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branches/UKMO/dev_r5518_GC3p0_package/DOC/TexFiles/Chapters/Chap_TRA.tex
r5102 r6440 1 1 % ================================================================ 2 % Chapter 1 �Ocean Tracers (TRA)2 % Chapter 1 ——— Ocean Tracers (TRA) 3 3 % ================================================================ 4 4 \chapter{Ocean Tracers (TRA)} … … 36 36 (BBL) parametrisation, and an internal damping (DMP) term. The terms QSR, 37 37 BBC, BBL and DMP are optional. The external forcings and parameterisations 38 require complex inputs and complex calculations ( e.g.bulk formulae, estimation38 require complex inputs and complex calculations ($e.g.$ bulk formulae, estimation 39 39 of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and 40 40 described in chapters \S\ref{SBC}, \S\ref{LDF} and \S\ref{ZDF}, respectively. 41 Note that \mdl{tranpc}, the non-penetrative convection module, although 42 (temporarily) located in the NEMO/OPA/TRA directory, is described with the 43 model vertical physics (ZDF). 44 %%% 45 \gmcomment{change the position of eosbn2 in the reference code} 46 %%% 41 Note that \mdl{tranpc}, the non-penetrative convection module, although 42 located in the NEMO/OPA/TRA directory as it directly modifies the tracer fields, 43 is described with the model vertical physics (ZDF) together with other available 44 parameterization of convection. 47 45 48 46 In the present chapter we also describe the diagnostic equations used to compute 49 the sea-water properties (density, Brunt-V ais\"{a}l\"{a} frequency, specific heat and47 the sea-water properties (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and 50 48 freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}). 51 49 … … 56 54 found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory. 57 55 58 The user has the option of extracting each tendency term on the rhsof the tracer59 equation for output (\ key{trdtra} is defined), as described in Chap.~\ref{MISC}.56 The user has the option of extracting each tendency term on the RHS of the tracer 57 equation for output (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}~=~true), as described in Chap.~\ref{DIA}. 60 58 61 59 $\ $\newline % force a new ligne … … 125 123 \end{description} 126 124 In all cases, this boundary condition retains local conservation of tracer. 127 Global conservation is obtained in both rigid-lid and non-linear free surface128 cases, but not in the linear free surface case. Nevertheless, in the latter 129 case,it is achieved to a good approximation since the non-conservative125 Global conservation is obtained in non-linear free surface case, 126 but \textit{not} in the linear free surface case. Nevertheless, in the latter case, 127 it is achieved to a good approximation since the non-conservative 130 128 term is the product of the time derivative of the tracer and the free surface 131 129 height, two quantities that are not correlated (see \S\ref{PE_free_surface}, … … 133 131 134 132 The velocity field that appears in (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco}) 135 is the centred (\textit{now}) \textit{eulerian} ocean velocity (see Chap.~\ref{DYN}). 136 When eddy induced velocity (\textit{eiv}) parameterisation is used it is the \textit{now} 137 \textit{effective} velocity ($i.e.$ the sum of the eulerian and eiv velocities) which is used. 133 is the centred (\textit{now}) \textit{effective} ocean velocity, $i.e.$ the \textit{eulerian} velocity 134 (see Chap.~\ref{DYN}) plus the eddy induced velocity (\textit{eiv}) 135 and/or the mixed layer eddy induced velocity (\textit{eiv}) 136 when those parameterisations are used (see Chap.~\ref{LDF}). 138 137 139 138 The choice of an advection scheme is made in the \textit{\ngn{nam\_traadv}} namelist, by … … 146 145 147 146 Note that 148 (1) cen2 , cen4and TVD schemes require an explicit diffusion147 (1) cen2 and TVD schemes require an explicit diffusion 149 148 operator while the other schemes are diffusive enough so that they do not 150 149 require additional diffusion ; 151 (2) cen2, cen4,MUSCL2, and UBS are not \textit{positive} schemes150 (2) cen2, MUSCL2, and UBS are not \textit{positive} schemes 152 151 \footnote{negative values can appear in an initially strictly positive tracer field 153 152 which is advected} … … 189 188 temperature is close to the freezing point). 190 189 This combined scheme has been included for specific grid points in the ORCA2 191 and ORCA4 configurationsonly. This is an obsolescent feature as the recommended190 configuration only. This is an obsolescent feature as the recommended 192 191 advection scheme for the ORCA configuration is TVD (see \S\ref{TRA_adv_tvd}). 