762 When the penetrative solar radiation option is used (\np{ln\_flxqsr}=true), 761 the solar radiation penetrates the top few 10 meters of the ocean, otherwise762 all the heat flux is absorbed in the first ocean level (\np{ln\_flxqsr}=false).763 the solar radiation penetrates the top few tens of meters of the ocean. If it is not used 764 (\np{ln\_flxqsr}=false) all the heat flux is absorbed in the first ocean level. 763 765 Thus, in the former case a term is added to the time evolution equation of 764 temperature \eqref{Eq_PE_tra_T} whilstthe surface boundary condition is766 temperature \eqref{Eq_PE_tra_T} and the surface boundary condition is 765 767 modified to take into account only the non-penetrative part of the surface 766 768 heat flux: … … 780 782 The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. 781 783 The ocean is strongly absorbing for wavelengths longer than 700~nm and these 782 wavelengths contribute to heating the upper few 10centimetres. The fraction of $Q_{sr}$784 wavelengths contribute to heating the upper few tens of centimetres. The fraction of $Q_{sr}$ 783 785 that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified 784 786 through namelist parameter \np{rn\_abs}). It is assumed to penetrate the ocean 785 followinga decreasing exponential profile, with an e-folding depth scale, $\xi_0$,786 of a few 10 centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsr namlist).787 with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 788 of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsr namelist). 787 789 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy 788 propagates to depths where it contributes to a penetrating flux of solar energy and thus789 to local heating below the surface.790 The way this second part of the solar energy penetrates in the ocean depends on791 which formulation is chosen. In the simple 2-waveband s light penetration(\np{ln\_qsr\_2bd}=true)792 a chlorophyll-independent monochromatic formulation is alsochosen for the shorter wavelengths,790 propagates to larger depths where it contributes to 791 local heating. 792 The way this second part of the solar energy penetrates into the ocean depends on 793 which formulation is chosen. In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}=true) 794 a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 793 795 leading to the following expression \citep{Paulson1977}: 794 796 \begin{equation} \label{Eq_traqsr_iradiance} 795 797 I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] 796 798 \end{equation} 797 where $\xi_1$ is the second extinction length scale s associated with the shorter wavebands.798 It is usually chosen to be 23~m through\np{rn\_si0} namelist parameter.799 where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. 800 It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter. 799 801 The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in 800 802 Jerlov's (1968) classification (oligotrophic waters). … … 802 804 Such assumptions have been shown to provide a very crude and simplistic 803 805 representation of observed light penetration profiles (\cite{Morel_JGR88}, see also 804 Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on the805 partic ules concentration and itis spectrally selective. \cite{Morel_JGR88} has shown806 Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on 807 particle concentration and is spectrally selective. \cite{Morel_JGR88} has shown 806 808 that an accurate representation of light penetration can be provided by a 61 waveband 807 809 formulation. Unfortunately, such a model is very computationally expensive. 808 810 Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this 809 formulation in which visible light is split tedinto three wavebands: blue (400-500 nm),810 green (500-600 nm) and red (600-700nm). For each wave-band, the chlorophyll-depend ant811 formulation in which visible light is split into three wavebands: blue (400-500 nm), 812 green (500-600 nm) and red (600-700nm). For each wave-band, the chlorophyll-dependent 811 813 attenuation coefficient is fitted to the coefficients computed from the full spectral model 812 of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}) assuming813 the same power-law expression. As shown on Fig.\ref{Fig_traqsr_irradiance},814 this formulation, called RGB (Re ed-Green-Blue), reproduces quite closely815 the light penetration profiles predicted by the full spectal model with much faster816 comput ing efficiently, in contrast with the 2-bands formulation.814 of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}), assuming 815 the same power-law relationship. As shown in Fig.\ref{Fig_traqsr_irradiance}, 816 this formulation, called RGB (Red-Green-Blue), reproduces quite closely 817 the light penetration profiles predicted by the full spectal model, but with much greater 818 computational efficiency. The 2-bands formulation does not reproduce the full model very well. 817 819 818 820 The RGB formulation is used when \np{ln\_qsr\_rgb}=true. The RGB attenuation coefficients 819 ($i.e.