193 192 … … 196 195 have this order of accuracy. \gmcomment{Note also that ... blah, blah} 197 196 198 % -------------------------------------------------------------------------------------------------------------199 % 4nd order centred scheme200 % -------------------------------------------------------------------------------------------------------------201 \subsection [$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4})]202 {$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=true)}203 \label{TRA_adv_cen4}204 205 In the $4^{th}$ order formulation (to be implemented), tracer values are206 evaluated at velocity points as a $4^{th}$ order interpolation, and thus depend on207 the four neighbouring $T$-points. For example, in the $i$-direction:208 \begin{equation} \label{Eq_tra_adv_cen4}209 \tau _u^{cen4}210 =\overline{ T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2}211 \end{equation}212 213 Strictly speaking, the cen4 scheme is not a $4^{th}$ order advection scheme214 but a $4^{th}$ order evaluation of advective fluxes, since the divergence of215 advective fluxes \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. The phrase ``$4^{th}$216 order scheme'' used in oceanographic literature is usually associated217 with the scheme presented here. Introducing a \textit{true} $4^{th}$ order advection218 scheme is feasible but, for consistency reasons, it requires changes in the219 discretisation of the tracer advection together with changes in both the220 continuity equation and the momentum advection terms.221 222 A direct consequence of the pseudo-fourth order nature of the scheme is that223 it is not non-diffusive, i.e. the global variance of a tracer is not preserved using224 \textit{cen4}. Furthermore, it must be used in conjunction with an explicit225 diffusion operator to produce a sensible solution. The time-stepping is also226 performed using a leapfrog scheme in conjunction with an Asselin time-filter,227 so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer.228 229 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an230 additional hypothesis must be made to evaluate $\tau _u^{cen4}$. This231 hypothesis usually reduces the order of the scheme. Here we choose to set232 the gradient of $T$ across the boundary to zero. Alternative conditions can be233 specified, such as a reduction to a second order scheme for these near boundary234 grid points.235 197 236 198 % ------------------------------------------------------------------------------------------------------------- … … 270 232 used for the diffusive part. 271 233 234 An additional option has been added controlled by \np{ln\_traadv\_tvd\_zts}. 235 By setting this logical to true, a TVD scheme is used on both horizontal and vertical direction, 236 but on the latter, a split-explicit time stepping is used, with 5 sub-timesteps. 237 This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}. 238 Note that in this case, a similar split-explicit time stepping should be used on 239 vertical advection of momentum to ensure a better stability (see \np{ln\_dynzad\_zts} in \S\ref{DYN_zad}). 240 241 272 242 % ------------------------------------------------------------------------------------------------------------- 273 243 % MUSCL scheme … … 296 266 297 267 For an ocean grid point adjacent to land and where the ocean velocity is 298 directed toward land, two choices are available: an upstream flux 299 (\np{ln\_traadv\_muscl}=true) or a second order flux 300 (\np{ln\_traadv\_muscl2}=true). Note that the latter choice does not ensure 301 the \textit{positive} character of the scheme. Only the former can be used 302 on both active and passive tracers. The two MUSCL schemes are implemented 303 in the \mdl{traadv\_tvd} and \mdl{traadv\_tvd2} modules. 268 directed toward land, two choices are available: an upstream flux (\np{ln\_traadv\_muscl}=true) 269 or a second order flux (\np{ln\_traadv\_muscl2}=true). 270 Note that the latter choice does not ensure the \textit{positive} character of the scheme. 271 Only the former can be used on both active and passive tracers. 272 The two MUSCL schemes are implemented in the \mdl{traadv\_tvd} and \mdl{traadv\_tvd2} modules. 273 274 Note that when using np{ln\_traadv\_msc\_ups}~=~true in addition to \np{ln\_traadv\_muscl}=true, 275 the MUSCL fluxes are replaced by upstream fluxes in vicinity of river mouths. 304 276 305 277 % ------------------------------------------------------------------------------------------------------------- … … 416 388 direction (as for the UBS case) should be implemented to restore this property. 417 389 418 419 % -------------------------------------------------------------------------------------------------------------420 % PPM scheme421 % -------------------------------------------------------------------------------------------------------------422 \subsection [Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm})]423 {Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=true)}424 \label{TRA_adv_ppm}425 426 The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984)427 \sgacomment{reference?}428 is based on a quadradic piecewise construction. Like the QCK scheme, it is associated429 with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented430 in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference431 version 3.3.432 390 433 391 % ================================================================ … … 464 422 surfaces is given by: 465 423 \begin{equation} \label{Eq_tra_ldf_lap} 466 D_T^{lT} =\frac{1}{b_t T} \left( \;424 D_T^{lT} =\frac{1}{b_t} \left( \; 467 425 \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right] 468 426 + \delta _{j}\left[ A_v^{lT} \; \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right] \;\right) … … 661 619 the thickness of the top model layer. 662 620 663 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components ($i.e.$ atmosphere, sea-ice, land), 664 the change in the heat and salt content of the surface layer of the ocean is due both 665 to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) 666 and to the heat and salt content of the mass exchange. 667 \sgacomment{ the following does not apply to the release to which this documentation is 668 attached and so should not be included .... 669 In a forthcoming release, these two parts, computed in the surface module (SBC), will be included directly 670 in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux. 671 The specification of these fluxes is further detailed in the SBC chapter (see \S\ref{SBC}). 672 This change will provide a forcing formulation which is the same for any tracer (including temperature and salinity). 673 674 In the current version, the situation is a little bit more complicated. } 621 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components 622 ($i.e.$ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer 623 of the ocean is due both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) 624 and to the heat and salt content of the mass exchange. They are both included directly in $Q_{ns}$, 625 the surface heat flux, and $F_{salt}$, the surface salt flux (see \S\ref{SBC} for further details). 626 By doing this, the forcing formulation is the same for any tracer (including temperature and salinity). 675 627 676 628 The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following … … 679 631 $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 680 632 (i.e. the difference between the total surface heat flux and the fraction of the short wave flux that 681 penetrates into the water column, see \S\ref{TRA_qsr}) 682 683 $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) 684 685 $\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchange 686 687 $\bullet$ \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) 688 689 The $\textit{emp}_S$ field is not simply the budget of evaporation-precipitation+freezing-melting because 690 the sea-ice is not currently embedded in the ocean but levitates above it. There is no mass 691 exchanged between the sea-ice and the ocean. Instead we only take into account the salt 692 flux associated with the non-zero salinity of sea-ice, and the concentration/dilution effect 693 due to the freezing/melting (F/M) process. These two parts of the forcing are then converted into 694 an equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess, 695 the surface boundary condition on temperature and salinity is applied as follows: 696 697 In the nonlinear free surface case (\key{vvl} is defined): 633 penetrates into the water column, see \S\ref{TRA_qsr}) plus the heat content associated with 634 of the mass exchange with the atmosphere and lands. 635 636 $\bullet$ $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 637 638 $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) 639 and possibly with the sea-ice and ice-shelves. 640 641 $\bullet$ \textit{rnf}, the mass flux associated with runoff 642 (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 643 644 $\bullet$ \textit{fwfisf}, the mass flux associated with ice shelf melt, (see \S\ref{SBC_isf} for further details 645 on how the ice shelf melt is computed and applied).\\ 646 647 In the non-linear free surface case (\key{vvl} is defined), the surface boundary condition 648 on temperature and salinity is applied as follows: 698 649 \begin{equation} \label{Eq_tra_sbc} 650 \begin{aligned} 651 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ 652 & F^S =\frac{ 1 }{\rho _o \, \left. e_{3t} \right|_{k=1} } &\overline{ \textit{sfx} }^t & \\ 653 \end{aligned} 654 \end{equation} 655 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps 656 ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the 657 divergence of odd and even time step (see \S\ref{STP}). 658 659 In the linear free surface case (\key{vvl} is \textit{not} defined), 660 an additional term has to be added on both temperature and salinity. 661 On temperature, this term remove the heat content associated with mass exchange 662 that has been added to $Q_{ns}$. On salinity, this term mimics the concentration/dilution effect that 663 would have resulted from a change in the volume of the first level. 664 The resulting surface boundary condition is applied as follows: 665 \begin{equation} \label{Eq_tra_sbc_lin} 699 666 \begin{aligned} 700 667 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } … … 702 669 % 703 670 & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 704 &\overline{ \left( (\textit{emp}_S - \textit{emp})\;\left. S \right|_{k=1} \right) }^t & \\671 &\overline{ \left( \;\textit{sfx} - \textit{emp} \;\left. S \right|_{k=1} \right) }^t & \\ 705 672 \end{aligned} 706 673 \end{equation} 707 708 In the linear free surface case (\key{vvl} not defined): 709 \begin{equation} \label{Eq_tra_sbc_lin} 710 \begin{aligned} 711 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ 712 % 713 & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 714 &\overline{ \left( \textit{emp}_S\;\left. S \right|_{k=1} \right) }^t & \\ 715 \end{aligned} 716 \end{equation} 717 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps 718 ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the 719 divergence of odd and even time step (see \S\ref{STP}). 