$ the inverse of the extinction length scales) are tabulated over 61 nonuniform821 ($i.e.$ the inverses of the extinction length scales) are tabulated over 61 nonuniform 820 822 chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} 821 in \mdl{trc\_oce} module). Three type of chlorophyll can be usedin the RGB formulation:822 (1) a constant 0.05 g.Chl/L value everywhere (\np{nn\_chdta}=0) ; (2) observed823 time varying chlorophyll (\np{nn\_chdta}= 0) ; (3) simulated time varying chlorophyll824 by TOP biogeochemical model (\np{ln\_qsr\_bio}=true). In the lat er case, the RGB825 formulation is used to calculate dboth the phytoplankton light limitation in PISCES823 in \mdl{trc\_oce} module). Three types of chlorophyll can be chosen in the RGB formulation: 824 (1) a constant 0.05 g.Chl/L value everywhere (\np{nn\_chdta}=0) ; (2) an observed 825 time varying chlorophyll (\np{nn\_chdta}=1) ; (3) simulated time varying chlorophyll 826 by TOP biogeochemical model (\np{ln\_qsr\_bio}=true). In the latter case, the RGB 827 formulation is used to calculate both the phytoplankton light limitation in PISCES 826 828 or LOBSTER and the oceanic heating rate. 827 829 … … 829 831 is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. 830 832 831 When $z$-coordinate is preferred to$s$-coordinate, the depth of $w-$levels does833 When the $z$-coordinate is preferred to the $s$-coordinate, the depth of $w-$levels does 832 834 not significantly vary with location. The level at which the light has been totally 833 835 absorbed ($i.e.$ it is less than the computer precision) is computed once, 834 and the trend associated with the penetration of the solar radiation is only added untilthat level.836 and the trend associated with the penetration of the solar radiation is only added down to that level. 835 837 Finally, note that when the ocean is shallow ($<$ 200~m), part of the 836 838 solar radiation can reach the ocean floor. In this case, we have … … 841 843 \begin{figure}[!t] \label{Fig_traqsr_irradiance} \begin{center} 842 844 \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_Irradiance.pdf} 843 \caption{Penetration profile of the Downward solar irradiance844 calculated by four models. Two waveband s chlorophyll-independant formulation (blue),845 a chlorophyll-depend ant monochromatic formulation (green), 4 waveband RGB formulation (red),845 \caption{Penetration profile of the downward solar irradiance 846 calculated by four models. Two waveband chlorophyll-independent formulation (blue), 847 a chlorophyll-dependent monochromatic formulation (green), 4 waveband RGB formulation (red), 846 848 61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of 847 849 (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. From \citet{Lengaigne_al_CD07}.} … … 873 875 earth cooling. This flux is weak compared to surface fluxes (a mean 874 876 global value of $\sim0.1\;W/m^2$ \citep{Stein_Stein_Nat92}), but it is 875 systematically positive and acts on the densest water masses.877 systematically positive \sgacomment{positive definite?} and acts on the densest water masses. 876 878 Taking this flux into account in a global ocean model increases 877 879 the deepest overturning cell ($i.e.$ the one associated with the Antarctic 878 880 Bottom Water) by a few Sverdrups \citep{Emile-Geay_Madec_OS09}. 879 881 880 The presence o r not of geothermal heating is controlled by the namelist882 The presence of geothermal heating is controlled by the namelist 881 883 parameter \np{nn\_geoflx}. When this parameter is set to 1, a constant 882 884 geothermal heating is introduced whose value is given by the … … 904 906 and volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, 905 907 or North Atlantic Deep Water. $z$-coordinate models tend to overestimate the 906 entrainment, because the gravity flow is mixed downvertically by convection908 entrainment, because the gravity flow is mixed vertically by convection 907 909 as it goes ''downstairs'' following the step topography, sometimes over a thickness 908 910 much larger than the thickness of the observed gravity plume. A similar problem 909 occurs in the $s$-coordinate when the thickness of the bottom level varies in large910 proportionsdownstream of a sill \citep{Willebrand_al_PO01}, and the thickness911 occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly 912 downstream of a sill \citep{Willebrand_al_PO01}, and the thickness 911 913 of the plume is not resolved. 912 914 … … 914 916 \citet{Beckmann_Doscher1997}, is to allow a direct communication between 915 917 two adjacent bottom cells at different levels, whenever the densest water is 916 located above the less dense water. The communication can be by a diffusive 917 (diffusive BBL), a dvective fluxes(advective BBL), or both. In the current918 located above the less dense water. The communication can be by a diffusive flux 919 (diffusive BBL), an advective flux (advective BBL), or both. In the current 918 920 implementation of the BBL, only the tracers are modified, not the velocities. 