720 721 The two set of equations, \eqref{Eq_tra_sbc} and \eqref{Eq_tra_sbc_lin}, are obtained 722 by assuming that the temperature of precipitation and evaporation are equal to 723 the ocean surface temperature and that their salinity is zero. Therefore, the heat content 724 of the \textit{emp} budget must be added to the temperature equation in the variable volume case, 725 while it does not appear in the constant volume case. Similarly, the \textit{emp} budget affects 726 the ocean surface salinity in the constant volume case (through the concentration dilution effect) 727 while it does not appears explicitly in the variable volume case since salinity change will be 728 induced by volume change. In both constant and variable volume cases, surface salinity 729 will change with ice-ocean salt flux and F/M flux (both contained in $\textit{emp}_S - \textit{emp}$) without mass exchanges. 730 731 Note that the concentration/dilution effect due to F/M is computed using 732 a constant ice salinity as well as a constant ocean salinity. 733 This approximation suppresses the correlation between \textit{SSS} 734 and F/M flux, allowing the ice-ocean salt exchanges to be conservative. 735 Indeed, if this approximation is not made, even if the F/M budget is zero 736 on average over the whole ocean domain and over the seasonal cycle, 737 the associated salt flux is not zero, since sea-surface salinity and F/M flux are 738 intrinsically correlated (high \textit{SSS} are found where freezing is 739 strong whilst low \textit{SSS} is usually associated with high melting areas). 740 741 Even using this approximation, an exact conservation of heat and salt content 742 is only achieved in the variable volume case. In the constant volume case, 743 there is a small imbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$. 744 Nevertheless, the salt content variation is quite small and will not induce 745 a long term drift as there is no physical reason for $(\partial_t\eta - \textit{emp})$ 746 and \textit{SSS} to be correlated \citep{Roullet_Madec_JGR00}. 747 Note that, while quite small, the imbalance in the constant volume case is larger 674 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. 675 In the linear free surface case, there is a small imbalance. The imbalance is larger 748 676 than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. 749 This is the reason why the modified filter is not applied in the constant volume case.677 This is the reason why the modified filter is not applied in the linear free surface case (see \S\ref{STP}). 750 678 751 679 % ------------------------------------------------------------------------------------------------------------- … … 821 749 ($i.e.$ the inverses of the extinction length scales) are tabulated over 61 nonuniform 822 750 chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} 823 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 828 or LOBSTER and the oceanic heating rate. 829 751 in \mdl{trc\_oce} module). Four types of chlorophyll can be chosen in the RGB formulation: 752 \begin{description} 753 \item[\np{nn\_chdta}=0] 754 a constant 0.05 g.Chl/L value everywhere ; 755 \item[\np{nn\_chdta}=1] 756 an observed time varying chlorophyll deduced from satellite surface ocean color measurement 757 spread uniformly in the vertical direction ; 758 \item[\np{nn\_chdta}=2] 759 same as previous case except that a vertical profile of chlorophyl is used. 760 Following \cite{Morel_Berthon_LO89}, the profile is computed from the local surface chlorophyll value ; 761 \item[\np{ln\_qsr\_bio}=true] 762 simulated time varying chlorophyll by TOP biogeochemical model. 763 In this case, the RGB formulation is used to calculate both the phytoplankton 764 light limitation in PISCES or LOBSTER and the oceanic heating rate. 765 \end{description} 830 766 The trend in \eqref{Eq_tra_qsr} associated with the penetration of the solar radiation 831 767 is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. … … 859 795 \label{TRA_bbc} 860 796 %--------------------------------------------nambbc-------------------------------------------------------- 861 \namdisplay{nam tra_bbc}797 \namdisplay{nambbc} 862 798 %-------------------------------------------------------------------------------------------------------------- 863 799 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 1103 1039 \subsection[DMP\_TOOLS]{Generating resto.nc using DMP\_TOOLS} 1104 1040 1105 DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$. Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled and run on the same machine as the NEMO model. A mesh\_mask.nc file for the model configuration is required as an input. This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1. The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work. The \nl{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for the restoration coefficient. 1041 DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$. 1042 Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled 1043 and run on the same machine as the NEMO model. A mesh\_mask.nc file for the model configuration is required as an input. 1044 This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1. 1045 The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work. 1046 The \nl{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for the restoration coefficient. 1106 1047 1107 1048 %--------------------------------------------nam_dmp_create------------------------------------------------- … … 1111 1052 \np{cp\_cfg}, \np{cp\_cpz}, \np{jp\_cfg} and \np{jperio} specify the model configuration being used and should be the same as specified in \nl{namcfg}. The variable \nl{lzoom} is used to specify that the damping is being used as in case \textit{a} above to provide boundary conditions to a zoom configuration. In the case of the arctic or antarctic zoom configurations this includes some specific treatment. Otherwise damping is applied to the 6 grid points along the ocean boundaries. The open boundaries are specified by the variables \np{lzoom\_n}, \np{lzoom\_e}, \np{lzoom\_s}, \np{lzoom\_w} in the \nl{nam\_zoom\_dmp} name list. 1112 1053 1113 The remaining switch namelist variables determine the spatial variation of the restoration coefficient in non-zoom configurations. \np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain. \np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea for the ORCA4, ORCA2 and ORCA05 configurations. If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as a function of the model number. This option is included to allow backwards compatability of the ORCA2 reference configurations with previous model versions. \np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines. This option only has an effect if \np{ln\_full\_field} is true. \np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer. Finally \np{ln\_custom} specifies that the custom module will be called. This module is contained in the file custom.F90 and can be edited by users. For example damping could be applied in a specific region. 1114 1115 The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \np{nn\_hdmp}. Equatorward of this latitude the restoration coefficient will be zero with a smooth transition to the full values of a 10$^{\circ}$ latitud band. This is often used because of the short adjustment time scale in the equatorial region \citep{Reverdin1991, Fujio1991, Marti_PhD92}. The time scale associated with the damping depends on the depth as a hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}. 1054 The remaining switch namelist variables determine the spatial variation of the restoration coefficient in non-zoom configurations. 1055 \np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain. 1056 \np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea 1057 for the ORCA4, ORCA2 and ORCA05 configurations. 1058 If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as 1059 a function of the model number. This option is included to allow backwards compatability of the ORCA2 reference 1060 configurations with previous model versions. 1061 \np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines. 1062 This option only has an effect if \np{ln\_full\_field} is true. 1063 \np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer. 1064 Finally \np{ln\_custom} specifies that the custom module will be called. 1065 This module is contained in the file custom.F90 and can be edited by users. For example damping could be applied in a specific region. 1066 1067 The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \np{nn\_hdmp}. 1068 Equatorward of this latitude the restoration coefficient will be zero with a smooth transition to 1069 the full values of a 10$^{\circ}$ latitud band. 1070 This is often used because of the short adjustment time scale in the equatorial region 1071 \citep{Reverdin1991, Fujio1991, Marti_PhD92}. The time scale associated with the damping depends on the depth as a 1072 hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}. 1116 1073 1117 1074 % ================================================================ … … 1167 1124 % Equation of State 1168 1125 % ------------------------------------------------------------------------------------------------------------- 1169 \subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)}1126 \subsection{Equation Of Seawater (\np{nn\_eos} = -1, 0, or 1)} 1170 1127 \label{TRA_eos} 1171 1128 1172 It is necessary to know the equation of state for the ocean very accurately 1173 to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency), 1174 particularly in the deep ocean. The ocean seawater volumic mass, $\rho$, 1175 abusively called density, is a non linear empirical function of \textit{in situ} 1176 temperature, salinity and pressure. The reference equation of state is that 1177 defined by the Joint Panel on Oceanographic Tables and Standards 1178 \citep{UNESCO1983}. It was the standard equation of state used in early 1179 releases of OPA. However, even though this computation is fully vectorised, 1180 it is quite time consuming ($15$ to $20${\%} of the total CPU time) since 1181 it requires the prior computation of the \textit{in situ} temperature from the 1182 model \textit{potential} temperature using the \citep{Bryden1973} polynomial 1183 for adiabatic lapse rate and a $4^th$ order Runge-Kutta integration scheme. 1184 Since OPA6, we have used the \citet{JackMcD1995} equation of state for 1185 seawater instead. It allows the computation of the \textit{in situ} ocean density 1186 directly as a function of \textit{potential} temperature relative to the surface 1187 (an \NEMO variable), the practical salinity (another \NEMO variable) and the 1188 pressure (assuming no pressure variation along geopotential surfaces, $i.e.