919 921 Furthermore, it only connects ocean bottom cells, and therefore does not include 920 one of the improvmentintroduced by \citet{Campin_Goosse_Tel99}.922 the improvements introduced by \citet{Campin_Goosse_Tel99}. 921 923 922 924 % ------------------------------------------------------------------------------------------------------------- … … 927 929 928 930 When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), 929 the diffusive flux between two adjacent cells living at the ocean bottomis given by931 the diffusive flux between two adjacent cells at the ocean floor is given by 930 932 \begin{equation} \label{Eq_tra_bbl_diff} 931 933 {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T … … 933 935 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 934 936 and $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{Beckmann_Doscher1997}, 935 the latter is prescribed with a spatial dependence, $ e.g.$ in the conditional form937 the latter is prescribed with a spatial dependence, $i.e.$ in the conditional form 936 938 \begin{equation} \label{Eq_tra_bbl_coef} 937 939 A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l} … … 943 945 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist 944 946 parameter \np{rn\_ahtbbl} and usually set to a value much larger 945 than the one used on lateral mixing inopen ocean. The constraint in \eqref{Eq_tra_bbl_coef}946 implies that sigma-like diffusion only occurs when density above the sea floor, at the top of947 than the one used for lateral mixing in the open ocean. The constraint in \eqref{Eq_tra_bbl_coef} 948 implies that sigma-like diffusion only occurs when the density above the sea floor, at the top of 947 949 the slope, is larger than in the deeper ocean (see green arrow in Fig.\ref{Fig_bbl}). 948 950 In practice, this constraint is applied separately in the two horizontal directions, … … 951 953 \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta \,\nabla_\sigma S 952 954 \end{equation} 953 where $\rho$, $\alpha$ and $\beta$ are function of $\overline{T}^\sigma$,954 $\overline{S}^\sigma$ ,$\overline{H}^\sigma$, the along bottom mean temperature,955 where $\rho$, $\alpha$ and $\beta$ are functions of $\overline{T}^\sigma$, 956 $\overline{S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature, 955 957 salinity and depth, respectively. 956 958 … … 961 963 \label{TRA_bbl_adv} 962 964 965 \sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following 966 if this is not what is meant then "downwards sloping flow" is also a possibility"} 963 967 964 968 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 965 969 \begin{figure}[!t] \label{Fig_bbl} \begin{center} 966 970 \includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_BBL_adv.pdf} 967 \caption{Advective/diffusive Bottom Boundary Layer. The BB lparameterisation is971 \caption{Advective/diffusive Bottom Boundary Layer. The BBL parameterisation is 968 972 activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$. 969 973 Red arrows indicate the additional overturning circulation due to the advective BBL. 970 The transport of the downslop ingflow is defined either as the transport of the bottom974 The transport of the downslope flow is defined either as the transport of the bottom 971 975 ocean cell (black arrow), or as a function of the along slope density gradient. 972 The green arrow indicates the diffusive BBL flux connecting directly$kup$ and $kdwn$976 The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$ 973 977 ocean bottom cells. 974 978 connection} … … 988 992 water to move down the slope. 989 993 990 \np{nn\_bbl\_adv} = 1 : the downslop ingvelocity is chosen to be the Eulerian994 \np{nn\_bbl\_adv} = 1 : the downslope velocity is chosen to be the Eulerian 991 995 ocean velocity just above the topographic step (see black arrow in Fig.\ref{Fig_bbl}) 992 996 \citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection … … 995 999 greater depth ($i.e.$ $\vect{U} \cdot \nabla H>0$). 996 1000 997 \np{nn\_bbl\_adv} = 2 : the downslop ingvelocity is chosen to be proportional to $\Delta \rho$,998 the density difference between the top and downcell densities \citep{Campin_Goosse_Tel99}.1001 \np{nn\_bbl\_adv} = 2 : the downslope velocity is chosen to be proportional to $\Delta \rho$, 1002 the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}. 999 1003 The advection is allowed only if dense water overlies less dense water on the slope ($i.e.$ 1000 1004 $\nabla_\sigma \rho \cdot \nabla H<0$). For example, the resulting transport of the 1001 downslop ingflow, here in the $i$-direction (Fig.\ref{Fig_bbl}), is simply given by the1005 downslope flow, here in the $i$-direction (Fig.\ref{Fig_bbl}), is simply given by the 1002 1006 following expression: 1003 1007 \begin{equation} \label{Eq_bbl_Utr} 1004 1008 u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o} e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right) 1005 1009 \end{equation} 1006 where $\gamma$, expressed in second , is the coefficient of proportionality1010 where $\gamma$, expressed in seconds, is the coefficient of proportionality 1007 1011 provided as \np{rn\_gambbl}, a namelist parameter, and \textit{kup} and \textit{kdwn} 1008 are the vertical index of the top and bottomcells, respectively.1012 are the vertical index of the higher and lower cells, respectively. 