$ 1189 the pressure in decibars is approximated by the depth in meters). 1190 Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state 1191 have exactly the same except that the values of the various coefficients have 1192 been adjusted by \citet{JackMcD1995} in order to directly use the \textit{potential} 1193 temperature instead of the \textit{in situ} one. This reduces the CPU time of the 1194 \textit{in situ} density computation to about $3${\%} of the total CPU time, 1195 while maintaining a quite accurate equation of state. 1196 1197 In the computer code, a \textit{true} density anomaly, $d_a= \rho / \rho_o - 1$, 1198 is computed, with $\rho_o$ a reference volumic mass. Called \textit{rau0} 1199 in the code, $\rho_o$ is defined in \mdl{phycst}, and a value of $1,035~Kg/m^3$. 1129 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship 1130 linking seawater density, $\rho$, to a number of state variables, 1131 most typically temperature, salinity and pressure. 1132 Because density gradients control the pressure gradient force through the hydrostatic balance, 1133 the equation of state provides a fundamental bridge between the distribution of active tracers 1134 and the fluid dynamics. Nonlinearities of the EOS are of major importance, in particular 1135 influencing the circulation through determination of the static stability below the mixed layer, 1136 thus controlling rates of exchange between the atmosphere and the ocean interior \citep{Roquet_JPO2015}. 1137 Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{UNESCO1983}) 1138 or TEOS-10 \citep{TEOS10} standards should be used anytime a simulation of the real 1139 ocean circulation is attempted \citep{Roquet_JPO2015}. 1140 The use of TEOS-10 is highly recommended because 1141 \textit{(i)} it is the new official EOS, 1142 \textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and 1143 \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature 1144 and practical salinity for EOS-980, both variables being more suitable for use as model variables 1145 \citep{TEOS10, Graham_McDougall_JPO13}. 1146 EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility. 1147 For process studies, it is often convenient to use an approximation of the EOS. To that purposed, 1148 a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available. 1149 1150 In the computer code, a density anomaly, $d_a= \rho / \rho_o - 1$, 1151 is computed, with $\rho_o$ a reference density. Called \textit{rau0} 1152 in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. 1200 1153 This is a sensible choice for the reference density used in a Boussinesq ocean 1201 1154 climate model, as, with the exception of only a small percentage of the ocean, 1202 density in the World Ocean varies by no more than 2$\%$ from $1,035~kg/m^3$ 1203 \citep{Gill1982}. 1204 1205 Options are defined through the \ngn{nameos} namelist variables. 1206 The default option (namelist parameter \np{nn\_eos}=0) is the \citet{JackMcD1995} 1207 equation of state. Its use is highly recommended. However, for process studies, 1208 it is often convenient to use a linear approximation of the density. 1155 density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}. 1156 1157 Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} 1158 which controls the EOS used (=-1 for TEOS10 ; =0 for EOS-80 ; =1 for S-EOS). 1159 \begin{description} 1160 1161 \item[\np{nn\_eos}$=-1$] the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used. 1162 The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, 1163 but it is optimized for a boussinesq fluid and the polynomial expressions have simpler 1164 and more computationally efficient expressions for their derived quantities 1165 which make them more adapted for use in ocean models. 1166 Note that a slightly higher precision polynomial form is now used replacement of the TEOS-10 1167 rational function approximation for hydrographic data analysis \citep{TEOS10}. 1168 A key point is that conservative state variables are used: 1169 Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: $\degres C$, notation: $\Theta$). 1170 The pressure in decibars is approximated by the depth in meters. 1171 With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. It is set to 1172 $C_p=3991.86795711963~J\,Kg^{-1}\,\degres K^{-1}$, according to \citet{TEOS10}. 1173 1174 Choosing polyTEOS10-bsq implies that the state variables used by the model are 1175 $\Theta$ and $S_A$. In particular, the initial state deined by the user have to be given as 1176 \textit{Conservative} Temperature and \textit{Absolute} Salinity. 1177 In addition, setting \np{ln\_useCT} to \textit{true} convert the Conservative SST to potential SST 1178 prior to either computing the air-sea and ice-sea fluxes (forced mode) 1179 or sending the SST field to the atmosphere (coupled mode). 1180 1181 \item[\np{nn\_eos}$=0$] the polyEOS80-bsq equation of seawater is used. 1182 It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized 1183 to accurately fit EOS80 (Roquet, personal comm.). The state variables used in both the EOS80 1184 and the ocean model are: 1185 the Practical Salinity ((unit: psu, notation: $S_p$)) and Potential Temperature (unit: $\degres C$, notation: $\theta$). 1186 The pressure in decibars is approximated by the depth in meters. 1187 With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, 1188 salinity and pressure \citep{UNESCO1983}. Nevertheless, a severe assumption is made in order to 1189 have a heat content ($C_p T_p$) which is conserved by the model: $C_p$ is set to a constant 1190 value, the TEOS10 value. 1191 1192 \item[\np{nn\_eos}$=1$] a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen, 1193 the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.) 1194 (see also \citet{Roquet_JPO2015}). It provides a simplistic linear representation of both 1195 cabbeling and thermobaricity effects which is enough for a proper treatment of the EOS 1196 in theoretical studies \citep{Roquet_JPO2015}. 1209 1197 With such an equation of state there is no longer a distinction between 1210 \textit{in situ} and \textit{potential} density and both cabbeling and thermobaric 1211 effects are removed. 1212 Two linear formulations are available: a function of $T$ only (\np{nn\_eos}=1) 1213 and a function of both $T$ and $S$ (\np{nn\_eos}=2): 1214 \begin{equation} \label{Eq_tra_eos_linear} 1198 \textit{conservative} and \textit{potential} temperature, as well as between \textit{absolute} 1199 and \textit{practical} salinity. 1200 S-EOS takes the following expression: 1201 \begin{equation} \label{Eq_tra_S-EOS} 1215 1202 \begin{split} 1216 d_a(T) &= \rho (T) / \rho_o - 1 = \ 0.0285 - \alpha \;T \\ 1217 d_a(T,S) &= \rho (T,S) / \rho_o - 1 = \ \beta \; S - \alpha \;T 1203 d_a(T,S,z) = ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a \\ 1204 & + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a \\ 1205 & - \nu \; T_a \; S_a \; ) \; / \; \rho_o \\ 1206 with \ \ T_a = T-10 \; ; & \; S_a = S-35 \; ;\; \rho_o = 1026~Kg/m^3 1218 1207 \end{split} 1219 1208 \end{equation} 1220 where $\alpha$ and $\beta$ are the thermal and haline expansion 1221 coefficients, and $\rho_o$, the reference volumic mass, $rau0$. 1222 ($\alpha$ and $\beta$ can be modified through the \np{rn\_alpha} and 1223 \np{rn\_beta} namelist variables). Note that when $d_a$ is a function 1224 of $T$ only (\np{nn\_eos}=1), the salinity is a passive tracer and can be 1225 used as such. 1226 1227 % ------------------------------------------------------------------------------------------------------------- 1228 % Brunt-Vais\"{a}l\"{a} Frequency 1229 % ------------------------------------------------------------------------------------------------------------- 1230 \subsection{Brunt-Vais\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)} 1209 where the computer name of the coefficients as well as their standard value are given in \ref{Tab_SEOS}. 1210 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing 1211 the associated coefficients. 1212 Setting to zero the two thermobaric coefficients ($\mu_1$, $\mu_2$) remove thermobaric effect from S-EOS. 1213 setting to zero the three cabbeling coefficients ($\lambda_1$, $\lambda_2$, $\nu$) remove cabbeling effect from S-EOS. 1214 Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. 1215 1216 \end{description} 1217 1218 1219 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1220 \begin{table}[!tb] 1221 \begin{center} \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|} 1222 \hline 1223 coeff. & computer name & S-EOS & description \\ \hline 1224 $a_0$ & \np{rn\_a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline 1225 $b_0$ & \np{rn\_b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline 1226 $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline 1227 $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline 1228 $\nu$ & \np{rn\_nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline 1229 $\mu_1$ & \np{rn\_mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline 1230 $\mu_2$ & \np{rn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline 1231 \end{tabular} 1232 \caption{ \label{Tab_SEOS} 1233 Standard value of S-EOS coefficients. } 1234 \end{center} 1235 \end{table} 1236 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1237 1238 1239 % ------------------------------------------------------------------------------------------------------------- 1240 % Brunt-V\"{a}is\"{a}l\"{a} Frequency 1241 % ------------------------------------------------------------------------------------------------------------- 1242 \subsection{Brunt-V\"{a}is\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)} 1231 1243 \label{TRA_bn2} 1232 1244 1233 An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a} 1234 frequency) is of paramount importance as it is used in several ocean 1235 parameterisations (namely TKE, KPP, Richardson number dependent 1236 vertical diffusion, enhanced vertical diffusion, non-penetrative convection, 1237 iso-neutral diffusion). In particular, one must be aware that $N^2$ has to 1238 be computed with an \textit{in situ} reference. The expression for $N^2$ 1239 depends on the type of equation of state used (\np{nn\_eos} namelist parameter). 