1009 1013 The parameter $\gamma$ should take a different value for each bathymetric 1010 step . But,for simplicity, and because no direct estimation of this parameter is1011 available, a uniform value has been retained. The possible values for $\gamma$1014 step, but for simplicity, and because no direct estimation of this parameter is 1015 available, a uniform value has been assumed. The possible values for $\gamma$ 1012 1016 range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}. 1013 1017 1014 The scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl} )$1018 Scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl} )$ 1015 1019 using the upwind scheme. Such a diffusive advective scheme has been chosen 1016 to mimic the entrainment between the downslop ingplume and the surrounding1017 water at intermediate depth . The entrainment is replaced by the vertical mixing1018 i ncludedin the advection scheme. Let us consider as an example the1019 case display in Fig.\ref{Fig_bbl} where the density at level $(i,kup)$ is1020 to mimic the entrainment between the downslope plume and the surrounding 1021 water at intermediate depths. The entrainment is replaced by the vertical mixing 1022 implicit in the advection scheme. Let us consider as an example the 1023 case displayed in Fig.\ref{Fig_bbl} where the density at level $(i,kup)$ is 1020 1024 larger than the one at level $(i,kdwn)$. The advective BBL scheme 1021 1025 modifies the tracer time tendency of the ocean cells near the 1022 topographic step by the downslop ingflow \eqref{Eq_bbl_dw},1026 topographic step by the downslope flow \eqref{Eq_bbl_dw}, 1023 1027 the horizontal \eqref{Eq_bbl_hor} and the upward \eqref{Eq_bbl_up} 1024 1028 return flows as follows: … … 1062 1066 are given temperature and salinity fields (usually a climatology). 1063 1067 The restoring term is added when \key{tradmp} is defined. 1064 It also requires that both \key{ temdta} and \key{saldta} are defined1068 It also requires that both \key{dtatem} and \key{dtasal} are defined 1065 1069 ($i.e.$ that $T_o$ and $S_o$ are read). The restoring coefficient 1066 $ S_o$ is a three-dimensional array initialized by the user in routine1070 $\gamma$ is a three-dimensional array initialized by the user in routine 1067 1071 \rou{dtacof} also located in module \mdl{tradmp}. 1068 1072 … … 1074 1078 field for a passive tracer study). The first case applies to regional 1075 1079 models that have artificial walls instead of open boundaries. 1076 In the vicinity of these walls, $ S_o$ takes large values (equivalent to1080 In the vicinity of these walls, $\gamma$ takes large values (equivalent to 1077 1081 a time scale of a few days) whereas it is zero in the interior of the 1078 1082 model domain. The second case corresponds to the use of the robust 1079 1083 diagnostic method \citep{Sarmiento1982}. It allows us to find the velocity 1080 field consistent with the model dynamics whilst having a $T$ -$S$ field1081 close to a given climatological field ($T_o -S_o$). The time scale1084 field consistent with the model dynamics whilst having a $T$, $S$ field 1085 close to a given climatological field ($T_o$, $S_o$). The time scale 1082 1086 associated with $S_o$ is generally not a constant but spatially varying 1083 1087 in order to respect other properties. For example, it is usually set to zero … … 1085 1089 \citep{Madec_al_JPO96} and in the equatorial region 1086 1090 \citep{Reverdin1991, Fujio1991, Marti_PhD92} since these two regions 1087 have a short time scale of adjustment; while smaller $ S_o$ are used1091 have a short time scale of adjustment; while smaller $\gamma$ are used 1088 1092 in the deep ocean where the typical time scale is long \citep{Sarmiento1982}. 1089 1093 In addition the time scale is reduced (even to zero) along the western … … 1091 1095 structure in equilibrium with its physics. 1092 1096 The choice of the shape of the Newtonian damping is controlled by two 1093 namelist parameters \np{ nn\_zdmp}. The former allows to specifiedthe1094 width of the equatorial band in which no damping is applied as well asa decrease1095 in vicinity of the coast and a damping everywhere in the Red and Med Seas,1096 whereas the latter set a damping actingin the mixed layer or not.1097 namelist parameters \np{??} and \np{nn\_zdmp}. The former allows us to specify: the 1098 width of the equatorial band in which no damping is applied; a decrease 1099 in the vicinity of the coast; and a damping everywhere in the Red and Med Seas. 1100 The latter sets whether damping should act in the mixed layer or not. 1097 1101 The time scale associated with the damping depends on the depth as 1098 1102 a hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as
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