1240 1241 For \np{nn\_eos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987} 1242 polynomial expression is used (with the pressure in decibar approximated by 1243 the depth in meters): 1245 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} 1246 frequency) is of paramount importance as determine the ocean stratification and 1247 is used in several ocean parameterisations (namely TKE, GLS, Richardson number dependent 1248 vertical diffusion, enhanced vertical diffusion, non-penetrative convection, tidal mixing 1249 parameterisation, iso-neutral diffusion). In particular, $N^2$ has to be computed at the local pressure 1250 (pressure in decibar being approximated by the depth in meters). The expression for $N^2$ 1251 is given by: 1244 1252 \begin{equation} \label{Eq_tra_bn2} 1245 N^2 = \frac{g}{e_{3w}} \; \beta \1246 \left( \alpha / \beta \ \delta_{k+1/2}[T] - \delta_{k+1/2}[S] \right)1247 \end{equation}1248 where $\alpha$ and $\beta$ are the thermal and haline expansion coefficients.1249 They are a function of $\overline{T}^{\,k+1/2},\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$,1250 and $z_w$, with $T$ the \textit{potential} temperature and $\widetilde{S}$ a salinity anomaly.1251 Note that both $\alpha$ and $\beta$ depend on \textit{potential}1252 temperature and salinity which are averaged at $w$-points prior1253 to the computation instead of being computed at $T$-points and1254 then averaged to $w$-points.1255 1256 When a linear equation of state is used (\np{nn\_eos}=1 or 2,1257 \eqref{Eq_tra_bn2} reduces to:1258 \begin{equation} \label{Eq_tra_bn2_linear}1259 1253 N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T] \right) 1260 1254 \end{equation} 1261 where $\alpha$ and $\beta $ are the constant coefficients used to 1262 defined the linear equation of state \eqref{Eq_tra_eos_linear}. 1263 1264 % ------------------------------------------------------------------------------------------------------------- 1265 % Specific Heat 1266 % ------------------------------------------------------------------------------------------------------------- 1267 \subsection [Specific Heat (\textit{phycst})] 1268 {Specific Heat (\mdl{phycst})} 1269 \label{TRA_adv_ldf} 1270 1271 The specific heat of sea water, $C_p$, is a function of temperature, salinity 1272 and pressure \citep{UNESCO1983}. It is only used in the model to convert 1273 surface heat fluxes into surface temperature increase and so the pressure 1274 dependence is neglected. The dependence on $T$ and $S$ is weak. 1275 For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$ 1276 when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has 1277 been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$. 1278 Its value is set in \mdl{phycst} module. 1279 1255 where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS, 1256 and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 1257 The coefficients are a polynomial function of temperature, salinity and depth which expression 1258 depends on the chosen EOS. They are computed through \textit{eos\_rab}, a \textsc{Fortran} 1259 function that can be found in \mdl{eosbn2}. 1280 1260 1281 1261 % ------------------------------------------------------------------------------------------------------------- … … 1298 1278 sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent 1299 1279 terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The freezing 1300 point is computed through \textit{ tfreez}, a \textsc{Fortran} function that can be found1280 point is computed through \textit{eos\_fzp}, a \textsc{Fortran} function that can be found 1301 1281 in \mdl{eosbn2}. 1302 1282 … … 1308 1288 \label{TRA_zpshde} 1309 1289 1310 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"} 1311 1312 With partial bottom cells (\np{ln\_zps}=true), in general, tracers in horizontally 1290 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, 1291 I've changed "derivative" to "difference" and "mean" to "average"} 1292 1293 With partial cells (\np{ln\_zps}=true) at bottom and top (\np{ln\_isfcav}=true), in general, tracers in horizontally 1313 1294 adjacent cells live at different depths. Horizontal gradients of tracers are needed 1314 1295 for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure 1315 1296 gradient (\mdl{dynhpg} module) to be active. 1316 1297 \gmcomment{STEVEN from gm : question: not sure of what -to be active- means} 1298 1317 1299 Before taking horizontal gradients between the tracers next to the bottom, a linear 1318 1300 interpolation in the vertical is used to approximate the deeper tracer as if it actually … … 1390 1372 \gmcomment{gm : this last remark has to be done} 1391 1373 %%% 1374 1375 If under ice shelf seas opened (\np{ln\_isfcav}=true), the partial cell properties 1376 at the top are computed in the same way as for the bottom. Some extra variables are, 1377 however, computed to reduce the flow generated at the top and bottom if $z*$ coordinates activated. 1378 The extra variables calculated and used by \S\ref{DYN_hpg_isf} are: 1379 1380 $\bullet$ $\overline{T}_k^{\,i+1/2}$ as described in \eqref{Eq_zps_hde} 1381 1382 $\bullet$ $\delta _{i+1/2} Z_{T_k} = \widetilde {Z}^{\,i}_{T_k}-Z^{\,i}_{T_k}$ to compute 1383 the pressure gradient correction term used by \eqref{Eq_dynhpg_sco} in \S\ref{DYN_hpg_isf}, 1384 with $\widetilde {Z}_{T_k}$ the depth of the point $\widetilde {T}_{k}$ in case of $z^*$ coordinates 1385 (this term = 0 in z-coordinates)
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